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Abstract
The Rayleigh resolution criterion sets the minimum separation for two-point objects to be distinguishable in a classical optical imaging system. We demonstrate that the sub-Rayleigh resolution can be achieved in a telecentric imaging system with the help of a partially coherent illumination whose spatial coherence has lattice-like distribution. We show that the orientation-selective sub-Rayleigh imaging can be realized by controlling the spatial distribution of the coherence lattice into different symmetries. We carry out a proof-of-principle experiment to demonstrate the orientation-selective sub-Rayleigh imaging for a 1951 USAF resolution target. Our results indicate a flexible orientation-selective high-resolution imaging with spatial coherence engineering of the partially coherent light.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The spatial resolution of the imaging systems is limited by the size of their diffraction-limited point-spread function. To quantify this limited resolution, the Rayleigh criterion has been proposed and widely used [1]. The Rayleigh resolution criterion is quantified by the minimal separation between two points of interest that can be achieved for a given wavelength of light and the specific imaging system. How to overcome the Rayleigh criterion has become one of the most important tasks in diverse imaging applications, including optical microscopy, optical measurements, optical trapping, lithography, and optical information storage [2–7]. Different methods have be proposed to realize the sub-Rayleigh imaging. For example, the subtle optical components, such as the superlens [8] and microspheres [9] have been used to overcome the classical resolution limit by collecting the near-field evanescent waves. The schemes with the aid of the fluorescent labeling of sample [10] and reconstruction algorithms [11] have also been widely used in super-resolution microscopic imaging. Furthermore, with the help of multi-photon detection array [12] the classic Rayleigh resolution limit can be broken. Another strategy for sub-Rayleigh imaging is through the modulation of the physical properties of the illumination light. In 2006, Tamburini and coauthors showed that a sub-Rayleigh separation distance about one order magnitude below the Rayleigh criterion can be achieved by using two mutually incoherent overlapping optical vortices with different topological charges [13]. By modulating the intensity distribution of the illumination light, e.g., into the customized random speckles [14] or into the superoscillation distributions [15], the superresolution imaging can be obtained. In addition, the sub-Rayleigh imaging can be attained by employing the prescribed polarization states such as the cylindrical polarization states [16].
Meanwhile, the effect of spatial coherence of the illuminating light on the process of image formation in optical system is well studied in the past decades [17–23]. More recently, it has been found that the partially coherent light with unconventional spatial coherence structure [24,25] can act as the illuminating light for the sub-Rayleigh imaging [26–29]. In 2012, Tong and Korotkova showed theoretically that the partially coherent beam endowed with the second-order twist phase can produce images with a resolution overcoming the Rayleigh limit by an order of magnitude [26]. Later, we demonstrated both theoretically and experimentally the sub-Rayleigh imaging with the partially coherent Schell-model beam having Laguerre-Gaussian correlation distribution [27]. However, up to now, all the sub-Rayleigh imaging under the partially coherent illuminations is isotropic, i.e., the resolutions along different directions cannot be distinguished, which limits their application scenarios.
In this work, we introduce an efficient approach to realize the orientation-selective sub-Rayleigh imaging with the help of the spatial coherence lattices. The spatial coherence lattices denote the spatial oscillations in the degree of spatial coherence of a partially coherent beam, which have been investigated both theoretically and experimentally [30–32] and have been used in the optical information encoding [32,33] and the resistance of the negative effects induced by the turbulent atmosphere [34–36]. Here, we show that with the rotation of the one-dimensional spatial coherence lattice, the sub-Rayleigh imaging can be selectively achieved along different directions. In addition, we propose that with the two-dimensional spatial coherence lattice, the sub-Rayleigh imaging along both the horizontal and the vertical directions can be obtained. We carry out a proof-of-principle experiment to realize the orientation-selective sub-Rayleigh imaging for the 1951 USAF resolution target with the minimum resolvable distance $0.8d_R$ being achieved, where $d_R$ is defined as the minimum resolvable distance between two points under incoherent light illumination. The experimental results are consistent well with our predictions.
This work is organized as follows. In Sec. 2, we present our basic principle for orientation-selective sub-Rayleigh imaging in a telecentric imaging system with the spatial coherence lattices. The method based on the generalized van Cittert-Zernike theorem for synthesizing different spatial coherence lattices has also been discussed in this section. In Sec. 3, we present our experimental verification of the orientation-selective sub-Rayleigh imaging for the 1951 USAF resolution target. The experimental results and the corresponding discussions are also included. We summarize our findings in Sec. 4.
2. Principle
2.1 Telecentric imaging system
The schematic diagram of a telecentric imaging system, composed by two thin lenses with the same focal length $f$, is shown in Fig. 1. The object with transmission function $O(\mathbf {r})$ is placed in the front focal plane of the first thin lens. The imaging plane is located in the rear focal plane of the second thin lens. In the rear focal plane of the first thin lens or the front focal plane of the second thin lens (or the frequency plane of the system), a circular aperture is placed to act as the pupil of the system. Due to the ubiquitous diffraction effects in the imaging system, we remark that the circular aperture is used to form a classic diffraction-limited imaging system. We now consider the illuminating light to be a partially coherent beam with all its (second-order) statistical properties involved in a cross-spectral density function [37,38], $W(\mathbf {r}_1, \mathbf {r}_2) = \langle E^\ast (\mathbf {r}_1) E(\mathbf {r}_2) \rangle$, where $\mathbf {r}_1$ and $\mathbf {r}_2$ are two arbitrary spatial points in the object plane, $E(\mathbf {r})$ is a random field realization, the asterisk and the angular brackets denote the complex conjugate and the ensemble average over all the field realizations, respectively. Taking the transmission function of the object into account, the cross-spectral density function in the imaging plane can be obtained by
To show the sub-Rayleigh resolution imaging of the system, we first consider the object as an opaque screen with the two-pinhole opening separated by distance $d$. The pinholes are placed along the horizontal axis and are symmetric with the vertical axis. Its transmission function can be expressed as
where $\delta (\cdot )$ is a Dirac delta function. Taking Eqs. (2) and (3) into Eq. (1) and assuming the illuminating light is a Schell-model type partially coherent beam, i.e., the cross-spectral density function $W(\mathbf {r}_1, \mathbf {r}_2) = [I(\mathbf {r}_1)I(\mathbf {r}_2)]^{1/2} \mu (\mathbf {r}_1 - \mathbf {r}_2)$, where $I(\mathbf {r}) = W(\mathbf {r}, \mathbf {r})$ denotes the average intensity of the partially coherent beam at point $\mathbf {r}$ and $\mu (\mathbf {r}_1 - \mathbf {r}_2) = W(\mathbf {r}_1, \mathbf {r}_2) /[I(\mathbf {r}_1)I(\mathbf {r}_2)]^{1/2}$ is the (complex) degree of coherence between points $\mathbf {r}_1$ and $\mathbf {r}_2$, we obtain that the intensity distribution of the field in the imaging plane can be written as2.2 Effect of spatial coherence
The ‘interference’ term $2 I_+(\boldsymbol {\rho }) I_-(\boldsymbol {\rho }) \mathrm {Re}[\mu (\mathbf {P}_1 -\mathbf {P}_2)]$ in Eq. (4), or more specifically, the real part of degree of coherence of illuminating light plays a critical role in the determination of the imaging resolution of two separated pinholes object. For the classical incoherent light, i.e., $\mathrm {Re}[\mu (\mathbf {P}_1 -\mathbf {P}_2)] = 0$, the minimum resolvable separation between two pinholes can be evaluated according to Rayleigh resolution criterion [1]. When the first zero of the Airy pattern generated by a single pinhole coincides with the principal maximum of the Airy pattern generated by the other pinhole, the two incoherently illuminated points can be identified as just resolved. As shown in Fig. 2 (black curve), the minimum resolvable separation of two pinholes is $d_R = 0.61 \lambda f /R$, which is also named the Rayleigh resolution limit. The ratio of the intensity at the image center to the maximum intensity is about 0.735, i.e., $I(0)/[I(\boldsymbol {\rho })]_\text {max} = 0.735$.
Fig. 2. Image of two illuminated pinholes for different degrees of spatial coherence $\mathrm {Re}[\mu (\mathbf {P}_1 -\mathbf {P}_2)]$. The separation between two pinholes is $d_R = 0.61 \lambda f /R$. The intensities are normalized with respect to the maximum value of the intensity for the incoherent illumination.
To show the dependence of the the minimum resolvable separation on the degree of coherence of illuminating light, the intensity distributions of the image for different values of $\mathrm {Re}[\mu (\mathbf {P}_1 -\mathbf {P}_2)]$ are shown in Fig. 2. In the calculation, the separation between the two pinholes in the object plane is fixed to be $d_R$. From the calculated results, it is found that with the increase of $\mathrm {Re}[\mu (\mathbf {P}_1 -\mathbf {P}_2)]$, i.e., the value becomes positive, the dip in the composite intensity curve almost disappears and the two-point object can not be identified anymore. However, with the decrease of $\mathrm {Re}[\mu (\mathbf {P}_1 -\mathbf {P}_2)]$, i.e., the value becomes negative, the dip in the composite curve of the intensity distribution at the center becomes more obvious. Thus, the Rayleigh resolution limit for the incoherent light illumination is broken with the partially coherent illumination having negative degree of coherence. For $\mathrm {Re}[\mu (\mathbf {P}_1 -\mathbf {P}_2)] = -1$ (red curve in Fig. 2), the image intensity at center vanishes and the two-point object can be resolved perfectly.
We remark that in Fig. 2 two-pinhole object is located horizontally along the $x$ axis. Thus, the partially coherent illuminating light having negative degree of spatial coherence along the same axis can realize the sub-Rayleigh imaging in the telecentric imaging system. In contrast, when the partially coherent light having negative degree of spatial coherence along the $y$ axis, the resolution of the object along vertical direction can be enhanced. To overcome the Rayleigh resolution criterion both along the horizontal and the vertical directions, the partially coherent light beam having negative degrees of spatial coherence along both the $x$ and the $y$ axes is required for the illumination. Therefore, the results in Fig. 2 also indicate the application of orientation-selective sub-Rayleigh imaging with the spatial coherence engineering of the partially coherent illuminating light.
2.3 Spatial coherence lattices
To realize the orientation-selective sub-Rayleigh resolution imaging, we use the recently introduced partially coherent beam with lattice-like spatial coherence distribution [31] as the illuminating light. The spatial distribution of the coherence lattice can be controlled flexibly in the experiment with the help of the generalized van Cittert-Zernike theorem [39]. Figure 3 shows the method to synthesize and manipulate the spatial coherence lattice from a structured incoherent light with its intensity being expressed by the nonnegative function $p(\mathbf {v})$. As shown in Fig. 3(a), the incoherent light propagates through a distance $f$ in free space and then is collected by a thin lens of focal distance $f$. Based on the van Cittert-Zernike theorem, the degree of spatial coherence of the output partially coherent beam can be expressed as [24,40,41]
Fig. 3. (a) Generation of the partially coherent illuminating light with degree of spatial coherence $\mu (\mathbf {r}_1, \mathbf {r}_2)$ by propagating an incoherent light source with intensity $p(\mathbf {v})$ through a distance $f$ in free space and a thin lens of focal length $f$. (b)–(d) Three cases for the distributions of $p(\mathbf {v})$.
We find the degree of spatial coherence $\mu (\mathbf {r}_1, \mathbf {r}_2)$ for the generated partially coherent beam and the intensity $p(\mathbf {v})$ for the incoherent light form a Fourier transform pair. Thus, the spatial coherence of the partially coherent light can be controlled by the $p(\mathbf {v})$ function. The spatial coherence lattice can be generated when the $p(\mathbf {v})$ function is composed by $M \geq 2$ off-axis circular functions, i.e.,
Taking the above three cases for $p(\mathbf {v})$ into Eq. (6), the spatial distributions for $\mu (\mathbf {r}_1, \mathbf {r}_2)$ can be obtained. The simulation results are shown in Fig. 4, in which the cross lines both along the horizontal and the vertical directions are also present (bottom panels). In the simulation, the parameters are $\lambda = 532$ nm, $f=100$ mm, $\sigma _0 = 0.15$ mm, and the distance between circles $l=4$ mm for the first and the second cases, $l = 3$ mm for the case 3. It is found from Fig. 4 that when the two circles in $p(\mathbf {v})$ are placed along the horizontal direction, the spatial distribution of $\mu (\mathbf {r}_1, \mathbf {r}_2)$ shows the vertical fringes distribution (one-dimensional coherence lattice). As shown in the cross lines [Fig. 4(d)], the degree of coherence along horizontal direction oscillates rapidly in the range from $-1$ to $+1$, while the degree of coherence along vertical direction are relatively uniform and its value is always large than $0.5$. Thus, we can predict that the partially coherent illuminating light with such spatial coherence distribution can be used to enhance the image resolution along the horizontal direction. For the second case, the spatial distribution of $\mu (\mathbf {r}_1, \mathbf {r}_2)$ rotates $90^\circ$ with respect to that for the first case. Thus, as shown in Fig. 4(e) the degree of coherence along vertical direction oscillates rapidly in the range from $-1$ to $+1$, while along horizontal direction the degree of coherence becomes positive and relatively uniform. The image resolution along the vertical direction, therefore, can be enhanced with the spatial coherence shown in Fig. 4(b).
Fig. 4. (a)–(c) The spatial distributions of the degree of spatial coherence $\mu (\mathbf {r}_1, \mathbf {r}_2)$ for three cases of $p(\mathbf {v})$ shown in Figs. 3(b)–3(d). (d)–(f) The corresponding cross lines along the horizontal and the vertical directions.
We note that when $\Delta x = 0.0135$ mm and $\Delta y = 0.0135$ mm in the first and the second cases, respectively, the degree of coherence $\mu$ reaches to the first minimum with the value being $-1$. When $\Delta x = 0.027$ mm and $\Delta y = 0.027$ mm, the degree of coherence $\mu$ reaches to the first maximum value [see in Figs. 4(d) and 4(e)]. Moreover, we remark here that the distance $\Delta \mathbf {r} = 0.0135$ mm is very close to the Rayleigh distance $d_R$. Thus, in such case the two-point object with point-separation $d = d_R$ can be resolved perfectly, see in Figs. 5(a) and 5(d). With the decrease of the point-separation $d$ in the two-point object, e.g., when $d = 0.8 d_R$, the two-point object can still be resolved, see in Figs. 5(b) and 5(e). When $d = 0.7 d_R$, the two-point object is just resolved, i.e. the ratio of the intensity at the image center to the maximum intensity is 0.735. To further increase the resolution ability, we can decrease the spatial period in the degree of coherence, e.g., with the increase of $l$ in the $p(\mathbf {v})$ function as shown in Figs. 6(a). Figure 6(b) shows the minimum separation of the two-point object could be resolved with different values of $l$. It is found that with the increase of $l$, the resolution ability is enhanced. When $l=20$ mm, the minimum resolvable separation is $d_\text {min} = 0.2 d_R$. While, with the Laguerre-Gaussian coherence, the minimum resolvable separation is about $0.8 d_R$ [27], indicating the superiority in the superresolution imaging with the spatial coherence lattices.
Fig. 5. Images for two-point objects with point-separation (a) $d = d_R$, (b) $d = 0.8d_R$, and (c) $d = 0.7 d_R$ under the illumination of partially coherent beam having degree of coherence shown in Fig. 4(a). (d)–(f) show the corresponding cross lines along $\rho _x$ direction ($\rho _y = 0$).
Fig. 6. (a) Spatial distributions of the degree of coherence $\mu (\Delta x, 0 )$ of the one-dimensional spatial coherence lattices for $l = 4$ mm and $l = 6$ mm. (b) The minimum resolvable separation $d_\text {min}$ with different values of $l$.
In spatial degree of coherence for the third case is shown in Fig. 4(c). It is found that $\mu (\mathbf {r}_1, \mathbf {r}_2)$ has a two-dimensional lattice-like distribution. From the cross lines shown in Fig. 4(f), we find the spatial distributions of the degree of coherence along both the horizontal and the vertical directions oscillate rapidly in the range from a negative value to a positive value. Thus, the spatial coherence lattice shown in Fig. 4(c) can be used to realize the sub-Rayleigh imaging both along the horizontal and the vertical directions. We remark that the minimum value of $\mu (\mathbf {r}_1, \mathbf {r}_2)$ in the case 3 is about $-0.5$, which is larger than that for the case 1 and case 2, indicating a better resolution capability for the first two cases.
2.4 Orientation-selective sub-Rayleigh imaging
To show the orientation-selective sub-Rayleigh imaging, we now let the partially coherent beams with the spatial coherences shown in Fig. 4 to illuminate the telecentric imaging system. The object is shown in Fig. 7(a), which is extracted from a 1951 USAF resolution target (group number 6 and element number 2). The separations between two closing rectangular slits in the object are very close to the Rayleigh resolution limit $d_R$, and the separation distance is 0.0135 mm, which is exacting the same as the value of $\Delta \mathbf {r}$ for the first minimum of the degree of coherence of the illuminating beam shown in Figs. 4(d) and 4(e). It is noted that in the simulation, the wavelength of the illuminating light is $532$ nm. In Figs. 7(b), 7(c), and 7(d), we show the simulation results of the image for three different cases, respectively. For the first case, it is found that the vertically placed rectangular slits can be resolved well and the horizontally placed rectangular slits can not be identified anymore, which indicates that the image resolution along the horizontal direction is indeed enhanced with the spatial coherence shown in Fig. 4(a). By switching the spatial coherence into that shown in Fig. 4(b), we find in Fig. 7(c) that the horizontally placed rectangular slits can be resolved well, while the vertically placed rectangular slits can not be identified anymore, indicating that only the image resolution along the vertical direction is enhanced. When the spatial coherence is switched into the third case shown in Fig. 4(c), it is found in Fig. 7(d) that both the horizontally placed and the vertically placed rectangular slits can be resolved in the imaging plane, which indicates that the Rayleigh resolution limit is broken both in the horizontal and the vertical directions. From the simulation results shown in Fig. 7, it is also found that the image resolutions obtained in the cases 1 and 2 are better than that in the case 3, which is consistent with our prediction shown above. In addition, we find two non-neighboring bars with separation being 0.027 mm, which is equal to $\Delta \mathbf {r}$ when the first maximum of the degree of coherence is reached in Figs. 4(a) and 4(b), can still be resolved when the degree of coherence between the bars becomes positive (the value is close to one), since the separation between bars are twice larger than the Rayleigh resolution distance $d_R$.
Fig. 7. (a) Object in the telecentric imaging system. (b)–(d) the simulation results of the images for the partially coherent illuminations with the degree of spatial coherence shown in Figs. 4(a)–4(c), respectively.
3. Experiment
In this section, we carry out a proof-of-principle experiment to examine the orientation-selective sub-Rayleigh imaging with the spatial coherence lattices. Figure 8 shows our experimental setup for creating the partially coherent beam with different spatial coherence distributions and optical imaging through the telecentric imaging system. A fully coherent laser beam of wavelength $\lambda = 532$ nm is expanded by a beam expander (BE) and reflected by a reflective mirror (RM). The collimated beam then goes into a spatial light modulator (SLM) which acts as an opaque screen with two or three circular opening pinholes. The output intensity is controlled to have the spatial distribution of $p(\mathbf {v})$ shown in Figs. 3(b)–3(d). An aperture is placed after the SLM to filter away the unwanted beam spot. The shaped coherent beam transmitted from the SLM and the aperture then illuminates on a rotating ground-glass disk (RGGD) to produce an incoherent light beam having the same intensity distribution of the fully coherent light beam, i.e., the intensity of the output incoherent beam is $p(\mathbf {v})$ as well. We note that the output beam from the RGGD can be regarded as the spatially incoherent source under the condition that the diameter of the beam spot on the RGGD is larger than the inhomogeneity scale of the RGGD [42]. This condition holds in our experiment. The output incoherent beam is collected by a thin lens L$_1$ of focal length $f_1 = 100$ mm. After L$_1$, the partially coherent light beam with the lattice-like degree of spatial coherence is created. The generated partially coherent beam then illuminates onto a telecentric imaging system composed by two thin lens L$_2$ and L$_3$. The focal distances $f_2$ and $f_3$ are both equal to 100 mm. The 1951 USAF resolution target is placed in the input plane of the telecentric imaging system. A circular aperture (CA) with radius $R=2.5$ mm is placed in the frequency plane of the system. In the output plane, a charge-coupled device (CCD) is used to capture the image.
Fig. 8. Experimental setup for the orientation-selective sub-Rayleigh imaging with the spatial coherence lattices. BE: beam expander; RM: reflective mirror; SLM: spatial light modulator; RGGD: rotating ground-glass disk; L$_1$, L$_2$, and L$_3$: thin lenses; RT: resolution target; CA: circular aperture; CCD: charge-coupled device.
In Fig. 9, we show the experimental results for the images of the 1951 USAF resolution target under the illumination of the partially coherent beams having degree of spatial coherence shown in Fig. 4. It is found that with a certain spatial coherence lattice, the structures with various separations can be resolved. It is also found that not only the bars, but also other complex structures, such as numbers can be resolved with the spatial coherence lattices. In addition, we find the anisotropic resolution enhancement in Figs. 9(b) and 9(c), while in Fig. 9(d) the resolution enhancement is isotropic. The results are consistent with our theoretical predictions. To show the sub-Rayleigh effect, the second and the fourth elements in group 6 (i.e., 6-2 and 6-4) of the resolution target are captured in the imaging plane. The distances between the adjacent slits in 6-2 and 6-4 elements are close to the Rayleigh resolution limit $d_R$ and $0.8d_R$, respectively. We now modulate the computer-generated holography loaded in the SLM to generate a partially coherent light beam having the degree of spatial coherence shown in Fig. 4(a). In such case, the captured images for the 6-2 and 6-4 elements are shown in Fig. 10. For a better comparison, the simulation results are also displayed. It is found from the experimental result [Fig. 10(a)] that the vertically placed rectangular slits can be well resolved in the 6-2 element of the resolution target. However, the horizontally placed rectangular slits can not be resolved as we have expected. From the cross line shown in Fig. 10(d), we find the intensity at center between two slits effectively vanishes, indicating the Rayleigh resolution limit along the horizontal direction is perfectly overcome. While, from the cross line shown in Fig. 10(c), we find the slits can not be identified and the resolution is not enhanced along the vertical direction. The reason behind is due to the degree of spatial coherence of the illuminating light between two vertically placed slits are $\mu \to -1$ and between two horizontally placed slits are $\mu \to 1$ [see in Fig. 4(d)].
Fig. 9. (a) The 1951 USAF resolution target in the object plane. (b)–(d) Experimental results for the images of a 1951 USAF resolution target under the illumination of the partially coherent beams having degree of spatial coherence shown in Figs. 4(a), 4(b), and 4(c), respectively. The scale bar is 0.2 mm.
Fig. 10. (a) and (e) Experimental results for the image of the 6-2 and 6-4 elements of the 1951 USAF resolution target under the illumination of a partially coherent beam having degree of spatial coherence shown in Fig. 4(a). (b) and (f) The corresponding simulation results. (c) and (g) show the cross line along vertical direction. (d) and (h) show the cross line along horizontal direction.
To switch the degree of spatial coherence of the illuminating light into the form shown in Fig. 4(b), the computer-generated holography loaded in the SLM is modified. Figure 11 shows the experimental results for the images under the illumination of the partially coherent beam having the degree of spatial coherence shown in Fig. 4(b). It is found from Fig. 11(a) that the horizontally placed rectangular slits become resolvable and the vertically placed slits can not be recognized anymore. Compared Fig. 11(a) with Fig. 11(b), it is found that the experimental results are consistent with the simulation results. From the cross lines shown in Figs. 11(c) and 11(d), we find that the Rayleigh resolution limit along the vertical direction is overcome nearly perfectly as the degree of spatial coherence of the illuminating light between two neighbouring vertically placed slits are approximate to $-1$, while the resolution along the horizontal direction is not enhanced by the positive degree of spatial coherence as shown in Fig. 4(e). Figures 11(e)–11(h) show the image of the element 6-4 of the 1951 USAF resolution target. We find that the horizontally placed slits with neighbouring distance being $0.8d_R$ can still be resolved.
Fig. 11. (a) and (e) Experimental results for the image of the 6-2 and 6-4 elements of the 1951 USAF resolution target under the illumination of a partially coherent beam having degree of spatial coherence shown in Fig. 4(b). (b) and (f) The corresponding simulation results. (c) and (g) show the cross line along vertical direction. (d) and (h) show the cross line along horizontal direction.
Up to now, we have demonstrated the orientation-selective sub-Rayleigh imaging through rotating the spatial distribution of the degree of coherence of the partially coherent illuminating light. To overcome the Rayleigh resolution limit along both the horizontal and the vertical directions, we now modulate the computer-generated holography loaded in the SLM to create the partially coherent beam having the degree of spatial coherence shown in Fig. 4(c) (i.e., two-dimensional spatial coherence lattice). The output images of the 6-2 and 6-3 elements of the resolution target under such illumination are shown in Fig. 12. It is found that both the horizontal placed and the vertical placed slits in the 6-2 and 6-3 elements can be resolved, which indicates that the Rayleigh resolution limit along both the horizontal and the vertical directions is overcome. Meanwhile, we find from the experimental results that under such illumination the intensity at center between two neighbouring slits does not vanish, indicating that a poorer resolution compared with the illuminations shown in Fig. 10 and Fig. 11. This is because the negative value of the degree of spatial coherence of the illuminating light between two neighbouring slits can only reach to $-0.5$ [see in Fig. 4(f)].
Fig. 12. (a) and (e) Experimental results for the image of the 6-2 and 6-3 elements of the 1951 USAF resolution target under the illumination of a partially coherent beam having degree of spatial coherence shown in Fig. 4(c). (b) and (f) The corresponding simulation results. (c) and (g) show the cross line along vertical direction. (d) and (h) show the cross line along horizontal direction.
4. Conclusions
In summary, we introduced a novel method to realize the orientation-selective sub-Rayleigh imaging in a classical $4f$ telecentric imaging system with the spatial coherence engineering of a partially coherent illuminating light beam. We demonstrated both theoretically and experimentally that by rotating the one-dimensional spatial coherence lattice of the partially coherent illumination, the Rayleigh resolution limitation along the horizontal and the vertical directions can be selectively overcome. In the experiment, we showed that the minimum resolvable distance is $0.8d_R$ with $d_R$ being the Rayleigh resolution limitation under the classic incoherent illumination. Meanwhile, we constructed a two-dimensional spatial coherence lattice that can overcome the Rayleigh resolution limitation along the horizontal and the vertical directions simultaneously. Finally, we remark that the minimum resolution distance can be further decreased by reducing the beating period in the spatial coherence lattices. The perfect optical coherence lattices [43] as well as the vector optical coherence lattices [44] may provide more degrees of freedom to control the sub-Rayleigh imaging. Our results may find particular application in low-coherence, speckle-free, and high-resolution microscopy imaging.
Funding
National Key Research and Development Program of China (2019YFA0705000); National Natural Science Foundation of China (11874046, 11904247, 11974218, 12104263, 12174279, 12192254); Innovation Group of Jinan (2018GXRC010); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX21_2935).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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