Abstract
1. INTRODUCTION
An inevitable problem in an imaging process is that the resolution (the minimal distance between two points that can be discriminated) of conventional optical imaging systems is limited by diffraction, as a consequence of the loss in the far-field of evanescent waves carrying information on the high spatial-frequency components. The need to overcome the diffraction limit has led to the invention of the scanning near-field optical microscope (SNOM), which achieves subwavelength optical resolution [1] and has inspired tremendous works in subwavelength imaging in recent years [2–8]. For instance, an intriguing idea to overcome the diffraction limit is using the “superlens” made of silver, which was proposed by Pendry in 2000 [2]. Later the idea was realized experimentally through a planar left-handed lens [3] and material with negative permittivity or permeability [4]. Furthermore, the “hyperlens” was proposed and developed [5–7]. Besides, thanks to the prosperous development of laser technology for decades, super-resolution fluorescence microscopy has obtained magnificent achievements [9–15], starting with stimulated emission depletion fluorescence microscopy proposed by Hell [9,10], followed by other proposals such as reversible saturable optical linear fluorescence transitions [11], stochastic optical reconstruction microscopy [12], and photoactivated localization microscopy [13]. However, all of these works about overcoming the diffraction limit have been considered in flat space without gravitational fields so that the size of the diffraction spot was only decided by the wavelength of light and the characteristics of the imaging system. It is well known from Einstein’s general relativity (GR) theory that light follows curved trajectories and unusual dynamics in the vicinity of a black hole. In this work, we investigate the effect of spatial curvature on diffraction limit, starting with Fraunhofer diffraction in two-dimensional (2D) curved space.
Wave dynamics of light in gravitational field has attracted much attention in recent years. Since measurable gravitational effect can be rarely obtained in a laboratory, researchers have introduced several analog systems to simulate gravitational fields with laboratory devices [16–27], such as Bose-Einstein condensates [16–18], extremely low group velocity electromagnetic fluid [19], metamaterials with inhomogeneous refractive index distribution [20,21], optical black-hole cavities [22], and optical fibers [23]. Among these methods, one approach is to study the behavior of light in curved space by directly constructing spatial geometries that mimic distortion by gravitational field [28–43]. This field of research originates from Batz and Peschel’s study of the nonlinear Schrödinger equation and propagation of a Gaussian beam in 2D curved space [28], inspired itself by Costa’s dynamics of particle constraint on the surface [29]. Subsequently a series of theoretical [30–38] and experimental [39–43] investigations have been carried out, investigating the Wolf effect in 2D curved space [30,31], spatial accelerating wave packets [32], Gouy phase shift [33], topological phase of photonic crystal [34], nonlinear dynamics [35–37], and, experimentally, the observation of self-focusing, diffraction-less propagation [39], accelerating wave packets [40], group and phase velocity [41], the Hanbury Brown and Twiss effect [42], and the surface plasmon polariton [43]. For the convenience and feasibility of calculation, the curved spatiotemporal geometries usually considered were often surfaces of revolution (SORs) embedded in the three-dimensional space because of their rotational symmetry.
In this work, we take advantage of this paradigm to investigate the diffraction limit of light on SORs. The Gaussian (or intrinsic) curvature of an SOR is defined as , where are the two principal curvatures associated with two tangent circles with the maximal/minimal radii for every regular point on the surface. Then the Gaussian curvature is later used to study how curved surfaces affect the optical effect. Starting with Fraunhofer and wave equations, we analytically give the expression of intensity distribution of light diffracted by a slit on SORs with constant Gaussian curvature, and we investigate the intensity distribution on the geodesic perpendicular to the propagation direction, as the mean to explore the diffraction limit, or equivalently the size of Fraunhofer diffraction spot. Then the effect of variable Gaussian curvature of curved space on the diffraction limit is explored. When the propagation direction is not along the longitudinal direction on SORs, we calculate the intensity distribution of the diffraction field through the Huygens principle in curved space. This method allows us to study propagation with arbitrary initial position and direction, instead of simply taking the equator as the input plane and longitude as the propagation direction. We also apply this method to other surfaces with variable Gaussian curvature. Here we consider different propagation distances and directions to calculate the diffraction intensity distribution and compare them with the case of flat space. We demonstrate that the positive spatial curvature decreases the diffraction effects to get a better optical resolution, which gives a new method to super-resolution. Moreover, the non-uniform distribution of spatial curvature varies the optical resolution in different directions, indicating that the anisotropy of space can be investigated by observing the variation of optical resolution in different directions. The results presented here may also stimulate imaging technologies in planar waveguides with transformation optics and stir up electron optics in 2D curved material.
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2. DIFFRACTION ON SORS WITH CONSTANT GAUSSIAN CURVATURE
First, we consider a family of SORs with constant Gaussian curvature (CGC) (see Fig. 1), which are described by the following metric:
Figure 1.SORs with constant (a)–(c) positive and (d) negative Gaussian curvature
Now, one can obtain the point spread function (PSF) of 2D curved space by solving Eq. (3) under paraxial approximation [42],
Clearly, from Eq. (5), the width of the center fringe of single-slit diffraction is fully determined by the zeros of the sinc function, while the double-slit interference pattern from Eq. (6) is modulated by the sinc function and its fine structure is further modulated by the cosine function. From Eq. (5), if one assumes the observation plane along the lines of latitude, one can obtain a formula similar to Abbe resolution, , to estimate the optical resolution in curved space, where is the numerical aperture on curved surfaces, and is the numerical aperture in flat space. Obviously, in the SORs with , we have , and , which tells that the optical resolution is improved in the SORs with positive Gaussian curvatures. In contrast, in the SORs with negative , we have , , corresponding to a decreased resolution.
However, when the propagation of light fields is considered in curved space, its intensity distribution must be considered along a geodesic (the equivalent of a straight line in flat space) perpendicular to the direction of propagation [38]. Thus, in order to obtain the diffraction pattern along a geodesic in curved space, we need to know the geodesics perpendicular to the propagation direction and then calculate the intensity of each point on the corresponding geodesics to observe. It is known that the geodesic in curved space satisfies the following equation [38]:
Using the above Eqs. (9) and (10), for a given proper length distanced from the on-axis point (, ), in turn we can obtain the value of the coordinate . Then using Eq. (8), we finally obtain the coordinate information of every point (, ) on the output plane along the geodesics (perpendicular to the optical axis). For the SORs with , we have
Since the other three coordinates , , and are known, can be obtained from Eq. (11) or Eq. (12) directly. Substituting the coordinates and into Eqs. (5) and (6), we obtain the intensity of a point with proper length on the output plane. Note that the sign of every proper length is defined by the sign of . In this way, we can calculate the intensities of all the points with different , and finally the intensity distribution along the different output planes can be obtained.
Figures 2(a1) and 2(a2), respectively, show the intensity distributions of diffraction through a single-slit and of interference through a double-slit at distance on SORs with positive, negative, and zero curvature (flat space). Compared with the cases of flat space, the widths of the main-spot peaks (i.e., the main-spot size of the central fringes for both single-slit and double-slit cases) in the cases of are smaller, which means that optical resolution has been improved compared to that in flat space. As illustrated in Fig. 2(a), if the main-spot size between the two zeros of the central fringe at distance for the single-slit case is, e.g., in flat space, it becomes 0.861 mm in the case of ; in the double-slit case, if the size of the central fringe is, e.g., 0.30 mm in flat space, it reduces to 0.215 mm in the case of . In both cases, the spot size has been reduced by 28.3%. In contrast, on the SORs with , the width of the main-spot peaks becomes larger, indicating a degraded resolution. It would be 1.608 mm for the single-slit and 0.402 mm for the double-slit case. To investigate the evolution of the width of the central fringe as light propagates along the line of longitude, we consider the interval between the first two zeros of the sinc function around the central fringe. Here, the superscripts “” and “” denote single-slit and double-slit cases, respectively. Figures 2(b1) and 2(b2) demonstrate the impact of the spatial curvature on the evolution of with increasing distance. We can see that the size of the central fringe for both single-slit and double-slit cases in the curved space with positive is always smaller than that in flat space. This effect is enhanced as the propagation distance increases.
Figure 2.Diffraction and interference on curved space. (a) Intensity distributions of Fraunhofer (a1) single-slit diffraction and (a2) double-slit interference of light at
Figures 2(c) and 2(d) display the evolutions of diffraction and interference light fields in different spaces. It is seen that, in flat space, the diffraction and interference fields propagate and expand along straight lines; however, the diffraction process of light fields is suppressed/enhanced in curved space with constant positive/negative Gaussian curvature. These evolution features correspond to the narrowing or expanding effect of the central fringe width in the curved spaces with positive or negative values of , compared with the case in flat space.
For conveniently discussing the influence of spatial curvature on the central fringe widths of the diffraction pattern in far-field regions, we define a relative quantity, , to quantify the narrowing or expanding effect of the central fringe width (i.e., the main-spot width) of light fields as the change of the diffraction limit in curved space. Here and are the widths for the central peak in curved and flat space, respectively. Clearly, indicates that the optical resolution has improved, whereas for it has degraded. Figure 3 shows the non-uniform dependence of with propagation distance and spatial curvature. The large positive/negative Gaussian curvature and long propagation distance will apparently decrease/increase the diffraction limit, respectively. Spatial curvature offers therefore a new degree of freedom to control the optical resolution and diffraction limit. This important result can be put into perspective in a practical way. Indeed, it was demonstrated that light dynamics on curved surfaces is equivalent to propagation in 2D planar waveguide structures with nonuniform distributions of the refractive index [36]. Transformation optics has been proposed to construct the refractive index distribution corresponding to a particular curved surface. The results we demonstrated here on curved surfaces apply therefore to transformed planar waveguides since these two systems share the same light dynamics. Therefore, a planar waveguide structure with designed refractive-index distribution can be used to control the optical resolution beyond the usual diffraction limit.
Figure 3.Effect of Gaussian curvature
However, the analytical solution we obtained above is not applicable when light propagation is not along the line of longitude, or the input plane is not along the equator anymore. One must call on the Huygens principle to solve the general case of light propagation on arbitrary curved surfaces along arbitrary propagation direction [38]. According to Ref. [38], the complex amplitude of the light field at the output plane on a 2D curved surface can be expressed by
Figure 4 further demonstrates the intensity evolution of light via a single-slit diffraction for three kinds of constant-Gaussian-curvature SORs, when the incident plane is at a different angle with the longitude direction (i.e., the propagation direction is at an angle of to the longitude direction). Clearly, although the propagation direction is not along the longitude direction anymore, the diffracting fields will focus in curved space with positive Gaussian curvature. The propagation direction does not affect at all the diffraction intensity distribution of light beam when Gaussian curvature is constant [note that the slight difference in intensity patterns is presented due to the calculation errors of the integral in Eq. (13)]. The effects induced by spatial curvature are more apparent at longer propagation distances.
Figure 4.Diffraction of light fields along different propagation directions on (a) spindle, (b) sphere, and (c) hyperboloid. The propagation direction is described by the angle
3. DIFFRACTION ON OTHER TYPICAL SORS WITH VARYING SPATIAL CURVATURE
A. Two-Dimensional Schwarzschild Metric
Now let us consider the diffraction effect of light in a typical model, such as a Schwarzschild black hole, which is a typical solution of Einstein field equation, used for mimicking a light bending near a black hole. In such a space, the spatial curvature changes as spatial position varies. The spacetime metric of a Schwarzschild black hole is characterized by the line element as follows [44]:
Figure 5.Diffraction of light fields along different propagation directions on (a) an FP surface, (b) a
B. Two-Dimensional Schwarzschild-de Sitter Metric
It is well known that the spacetime near a black hole may be modified due to the presence of some matter-energy distributions [44]. One possible situation is that a black hole may be immersed in the background of dark energy which is characterized by a cosmological constant (). Then Eqs. (14) and (15) are modified with the existence of positive cosmological constant in the above subsection (Sec. 3.A), and the metric becomes the Schwarzschild-de Sitter () metric. By similarly taking a 2D slice, the line element is given by
C. Diffraction on the Surface with a Periodic Peanut-Shell Shape
In this subsection, a special SOR with periodic structures [see Fig. 5(c)], called periodic peanut-shell shape (PPSS) [31,45], is introduced out of our interest. For convenience, we apply longitudinal proper length and rotational angle coordinates system (, ) to describe this surface. The metric of PPSS is given by [31]
Finally, in Fig. 6, we plot how the propagation direction on different SORs affects the diffraction limit at different propagation distances. It is found that on the SORs with constant Gaussian curvature, the diffraction limit does not vary with the propagation direction at fixed propagation distances. Clearly, as discussed above, the diffraction limit is smaller for positive and larger for negative , compared with the flat-space case. However, the relative quantity, , changes with propagation direction for other families of SORs with non-constant spatial curvature. On the surface of FP, it is seen that, as the angle increases, the diffraction limit tends to decrease and approach its value in flat space since the propagation of the light beam gradually deviates from the Schwarzschild black hole. Meanwhile, the propagation distance also impacts the diffraction limit. On the SORs of the metric, the change of the diffraction limit as a function of the propagation direction is similar to that on FP, but the value of the related quantity can change from positive to negative and the sign of is dependent on the local spatial curvature. Similarly, on PPSS, the relative quantity oscillates as grows, which indicates how optical resolution deteriorates or improves compared with that of flat space. From our result, since the non-uniform distribution of spatial curvature varies the optical resolution, the isotropy and anisotropy of space may be reflected by the observation of the variation of optical resolution. Thus, we may detect the presence of anisotropic universe and spacetime by observing the variation of optical resolution in different propagation directions and distances.
Figure 6.Dependence of the relative quantity
4. CONCLUSION
In summary, we have investigated the effect on single-slit diffraction or double-silt interference of light in curved space with constant and variable Gaussian curvature. The result demonstrates that the width of the main (central) fringe (indicating the optical imaging resolution) in diffraction patterns becomes narrower when Gaussian curvature is positive and wider when Gaussian curvature is negative, compared with that in flat space, which gives a new possibility to improve the imaging resolution using a curved surface with positive curvature. Using the transformation from a curved to a flat plane [36,37], one can use 2D planar waveguides with an equivalent refractive index profile to improve imaging resolution. For example, by conformally transforming a sphere to a plane with nonuniform refractive index distribution [37], the results we obtained on spheres should be also emerged in new transformed planes since they share same dynamics of light [36]. This provides a more practical method to realize optical super-resolution through a planar waveguide with the designed refractive index distribution. In addition, both the design of curved surfaces and refractive index profiles can be considered in freeform optics [46–49] to enhance imaging performance. Moreover, the diffraction limit may vary when the spatial curvature of the SORs is no longer a constant, and it may oscillate when the curvature alternates between positive and negative values, like on the SORs of PPSS. The effect increases when the magnitude of Gaussian curvature is larger and the propagation distance is longer. We also show that the resolution keeps constant for the arbitrary propagation direction of light when the spatial curvature is constant, and it varies as the local spatial curvature changes. This tells us that the uniformity of space may be reflected by the change of the diffraction limit. Our results will help in the control of optical resolution on curved surfaces and may also probe the non-uniform spacetime in universe space, or—in a more realistic perspective—one can expect a method to realize the optical super-resolution in a planar integrated waveguide structure in the principle of transformation optics. Moreover, our results may also provide inspiration for electron optics [50,51] extending into synthetic curved spaces [52] or non-Euclidean surfaces [53].
References
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[52] C. Lv, R. Zhang, Z. Zhai. Curving the space by non-Hermiticity. Nat. Commun., 13, 2184(2022).
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