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Abstract
When it comes to the high-spatial-frequency electromagnetic waves, we usually think of them as the evanescent waves which are bounded at the near-field surface and decay along with propagation distance. A conventional wisdom tells us that the high-spatial-frequency waves cannot exist in the far field. In this work, we show, however, that these high-spatial-frequency waves having wavenumbers larger than the incident one can propagate freely to the far-field regions. We demonstrate theoretically a technique, based on an abrupt truncation of the incident plane wave, to generate these intriguing waves. The truncation functions describing the slit and the complementary slit are considered as typical examples. Our results show that both the slit structures are able to produce the high-spatial-frequency wave phenomena in the far field, manifested by their interference fringes of the diffracted waves. This work introduces the high-spatial-frequency propagating waves. Therefore, it may trigger potential investigations on such an interesting subject, e.g., one may design delicate experiment to confirm this prediction. Besides, it would stimulate potential applications such as in superresolution and precise measurement.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As is known, the high-spatial-frequency electromagnetic wave processes a wavevector which is larger than the initial wavevector [1]. When it comes to the high-spatial-frequency wave, we usually think of it as a surface wave, because this wave decays along propagation direction and exhibits a transverse wavevector along the surface. These evanescent waves carry fine information about the objects. The fundamental reason why the spatial resolution of a conventional optical system is limited to half wavelength, i.e., Abbe diffraction limit [2], is just the absence of the high-spatial-frequency waves. In other words, the high-spatial-frequency waves can be exploited to break the diffraction limit, providing a mechanism for optical super-resolution. Currently, researchers have proposed many techniques to generate and make full use of the high-spatial-frequency waves, in order to achieve a higher spatial resolution [3–6].
Typical examples include the surface plasmon polaritons (SPPs) [7] and the Bloch surface waves (BSWs) [8]. These high-spatial-frequency surface waves can be excited by prism coupling [9], grating coupling [10] or multi-mode fiber mode coupling [11]. Let us first take the SPPs into account. The circular or the elliptical structure imprinted in a metallic film not only can excite the SPPs that exhibit higher wavevector than the incident one ${k_0}$; but also they play as a plasmonic lenses, able to focus the electromagnetic energy of the SPPs [12], with resolution breaking the diffraction limit. While for the BSWs, a stack of high and low refractive index layers can excite the BSWs with high spatial frequency, which allows for generating a subwavelength spot when they are arranged to focus [13,14]. Besides, it is worth mentioning that these surface waves (or the evanescent waves) can be transformed into propagating waves using specially designed optical devices, such as the negative-index superlens [15], the micro-sphere-based nanolens [16] and the hyperlens [17]. However, these elements face technological challenges of near-field operations and fabrications. As a result, if one wants to use these elements to resolve an object with ultrahigh resolution, the object to be imaged should be placed just several tens of nanometers away from the lenses [18,19]. Currently, the high-spatial-frequency waves only exist at the surface. A fundamental question is whether the high-spatial-frequency propagating wave exists in free space in the far field.
Recently, the concept of superoscillation has attracted much interest [20–22]. It is a wave phenomenon that a band limited wave function oscillates at a local spatial frequency exceeding its highest Fourier components. Therefore, a superoscillatory wave field usually associates with the fine spatial structure. A remarkable advantage of the superoscillatory wave field is capable of creating, in principle, arbitrary small feature size of light in the far field without the participation of the evanescent waves. However, the realization of the superoscillatory wave field relies on a superoscillatory lens (SOL) [21,23–26], that is a specially designed nanostructured mask. Such a SOL is complicated in designs and fabrications since it requires optimization algorithm for the design. On the other hand, the SOL-generated superoscillatory spot is actually a result of interference of many low-spatial-frequency propagating waves emitting from the mask. So far, direct generation of a freely propagating wave that processes high-spatial-frequency wavevector is still exclusive.
In this article, we predict theoretically that the high-spatial-frequency propagating wave does exist in the far field. To this end, we propose a simple method based on an edge truncation of the incident wave to generate the underlying high-spatial-frequency propagating waves. A remarkable phenomenon is that, different from the evanescent wave, the excited high-spatial-frequency waves can freely propagate to the far-field region. Such a high-spatial-frequency wave phenomenon is manifested by the far-field interference fringes. We demonstrate a theoretical model to govern these intriguing wave phenomena. Our demonstrations may trigger potential investigations on manipulating the high-spatial-frequency waves in the far field.
2. Principle and theoretical model
We start by considering the principle of generation of the propagating waves with high-spatial-frequency wavevectors. This is an extremely challenging issue. The reason is two folds: first, the light field emitted or diffracted from an ordinary object contains negligible propagating waves with high spatial frequency (the mostly high-spatial-frequency waves are the evanescent waves which decay exponentially with propagation distance as mentioned before); second, owing to the irregular diffractive geometry, the distribution of the diffraction-induced high-spatial-frequency wavevectors in the reciprocal space, generally speaking, is out-of-phase, which gives rise to destructive interference patterns in the far field. These two issues hinder the generation and observation of the high-spatial-frequency propagating waves in the far field. To overcome these problems, a planar structure with sharp edge and regular spatial geometry is required. When the sharp edge of the planar structure truncates the incident wave, it can induce significant diffraction components having different wavevectors. The geometrical shape of the structure is therefore used to manipulate the distribution of these diffraction components in the reciprocal space, so that they are in phase in a way to enhance the high-spatial-frequency wave signature. In such a manner, it is possible to generate the high-spatial-frequency propagating waves in the far field.
For example, we consider a simple planar structure, i.e., a one-dimension slit structure as illustrated in Fig. 1(a). An incident plane wave is diffracted seriously when it hits the edge of the slit structure. The transmission function is described by a $rect$ function ${t_1}(x )= rect(x )$, as illustrated in Fig. 1(b), which represents a binary modulation of the amplitude of the incident light at the slit edge. In the reciprocal space, the $rect$ function corresponds to a $sinc$ function of ${k_x}$ in the Fourier space, i.e., $\mathrm{{\cal F}}({{t_1}(x )} )= sinc({{k_x}} )$, where ${k_x}$ is spatial frequency with respect to x coordinate. Figure 1(c) illustrates the corresponding high-order frequency components of the $rect$ function, indicating significant high-order wavevectors ${k_1}$, ${k_2}$ and ${k_3}$ or ${k_1}^{\prime}$, ${k_2}^{\prime}$ and ${k_3}^{\prime}$, as marked by the red dots in the figures. Figure 1(d) shows three different cases of the generated wavevectors marked as ${k_1}$, ${k_2}$ and ${k_3}$, obviously deviating from the original one ${k_0}$, where ${k_0} = 2\pi /\lambda $, with $\lambda $ being the incident wavelength. Note that the higher-frequency wavevector would give rise to larger deviation angle, however, the energy carried by such a wavevector would become relatively smaller, as shown in Fig. 1(d). Owing to the symmetric edges with respect to the slit center, the induced diffractive waves having opposite transverse wavevectors (${k_x}$ and $- {k_x}$) coherently interfere at the slit center. Therefore, the high-spatial-frequency wave phenomena may be manifested by the resultant interference fringes.
Fig. 1. Principle of generating high-spatial-frequency propagating waves by the slit and the complementary slit structures. (a) and (e) Schematic diagrams of two types of slits. (b) and (f) The truncation functions used to describe the two slit structures in (a) and (e). (c) and (g) The corresponding spectrums of (b) and (f) in the Fourier space, which show significant high-order wavevectors, e.g., marked as ${k_1}$, ${k_2}$ and ${k_3}$ or ${k_1}^{\prime}$, ${k_2}^{\prime}$ and ${k_3}^{\prime}$. (d) and (h) Schematic illustrations of edge diffraction of (a) and (e) show that the high-order wavevectors deviate from original one ${k_0}$.
In another scenario, we consider a complementary obstructive slit, as shown in Fig. 1(e), with its transmission function described by ${t_2}(x )= 1 - {t_1}$, i.e., two built-up $step$ function $[{step(x )+ step({ - x} )} ]$, as illustrated in Fig. 1(f). Figure 1(g) and 1(h) show the distribution of the induced high-frequency wavevectors and their deviation directions, respectively. It is worth noting that these high-spatial-frequency wavevectors are able to bypass the structure edges and reach to the area on the back. Similar to the slit structure, the symmetric edges of the obstructive slit give rise to constructive interference pattern of the diffracted waves. Therefore, both the two simple cases of the structure edges allow to reveal the high-spatial-frequency propagating waves.
To verify the analysis, we theoretically investigate the far-field diffraction patterns with the above two types of planar structures. The propagation dynamics of the diffracted waves beyond the planar structures (we placed them at the plane of $z = 0$) can be governed by the paraxial Helmholtz wave equation
We first consider the transmission function of a slit structure. Since the thickness of the structure is negligible in comparison the width of the slit structure, here we adopt the Kirchhoff boundary condition, i.e., we assume that an incident plane wave passing through the slit can be expressed as
where ${E_0}$ is the plane wave amplitude, and d denotes the width of the slit. Accordingly, we find the solution written asWith this formula, Eq. (4) can be reduced to
We proceed to reveal the diffractive wave phenomena. According to the character of Cornu Spiral [28], which represents the function image of the Fresnel integrals, we obtain relation ${F_c} = {F_s}$ at the maximum and minimum points of $F(\omega )$. At these extreme points, we have
accordingly, we getWhen n is even value, we can obtain the maximum of $F(\omega )$; however, when n is odd value, we find the minimum of $F(\omega )$.
From these conditions, substituting the integral boundaries of Eq. (4) into $\omega $, we derive a set of locations at which the diffracted field has maximum or minimum values. Specifically, $E({z,x} )$ has the maximum value at
while $E({z,x} )$ has the minimum value atThe above theory can be applied to the complementary obstructive slit. In this case, the plane wave field right behind the obstructive slit is expressed as
The resultant diffracted filed after a propagation distance of z can be expressed as
It is clear that owing to the complementarity between these two slit structures, the diffracted light field in Eq. (12) is also complementary to that shown in Eq. (6). Namely, when $Re[{E({z,x} )} ]$ has maximum (minimum) value at the location $({x,\; z} )$, ${|{{E_c}({z,x} )} |^2}$ has maximum value at the same place.
3. Simulated results and discussion
Based on the above theoretical analysis, we carried out simulations with a plane wave condition with wavelength $\lambda = 632.8\; nm$ (not limited to this, other wavelengths are also applicable). In the simulations, a typical slit with the width setting as $d = 9\; \mu m$ is considered. Figure 2(a) shows the simulated intensity distribution of the diffracted light fields in the $x - z$ plane, according to Eq. (6). As expected, it shows the elaborate interference behavior of the diffracted waves. The intensity of the interference fringes is increasing with an increase of the propagation distance. This effect confirms the above assertion that the energy carried by the diffracted wavevectors is increasing with the propagation distance. On the other hand, we present the constructive and destructive interference regions, corresponding to the maximum and minimum values of the diffracted light field based on Eqs. (9) and (10), with result shown in Fig. 2(b). Note that the intersections between the red solid and dotted lines represent the maximum points; while the intersections between the black solid and dotted lines represent the minimum points. Clearly, the distributions of these extreme points in the $x - z$ plane are the same as those shown in Fig. 2(a), confirming the accuracy of the theoretical models in Eqs. (9) and (10).
Fig. 2. (a) and (c) The simulated intensity distributions of the diffracted light fields in the $x - z$ plane, for two different edge structures: (a) the slit structure; and (b) the complementary slit structure. Both the slit width is setting as $d = 9\; \mu m$. (b) The distribution of the extreme points of the diffracted light fields shown in (a). The intersections between the red lines represent the maximum points; while the intersections between the black lines represent the minimum points. (d) Lines representing the constructive interference fringes shown in (c).
Similarly, according to Eq. (12), we also obtain the distribution of diffracted light field by the obstructive slit as shown in Fig. 2(c). These interference fringes are generated at the back of the obstructive slit. It is obvious that the interference fringes near the slit ($z$ is relatively small) are dense and their intensities are relatively weak, manifesting the high-spatial-frequency wave phenomenon. This result in Fig. 2(c) is similar to the outcome shown by Fig. 2(a). The crossings of the parabolas maxima of the intensity pattern are also presented in Fig. 2(d), in consistence with result in Fig. 2(c). The only difference between these two structures is that the slit structure [Fig. 1(a)], owing to spatial confinement of the light field, gives rise to a serious of discrete extreme points in the $x - z$ plane, while the obstructive slit structure produces a continuous distribution of the interference fringes along the $x - z$ plane. This effect fundamentally originates from their corresponding discrete frequency band and continuous frequency band, as shown in Fig. 1(c) and 1(g), respectively.
In order to reveal the underlying high-spatial-frequency wave phenomena, we present the amplitude distribution of the diffracted light field along transverse axis x at different propagation distances. Three different cases of slit widths are considered here. For $d = 11\; \mu m$, we plot amplitude profiles (red curves) of the high-spatial-frequency waves along x axis, at propagation distances of $z = 4.5\; \mu m$, $z = 5.5\; \mu m$ and $z = 7.0\; \mu m$, as shown in Fig. 3(c), 3(b) and 3(a), respectively. In comparison, the amplitude oscillating profile of a propagating wave whose wavevector is ${k_0}$ is also plotted, see the blue curves. With these conditions, we can quantificationally determine the spatial frequency ${k_x}$ of the diffracted light wave. For example, when the propagation distance is at $z = 5.5\; \mu m$, the oscillation frequency, i.e., the spatial frequency ${k_x}$ of the diffracted light field, is exactly equal to ${k_0}$, as seen in Fig. 3(b). At this critical point, we find that the propagation distance is exactly half of the slit width, i.e., $z = d/2$. So, we define this propagation distance as the critical distance ${z_c}$. When the propagation distance $z > {z_c}$, the oscillation frequency is ${k_x} < {k_0}$, as shown by the result in Fig. 3(a); when the propagation distance is small, i.e., $z < {z_c}$, the oscillation frequency is ${k_x} > {k_0}$, which means that we do generate the high-spatial-frequency waves. Clearly, this wave is not bounded at the surface, but able to propagate to the far field, in contrast to the evanescent wave (even though its spatial frequency is high). Further, we measure numerically the value of ${k_x}$, by the formula ${k_x} = \frac{{2\pi }}{{{\lambda _x}}} = {k_0}\frac{\lambda }{{{\lambda _x}}}$. Here $\lambda $ is the amplitude oscillation period of the initial light wave, i.e., $\lambda = 632.8\; nm$, and ${\lambda _x}$ represents the amplitude oscillation period of the diffracted light field. For example, when the propagation distance $z = 4.5\; \mu m$, ${k_x} \approx 1.2{k_0}$.
Fig. 3. The amplitude distributions (red curves) of the light fields at different propagation distances, diffracted by the slit with different widths: (a)-(c) $d = 11\; \mu m$, (d)-(f) $d = 16\; \mu m$, and (g)-(i) $d = 21\; \mu m$. In comparison, blue curves represent the amplitude distributions of the wave whose wavenumber is ${k_0}$.
We also present the results for another two cases of slit widths, as shown in Fig. 3(d)-(f) and Fig. 3(g)-(i), respectively. The similar oscillatory features can be found. The critical distance ${z_c}$ is found as ${z_c} = 8.0\; \mu m$ and ${z_c} = 10.5\; \mu m$, for $d = 16\; \mu m$ and $d = 21\; \mu m$, respectively, both satisfying the relationship ${z_c} = d/2$. It is worth noting that, these high-spatial-frequency wave phenomena can be also revealed by the obstructive slit, as shown by the results in Fig. 4. For instance, we plot the amplitude profiles of the diffracted light field along x coordinate, at different propagation distances. Similarly, it can be found that the high-frequency components having ${k_x} > {k_0}$ are induced by the obstructive slit structure at $z = 4.5\; \mu m$, $z = 6.9\; \mu m$, and $z = 9.3\; \mu m$, as seen from Fig. 4(c), 4(f) and 4(i), respectively. And the critical distance also satisfies the relationship ${z_c} = d/2$ for both the considered cases of the obstructive slits.
To verify the finding of ${z_c} = d/2$, which is the boundary between the high- and low-spatial- frequency waves, we examine the diffractive wave phenomena using both the slit and the obstructive slit structures. We study the situation where the slit width is ranging from $d = 6\; \mu m$ to $24\; \mu m$, and obtain their corresponding critical distances ${z_c}$. The data points are plotted in Fig. 5(a) and 5(b), respectively, which obviously satisfy the formula of ${z_c} = d/2$. In order to find out the reason, we conducted a detailed mathematical analysis. Accordingly, Eq. (4) can be written as
Fig. 4. The amplitude distributions (red curves) of the light fields at different propagation distances diffracted by the complementary obstructive slit with different widths: (a)-(c) $d = 11\; \mu m$, (d)-(f) $d = 16\; \mu m$, and (g)-(i) $d = 21\; \mu m$. In comparison, blue curves represent the amplitude distributions of the wave whose wavenumber is ${k_0}$.
Fig. 5. The relation between the critical propagation distance ${z_c}$ and the width d of the slit (a) and the complementary obstructive slit (b). The data points in both figures are obtained numerically based on Eqs. (6) and (12), while the solid lines are based on the formula ${z_c} = d/2$.
It is obvious that the spatial frequency of $E({z,x} )$ depends only on the term $\mathop \smallint \nolimits_{ - d/2}^{d/2} exp \left[ {\frac{{i{k_0}}}{{2z}}{{({x - x^{\prime}} )}^2}} \right]dx^{\prime}$. Let's write it as
To obtain generalized relationship between the critical distance ${z_c}$ and the slit width d, we take the derivative of Eq. (14) with respect to the slit width. In this case, we have
And take the real part of Eq. (15), we finally get
It can be seen from Eq. (16) that the high-order term ($sin\left[ {\frac{{{k_0}}}{{2z}}\left( {{x^2} + \frac{{{d^2}}}{4}} \right)} \right]$) can be ignored near $x = 0$. Therefore, we focus on $sin\left( {\frac{{{k_0}d}}{{2z}}x} \right)$ and find that when $z = d/2$, the spatial frequency of the diffracted light field near the region of $x\sim 0$ is always equal to the frequency of the incident plane wave, i.e., ${k_x} = {k_0}$, regardless of the variation of slit width d. Namely, for a slit structure with the arbitrary width, this critical distance is always equal to half of the slit width, i.e., ${z_c} = d/2$, and not anywhere else. On the other hand, similarly, for the complementary obstructive slit structure, the light field diffracted by it can be rewritten as
According to the analysis above, we can still get the same conclusion, namely, ${z_c} = d/2$. This finding is useful as we can use a larger size of slit structure to generate the high-spatial-frequency waves in a longer distance that is many times wavelength from the sample.
4. Conclusion
In conclusion, we have verified theoretically that the high-spatial-frequency propagating wave does exist in free space. We have proposed theoretically a straightforward method to generate these high-spatial-frequency waves that can propagate in free space. Such waves are contrary to the well-known evanescent waves which are bounded at the near-field surface and decay along with propagation distance. Specially, we realize the high-spatial-frequency propagating waves by a binary modulation of the incident plane wave using a slit and a complementary slit structure. In this way, the edge of the structure induces a large number of the high-frequency components propagating to the far field, and these high-frequency components are arranged to interfere constructively near the slit center. The high-spatial-frequency wave phenomenon is therefore manifested by the oscillating signal of the diffracted waves in the far field. It is worth to mention that a critical condition is found for these high-spatial-frequency wave phenomena. Our study will open new possibilities to manipulate the high-spatial-frequency waves in the far-field regions, e.g., it can be naturally generalized to two-dimensional planar structures [29,30], and we believe this simple and easy-to-perform technique may find interesting applications such as in superresolution imaging [21], and nanoparticle manipulation [31].
Funding
National Natural Science Foundation of China (11974146, 62175091); Guangdong Provincial Pearl River Talents Program (2017GC010280); Special Project for Research and Development in Key areas of Guangdong Province (2020B090922006); China Postdoctoral Science Foundation (2021M701436).
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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