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Abstract
Inspired by the capability of structured illumination microscopy (SIM) in subwavelength imaging, many researchers devoted themselves to investigating this methodology. However, due to the free-propagating feature of the traditional structured illumination fields, the resolution can be only improved up to two-fold of the diffraction-limited microscopy. Besides, most of the previous studies, relying on incoherent illumination sources, are restricted to fluorescent samples. In this work, a subwavelength non-fluorescent imaging method is proposed based on the illumination of terahertz traveling waves and plasmonics. Excited along with a metal grating, the spoof surface plasmons (SSPs) are employed as one of the illuminating sources. When the scattering waves with the SSPs illumination are captured, the sample’s high-order spatial frequencies (SF) components are already encoded into the obtainable low-order ones. Then, a modified post-processing algorithm is exploited to shift the modulated SF components to their actual positions in the SF domain. In this manner, the fine information of samples is introduced to reconstruct the desired imaging, leading to an enhancement of the resolution up to 0.12λ0. Encouragingly, the resolution can be further enhanced by attaching extra illumination of SSPs with an elaborately selected frequency. This method holds promise for some important applications in terahertz non-fluorescent microscopy and sample detection with weak scattering.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Owing to the superiorities in negligible invasiveness [1], optical microscopy has impacted numerous fields, including biological imaging, material identification, precision instrument inspection, etc. However, in conventional far-field microscopy, one of the major limitations is the insufficient resolution, which is constrained to approximately half the illumination wavelength [2]. This phenomenon is termed the Rayleigh limit or classical diffraction limit [3]. Specifically, the samples’ fine information is encompassed in the high-order spatial frequency (SF) components, manifesting as the evanescent waves. The evanescent waves attenuate exponentially perpendicular to the propagating direction and permanently vanish in the far-field [4]. Thus, the fine information of the sample is absent in the image reconstruction, posing a wavelength level resolution under optimal conditions [5].
To overcome this resolution limitation, a series of attempts have been developed over the past decades. For instance, the super-resolution fluorescence microscopy techniques, including stimulated emission depletion [6,7], stochastic optical reconstruction [8,9], structured illumination microscopies (SIM) [10–12], have endowed a breakthrough in subwavelength resolution imaging. Among the myriad fluorescence super-resolution approaches, SIM is considerably remarkable due to its fast and highly parallelizable feature [13]. In SIM, an elaborate structured illumination pattern is applied to impinge the sample. In this case, the high-order SF components (corresponding to the fine detail of the sample) can be down-modulated into the passband of the objective and turn out to be detectable. By post-processing, the high-order SF information can be precisely shifted and located at the true position in the SF domain (k-domain or Fourier domain). As a result, one can defeat the diffraction barrier and achieve a subwavelength resolution.
However, owing to the free-propagating feature of the structured illumination fields [14], the spatial resolution of traditional SIM is restricted to a two-fold enhancement in comparison with the diffraction-limited techniques [15]. Aside from the resolution, the application scenarios of SIM should be taken into consideration. Most of the aforementioned researches belong to incoherent imaging and depend heavily on fluorescent markers, which become out-of-operation for label-free imaging [15–20]. From the perspective of the terahertz range, the situation is similar. The existing researches focus on fluorescent samples and realize an excellent subwavelength resolution [21]. Although the SIM based on fluorescent labeling develops rapidly, the coherent and non-fluorescent imaging method can be also desired for the applications. However, to the best of our knowledge, the terahertz label-free subwavelength imaging with structured illumination method has not been reported.
In this work, a terahertz subwavelength imaging method of label-free samples is proposed based on composite photonics-plasmonics structured illumination. The spoof surface plasmons (SSPs) subsisted on a metal grating are employed as the plasmonics illumination. When the sample is impinged by the SSPs, the scattering field is encoded. That is, the relatively high-order SF components (carrying fine information of the sample) are down-transformed to the low-order and obtainable SF ones. Then, the scattering field intensity is caught to get the sample’s fine information. As for the low-order SF information of the sample, vertical and oblique incidences of traveling waves (photonics illuminations) are applied. Subsequently, a modified iterative algorithm is applied to extract and locate all the SF components in their true positions. In this way, a wide SF spectrum is generated, which is imperative for subwavelength imaging. Based on this principle, the simulated resolution of the proposed method reaches up to 0.12λ0. Note that the resolution can be further improved by adding other SSPs illumination with elaborately selected working frequencies. This method can be beneficial for terahertz nonfluorescent microscopy and the detection of weak scattering samples.
2. Theory
2.1 Imaging principle
The imaging principle in this work relies on the coherent scattering from the label-free samples. Before elaborating on the imaging principle, there are several settings to be clarified. Firstly, this process is between the spatial domain (x-y) and the k-domain, where k is the SF vector. Here, we consider the condition where the sample is positioned along the x-direction and infinite in the y-direction. The incident wave is on the x-z plane, lacking the SF component ky. Secondly, we postulate that when an incident wave impinges on a thin subwavelength sample, the scattered wave can be illustrated as s(x) = o(x)exp(iktx). o(x) refers to the complex field of the sample [2] and kt is the SF tangential component of the incident wave. Thirdly, the coherent point spread function (PSF) hc(x) of our imaging system is defined via its Fourier transform Hc(kx). Hc(kx) is modeled by an abrupt rectangular low-pass filter, which has a width of 2kc (kc is the cut-off frequency). Since the imaging system of this work is in the free space, kc can be substituted by k0 (the wavenumber of vacuum). Thereby, Hc(kx) is rendered into gray in Fig. 1 (a), where C denotes the module value of the SF spectrum. On this basis, the detected intensity imaging i(x) of the sample is derived as the following equation [2,22]
where $\otimes$ is the convolution operator. Equation (1) can be written in the k-domain [23], which is signified with Eq. (2) where S(kx) and I(kx) are the SF distributions of the sample’s scattering field and the detected image intensity, respectively. It should be noticed that S(kx) = O(kx − kt), where O(kx) is the Fourier transform of o(x). Equation (2) thus can be derived as In this manner, the detected image intensity I(kx) contains the attainable SF spectrum, which is governed by | kx − kt | < kc. Since kc = k0, it is written as −k0 + kt < kx < k0 + kt.
Fig. 1. Detectable SF ranges under different illumination patterns. (a) Vertical illumination. (b) Oblique illuminations. (c) Evanescent illuminations.
According to the aforementioned principle, the range of the obtainable SF spectrum is attributed to the tangential component kt of the illuminating wave. When a plane wave impinges the sample vertically, the obtained SF region is −k0 < kx < k0 (the same as the system’s passband), as shown in Fig. 1(a). Analogously, a plane wave with left oblique illumination leads to the acquisition of the SF spectrum in −k0 − k0sinθ < kx < k0 − k0sinθ (θ is the incident angle). For the right oblique incidence, the acquired range is −k0 + k0sinθ < kx < k0 + k0sinθ (Fig. 1(b)). Since θ < 90°, the detectable region is limited by ±2k0.
To further extend the observable region, the evanescent wave is also exploited as the illuminated wave (along both x and −x-directions). It should be noted that the tangential component keva of the evanescent wave is larger than k0. Thus, the attainable SF regime (−k0 ± keva < kx < k0 ± keva) is enlarged further, as depicted in Fig. 1(c).
Finally, through judiciously tailoring the incident angle of the plane wave and the tangential component of the evanescent wave, a wide integrated k-domain spectrum can be spliced with no empty regions. It is noted that the wide information in the k-domain is imperative for high-resolution imaging. Consequently, utilizing the inverse Fourier transform, a subwavelength spatial image of the sample is restored.
2.2 Reconstruction algorithm
After gathering all of the raw intensity images under different illuminated patterns, a meliorated Gerchberg-Saxton algorithm [24,25] is exploited to iteratively generate the wide k-domain spectrum. In this way, a subwavelength-resolution image can be realized. The procedure involves five steps.
Step 1. An initial value Is1/2eiφ of the subwavelength sample in the spatial domain is assumed, using φ = 0 and Is being a constant value (detected data group is also available [2]). Applying the Fourier transform to the initial value, a wide spectrum W(kx) can be presented in the k-domain
Step 2. The low-order SF components are restored with the image under vertical illumination, i.e., oblique illumination with an incident angle of 0°. A sub-region of the wide k-domain spectrum is extracted and dealt with the invert Fourier transform, producing a new temporary image It1/2eiφt. The selected area in the k-domain hinges on the low-pass filter Hc(kx). The location of the low-pass filter depends on the tangential wavenumber of the illumination pattern. Under the vertical incidence (tangential wavenumber being zeros), the position of the filter’s center is kx = 0. Thus, this sub-region is −k0< kx < k0. Resultantly, the aforementioned procedure is explicated asStep 4. Repeating steps 2 and 3 until a self-consistent solution is realized.
Step 5. Applying inverse Fourier transform to the converged solution Wsolved(kx), a spatial image is obtained. After an origin calibration, the final subwavelength image isolved(x) is recovered. In section A of Supplement 1, pseudocodes of these procedures are demonstrated. Compared with the Gerchberg-Saxton algorithm, the slightly modified parts are also presented.
Aside from the aforementioned reconstruction algorithm, there are various reconstruction methods for SIM imaging. For example, the authors in [15] change the phase of the illumination patterns several times and capture the raw images under these illumination patterns to promote their image reconstruction. In this work, based on this iteration algorithm, it is not necessary to concern the phase of the illumination waves.
2.3 SSPs illumination field
As mentioned above, evanescent waves are applied for illumination to achieve a down-modulation of the high SF information. In the optical region, the authors in [15] and [16] apply the surface plasmon polaritons (SPPs) and localized plasmons (LPs) as the illumination evanescent field, respectively. Both the SPPs and the LPs have the intrinsic properties of strong near-field enhancement and subwavelength confinement . To be specific, the metals’ permittivities have negative real components in the optical range, which leads to the evanescently confined fields in the metal and dielectric [26]. Thus, the SPPs and LPs can be employed as evanescent wave illumination [27,28]. However, metals display the ideal electric conductor property in the terahertz region, as a result of which natural SPPs are severely declined [29]. To solve this problem, the SSPs are applied for the evanescent wave illumination in this work. Compared with the SPPs and LPs illumination, the SSPs aim to mimic the properties of nature SSPs in the terahertz region. In this case, the SSPs propagate on a metal grating surface and decay exponentially along the vertical direction [30,31]. Via manipulating the working frequencies of the SSPs, a wealth range of tangential wavenumbers (keva) are available, benefiting the system design.
The schematic diagram of the proposed approach is shown in Fig. 2, wherein a metal grating is applied to sustain the SSPs on the surface. Two thin samples with different relative permittivities are deposited above the grating in a subwavelength distance. Leveraging on a coupler between the SSPs and traveling waves, the SSPs are excited adequately in this device. Then, the SSPs propagate along the grating and impinge on the samples, producing a detectably scattering wave. After that, the SSPs propagate forward and exit as a traveling wave through another coupler. It is worth emphasizing that the SSPs can transmit along both the x-direction and −x-direction, just depending on which coupler it is excited on.
Fig. 2. Schematic diagram of the SSPs illumination, where Ex is the x-direction component of the scattering field.
Specifically, the grating’s unit cell is presented in Fig. 3(a), where h = 0.049 mm, p = 0.02 mm, and d = 0.01 mm (the same size in all simulations). The relationship between the tangential wavenumbers (along the x-direction) of SSPs and their working frequencies is demonstrated by the dispersion equation [32]
Fig. 3. Dispersion of the grating and the SSPs field of different working frequencies. (a) Unit structure and fundamental mode dispersion curves of the grating. (b) SSPs field at f1. (c) SSPs field at f2. (d) SSPs field at f3.
In this manner, selecting the working frequencies of the SSPs is almost equal to choosing the tangential wavenumbers. The tangential wavenumbers decide the position of the obtainable k-spectrum under the corresponding SSPs illumination. It should be noticed that this image reconstruction method requires overlap between the adjacent k-spectrum encompassed by each raw image [24]. Thus, to promote the reconstruction process, the working frequencies are elaborately selected to ensure a suitable overlap ratio. Through an optimization of the overlap, the tangential wavenumbers of SSPs are presented as keva1 = 1.387k0, keva2 = 2.082k0, keva3 = 3.539k0, corresponding to the working frequencies f1 = 1.1 THz, f2 = 1.23 THz, and f3 = 1.33 THz. k0 is calculated as 2πf0/c, where c is the velocity of light. f0 is the working frequency of the vertical and oblique plane waves, set as 1.185 THz. As the working frequencies increasing, keva becomes larger, resulting in a more confined SSP field [Figs. 3(b), (c), and (d)]. Based on this rule, further promotion of the imaging resolution is feasible by adding other SSPs illumination with a working frequency higher than f3. With a higher working frequency, larger SF components can be down-modulated into the passband of the system. Thus, it leads to a larger effective numerical aperture NA = (max{keva1, keva2, keva3, keva4, … } + max{feva1, feva2, feva3, feva4, … } / f0)k0 and a higher resolution [calculated as λ0/(2NA)]. Since more illumination patterns cause a more sophisticated post-process, a balance between the resolution and number of illumination patterns should be elaborately determined.
3. Simulation and results
Hereto, the procedure of how to recover a subwavelength image with the composite photonics-plasmonics structured illumination is perspicuous. First, various illumination patterns are exploited to encode the SF spectrum of the sample. To be specific, there are nine detected raw images for the reconstruction, corresponding to the vertical illumination, right and left oblique illuminations, and the SSPs illuminations at three frequencies (f1, f2, and f3). At each frequency, the SSPs illuminate twice with opposite propagating directions (propagating along the x and −x-directions, respectively). Then, for the data collection, the scattering field intensity of the sample is directly detected, which avoids extra laser to excite the sample [21]. Finally, a post-process algorithm is applied to reconstruct images. In the following, the aforementioned processes are demonstrated by the simulation and the final image is retrieved.
The simulation procedure is established in the commercial software COMSOL Multiphysics 5.4a [34]. When the program is initialized, the physic field and the study are set as “wave optics, electromagnetic waves, frequency domain” and “frequency domain”, respectively. After the parameters and geometric shapes are set up, the sample and the grating are set as dielectric and copper, respectively. The remaining solution area is assigned to be air. Besides, the boundary of the simulation area is set as the “scattering boundary condition”. For the propagating illumination, a background electric field is added to the program. The polarization of the electric field is along the x-direction for the vertical incidence. The oblique incidence is realized via changing the propagating directions of the background field. For the data collection, a post-process calculation is applied to obtain the x-polarized scattering electric field (presented as relEx). As for the SSPs illumination, the main change is the illumination setting. In this case, a grating with two couplers is added to the program. Two “ports” are imposed at the entrances of the couplers. The polarization of the incident electric field at the port is along the y-direction. The propagating directions of the SSPs can be controlled by turning on or off the wave excitation at the two ports.
Thus, with the assistance of the commercial software COMSOL Multiphysics 5.4a [34], the scattering field intensity is investigated under diversiform illuminations. Take the single sample (0.15λ0×0.1λ0 on the x-z plane) imaging as an example, for the vertical illumination, the sample is directly impinged by a plane wave [see Fig. 4(a)]. Similarly, plotted by Figs. 4(b) and (c), the oblique patterns employ plane waves with ±66° incident angles. The corresponding tangential wavenumbers are ±0.914k0. For the evanescent illumination, the sample is placed above the grating surface with an extremely close distance, which is approximately0.024λ0−0.12λ0. Figures 4(d) and (e) demonstrate the scattering field intensity with the SSPs propagating along x and –x-directions (working frequency is f1). In addition, the SSPs illuminations at f2 and f3 are also simulated. Consequently, the tangential wavenumbers along the x-direction of the nine illumination waves are [0, 0.914k0, −0.914k0, 1.387k0, −1.387k0, 2.082k0, −2.082k0, 3.539k0, −3.539k0]. In Fig. 4(f), the nine obtained SF spectrums are presented. Since the working frequencies of traveling waves and SSPs are distinct, the corresponding width of the detectable SF regions are derived as 2 (fi / f0) k0 = 4πfi / c,Th i = 0, 1, 2, and 3.
Fig. 4. Scattering fields intensity with various illuminations for the corresponding SF regions. (a) Scattering field intensity with the vertical illumination. (b) Right oblique illumination. (c) Left oblique illumination. (d) SSPs illumination along with the x-direction (working frequency is f1). (e) SSPs illumination along with the −x-direction (working frequency is f1). (f) Corresponding detectable SF regimes of diverse illuminated patterns.
Next, the scattering field intensity under these nine illumination patterns is directly collected, without the laser excitation in [21]. For the resolution investigation, the length of the detective line on the x-z plane is 2540 μm, which is 268 μm away from the sample. With the post-process algorithm, a wide k spectrum is reconstructed. Then the k spectrum is processed by the apodization and zero-padding function to extract the critical SF information and reveal smooth results, respectively.
The first recovered result is the imaging of a single, subwavelength sample, whose dimension is 0.15λ0 × 0.1λ0 on the x-z section. The relative permittivity of the sample is ε = 2.5, characterizing the dielectric property. The reconstructed images of this sample are depicted in Fig. 5(a), which contain various illumination groups. The black curve illustrates the result generated by the whole nine illuminations (group1), presenting a 0.11λ0 full width at half maximum (FWHM). When the SSPs illuminations at f2 and f3 are absent in the reconstruction process (group2), the FWHM becomes larger than before, as designated by the blue curve. Ulteriorly, if only the vertical and oblique illuminations (group3) remain without any SSPs illumination, the FWHM increases further. Corresponding to these illumination combinations, the modulus of the k spectrum is plotted in Fig. 5(b). The white region (around k0 = 0) represents the attainable SF spectrum only with vertical and oblique patterns. As the SSPs illumination at f1 is added, the achieved SF spectrum is the white plus orange regions. When all of the SSPs illuminations participate, the homologous SF region includes the white, orange, and gray pieces. Apparently, high SF information in the k-domain is obtained with the relevant SSPs illumination, leading to a small FWHM. From the point of Abbe's criterion (imaging a single sample) [35], this phenomenon is rational. According to the criterion, smaller FWHM means higher resolution, which is attained through collecting more fine information (high SF regions). Since the SSPs illumination brings high SF in k-domain, employing them naturally causes the high resolution and small FWHM. Finally, based on Abbe’s criterion, this work can realize a 0.11λ0 resolution.
Fig. 5. Simulated results of an isolated sample. (a) Images with three illuminating wave groups (group 1, 2, and 3). (b) Modulus value of the Fourier spectrum. (c) Images with ±0.5λ0 displacement. (d) Images with different relative permittivities.
Besides, to verify the feasility of this method, the sample is repositioned along the x-axis. Assuming the sample is placed at the origin for the aforementioned discussion. Whereupon we move it to ±0.5λ0 positions, keeping the other conditions fixed. The result is displayed in Fig. 5(c), testifying the method’s stability with disparate locations. There exists a slight difference between the two curves, which is mainly caused by the personalized iteration number when the convergent solution is generated. Another issue that needed to be explored is whether the sample’s dielectric feature has an impact on the reconstruction. The relative permittivity is set as 1.5, 2.5, 3.5, and 4.5 with the other conditions being identical. Comfortingly, Fig. 5(d) illustrates that the restored imaging keeps stable, meaning that the method is robust to the variation of the relative permittivity within a certain range.
Up to here, we have demonstrated the 0.11λ0 resolution and the robustness of this method. However, there are many other criteria to evaluate the imaging resolution in the literature. The most likely accepted is the Rayleigh criterion [3], which confirms the minimum resolvable distance via imaging two closely separated samples. In the following, the resolution in the view of the Rayleigh criterion is presented.
First, we consider two same samples with ε = 2.5, h = 0.1λ0, and width ws. They are separated by a gap, whose length is wg = ws. We recover the images with ws ranging from 0.15λ0 to 0.225λ0. Figures 6(a)-(c) depict the images under these conditions, verifying the method’s subwavelength imaging ability of two samples. Furthermore, in Fig. 6(d), the samples are set extremely small with h = ws = 0.02λ0 for the judgment of the highest resolution. The distance between the center of them is wc = 0.12λ0. In this case, the samples can be approximately regarded as two points with a separate distance approximately being 0.12λ0. To increase the interaction between the tiny samples and the illuminating waves, their relative permittivities are raised to 12. Under this condition, the result verifies that a resolution of 0.12λ0 can be achieved [black curve in Fig. 6(d)]. Subsequently, when the samples get closer to each other (0.11λ0), the image becomes inexplicit. As plotted in Fig. 6(d), one peak is almost a quarter lower than the other, illustrating that the method cannot distinguish two samples separated by 0.11λ0. Thus, on basis of the Rayleigh criterion, the ultimate resolution of this method is 0.12λ0. Actually, there exists a slight deviation about the resolution with the Abbe and Rayleigh criterion. One reason is that the samples in the Rayleigh criterion have a 0.02λ0 width, and the gap between them is actually shorter than 0.12λ0. Besides, the undesired reflection between the two samples conveys noise to the reconstruction and leads to a slight deviation.
Fig. 6. Verification of the imaging capability of two samples. (a) wg = ws = 0.225λ0. (b) wg = ws = 0.175λ0. (c) wg = ws = 0.15λ0. (d) ws = 0.02λ0, wc = 0.12λ0 (black curve and samples), and wc = 0.11λ0 (red curve and samples).
Hereto, the theoretical resolution and the simulated resolution can be exhibited and compared. For this work, the effective NA is decided by the maximum keva of the SSPs illumination, which is 3.539k0. Thus, the highest detectable SF in k-domain is ±(3.539 + f3/ f0) k0 = ±4.66k0, i.e., NA = 4.66. In this way, the theoretical resolution is approximately 0.107λ0. According to Abbe’s criterion, the simulated resolution is 0.11λ0. Considering the approximation error, it is almost consistent with the theoretical value. With the Rayleigh criterion, the established resolution is 0.12λ0, which has a slight excursion. The offset may be caused by the finite size of the samples and the undesired reflection between them. Finally, the imaging resolution is demonstrated as 0.12λ0, almost four-fold of the diffraction-limited imaging.
Besides, we also restore images of two samples with various relative permittivities. At the beginning, we employ two samples (h = 0.1λ0, ws = 0.15λ0, wg = 0.2λ0) with the same relative permittivity ε = 2.5 as a reference standard, presented in Fig. 7(a). Afterward, the relative permittivities of the left and right samples are stated as ε1 and ε2, respectively. In Fig. 7(b), it is displayed that when ε1 = 2.5 and ε2 = 4.5, the right peak’s height is larger than the left one. Analogously, for ε1 = 2.5 and ε2 = 1.5, the left peak is higher than the right peak, shown in Fig. 7(c). Actually, due to the sample’s different scattering power with diverse relative permittivities, there emerge unequal peaks. Intuitively, a large relative permittivity yields strong scattering under the impinging wave. To some extent, this method can distinguish two samples with distinct relative permittivities, which may be available for subwavelength dielectric determination. It should be mentioned that the ambiguous image in Fig. 6(d) also has unequal peaks, demonstrating the resolution limit. This means when one peak behaves lower than the other, the reason is not only the inconsistent ε. Thus, this function of the relative permittivity distinction can be exploited to two same-size samples with relatively large intervals.
Fig. 7. Discussion on imaging properties. (a) Image of two identical samples. (b) Image of two samples with ε1 = 2.5 and ε2 = 4.5. (c) Image of two samples with ε1 = 2.5 and ε2 = 1.5. (d) Image of three same samples with wg1 = 0.25λ0 and wg2 = 0.45λ0.
Finally, the imaging ability of three samples is also demonstrated in Fig. 7(d). Under this condition, the detected line is moderately adjusted to fit the extended space of the samples. There are three same samples (h = 0.1λ0, ws = 0.15λ0, ε = 2.5) with the gap distances wg1 = 0.25λ0 and wg2 = 0.45λ0. Consequently, they are successfully reconstructed.
4. Discussion
Firstly, the resolution can be further improved by adding other SSPs illuminations, which work at higher frequencies compared with the aforementioned simulation. In this manner, the effective NA is increased so that the resolution can be enhanced. Remarkably, the working frequencies need to be advisably chosen to ensure a wide k spectrum without an empty region [24].
Secondly, compared with the laser excitation method in [21], our method greatly simplifies the detection process. In this work, the sample’s scattering field intensity is collected straightly, circumventing the extra laser excitation process. Besides, compared with the incoherent imaging based on fluorescence sample in [15–16], this work realizes non-label and coherent imaging. Both the illumination fields and the retrieve methods between this work and those in [15,16] are different.
Thirdly, the noise in this system may cause side lobes and artifacts in the reconstructed imaging, which can be further reduced through the high fidelity algorithm [36]. Besides, the influence of couplers’ reflection and the higher harmonics on the grating can also be sufficiently suppressed to improve the imaging performance. Specifically, by optimizing the structure of the coupler, the reflection can be weakened. For the higher harmonics on the grating, it can be averted by increasing the distance between the samples and the grating.
Fourthly, a possible experimental scheme is described as follows, the sketch of which is presented in section B of Supplement 1. Before the experiment, the grating may be fabricated by microfabrication and micro-electro-mechanical technology. The coupler can be modified to match the actual situation (more details in section B of Supplement 1).
For the generation of a terahertz wave, a photoconductive antenna (PCA) is applied as the emitter [37]. The PCA is triggered by the pump pulse coming from a femtosecond laser and then emits a terahertz pulse. To some extent, the terahertz pulse can be manipulated by the lens, mirrors, polarizers, and so on to achieve this work’s requirement about the illumination fields. Then, the terahertz pulse can be sent to the sample (propagating wave illumination) or the coupler (SSPs illumination). A PCA can be also applied for the detection [38]. Similarly, the PCA is triggered by another probe pulse and then responds to the terahertz field. By tuning the length of the optical path, the terahertz field’s time domain is captured, leading to the amplitude and phase information [39]. To detect the x-polarized scattering field, the detector may be designed to be sensitive to a particular polarization direction [40]. The scattering electric field can be calculated by subtracting the background electric field from the total electric field. Besides, the generated terahertz pulse has a wide frequency bandwidth, which can be designed to cover the required working frequencies. In this manner, the data at each working frequencies can be captured with the aforementioned process. The proposed experimental setup may be further modified during the actual experiment process.
Finally, the method has the potential to be extended to two-dimensional imaging, the key step of which is to realize multidirectional SSPs illuminations on the x-y plane. As a possible solution, the grating may be rotated to obtain multidirectional SSPs illuminations. In this case, the slits of the grating can be filled with dielectric and the surface can be covered with a thin layer of dielectric. The sample is assumed to be directly put on the filled grating. When the orientations of the SSPs are needed to be changed, one can separate the sample and grating along the z-axis and then rotate the grating (or sample). After that, the sample can be reposited on the surface of the grating again to proceed with the next illumination. The specific analysis of this possible method is supplemented in section C of Supplement 1. In another potential way, the multidirectional SSPs illuminations may also be attainable based on the 2D isotropic SSPs. In future work, we aim to construct a possible platform to sustain the 2D isotropic SSPs. Besides, for the 3D imaging, other potential principles [41,42] may be introduced to optimize and enhance this work and make it possible to realize 3D imaging.
5. Conclusion
In conclusion, we propose a subwavelength imaging method for non-label samples in the terahertz region. A tunable composite photonics-plasmonics illumination pattern is used in the scheme. With the assist of SSPs on a metal grating, the plasmonics illumination pattern is obtained. When the SSPs impinge on the sample, the high-order SF information is down-modulated and then contained in the detected scattering field intensity. Meanwhile, under the traveling waves illumination, the relatively low-order SF information is also captured. In this manner, an integral wide Fourier spectrum of the sample is attained, thereby a subwavelength image is achieved with a revised post-process algorithm. In this manner, a resolution up to 0.12λ0 is verified against both Abbe’s and Rayleigh’s criteria. Besides, the imaging method is also effective when the relative permittivities or the positions of the samples are adjusted in a range. Finally, a potential experiment setup is described for furture work, and possible improvements in image quality and dimension are also discussed. The proposed method may provide an alternative way for biomedical imaging and semiconductor chip testing.
Funding
National Key Research and Development Program of China (2019YFA0210203); National Natural Science Foundation of China (61971013, 61531002).
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.
Supplemental document
See Supplement 1 for supporting content.
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