1. Introduction

Wavefront manipulation plays a fundamental and important role in optical applications, such as interference optics, communication, laser physics, etc. Relying on the intrinsic feature, wavefront manipulation of classical optical devices would result in a considerable volume [1]. While diffraction optical devices demonstrated could flexibly tailor the outgoing light [2], require at least a half-wavelength thickness space to accumulate the maximal phase of 2π. To satisfy the demand for miniaturization and integration in modern nanophotonics, the thickness should be reduced furtherly. In 2011, F. Capasso et al demonstrated that the discrete metasurface composed of well-designed nanoantennas was capable of bending light and generating optical vortices [3]. Since the phase in full region was generated from the resonance of the nanoantenna, the thickness could be smaller than λ₀/100. This method paved the way for flexibly manipulating phase profile. Later, various optical devices based on discrete metasurfaces were put forward, including ultrathin flat metalens [4,5], beam shaper [6], quarter-wave plates [7], optical holography [8,9], etc [10–14]. To realize high efficiency, four phase levels of the nanoantennas in each period for metasurface are required at least [15]. Otherwise, efficiency would degenerate. Therefore, we wonder the manipulation behavior when the phase level tends to infinity or the structure becomes geometrical continuity.

In this paper, we systematically introduce the approach of manipulating wavefront with meander line structures from the method, parameter optimization, simulation, and experiment. The designed sine-shaped meander line can split the incident light into two beams with conversed polarization and send them into the desired directions. It is found that the outgoing light within long wavelength region is mainly originated from the surface mode of TM component, while in short wavelength is dominantly from the oscillating guided mode of TE component, resulting in a broadened operating region. Additionally, we successfully verify the approach by deriving a special function from a given irradiation pattern, named backward process. Since the manipulation relies on the phase profile that is yielded from the structure geometry, the thickness of the meander line can be much smaller than the incident wavelength. Our approach gives an alternative to realize wavefront tailoring with an ultrathin optical profile and an insight into the light propagation in meander line structures.

2. Results

The sine function f(x) = A sin(πx) in rectangular coordinate is present, where A = 0.55 μm is the amplitude coefficient and the period P is 2 μm. For the convenient expression in the following section, we define a local u-v coordinates where u- and v-axes are normal or tangential to the sine function, as shown in Fig. 1(a). The function can be regarded as a transmittance function. The space-variant angle introduced by the function is expressed as ξ (x, y) = arctan (∂y/∂x), which is linked with the geometric phase φ via φ (x, y) = 2σξ (x, y), where σ = ± 1 stand for left circular polarization (LCP) and right circular polarization (RCP), respectively. Since the geometric phase is relative to the polarization state, we estimate the intensity of the outgoing light with polarization-conversed by fast Fourier transform, i.e., I = [FFT (e^{iφ (x, y)})]². Adopted the LCP as incident light, the intensity of RCP light as a function of incident wavelength λ and sin θ_m are depicted in Fig. 1(b), where θ_m denotes the diffraction angle of m^-th scattering order with respect to the specular direction, approximately satisfying |sin θ_m| = mλ / P.

 

Fig. 1 (a) Sine function in the rectangular coordinate. ξ (x, y) = df(x)/dx stands for the space-variant angle, i.e., the angle between x-axis and u-axis. (b) Intensities of polarization conversed outgoing light as a function of sin θ_m and incident wavelength λ, where θ_m is the m^-th diffraction angle. The beam splitting behavior is approximately described by the equation |sin θ_m | = λ/P with P being the period of the function. (c) Top view of the simulation model improved from the sine function. The inset is the side view, where tm is the thickness of the meander line. The bottom is the SEM image of the fabricated sample. The structure parameters are noted in the zoomed inset. The black rectangle is the unit cell used in the simulation. (d) Simulation results of RCP intensities (red solid lines) at wavelengths of 532 nm, 632.8 nm, and 1050 nm. The experimental result at 632.8 nm marked with blue circles, which agrees well with the calculation and the simulation ones.

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To verify the method, necessary decorations are needed to build electromagnetic simulation models and fabricate test samples. First, we shift the function along y-direction with a width of b to form a closed stripe, then repeat it along y-direction with a period of py. To suppress the light diffraction along yoz plane, py should be less than the incident wavelength λ. By extending the figuration along z-direction with the thickness of tm and replacing the stripe with metal (gold in this paper), the simulation model is built finally, as shown in Fig. 1(c). Note that the width of metallic wire along y-direction is equal to be b as expected, but along the sine curve the width of the metallic wire w is different, as well as the gap width defined as g = py − d. The minimal w is about b/$\sqrt{2}$. The influence of the decoration on the manipulation performance will be discussed in Section 3.

The simulation is based on the method of finite integral technique with the optimized structure parameters of py = 0.5 μm, b = 0.25 μm, tm = 0.1 μm for x varying from −15 μm to 15 μm. The incidence is LCP light and the diffraction behaviour of RCP light is monitored. The simulation results of the RCP intensity spectra are depicted in Fig. 1(d) for incident wavelengths of 428.6 nm, 532 nm, 632.8 nm, and 1050 nm, which agree well with the theoretical calculation in Fig. 1(b). The intensity of 428.6 nm incident is greatly suppressed due to the enhanced diffraction along yoz plane caused by small wavelength in comparison with py value. It is also found that the intensity reduces as wavelength increases due to the operation band broadening except 532 nm where the 1^st diffraction light is greatly suppressed. The reason will be discussed in Section 3 as well.

Hereafter, we fabricate the meander-line test sample by focused ion beam (Helios Nanolab 650, FEI Company) on a gold layer of 100 nm thickness, deposited on the top of a 1-mm-thick glass substrate. The scanning electron microscope (SEM) image is shown in Fig. 1(c) at the bottom. In the experiment, an input beam from a laser source is purified by a vertical direction polarizer and then impinges a 45° rotated quarter-wave plate to obtain an LCP incident light that illuminates the sample at normal direction. To avoid the disturbance of the directly penetrated LCP light, a −45° rotated quarter-wave plate and a horizontal polarizer are used to pick out the desired RCP light which will be received by a charge-coupled device (CCD, WinCamD-UCD15, DataRay Inc) fixed on an annular guide rail with the radius of 30 cm. More detail can be found in [16]. The experiment result is shown in Fig. 1(b) marked with blue circles for 0.6328 μm incidence, which is overlapped with the simulation one. Experiment results of the other wavelengths are coherent with theoretical ones as well. The efficiency defined as the power ratio of the received light and the incident light is about 6% at 0.6328 μm, which is considerable for the wavefront modulator aiming at transmission light.

3. Discussions

3.1. Propagation of the unit cell

Different from the literature [17], the presented meander line works in the visible region rather than the millimeter wave, where the metallic feature should be considered. From the aspect of morphology, the meander line can be regarded as a series of metal-insulator-metal (MIM) waveguide connected along the sine function with different space-variant angle ξ. We select a small segment of the meander line as the unit cell to investigate the polarization conversion ratio (PCR) of the incident LCP light. It is note that the transmission characteristics of the unit cell under linear polarization components are in u-v coordinates. The simulation is based on the finite element method (FEM) with Floquet (open) boundary perpendicular (parallel) to the propagation direction. Figure 2(a) depicts the PCR (red solid line) versus wavelength varying from 0.5 μm to 3 μm with a zoomed inset picture. The value holds around 0.5 when the wavelength is above 0.57 μm and sharply reduces while below it. To understand it, we investigate the transmittance characteristics by decomposing the LCP light into TE and TM components with the electric field tangential or normal to the sine function. The results are displayed in Fig. 2(a) marked with black dots and blue dash lines, respectively. Via the waveguide method, the propagation characteristics of the two components can be expressed as [18]:

 

Fig. 2 (a) Polarization conversion ratio (PCR, red solid line) under LCP light, transmittances under TM polarization (blue broken line) and TE polarization (black dot line) light as a function of incident wavelength. The inset is the zoomed picture ranging from 0.5 μm to 0.7 μm. The sketch map of the unit cell is displayed in the gray area. (i) and (ii) are the electric fields, (iii) and (iv) are the magnetic fields of 1.5 μm and 0.54 μm at yoz plane, respectively. (b) Propagation constants of oscillating modes respected to k₀. The top panel is the real parts of 0-order odd mode for TM (blue circle) and even mode for TE (black star) components. The bottom panel is the imaginary parts. (c) Same items for propagation constants of surface modes of TM component marked as TM-E.

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3.2. Impact of the structure parameter

The structure parameters may influence the light propagation. First, the impact of metallic film thickness is considered. The PCR versus incident wavelength is shown in Fig. 3(a), which is sensitive to tm value. To explain it, we probe the transmittance of the unit cell under TM and TE incidence, as shown in Figs. 3(b) and 3(c). When the thickness is smaller than the skin depth of the metal (~0.1 μm), for example, tm = 0.05 μm, the wavelength of SPP wave is red shifted. Beyond this thickness, SPP is independent of tm and just relies on the period of the meander line. Within the short wavelength region, PCR relies on the attribution of TE component. The transmittance of TE component reduces as tm value increases due to the propagation loss, leading to the low and the complex PCR. As wavelength increases, TM component is relevant to PCR. As the evidence, we see that maximal PCR values appear at the peak position of TM transmittance. To obtain a steady PCR in the broadband region, the thickness is adopted as 0.1 μm.

 

Fig. 3 Different items as a function of incident wavelength for varying film thicknesses tm. (a) PCR under LCP incident, (b) Transmittance under TM light, and (c) Transmittance under TE light. Red, green, and yellow areas stand for the cutoff region, short wavelength region, and long wavelength region.

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Figure 4 depicts PCR as a function of frequency and gap width g for tm = 0.1 μm and py = 0.5 μm. For the convenience of description, we schematically divide the picture into four areas, from I to IV. Within areas I and II, the incident wavelength is short. PCR values in area II are complex. The dark zone around 550 THz is the result of the bounded TM component and low transmittance of TE component. Electric field distributions at g = 1.67 μm are given in the bottom of the figure, where blank arrow and blue cross stand for TM and TE components. It is interesting that there is an obviously enhanced PCR around 508 THz. From the field analysis, we know that the harvested light propagation is originated from the coupling between the adjacent stripes. In area I, PCR relies on the transmitted TE component in the form of oscillating mode, as demonstrated in Section3.1. The wavelength is relatively longer in respect to the metallic width b in areas of III and IV. The schematic of the electric field at 300 THz suggests that almost all TE component is reflected and TM component mainly propagates through the meander line via the gap in the form of surface mode. Meanwhile, the crosstalk between the adjacent stripes reduces as g increases, agreeing well with the conclusion that none noticeable crosstalk is found at g = 0.25 μm for long wavelength. It is also seen that tiny fluctuations of the adopted structure parameters show ignorable influences on the PCR values.

 

Fig. 4 PCR as functions of incident frequency (f = c / λ, with c being light velocity) and gap width g. The picture is divided into four areas marked with I, II, III, and IV. The sketch maps of the electric field distribution at positions marked with stars are displayed in the bottom.

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3.3. Backward process

To verify the universality of the method, we derive a special function in rectangular coordinate from a given irradiation pattern, named backward process. The target pattern is a uniform transparent of RCP light within the range of ± 30° for a 0.6328 μm LCP incident light. The red solid line in Fig. 5(a) stands for the ideal frequency spectrum. By utilizing inverse fast Fourier transform, an optimized function is obtained. It is an irregular function with x varying within the preset range of [-10, 10] μm and the consequential y range of [-3, 3] μm, as Fig. 5(b) shows. The actual spectrum of the function is appended in Fig. 5(a) marked with the blue dash line, which is approximate to the ideal one. The same structure parameters are chosen to fabricate the sample. Figure 5(c) shows the SEM image with a detailed inset. The minimal w is about 50 nm.

 

Fig. 5 (a) Ideal radiation pattern (red solid line) and the actual radiation pattern obtained from the function in rectangular coordinate shown in (b) with the method of fast Fourier transform. (c) SEM image of the fabricated sample and the detail image.

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Figure 6(a) shows the simulation results of the 2D forward scattering power field at the distance of 0.03 m far away from the sample at a 0.6328 μm LCP incident light. The inset gives the experimental result. A quasi-continuous “light band” of RCP light is captured by a CCD camera. The data along the white broken line is abstracted and depicted in Fig. 6(b) marked with black circles. Distinct harvesting of transmission energy appears within the diffraction angle ranging from −14.5° to 14.5°, which agrees well with the simulation one (red solid line). The simulation results at 0.785 μm are shown in Figs. 6(c) and 6(d). The bandwidth of the “light band” is about 36°, coherent with the equation |sin θ_m| = mλ / P.

 

Fig. 6 (a) Simulation result of 2D forward scattering power field image at 632.8 nm. The inset gives the experimental result directly captured by a CCD camera. The white bar is the diffraction angle calculated from the position relationship bwteen CCD and sample. Based on this, we deduce the size of the “light band” is about 6°☓24°, coherent with the simulation results. (b) 1D results of the simulation (red solid line) and experiment (black circle). Simulation result of 2D forward scattering power field image (c) and 1D simulation result (d) at 785 nm.

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4. Conclusion

In summary, the approach of manipulation wavefront with the meander line structure is demonstrated successfully. The diffraction behavior of the polarization conversed outgoing light is relevant to wavelength of |sin θ_m| = mλ / P. The propagation characteristics analysis suggests that the manipulation in short wavelength region is originated from the oscillating mode of TE component rather than surface mode of TM component for long wavelength, leading to a broadening of operation region towards short wavelength. The backward process implies the universality of the approach. Our finding provides another option to realize arbitrary wavefront manipulation with an ultrathin optical device.