1. Introduction

The development of photonic crystal structures has prompted interest in Bloch surface states at the surface of semi-infinite super lattice structures [1]. The surface state is located in the bandgap region of one-dimensional photonic crystals [2, 3], and researchers have proposed numerous methods for the computation of mode [4–8]. The roundtrip phase of Bloch surface waves (BSWs) confined to the top layer is highly sensitive to the thickness of the cap layer, due to the effects of multiple reflections from periodic dielectric layers beneath. Thus, the thickness of the cap layer is an important factor in device tuning, with considerable influence over the field distribution and the mode propagation vector in the resulting devices [8]. The complex coupling effect between surface modes at the edges of a two-dimensional photonic crystal and continuum modes in a vacuum were discussed in [9, 10]. In [11, 12], the researchers conducted experiments to observe fine coupling effects between reflections from 1D photonic crystal layers and surface modes. This confined mode features superluminal propagation similar to that of surface plasmon waves at a metal/dielectric interface. Moreover, the all-dielectric environment greatly reduces loss and expands the operating bandwidth [13]. This makes it possible to implement designs involving two-dimensional flat optics utilizing all-dielectric surface states [7, 14–16]. Researchers have developed a wide range of applications involving resonant sensors, which have proven comparable to sensors using surface plasmon waves [17–19]. Near-field scanning optical microscopy has been used to examine Bloch surface waves directly [20]. In this study, we deposited a layer of fluorescent dye as an imaging layer in order to illustrate the propagation paths of BSWs using gratings as a coupling component. The outward emission of rays from the grating region indicates the direction of BSWs. The propagation constant of the proposed structure is based on the formulation proposed in [7]. Our experimental observations are in strong agreement with simulation results. The proposed imaging platform is suitable for the design of two-dimensional optical devices such as BSW waveguides and resonators [15]. This imaging method could also be used in the development of two-dimensional metasurface devices [14, 16].

2. Experiment setup

The structure of the samples used in this study are presented in Fig. 1. As shown in Fig. 1(a), the planar periodic structure comprises layers of TiO₂ (n = 2.58 for λ = 532nm) (73nm thickness) and SiO₂ (n = 1.5) (119nm thickness), as well as a cap layer of TiO₂ (196nm thickness). 1D poly(methyl methacrylate) (PMMA) gratings were produced using electron-beam lithography. As shown in Fig. 1(b), we fabricated two sets of circular gratings with a periodicity of 340nm. One circular PMMA grating was patterned on the 1D photonic crystal substrate and another was patterned on a film of Au (50nm thickness) over a regular glass substrate. A layer of 4-(dicyanomethylene)-2-t-butyl-6-(1,1,7,7-tetramethyljulolidyl-9-enyl)-4H-pyran (DCJTB) dissolved in Tris-(8-hydroxyquinoline) aluminum (Alq3) was deposited on the surface of the sample. The device was then examined using a fluorescence optical microscope (FOM). As shown in Fig. 1(a), we employed a pumping laser at a wavelength of 532nm with TM polarization incidental to the surface of the sample at a specified angle. The stripe grating was rotated until it met the quasi-phase matching conditions.

 

Fig. 1 One-dimensional photonic substrate using a single cap layer with a thickness different from that of the periodic layers on the substrate: (a) PMMA stripe grating on substrate; (b) circular PMMA grating on substrate

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3. Results and discussion

Figure 2 presents the outward emission of rays, wherein differences in pitch Λ indicate the various incident angles (θ, ϕ) required to meet phase matching conditions. The incident angle θ was set to 58°. The rotational angles ϕ are denoted in the figure caption with a ϕ error range of +/−2°. The outward emission angle is specified as ψ. The grating pitch range from 300nm to 550nm and the length of the grating region is 50μm. The gratings are indicated by white dashed squares and the emission patterns excited by the gratings are in red. The three white lines in the dashed squares indicate the orientation of the stripe gratings. The momentum matching conditions are illustrated in Fig. 2(g). According to the emission patterns, the propagation path of BSW is straight and directional, similar to the propagation effects induced by small defects excited by surface plasmon waves [21]. The coupling conditions are explored later in a discussion of the corresponding dispersion curves.

 

Fig. 2 Outward emission of rays through grating in which laser is rotated to various angles ϕ with θ set to 58°. The region within the white dashed lines includes the pattern from the stripe grating. TM-polarized laser light incidence and the measured direction of emissions are expressed as (ϕ,ψ) = (a) (56°,21.5°) (b) (38°,32.0°) (c) (27°,38.5°) (d) (18°, 42.0°) (e) (13°,40.5°) (f) (5°,40.0°) (g) momentum matching diagram.

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Figure 3 presents the optical paths through circular gratings excited by BSW and SPP modes, respectively. Figures (a) and (b) were obtained from a 1D PC substrate and an Au thin film substrate, respectively with incident angle θ set to 58°. The horizontal red line in Fig. 3(b) indicates a defect resulting from patterning bugs in the lithography controller. Nonetheless, the directionality of BSW propagation and SPP mode are clearly illustrated. Figure 3(c) presents the intensity profile of the white dashed line. The main paths are denoted as “1” and the unclear paths are denoted as “2”.

 

Fig. 3 Emission patterns from circular PMMA gratings on various substrates: (a) 1D PC substrate, (b) Au film substrate; (c) image intensity profile of region within white dashed line in (a) where path 1 is the first order mode of BSW and path 2 is the second order mode, with low field intensity in the fluorescent layer. Incident angle θ was set to 58°.

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We followed the simulation procedures used in [7] to obtain the correct propagation constant for the proposed substrate, the resolved dispersion curves of which are presented in Fig. 4. The dispersion curves include the curves of a 1D PC-Air interface, 1D PC-PMMA interface and an equivalent asymmetrical slab waveguide, which are later used for comparison. k_BSW-air is the propagation k-vector at the air/1D PC interface, whereas k_BSW-pmma is the propagation k-vector at the PMMA/1D PC interface. We were unable to obtain a solution for the 1st mode with a wavelength exceeding 610nm because a field with a longer wavelength cannot be fit within the space available in the cap. At 532nm, k_BSW-air = 1.18k_o (denoted as”dark star” in figure) and the composite k values (using the set incident angle and various grating pitches in Fig. 2) were fitted using Eq. (1) below. A directional beam was created when the composited k values (ranging from 1.17k_o~1.23k_o) fit to k_BSW-air of the 1st mode. Outward emission angle ψ (defined in Fig. 2) was derived according to vector analysis in the k-space, and a comparison of simulation and experiment results is presented in Fig. 5. The emission path is rather difficult to judge due to the blurred boundaries. Nonetheless, it is clear that the principle of momentum matching fits perfectly in this scenario.

 

Fig. 4 Resolved dispersion curves on various surfaces. The black squares indicate the light line in air. The red circles indicate the first mode at the interface between air and the 1D PC structure. The blue upright triangles indicate the first mode at the PMMA/1D PC interface. The pink downward triangles and the green diamonds indicate the 2nd mode at the air/1D PC and PMMA/1D PC interfaces, respectively. The center and the right most curves indicate dispersion associated with the 1st and 2nd modes of an equivalent asymmetrical slab waveguide. The “star” symbols are located at the pumping wavelength whereas the “cross” is located at 620nm, i.e., the emission wavelength.

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Fig. 5 Comparision between ψ formulated from vector analysis and ψ emission measured in Fig. 2.

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At the appearance of the directional beams, the images of pumping light scattering revealed nothing. This corresponds to the fact that the maximum field intensity in this design is located within the cap layer and does not necessarily extend into the air. Figure 6 presents the field profiles of the 1st mode at 532nm (k_BSW-air = 1.18k_o) and the 2nd mode at 620nm (k_BSW-air = 2.22k_o), i.e. the emission peak wavelength. We used the term “second” mode for the second set of curves despite the fact that these curves are the first mode for wavelength longer than 610nm. The 2nd mode profile at 532nm is similar to that shown in Fig. 6(b), showing very little field extension into the uniform upper medium. Thus, the k-vectors of the 2nd modes at the two types of interface (BSW-air and BSW-pmma) present similar k_x values. The different field distributions of the 1st mode at 532nm and 2nd mode at 620nm facilitate the observation of BSW modes. A higher field amplitude (> 40%) extends into the air region (Fig. 6(a)), thereby guaranteeing pumping light of higher intensity in the fluorescence layer. The lower field amplitude at the cap/ air interface @ 620nm (< 10%, Fig. 6(b)) results in most of the fluorescence being emitted back to the air due to low coupling efficiency in this BSW mode.

 

Fig. 6 H field profiles of (a) 1st mode at 532nm (b) 2nd mode at 620nm.

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One might question whether the proposed BSW modes in Fig. 6 are indeed similar to the modes of the slab waveguide based on a three-layer dielectric scheme. Calculations were performed on the asymmetrical slab waveguide in accordance with the methodology outlined in [22], from which we derived the dispersion curves of the 1st and 2nd modes, as shown in Fig. 4. The average optical refractive index of the multi-film substrate was ~1.94 ($(n_{Ti{O}_2}\cdot t_{Ti{O}_2}+n_{Si{O}_2}\cdot t_{Si{O}_2})/(t_{Ti{O}_2}+t_{Si{O}_2})$, where t indicates the thickness of the layer). A 197nm cap layer of TiO₂ is adjacent to the air. In Fig. 4, k_slab-1 = 1.93*k_o @532nm (denoted as “bright star”) does not fit our observations of the stripe or circular grating structures. A slab waveguide mode is therefore intrinsically different from a BSW mode in that the former requires total internal reflection at both boundaries whereas the latter is associated with multiple reflections (with a phase shift) from each periodic layer.

Figure 7 illustrates the phase matching conditions of the circular grating on the 1D PC substrate, where the red circle with radius “k_G” represents the momentum diagram of the circular gratings, which is shifted horizontally due to a tilt in the incident angle of θ = 58°. The two black circles represent the first two BSW modes @532nm at the Air/PC interface (1.18k_o and 2.37k_o respectively). The crossing points between the red circle and the two black circles indicate the propagation directions of the excited BSWs within the circular grating. The fitted angles between the propagation directions of the 1st order and 2nd order modes and the horizontal line are respectively designated as θ₁ (48°) and θ₂ (22°). These values are in good agreement with the emission patterns in Fig. 3(a) with θ₁ (49°+/−4°) and θ₂ (22.5°+/−1.5°). The ratio of the field exposed to air over the maximum field value within the cap layer is far larger for mode 1 than for mode 2. Thus, the fluorescence emission excited by mode 1 far exceeds that excited by mode 2. As for the SPP mode, the theoretical k_spp value at the air/Au film interface presents only one black circle with a radius of 1.09*k_o. In this case, the fitted path is at angle (θ₁ = 42°), such that the phase matching conditions allow for only two symmetrical directions. As shown in Fig. 3(b), θ₁ of the emission path is 41° with error range of +/−2°. Again, the experimental observations of the SPP mode are in good agreement with theoretical computations.

 

Fig. 7 Phase matching conditions of circular gratings on 1D PC substrate.

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4. Conclusions

This paper describes the directionality of Bloch surface waves on a 1D photonic crystal substrate, as observed from a layer of fluorescent material. We used the calculated dispersion relationship of BSWs based on the field continuity conditions located in the photonic bandgap region for the examination of emission patterns. We used stripe and circular gratings as coupling elements. This platform could be used to facilitate the design of other two-dimensional planar optical elements.