1. Introduction

Wavelength-dependent directional beaming is an important technological issue in various research fields and applications where the spatial separation of different spectral components is demanded [1–6]. For this purpose, various technologies based on diffraction gratings have been developed [7–11], which primarily rely on periodically varying optical structures. They are extensively utilized in fiber-optic applications as well, because fiber-optic systems often require accessing a wide range of spectral bands in combination or separately. For example, a variety of types of diffraction gratings are used for pulse shaping [7], chirped pulse amplification [8], optical sensing [9], laser scanning [10], multi-wavelength generation [11], etc. However, the use of diffraction gratings in bulk types requires a number of free-space coupling optics alongside fibers, whilst fibers themselves are in a very simple and compact form. Thus, the use of bulk diffraction gratings together with fibers tends to result in unnecessarily complex arrangements. This consequence considerably undermines the intrinsic advantages of fiber-optic devices, which includes compactness, flexibility, and cost-effectiveness, for example [12–15]. To overcome such drawbacks incurred by the use of conventional bulk diffraction gratings in fiber-optic systems, one can instead think of implementing plasmonic technologies onto fiber-optic platforms [16–19].

Surface plasmons (SPs) are collective oscillations of electrons at the interface of dielectric and metal. They can be excited from optical radiation and can give rise to unique out-coupling characteristics when they are obstructed by micro- or nano-scaled structures. For example, off-axis directional beaming has been achieved by combining metal-nano-slit-based sub-wavelength apertures with periodic grating structures [17–20]. It is worth noting that such off-axis beaming functions exhibit highly wavelength-dependent characteristics [18–20]. Hence, by exploiting these novel technologies, one can realize SP-based wavelength-dependent off-axis directional beaming (WODB) for fiber-optic applications.

In fact, metal-coated fiber-facet structures have frequently been utilized to hybridize unique SP functions onto optical fibers [21–24], although SP-based WODB in a fiber-optic platform has yet to be investigated extensively. Recently, the authors have investigated a grating-assisted metal nano-slit structure on a fiber facet and experimentally demonstrated its WODB functionality [25]. Unfortunately, such slit-based structures, including our recent work [25] and others [21–23], tend to exhibit very limited SP coupling efficiency when being excited by a fiber mode that has a relatively large mode-field size compared with the slit dimension. To overcome such an intrinsic limitation of nano-slit- or aperture-based structures, we here propose a novel SP coupling technique that, to the best of our knowledge, has not been considered so far: It is based on a metal-coated angled fiber facet (MCAFF) with “no aperture”, exploiting the so-called Kretschmann configuration for SP generation [26] implemented on a fiber-optic platform.

Unlike a number of previously investigated fiber-SP hybrid devices [21–25], the proposed scheme utilizes a simple metal layer coated on an angled fiber facet, on which a periodically corrugated structure is subsequently implemented with “no aperture”. The angle of the fiber-facet is carefully determined to match the incident fiber mode to a target SP mode via the law of momentum conservation. This initially enables efficient SP coupling without being blocked by the subwavelength-size aperture or considerably exciting either anti-propagating SPs or non-confined diffracted light. The periodically corrugated structure on the top of the metal layer subsequently out-couples the SP into a radiation mode out of the MCAFF, completing unidirectional WODB functionality in an all-fiberized format. We stress that unlike all other WODB configurations, which normally rely on combinations of grating and aperture structures [21–25], the proposed scheme does not incorporate any sub-wavelength-scaled aperture outlet. We also emphasize that the WODB mechanism of this scheme is entirely different from that of a simple assembly of an optical fiber and a periodically corrugated metallic grating, on account of the fact that the proposed scheme primarily relies on the excitation of SPs in the metal-air interface. Consequently, the resultant characteristics are unique and unmatched to those of conventional surface relief gratings, which will become evident in more detail later in the paper.

Thus, the rest of the paper is organized as follows: First, we discuss the schematic configuration of the proposed scheme of the MCAFF and its working principle for unidirectional fiber-to-SP coupling. Second, we discuss the design principle of the corrugation-assisted MCAFF (CA-MCAFF). Third, we further discuss the fine-tuning of the corrugation parameters in order to optimize the consequent WODB functionality. Fourth, we numerically investigate and discuss the characteristics of the CA-MCAFF in terms of WODB functionality. Finally, we present our conclusion and outlook on the CA-MCAFF technology.

2. MCAFF scheme for SP generation

In general, fiber-facet-based devices for SP generation often adopt subwavelength-scaled apertures or nano-slit structures [21–25]. Our previous work was also based on a rectangular nano-slit for SP generation, incorporating periodic corrugations for WODB functionality [25]. However, in this case the coupling efficiency between fiber and SP modes is substantially lowered by three factors, as depicted in Fig. 1(a): (1) The field-overlap between the fiber mode and the subwavelength-scaled aperture is so poor that the most of the incident radiation from the fiber core to the aperture tend to be blocked by the metallic aperture itself. (2) A considerable amount of the incident radiation is lost into the excitation of non-confined diffracted light [16,25]. (3) It also generates unwanted SPs propagating in the opposite direction from the target periodic corrugation structure on one side. Consequently, as briefly mentioned in the preceding section, we would like to resolve this issue by instead utilizing the Kretschmann configuration [26] with an angled fiber facet, which is schematically depicted in Fig. 1(b).

 

Fig. 1 Fiber-SP-mode coupling schemes: (a) Nano-slit coupling scheme and (b) MCAFF scheme.

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The prism coupling technique based on the Kretschmann configuration is one of the most frequently used methods to generate SPs from optical radiation [26,27]. This technique normally utilizes an oblique incidence towards a dielectric-metal interface where the real part of the dielectric material should be higher than that of the metal. It is worth noting that if the fiber has an angled facet, the angled facet itself can play exactly the same role as a prism to the optical mode guided in the core of the fiber. Without loss of generality, this MCAFF can generate SPs along the metal surface if the following phase-matching condition is satisfied:

Whilst the proposed scheme, in principle, relies on the well-known Kretschmann configuration, one distinct aspect is the fact that the incident wave is given not by a free-space beam or a plane wave, but by a fiber mode that is localized within the core of the fiber and its vicinity. This makes a very important difference, because the “localized” fiber mode can generate “localized” SPs (LSPs) in addition to the normal SPs that are determined only by the metal-dielectric interface [16,28]. If the metal layer is thin enough, the influences by the LSPs as insulator-metal-insulator (IMI: dielectric-metal-air) SPs become more considerable [16]. Consequently, the conventional approaches based on the rigorous coupled wave analysis (RCWA) that only takes account of normal SPs [26] may give rise to deficiency of accuracy in analyzing the proposed scheme. Hence, we instead choose to perform full numerical calculations based on the finite element method (FEM via COMSOL Multiphysics^®). This will eventually lead to a more comprehensive analysis on the proposed scheme, including the influences and effects by LSPs. Moreover, it is worth noting that the feasibility of this numerical modeling method has successfully been benchmarked, showing good agreement with experimental results when applied to the analysis of a similar plasmonic-fiber-optic device, i.e., a corrugation-assisted nano-slit structure for WODB [25]. In addition, it is also to note that although the proposed scheme is not limited to a specific spectral range, our investigation on the WODB functionality will hereinafter be in the visible wavelength range (i.e., 400 to 700 nm), simply because a lot of important SP-based laser applications lie within this range [16–23].

We consider silver for the metal layer from the viewpoint of the simplicity and cost-effectiveness to demonstrate the functionality of the proposed WODB in the visible range [25] (It is worth noting that gold or any other good conductors can also be utilized instead, depending on the purpose of the target application). The material parameters of silver and silica, i.e., its refractive index and extinction coefficient, are presented in detail in Ref. 29 and 30. We assume that the fiber is a step-index fiber with a core diameter of 3 μm and a numerical aperture of 0.1, which leads to the cutoff wavelength of ~392 nm, so that it will allow for strict single-mode operation in the whole visible wavelength range (i.e., 400 to 700 nm).

Initially, we determine the thickness of the metal layer to put upon the fiber facet and the fiber-facet angle relative to the fiber axis, i.e., θ, in order for the device to be suitable for broadband SP coupling in the given wavelength range. It is worth noting that in the conventional Kretschmann configuration with a prism, the thickness of the metal layer is normally given by 30 to 50 nm [26,27], and the incidence angle is externally determined and fine-tuned via free-space launching, depending on the wavelength of the incident beam. In contrast, the proposed scheme relies on the full fiber-optic scheme, so that the incidence condition is pre-determined by the angle of the fiber facet for any spectral radiation through it. Thus, we emphasize that it is very important and critical to choose a right value for the fiber-facet angle, because it primarily defines the spectral coverage of SP coupling.

In Fig. 2(a) we show the electric-field (E-field) pattern for an incident radiation at 600 nm to an MCAFF having the metal layer of 20 nm and the fiber-facet angle of 45°, illustrating how the fiber mode is coupled into SPs as a typical example. The incident fiber mode is assumed to be an x-polarized LP₀₁ mode, which is of a proper polarization to excite SPs on the metal surface in the given IMI geometry [see Fig. 1(b)] without loss of generality. One can see that when the x-polarized LP₀₁ mode is obliquely incident onto the IMI structure, a fraction of it is immediately reflected to the side of the optical fiber, and the rest of it is coupled into SPs. These SPs propagate parallel to the IMI structure, gradually undergoing attenuation. It is worth noting that there are two main mechanisms for the attenuation: One is the ohmic loss by the metal layer itself, and the other is the decoupling of the propagating SPs back into the dielectric, i.e., into the side of the optical fiber. For example, in the right-top-corner of the cladding region of the optical fiber indicated by a rounded rectangle in a dashed-line in Fig. 2(a), the noticeable optical radiation to the right side is mainly due to the latter effect. This is significantly distinguished from the direct reflection of the incident radiation occurring in the core region, because the incident optical radiation into the cladding region is inherently negligible if the fiber operates strictly in single mode. This decoupling effect of the propagating SPs can be avoided if the thickness of the metal layer increases; however, in this case turns up the ohmic loss instead.

 

Fig. 2 (a) Field pattern (E-field magnitude) for an incident radiation of 600 nm to an MCAFF having the metal layer of 20 nm and the fiber-facet angle of 45 °. (b) OCE_fiber-SP with respect to the thickness of the metal layer for different fiber-facet angles.

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Thus, we calculate the coupling efficiency between the fiber and SP modes, varying both the thickness of the metal layer and the angle of the fiber facet over the given wavelength range, and plot the result in Fig. 2(b). We define the overall coupling efficiency between the fiber and SP modes (OCE_fiber-SP) such that the coupling efficiency is averaged out over the entire visible wavelength range within the core region. Having mentioned in the preceding section, we are going to put a periodically corrugated structure upon the metal layer just above the core region, in order to out-couple the propagating SPs into optical radiation outside the MCAFF, thereby introducing WODB functionality to it. Thus, the OCE_fiber-SP within the core region is mainly of importance. As shown in Fig. 2(b), the OCE_fiber-SP increases to a considerable level as the thickness of the metal layer decreases down below 30 nm, which also slightly varies with the angle of the fiber facet around 45° [see Eq. (1)]. We stress that such efficiency levels are significantly high, compared with conventional nano-slit-based devices [31], even taking into account the fact that it is the value averaged out for the broad wavelength range from 400 to 700 nm. It is worth noting that the precise optimization of the fiber-facet angle is unnecessary for the moment, because a periodically corrugated structure will be added upon the metal layer for WODB functionality, and this structural modification will slightly alter the effective phase-matching condition of the coupling at the IMI interface [17,19]. Thus, we only determine the thickness of the metal layer at this stage.

In conventional Kretschmann configurations, the use of a thin metal layer of below 30 nm is not often chosen because of the SP-to-optical decoupling issue if SPs are allowed to propagate a considerably extended distance [26,27]. However, in our case one can expect it will not be a critical issue, because the periodically corrugated structure that is to be put upon the top surface of the metal layer for WODB will immediately out-couple the propagating SPs into optical radiation in free space. Therefore, we consider that the thickness of the metal layer below 30 nm should be acceptable without loss of generality. This feature will further be confirmed and verified in the section that follows. Notwithstanding, the use of too thin a metal layer of below the skin depth of the metal, which is in the range of approximately 10 nm for silver for optical radiation in the visible wavelength range, must be avoided, because it can cause direct tunneling of the optical fiber mode into free space. Consequently, we choose the thickness of the metal layer to be 20 nm, which is thick enough to prevent such a direct tunneling effect. Moreover, this thickness is fully compatible with the currently available electron-beam (e-beam) coating and focused-ion-beam (FIB) milling technologies [32,33]. Emphatically, this arrangement can offer substantially high OCE_fiber-SP in comparison with those conventional nano-slit-based coupling schemes do [31,34], as estimated in Fig. 2(b).

With the thickness of the metal layer fixed to 20 nm, we calculate the spectrum of the coupling efficiency between the fiber and SP modes for various fiber-facet angles around 45° in Fig. 3(a), where the estimated efficiency is the individual coupling efficiency between the fiber and SP modes (ICE_fiber-SP) before being averaged into the OCE_fiber-SP that is shown in Fig. 3(b) together with the 3-dB spectral bandwidth (SB_fiber-SP) for the given fiber facet angle. It is worth noting that the ICE_fiber-SP reaches up to 70% in some conditions. In addition, the OCE_fiber-SP is maximized when the fiber-facet angle is given by ~45°, and the given IMI structure offers a sufficiently broad spectral bandwidth, being able to cover the targeted visible wavelength range (i.e., 400 to 700 nm). We emphasize that these enhanced characteristics fully support the proposed scheme to be suitable for an SP-generating platform for WODB functionality, once a periodically corrugated structure is introduced on the metal-air boundary.

 

Fig. 3 (a) ICE_fiber-SP spectrum with respect to the angle of the fiber facet. (b) OCE_fiber-SP and SB_fiber-SP with respect to the angle of the fiber-facet.

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3. Design principle of a corrugation-assisted MCAFF

We now discuss the MCAFF incorporating a corrugated structure, which will eventually enable to put the whole conventional WODB arrangements into an all-fiberized form, as schematically depicted in Fig. 4(a). Having been described in the preceding section, the incident radiation from the optical fiber can excite SPs at the metal-air interface, which are now supposed to be out-coupled into optical radiation in free space, incorporating WODB characteristics. For this purpose, we implement periodical, rectangular-shaped corrugations in the metal layer as shown in Fig. 4(b).

 

Fig. 4 (a) Illustration of the working principles of a conventional WODB scheme and the proposed WODB scheme. (b) Schematic of the proposed MCAFF scheme. (c) Two different types of SP modes excited at the periodically corrugated metal surface.

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Before going into the detailed discussion on the WODB functionality of the proposed scheme, i.e., the corrugation-assisted MCAFF (CA-MCAFF), let us first clarify the characteristics of plasmonic oscillations that can be excited in the periodically corrugated metal layer doubly interfaced with the fiber facet and air. A magnified view of the structure is shown in Fig. 4(c) where one can see the incident optical radiation into the bottom of the metal layer. In the given configuration, the corrugated metal surface can support three different types of SP modes: The first SP mode is based on the LSP oscillation at the troughs of the corrugation where the metal layer is thinnest, as discussed in the preceding section [16,28]. It is represented by the black arrows in the regions highlighted by a dashed-line round-rectangle in Fig. 4(c). This type of mode is hereinafter called the LSP mode. The second SP mode is based on the non-localized, normal SP oscillation spread out along the whole metal-air interface of the periodical corrugation. It is represented by the red arrow in Fig. 4(c). This SP mode bounded at the corrugated-metal-air interface is hereinafter called the SP_CM/A mode. The third SP mode is based on the SP oscillation bounded at the corrugated-metal-fiber interface hereinafter called SP_CM/F mode, which is normally forbidden if the metal layer is plain and uniform. However, this parasitic SP_CM/F mode can exist locally at the given interface on account of the deep, periodic corrugation in the metal layer. The related effects and consequences will be discussed in more detail in the section that follows.

By the way, the local phase variation of the LSP mode primarily relies on the spatial phase distribution of the incident optical radiation projected onto the fiber-metal interface. This means the wavenumber of the LSP mode is primarily determined by

On the other hand, a fraction of the incident optical radiation can also be converted into an SP_CM/A mode that propagates through the whole corrugated surface in the form of propagating eigen-mode. The wavenumber of the SP_CM/A mode can be obtained once the effective permittivity of the corrugation, ε_C, is determined by utilizing the Maxwell-Garnet theory [35], which is given by

The LSP and SP_CM/A modes interact at the metal-air interface and can result in a maximized, combined SP oscillation if they overlap in phase. It is worth noting that the LSP mode can be regarded as a dominant mode in comparison with the SP_CM/A mode, because the former undergoes significantly less ohmic loss than the latter. This is mainly thanks to the fact that the former is localized and bounded mostly within the trough of the corrugation structure. In addition, the SP_CM/F mode is quickly attenuated, thereby not dominantly contributing to the out-coupling into optical radiation in free space.

We next discuss the out-coupling from the SP modes to the free-space radiation mode, as depicted in Fig. 4(b). In principle, it is governed by the phase-matching condition among the wave vectors of the SP modes and the reciprocal lattice vector of the periodic corrugation as follows [18]:

In fact, it is worth noting that there are seven adjustable design parameters in the proposed scheme: (1) the type of the fiber for the incident optical radiation, (2) the thickness of the basement of the metal layer (i.e., the thickness of the trough region of the corrugation in the metal layer), (3) the period of the corrugation, (4) the length of the corrugation, (5) the modulation depth of the corrugation (i.e., the height of the corrugation ridge), (6) the duty cycle of the corrugated ridge, and finally (7) the fiber-facet angle. Considering that the target operating wavelength of the proposed device is in the visible wavelength range, we use the same fiber type as specified in Section 2 designed for strict single-mode operation in the whole visible wavelength range, and choose the thickness of the basement metal layer of 20 nm and the corrugation period of 380 nm. Thus, we set the length of corrugation to be 6.46 μm, which results from 17 periods of the unit corrugation structure, so that the whole corrugation length is long enough to cover the core region even for the case of high fiber-facet angle, e.g., 60°. The other parameters—the modulation depth of the corrugation ridge, the duty cycle of the corrugation, and the fiber-facet angle—will be determined more precisely through optimization in the section that follows.

Figure 5 shows how the optical radiation takes place via an example CA-MCAFF, in which the out-coupled beam pointing to the top-left corner is of our interest for WODB. We here assume that the modulation depth and duty cycle of the corrugation, the fiber-facet angle, and the wavelength of the incident optical radiation are given by 40 nm, 50%, 49.5 °, and 500 nm, respectively. In particular, it is worth noting that the fiber-facet angle is set a bit higher than 45 ° that was the angle supposed to lead to the maximum OCE_fiber-SP when a plain metal layer of 20 nm was considered [see Fig. 2(b)]. We here choose a slightly increased fiber-facet angle because, with the inclusion of the corrugated metal structure, the wavenumber of the relevant SP mode (k_SP,CM/A) tends to shift from that of the SP mode for a plain metal surface, i.e., k_SP,PM/A [refer to Eqs. (1) and (3)]. In fact, this consequence is due to the fact that the total path length of the metal surface increases with the inclusion of the periodic corrugation, thereby being also considerably affected by the modulation depth and the duty cycle of the corrugation. Thus, assuming that the corrugation modulation depth is in a range of 30 to 60 nm and the duty cycle in a range of 45 to 55%, we approximately estimate that the fiber facet angle of θ should be set to 49.5 °, initially. Nevertheless, we will carry out fine-tuning of each specific parameter in the next section in order to optimize the overall performance of the proposed CA-MCAFF.

 

Fig. 5 Field pattern for an incident radiation of 500 nm to a CA-MCAFF with θ = 49.5°.

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In addition, it is worth noting that the origin of the out-coupled beam is slightly shifted to the right-top corner of the fiber-facet rather than locating at the center of the core. This is due to the fact that the excited SP modes intrinsically tend to propagate along the metal layer in that direction. According to Eq. (4), the angle of out-coupling ϕ, will increase in the counterclockwise direction with respect to the wavelength of the incident optical radiation.

4. Fine-tuning of the corrugation parameters for WODB functionality

We pay more attention to fine-tuning of the three parameters that were not specified in the preceding section, which are (1) the modulation depth and (2) the duty cycle of the periodic corrugation, and (3) the fiber-facet angle. It is worth noting that, to optimize these three parameters, we take sequential and iterative steps, carrying out fine-tuning of one parameter at a time while keeping the other two fixed to the initial values or the refreshed values obtained at the prior step, mainly because it would require too much computing power to carry out the optimization of them at the same time.

To provide criteria for the optimization of them, we introduce a figure of merit for the WODB functionality of the proposed CA-MCAFF, taking account of the fact that the device is primarily for offering efficient and broadband WODB functionality in the visible wavelength range. In this regard, the figure of merit is given by either the out-coupling efficiency or the spectral coverage of the device. For the former, we take overall out-coupling efficiency (OCE_out), such that individual out-coupling efficiency (ICE_out) is averaged out over the entire wavelength range of interest, in a similar way to defining the OCE_fiber-SP and ICE_fiber-SP in Section 2. For the latter, we take a spectral bandwidth (SB_out) specified by the full width at half maximum (FWHM; 3 dB) of the whole ICE_out spectra.

First, we optimize the modulation depth of the corrugation. It is directly linked to the out-coupling strength of the periodic corrugation [36], so that the higher modulation depth would lead to the larger out-coupling coefficient. However, too large a value for it would incur excessive ohmic loss to SP modes that are basically bounded along the metal/dielectric interface, so that there must be a trade-off in increasing the value. In Fig. 6(a), we show the calculated ICE_out spectra of the CA-MCAFF for different modulation-depth values varying from 15 to 65 nm, assuming that the corrugation is based on a rectangular shape and that all the other parameters are given by the initial values specified in Section 2. Relying on the ICE_out spectra, we can also obtain the OCE_out and the SB_out as shown in Fig. 6(b). Based on the results, we hereinafter assume that the modulation depth is set to 40 nm, because both the OCE_out and the SB_out of the given CA-MCAFF are sufficiently maximized around the value.

 

Fig. 6 (a) ICE_out spectrum with respect to the modulation depth of the periodic corrugation. (b) OCE_out and SB_out with respect to the modulation depth of the periodic corrugation.

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Second, it is worth noting that the ICE_out spectra shown in Fig. 6(a) have unexpected spectral dips at the wavelengths around ~440 and ~550 nm. We note that these spectral dips are related with a very special situation, such that the LSP mode results in exciting backward-propagating SP_CM/A and SP_CM/F modes, which are presumably bounded at the metal-air and metal-fiber interfaces, respectively. This is due to the fact that the periodic rectangular corrugation does include higher-harmonic components.

In such cases, the phase-matching condition previously represented by Eq. (4) can be rewritten into the following form:

 

Fig. 7 (a) Illustration of the coupling to back-propagating SP modes. (b) Field for an input radiation of ~550 nm to a CA-MCAFF with the duty cycle of the corrugation of 50%.

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In Fig. 7(b), we show the field distribution for an input radiation at ~550 nm, in which one can see that backward-propagating SP_CM/F modes are very strongly excited along the metal-fiber interfaces when the phase-matching condition of Eq. (5) is satisfied. This diminishes the out-coupling efficiency of the CA-MCAFF at a very specific wavelength, which must be an undesirable situation, because it eventually leads to split of the ICE_out spectra, as is already seen in Fig. 6(a).

However, this 2nd-order coupling effect can considerably be alleviated if the “effective” asymmetry of the corrugation is minimized via adjusting its duty cycle. In Fig. 8(a), we present the calculated ICE_out spectra with respect to the duty cycle of the rectangular corrugation. In fact, without loss of generality the spectral dip at ~550 nm is more critical than the other at ~440 nm, mainly because the former is located exactly in the middle of the wavelength range of our interest. We further calculate and plot in Fig. 8(b) the ICE_out at ~550 nm for the given CA-MCAFF structure, fine-tuning the duty cycle of the corrugation. One can see that the ICE_out at the wavelength of the spectral dip is maximized or the depth of the spectral dip is minimized when the duty cycle is adjusted to ~47%. Thus, we hereinafter assume the duty cycle is set to that value. (Actually, one can further reduce down the 2nd-order coupling effect if a more sophisticate form of the modulation function is devised for the periodic corrugation rather than using a simple, rectangular function [37]. However, considering the scope of our main discussion, we do not think that such a delicate consideration is really necessary for the moment.)

 

Fig. 8 (a) ICE_out spectrum with respect to the duty cycle of the periodic corrugation. (b) ICE_out at 550 nm (i.e., the 2nd-order spectral dip position) with respect to the duty cycle of the periodic corrugation.

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Third, we determine the fiber-facet angle θ, based on the refreshed values for the modulation depth and duty cycle in the preceding steps, which are now set to be 40 nm and 47%, respectively. In Fig. 9(a) and 9(b), we present, respectively, the calculated ICE_out spectrum and the corresponding OCE_out and SB_out with respect to the fiber-facet angle of 40 to 60°. One can see that the OCE_out and the SB_out are maximized at ~50° (θ_OCE) and ~47° (θ_SB), respectively. Thus, we hereinafter assume that the fiber-facet angle is set to either θ_OCE or θ_SB from the viewpoint of the individual figure of merit of the CA-MCAFF. In addition, we also take the fiber-facet angles of ~46° (θ_3dB,L) and ~52° (θ_3dB,H) as an additional reference group for further comparison, which are the lowest and highest angles that can lead to at least half the maxima in terms of both the OCE_out and the SB_out.

 

Fig. 9 (a) ICE_out spectrum with respect to the angle of the fiber facet of a CA-MCAFF. (b) OCE_out and SB_out with respect to the angle of the fiber facet of a CA-MCAFF.

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Eventually, all the design parameters of the proposed CA-MCAFF have now been determined, although they are not final. In order to optimize them further, one may iterate all the sequential steps described in this section. However, we do not consider taking additional iterative steps in our current discussion, mainly because even without doing so, we are already in a good position to discuss the unique characteristics of the WODB functionality of the proposed CA-MCAFF.

5. Characteristics of the CA-MCAFF in terms of WODB functionality

In particular, for the four different fiber-facet angles, i.e., θ_OCE, θ_SB, θ_3dB,L, and θ_3dB,H, we plot the far-field magnitude distribution of the out-coupled optical radiation from the CA-MCAFF in Fig. 10 with respect to the wavelength of the incident optical radiation and the azimuthal angle ϕ [see Fig. 4(b)]. (The results for other fiber-facet angles can also be found in the attached supplementary video clip, Visualization 1.) In each figure, a virtual line that may be drawn by joining the highest field points at the given wavelengths, running from the left-mid position to the right-top corner, represents the primary out-coupling angle of the optical radiation from the CA-MCAFF. We highlight that this virtual line has good linearity in the given wavelength range. In addition, the white solid line denotes the out-coupling-angle trace calculated based on the phase-matching condition between the LSP mode (k_LSP), and the free-space optical radiation (k₀) via Eq. (4). In comparison, the black dashed line denotes the out-coupling-angle trace calculated based on the phase-matching condition by the SP_CM/A mode (k_SP,CM/A) instead of the LSP mode. As we have already explained in Section 3 that the LSP mode is the dominant SP mode in the given structure, one can see that the virtual out-coupling angle is, in general, fitted better with the out-coupling-angle trace by the LSP mode (i.e., the LSP phase-matching curve) than with the curve by the SP_CM/A mode (i.e., the SP_CM/A phase-matching curve). In particular, one can also see that the ICE_out tends to be high when the LSP phase-matching curve coincides with the SP_CM/A phase-matching curve, thereby leading to a low wavenumber difference (i.e., |k_LSP – k_SP,CM/A|) between the two SP oscillations. In fact, one can intuitively say that the distance between the two curves and its change rate are roughly inverse-proportional to the ICE_out and the SB_out, respectively. (See Appendix for more details.)

 

Fig. 10 Numerical results of the far-field magnitude distribution (colormap) and the analytical results of the primary out-coupling angle (the white solid and black dashed lines for the LSP and SP_CM/A phase-matching curves, respectively) of the optical radiation from the CA-MCAFF with respect to the wavelength of the incident optical radiation and the azimuthal angle ϕ, regarding four typical cases of the fiber-facet angle: (a) θ_3dB,L = 46°, (b) θ_SB = 47°, (c) θ_OCE = 50°, and (d) θ_3dB,H = 52°. The red dash-dotted and blue dash-dotted curves represent the phase-matching conditions for the cases in which backward-propagating SP_CM/A and SP_CM/F components are resonantly generated, respectively.

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Although we have presumably focused on the 1st-order out-coupling effect (m = 1), which is the most dominant out-coupling mechanism from forward-propagating LSP and SP modes to free-space optical radiation [see Eq. (4)], one can also observe the 0th- and 2nd-order out-coupling effects, which are more easily noticeable in viewing Visualization 1. For example, for low facet angles (below 45°) at the bottom side of the images shown in Visualization 1, a long horizontal lobe exists, which is due to the 0th-order out-coupling effect. In the same images for high facet angles (over 50°), one can also observe that a portion of the optical radiation is coupled out in the case of short wavelengths and high azimuthal angles (i.e., the left-top corner of the image), which is on account of the 2nd-order out-coupling effect resonant at ~430 nm. Thus, it must be sensible to choose the fiber facet angle within the range between 45 to 50° in order to avoid the 0th- and 2nd-order effects that may eventually degrade the OCE_out.

In addition, one can observe the virtual line of the primary out-coupling angle has two slight, spectral dips, one located at ~450 nm and the other located at ~550 nm. These two are due to the unwanted backward propagating SP components (see Fig. 7). The phase-matching conditions for these spectral dips can be calculated analytically via Eq. (5) for m = −2, and the calculation results are also shown in Fig. 10: The red dash-dotted and blue dash-dotted curves represent the corresponding phase-matching conditions, respectively, indicating when backward-propagating SP_CM/A and SP_CM/F components are generated. In fact, when these two curves pass through the virtual line of the primary out-coupling angle, degradation of ICE_out is expected. Notwithstanding, it is worth noting that the spectral dip at ~550 nm is significantly shallow and its effect is considerably alleviated. This is because the duty-cycle of the given CA-MCAFF has been precisely adjusted in order to minimize the resonant coupling to the backward-propagating SP_CM/F mode particularly at 550 nm (see Fig. 8).

To justify the spatial beaming characteristics of the out-coupled optical radiation from the CA-MCAFF, we plot the field-magnitude patterns in Fig. 11 for a few typical incident wavelengths (i.e., 450, 550, and 650 nm) and fiber-facet angles (i.e., 46, 47, 50, and 52°). (The more detailed information and the results for other wavelengths can be found in the attached supplementary video clip, Visualization 2.) The diagonal black solid line in the figure indicates the location of the top surface of the CA-MCAFF, and the horizontal and vertical dashed lines denote the x and z axes as defined in Fig. 4. All the field-magnitude patterns explain on their own how the corresponding CA-MCAFFs function, depending on the given fiber-facet angle and incident wavelength, as having been discussed with the results shown in Figs. 9 and 10.

 

Fig. 11 Field magnitude patterns for the incident optical radiations at 450, 550, and 650 nm, regarding the four typical cases of the fiber-facet angle: (a) θ_3dB,L = 46°, (b) θ_SB = 47°, (c) θ_OCE = 50°, and (d) θ_3dB,H = 52°.

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In particular, comparing the field-magnitude patterns in the same horizontal row, i.e., for a fixed fiber-facet angle but with different wavelengths, one can clearly see that the primary out-coupling angle changes proportionally to the wavelength of the incident optical radiation, demonstrating its inherent WODB characteristics. In addition, comparing the field-magnitude patterns in the same vertical column, i.e., for a fixed wavelength but with different fiber-facet angles, one can see that the out-coupling angle against the fiber axis (i.e., the z axis), which is given by ϕ_F = θ + ϕ [see Fig. 4(b)] remains nearly unchanged, regardless of the value of the fiber-facet angle θ. For further clarification, varying the fiber-facet angle, we calculate ϕ_F’s for typical wavelengths from 400 to 700 nm by means of the analytical [via Eqs. (2) and (4)] and FEM-based numerical methods, and present them in Fig. 12. Although there are some discrepancies between the analytic and numerical results to some extent, one can clearly see that ϕ_F remains nearly constant, independent of the value of the fiber-facet angle θ, except for the one having some mismatch for θ = 45° at 400 nm. The exception is owing to the generation of backward-propagating SP mode (–k_SP,CM/A) as discussed earlier. The overall preservation of ϕ_F is a unique feature of the proposed CA-MCAFF, which is primarily thanks to the fact that the LSP mode plays the dominant role in converting the incident fiber mode into the free-space optical radiation. The out-coupling angles of other wavelengths are also presented in the supplemental video, Visualization 3.

 

Fig. 12 Out-coupling angle relative to the fiber axis (i.e., ϕ_F) with respect to the fiber-facet angle.

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Based on the FEM simulations, we finally calculate the OCE_out, SB_out, angular width of the out-coupled optical radiation, WODB sensitivity, and primary out-coupling angle for the four specifically defined fiber-facet angles of θ_3dB,L, θ_OCE, θ_SB, and θ_3dB,H (i.e., 46°, 47°, 50° and 52°) for four typical wavelengths of 400, 500, 600, and 700 nm. These are summarized in Table 1 together with the properties that can be obtained from a typical nano-slit-based device having a similar WODB functionality for the sake of comparison [25].

Table 1. Characteristics of CA-MCAFF schemes in comparison with a typical nano-slit-based WODB scheme.

View Table

It is worth noting that the angular width and the WODB sensitivity are defined by the angular spread of the 1st-order out-coupled component within the 3-dB intensity range and the change rate of the 1st-order out-coupling angle per unit wavelength, respectively. In particular, the characteristics of the CA-MCAFFs of the four different fiber-facet angles represent themselves well why we chose those angles specifically. For example, θ_SB was set to 47° for wide SB operation, and θ_OCE was set to 50° for high OCE_out operation (see Fig. 9).

Having been shown in Figs. 10 and 12, the linearity of the WODB and the primary out-coupling angle ϕ_F are well preserved even when the fiber-facet angle is significantly detuned from the optimal angles, such as θ_SB and θ_OCE. We emphasize that these features are really attractive and advantageous, considering that they will provide high tolerance from the viewpoint of the fabrication of the angled fiber facet. In addition, we also stress that the OCE_out of the proposed CA-MCAFFs can increase up to 30.4%, which is enhanced by an order of magnitude in comparison with that of a nano-slit-based device having a similar type of WODB functionality whose OCE_out is limited to ~2.37% at the maximum [25]. In terms of SB, the proposed CA-MCAFFs also exhibit substantial enhancement by an order of magnitude in comparison with that of a nano-slit-based device.

6. Conclusion

We have proposed a novel fiber-optic-plasmonic hybrid device for high efficiency WODB. In principle, the device is realized by the periodic corrugation of a thin metal layer put upon an angled fiber facet. In fact, the metal layer itself (i.e., MCAFF) couples the optical fiber mode to relevant SP modes, and the additional periodic corrugation decouples the SP modes into free-space optical radiation in different directions, depending on the wavelength of the incident optical radiation. Therefore, a novel SP-based WODB functionality is obtained in a simple, compact, and all-fiberized format. We highlight that to the best of our knowledge, this is the first numerical work on a plasmonic-fiber-optic hybrid device for WODB exploiting a “no-aperture” configuration. We also emphasize that the validity and feasibility of our numerical modeling method used in this work has already been verified in comparison with our previous experimental work on a similar WODB device, i.e., a corrugation-assisted nano-slit structure for WODB [25].

We initially investigated the characteristics of the MCAFF with no corrugation, which is a modified Kretschmann’s configuration, implemented onto a fiber-optic platform. The configuration offers the capability of generating unidirectional SPs as well as being able to minimize non-confined diffracted radiation loss. Therefore, the proposed scheme dramatically enhances the SP coupling efficiency, in comparison with nano-slit-based schemes that normally suffer from the aforementioned loss mechanisms. We justified the intrinsic characteristics of the proposed device in terms of the thickness of the metal layer and the angle of the fiber facet. This was carried out to see whether the proposed device is capable of yielding high OCE and broad SB, which are pivotally required for WODB functionality. Regarding the thickness of the metal layer, we found that the SP coupling efficiency is inversely proportional to it. However, since a thickness below 15 nm tends to make the optical fiber mode to directly penetrate the metal layer, we conservatively determined it to be 20 nm, also considering tolerable variations during its fabrication process, e.g., an E-beam or FIB process. With this thickness, we further analyzed the MCAFF in terms of the fiber-facet angle and verified that it can support high OCE (over 60%) and broad SB (over 200 nm) suitable for producing the WODB functionality in the visible range from 400 to 700 nm.

We then extended our investigation to the MCAFF with a periodic corrugation, i.e., the CA-MCAFF, and characterized its WODB functionality. In fact, the role of the periodic corrugation is to out-couple the SP modes bounded at the metal-air interface originally excited from the optical fiber mode, into free-space optical radiation in a direction that is significantly tuned by the wavelength of the incident optical radiation. We found that two types of SP modes play a crucial role in determining the WODB functionality of the CA-MCAFF, including the OCE_out and SB_out: One is the LSP mode that is a localized SP at the trough region of the periodic corrugation, directly driven by the incident optical fiber mode, and the other is the SP_CM/A mode that is the non-localized, eigen-mode bounded along the overall metal-air interface. We verified that although the LSP mode plays a dominant role in leading to the WODB functionality, its OCE_out is maximized when the LSP mode and the SP_CM/A mode are phase-matched or resonant. In particular, we paid more attention to two specific fiber-facet conditions, in which the CA-MCAFF is optimized in terms of OCE_out or SB_out. In the first condition given by θ_OCE = 50° for maximizing its OCE_out property, we obtained 30.4% and 230 nm for the OCE_out and SB_out, respectively. In the second condition given by θ_SB = 47° for maximizing its SB property, we obtained 24.5% and 245 nm for the OCE_out and SB_out, respectively. We emphasize that with this novel scheme, we were able to enhance the OCE_out and SB_out by an order of magnitude, in comparison with those of a nano-slit-based device having a similar type of WODB functionality. In fact, these substantial enhancements are mainly thanks to the novel implementation of the Kretschmann configuration on a fiber-optic platform. We also stress that the beaming characteristics of the CA-MCAFF are well preserved even with the detuning of the fiber-facet angle from 40° to 60°, which will eventually allow for high tolerance from the viewpoint of its fabrication.

We expect that the proposed CA-MCAFF scheme will be very useful for various plasmonic and optical applications where WODB functionality is required, thanks to its compactness, flexibility, and cost-effectiveness. In particular, the proposed MCAFF scheme (fiber-to-SP coupling scheme) has great potential to be an excellent, alternative SP generation method that can provide high efficiency, unidirectionality, and full compatibility with fiber-based optical sources. We believe that our study will broaden the fiber-optic and plasmonic research fields that are invariably seeking novel opportunities for combining the extraordinary characteristics of plasmonics and the outstanding flexibility of fiber optics.

Appendix

We further provide intuitive explanations of the coupling behaviors of the MCAFF or the CA-MCAFF in terms of the wavenumber of SP modes, reciprocal lattice vector of the periodic corrugation, and the attenuation coefficient of the metal, which we believe will be beneficial to understanding of the mode coupling principle in the given configuration.

Case 1. Dispersion analysis on the MCAFF

The characteristics of the OCE_fiber-SP and the SB_fiber-SP presented in Fig. 3(a) can also be explained in terms of the dispersion relations of the normal SP and LSP modes that are excited in the given structure, together with the attenuation property of the metal layer, which are shown in Fig. 13. In principle, one can think about two crucial factors that determine the OCE_fiber-SP and the SB_fiber-SP: One is how closely the normal SP mode (k_SP,PM/A) is matched with the LSP mode (k_LSP ≈ n_eff k₀ sin θ). In fact, the LSP mode can be regarded as the driving polarization, so that phase matching condition implies the condition for resonance between the normal SP mode and the driving LSP mode. Both dispersion curves can readily be obtained via Eq. (1) and are shown in Fig. 13. The other is how low the attenuation coefficient of the metal layer is, because the imaginary part of the wavenumber of the SP mode is approximately proportional to the magnitude of the attenuation coefficient of the metal layer. In fact, in Fig. 13 one can see that the phase matching or resonance condition between the normal SP mode and the LSP mode is well satisfied across the whole wavelength range for the given fiber facet angle of 45 °. It is worth noting that the amount of the phase mismatch between the normal SP mode and the LSP mode (i.e., Δk = k_SP – k_LSP) is relatively large at low wavelengths in comparison with those at high wavelengths, which may give rise to relatively lower OCE_fiber-SP via the first factor aforementioned. However, this effect is well alleviated via the second factor aforementioned, because the ohmic attenuation is relatively low at low wavelengths. Therefore, the consequence balanced by the two factors could lead to the considerably broad SB_fiber-SP as shown in Fig. 3(a).

 

Fig. 13 Dispersion curves of SP modes [k_SP,PM/A and k_LSP (θ = 45°)] and the attenuation coefficient.

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Case 2. Dispersion analysis on the CA-MCAFF

We would like to provide an additional discussion about the characteristics of the OCE_out and SB_out of the CA-MCAFF in terms of SP resonance and attenuation conditions in a similar manner as in Case 1. In Fig. 14 we plot k_SPCM/A, k_SP,CM/F, and k_LSP with respect to wavelength. Each dispersion curve can be obtained analytically via Eq. (2) and Eq. (3). In fact, the ICE_out tends to increase when k_LSP (driving polarization) and k_SP,CM/A (eigen polarization) are matched or Δk = 0. In a similar manner as mentioned in Case 1, the ICE_out is also dependent on the attenuation coefficient of the metal layer.

 

Fig. 14 Dispersion curves of SP modes [k_SP,CM/A, k_SP,CM/F and k_LSP (θ = 47° and θ = 50°)] and the attenuation coefficient.

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Incidentally, in Fig. 14, one can see that the trace of k_SP,CM/A is slightly shifted up in comparison with the case of k_SP,PM/A (the wavenumber of the SP bounded in the plain surface with no corrugation; see Fig. 13), which is due to the fact that the inclusion of the corrugation elongates the path length of the metal surface that the SP propagates, so that the effective wavenumber of the SP in the direction parallel of the metal surface increases accordingly [see Eq. (3)]. As a result, one can readily expect that, if the metal layer incorporates the periodic corrugation, the fiber-facet angle for high SP coupling efficiency should be adjusted from 45° such that the k_LSP is matched with the shifted k_SP,CM/A. However, it is worth noting that the k_LSP cannot be perfectly matched with the k_SP,CM/A over the entire wavelength range on account of their intrinsically different dispersion relations. Hence, a fiber-facet angle should be carefully determined, depending on which figure of merit is more critical in terms of the desired performance of the CA-MCAFF.

For example, if the fiber-facet angle is set to 50° (the black solid line), the k_LSP curve intersects or is perfectly matched with the k_SP,CM/A curve at ~ 500 nm (blue-shifted from the center of the entire spectral range), giving rise to moderate quantities of Δk for both the low and the high wavelength ranges. Thus, one can intuitively expect that the ICE_out would peak in the slightly low wavelength range where the attenuation coefficient is low. However, it would be reduced down gradually as the wavelength is further detuned, decreasing more rapidly for high wavelengths. Thus, this fiber-facet angle suits better for offering optimized performance in terms of OCE_out rather than SB_out, as was observed in Fig. 9(b). In contrast, if the fiber-facet angle is set to 47 °, the k_LSP curve intersects with the k_SP,CM/A curve at ~ 580 nm (red-shifted from the center of the entire spectral range), giving rise to enlarged Δk for the low wavelength range but significantly reduced Δk over the high wavelength range. One can expect that this situation would lead to high performance in terms of SB_out because the better phase-matching conditions in terms of Δk in the long wavelength range are considerably washed out by the higher attenuation, whereas the bad phase-matching condition in the low wavelength range is compensated by the low attenuation, although the OCE_out performance would be relatively degraded. Indeed, the dispersion relations together with the attenuation property presented in Fig. 14 qualitatively explain why and how the θ_OCE and θ_SB were determined as 50° and 47° in Fig. 9(b), respectively.

By means of the dispersion relations shown in Fig. 14, one can also verify whether and when the 2nd-order coupling into a backward-propagating SP mode takes place. In fact, this happens when the phase-matching condition of Eq. (5), i.e., k_LSP – 2k_C = –k_SP, is satisfied. This means that the mean value of k_LSP and k_SP (i.e., k_SP,CM/A or k_SP,CM/F) should be matched with k_C. In other words, the location of the k_C is equidistant to the corresponding k_LSP and k_SP. Based on the curves of k_LSP, k_SP,CM/A and k_SP,CM/F given in Fig. 14, one can graphically determine where the required phase-matching condition is satisfied: The locations are now indicated by the vertical dashed lines. Once the 2nd-order coupling into a backward-propagating SP mode bound at the metal-fiber or metal-air interface becomes resonant at such a wavelength, it would eventually give rise to substantial loss in the out-coupling efficiency. The resonant wavelengths estimated by the phase-matching condition via the dispersion relations illustrated in Fig. 14 are well matched with the wavelengths at which the ICE_out have spectral dips as shown in Fig. 9(a).

Funding

National Research Foundation of Korea (NRF) grant funded by the government of Korea (2014R1A1A2059418); Ministry of Trade, Industry and Energy (10065150); Ministry of Science, ICT and Future Planning (R01011600680003003); Brain Korea 21 Plus Program.

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