1. Introduction

In the last several decades, optical diode has drawn great attention of researchers [1]. The first proposed optical diode used the medium doped with a nonlinear material [2] and worked on asymmetric Bragg reflection. With the development of the study, wide varieties of materials have been employed in the design of optical diode, such as anisotropic [3], magneto-optic [4], left-handed [5] and random amplified materials [6]. There is also another kind of optical diode made of isotropic materials, which is fabricated asymmetrically in structure to implement the optical diode function [7,8]. Recently, two burgeoning new kinds of material with optical diode function have aroused great interest of researchers. One is the asymmetric photonic crystal [9,10]. The other is the asymmetric metal grating [11–15], operating on the surface plasmon enhanced effect.

The phenomenon of enhanced transmission through a subwavelength metal film perforated with hole arrays was observed by Ebbesen et al. in 1998 [16]. It evoked great interest of study on the metal film with various micro-nano structures, such as one-dimensional gratings [17], periodic cylinder arrays [16], periodic rectangle arrays [18], a single hole (or slit) surrounded by periodic grooves [19,20], and the asymmetric metal grating [11–15] mentioned above. The last one having optical absorption is featured of geometric asymmetry, which breaks the time reversal symmetry of the electromagnetic field and results in the nonreciprocal optical transmission, acting as the optical diodes. In this paper, we propose a simple metal micro-nano structure having the character of nonreciprocal optical transmission. The structure shown in Fig. 1 is a single conical air hole (CAH) in an Ag film, which has evidently geometric asymmetry. Using the method of numerical simulation (the 3D FDTD method, Meep software [21]), we analyze the forward and backward zero-order transmission spectra of this structure and discuss the influence of structural parameters on the spectra. Under optimized conditions, we find that the ratio of forward zero-order transmission to backward one can be larger than 9 at a proper wavelength in visible range. In numerical analyses, the conditions of linearly polarized incident light and uniaxial perfectly matched layer absorbing boundary [22] are used. And the effect of plasmon resonance of Ag surface is interpreted by the complex permittivity from the Lorentz-Drude model [23]. For simplification, we choose the medium outside of the Ag film to be air with a relative permittivity of 1. The normalized optical zero-order transmission through a CAH is defined as the ratio of the power flux integrated at the exit of hole to that through the cross section of the large end of hole. Furthermore, we set the forward direction along the axis of the cone, from the large end to the small one, and vice versa.

 

Fig. 1 The schematic of a single CAH in an Ag film.

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2. Analysis and discussion

Generally, the transmitted light through an Ag CAH mainly contains three parts: the diffracted light, the light stemmed from LSPs stimulated at the small end of hole, and that from guided mode related to the FPL effect. When the diameter of small end of hole is less than the wavelength of the incident light, the diffracted light barely exists. But the transmission can be greatly enhanced in certain wave bands due to the excitation of LSPs. And when the diameter of small end of hole is larger than the wavelength of the incident light, the diffraction effect will play an important role and the effect of LSPs is weak. It is known that there is no surface plasmon for IC under nonperiodic condition [24], so we can make a comparison of spectra between an Ag CAH and an IC one, to figure out the contributions of different physical effects and how they interact. We now consider two special series of structures with the same and unchanged film thickness. The first series is that, the diameter of small end of hole is evidently smaller than 400 nm and is kept unchanged, while the radius of large end of hole (R_L) is made variable. The second one is that, the diameter of small end of hole varies and is equal to or larger than 800 nm, and R_L is fixed. For the first series, the radius of small end of hole (R_S) and the film thickness (t) are fixed to be 100 nm and 1000 nm, respectively, and R_L takes the values of 400 nm, 500 nm, 600 nm, and 700 nm, respectively. The normalized zero-order transmission spectra of CAH are shown in Figs. 2(a)-2(d), where the dark and light blue lines are the forward and backward zero-order transmission spectra of Ag CAH, respectively; and the red and yellow lines are those of IC CAH, respectively. These conventions are always used below. In this series of structures, the forward transmitted light of IC CAH (ranging from 400 nm to 800 nm; the same below) is mainly from guided mode and is rather weak when the wavelength is larger than 600 nm, due to the backward reflection off the CAH [see Figs. 3(c) and 3(d)]. Comparing the spectra of IC CAH with those of Ag CAH in Figs. 2(a)-2(d), we can conclude that, the transmission of Ag CAH is mainly from LSPs since its peak is located at the edge of the transmission band of IC CAH. The electric field distributions shown in Figs. 2(i)-2(j) and those in Fig. 3 can confirm this further. For example, for the structure corresponding to Fig. 2(a) with an incident light of 550 nm, there is a maximal value of light field at the small end for Ag CAH [Figs. 2(i)-2(j)] but not for IC CAH [Fig. 3(b)]. And at 550 nm, the forward transmission of the Ag CAH is almost maximal while that of IC CAH is rather weak. From Figs. 2(a)-2(d), one would find that the transmission of Ag CAH in the short wave band is lower. This is due to that the guided wave of shorter wavelength in Ag CAH has a larger loss [smaller surface reflectivity of Ag; see Fig. 2(k)] and a weaker intensity reaching at the small end of Ag CAH yields weaker LSPs [see Fig. 3(e)]. The redshift of the forward transmission spectrum of Ag CAH shown in Figs. 2(a)-2(d) is also partly related to this effect. When the diameter of large end of Ag CAH grows, on one hand, the light experience more forward reflections on the wall of hole and the loss of short wave is enhanced; and on the other hand, the chance for long-wave light to reach the small end of CAH (for excitation of LSPs) increases and the long-wave forward transmission goes up. What we are especially interested in is that when the large end of hole grows, which enlarges the structural asymmetry, the difference between the forward and backward transmissions becomes large. As shown in Figs. 2(a)-2(d), the ratio of forward transmission to backward one rises from 4.32 to 9.24.

 

Fig. 2 (a)-(h): Normalized zero-order transmission spectra for different hole ends. (a)-(d): R_S = 100 nm, t = 1000 nm, and R_L is chosen to be 400 nm, 500 nm, 600 nm, and 700 nm respectively. (e)-(h): R_L = t = 1000 nm, and R_S is chosen to be 400 nm, 500 nm, 600 nm, and 700 nm respectively. (j)-(l): The electric field distributions when a 550nm plane wave is incident upon the Ag CAH, where (i) and (j) correspond to (a), and (k) and (l) to (e). (m): The normal reflectivity of Ag (dark blue) and ideal conductor (red) surfaces.

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Fig. 3 The electric field distributions for wavelengths of 450 nm, 550 nm, 650 nm, and 750 nm being incident upon the large ends of IC CAH [(a)-(d)] and Ag CAH [(e)-(h)], respectively. The structure is the same as that for Fig. 2(a).

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Figures 2(e)-2(h) show the zero-order transmission spectra of the second series of structures, in which R_L = t = 1000 nm, and R_S is chosen to be 400 nm, 500 nm, 600 nm, and 700 nm, respectively. Here the diameter of small end of hole is ≥ 800 nm, thus the transmitted light through the IC CAH consists of two parts from diffraction and guided mode. And for the Ag CAH, the transmitted light also contains an additional weaker part from LSPs [see the electric field distributions in Figs. 2(k)-2(l)]. One sees that there is no cutoff band in each spectrum of Figs. 2(e)-2(h), which is very different from Figs. 2(a)-2(d). This is due to that the diffracted light makes its contribution over the whole wave band considered. The contribution of diffracted light grows with the diameter of small end of hole. So the forward and backward transmissions increase when the diameter of small end of hole gets large. This can be seen clearly in Figs. 2(e)-2(h). The forward transmission increases with the diameter of small end is also due to the increasing contribution of guided mode since when the diameter of small end gets large the forward guided wave has a greater chance to escape from the CAH. The difference between the forward transmission of IC CAH and that of Ag CAH shown in Figs. 2(e)-2(h) is mainly due to the reflectivity difference of IC and Ag surfaces [see Fig. 2(k)]. One can see that there is comparability between difference of forward transmission in Fig. 2(g)/2(h) and that of reflectivity in Fig. 2(k). The forward transmission in Figs. 2(g)/2(h) appears positively related to the reflectivity. And this positive correlation is partly modified by the weak excitation of LSPs when the diameter of small end of Ag CAH is slightly larger than the wavelength of light [see Figs. 2(e) and 2(f)].

We further consider the influence of film thickness on the transmission spectrum. Two of structures discussed above are selected for further investigation. One corresponds to Fig. 2(a) and the other to Fig. 2(e). For these two structures, we reset t as 500 nm, 1000 nm, 1500 nm, and 2000 nm, respectively. Their zero-order transmission spectra are shown in Figs. 4(a)-4(h). It is expected that, when the diameter of small end of hole is less than 400 nm, with two end cross sections of hole kept unchanged, the bandwidth of forward transmission from guided mode will become narrow since the increase of the film thickness will multiply the forward reflections of light on the wall of hole. This phenomenon can be seen clearly from the forward transmission spectra of IC CAH shown in Figs. 4(a)-4(d). For the Ag CAH, the increase of reflections on the wall of hole will enhance the absorption to the forward traveling light. Therefore, the light intensity for exciting the LSPs decreases and the forward transmission through the Ag CAH decreases correspondingly. This is also clearly shown in Figs. 4(a)-4(d). We now focus on another kind of structures with diameters of small end of hole being ≥ 800 nm. For such structures, it is evident that, the contribution of guided mode resonance to the forward transmission is small if the film is thin enough since the reflections on the wall of hole are very few. Figure 4(e) is this case, from which we find that, the difference between the transmission spectra of Ag CAH and those of IC CAH is not so much and the forward transmissions are a little larger than the backward ones. These characters can be interpreted as follow: The light through the IC CAH contains only the diffracted part and that from guided mode. And according to the above discussion, when the diameter of small end of hole is ≥ 800 nm and the film is thin enough, the transmitted light of CAH is mainly the diffraction one. So the transmission spectrum of Ag CAH is similar to that of IC CAH since the excitation of LSPs is weak. And for the backward transmission, the light in CAH does not fully contact with the wall of hole and the guided mode resonance is weaker than that of forward transmission. So the forward transmissions of both Ag CAH and IC CAH are a little larger than the backward ones. In contrast, the forward traveling light experience more reflections on the wall of hole when the film thickness increases, the relative contribution of guided mode resonance to the forward transmission grows, hence the difference between forward transmission and backward one of the Ag CAH becomes large. For example, the average ratio of forward transmission to backward one of the Ag CAH rises from 1.3 to 4.2 when the film thickness increases from 500 nm to 2000 nm. On the other hand, the absorption loss increases when the thickness of Ag film grows, which leads to the increasing difference of forward transmissions between Ag CAH and IC CAH, shown in Figs. 4(e)-4(h). In a word, the increase of film thickness can obtain a greater difference between forward and backward transmissions but with a higher absorption loss in an Ag CAH, which should be taken into account in design.

 

Fig. 4 Normalized zero-order transmission spectra for different film thicknesses. (a)-(d): R_L = 400 nm, R_S = 100 nm, and t is chosen to be 500 nm, 1000 nm, 1500 nm, and 2000 nm respectively. (e)-(h): R_L = 1000 nm, R_S = 400 nm, and t is chosen to be 500 nm, 1000 nm, 1500 nm, and 2000 nm respectively.

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We also investigate how the transmission spectrum is changing when the CAH is pro rata enlarged (R_L, R_S, and t are enlarged by the same factor simultaneously). To do this, we choose two CAHs corresponding to Figs. 2(a) and 2(e) for enlarging. The enlarged factors for either of the CAHs are 1.0, 1.3, 1.5, and 2.0, respectively. Figures 5(a)-5(h) are the zero-order transmission spectra, where 5(a)-5(d) correspond to Fig. 2(a), and 5(e)-5(h) to Fig. 2(e). For the enlarged structures, it is expected that, the transmission from guided mode will have a redshift since it is favored for long-wave light to enter into and escape from the hole when the CAH is pro rata enlarged. Besides, the light of long wave in an enlarged Ag CAH has a lower absorption loss, which makes the forward transmission of Ag CAH rise in long-wave band. In Figs. 5(a)-5(h) the forward transmission spectrum of IC CAH really exhibits a redshift; and the forward transmission spectrum of Ag CAH shows a redshift besides rising in long-wave band when the hole is enlarged in proportion. The mentioned character for the forward transmission spectrum of Ag CAH shown in Fig. 5 is: (a)-(d), mainly due to the enhancement of long-wave light excitation of LSPs and, (e)-(h), due to the enhanced transmission of long-wave guided mode. One would find that the forward transmission spectrum of Ag CAH in Figs. 5(a)-5(d) has a decrease in short wave band when the CAH is pro rata enlarged. It is because the short wave light excitation of LSPs will become weak when the diameter of small end of hole gets large. We find from Fig. 5 that, the maximum difference between forward and backward transmissions of Ag CAH is amplified when the CAH is enlarged. So an effective way to increase the ratio of forward transmission to backward one is pro rata enlarging the dimension of Ag CAH, which is superior to that of just thickening the film.

 

Fig. 5 Normalized zero-order transmission spectra for different enlarged factors. Series (a)-(d) correspond to Fig. 2(a), and series (e)-(h) to Fig. 2(e). The structurally enlarged factors for either of these two series are 1.0, 1.3, 1.5, and 2.0, respectively.

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In brief, when the diameter of small end of hole is evidently smaller than 400 nm, the diffraction effect of light in visible range barely exists. The forward transmitted light through the Ag CAH contains mainly the components from LSPs and guided mode. The backward transmitted light through the Ag CAH contains also these two components but they are far weaker than the forward ones. For example, for the Ag CAH with the parameters of radii of two ends of hole and film thickness being 600 nm, 150 nm, and 1500 nm, respectively, the ratio of forward transmission to backward one at 605 nm is about 9.0 [see Fig. 5(c)], which shows a rather well nonreciprocal optical transmission. When the diameter of small end of hole is ≥ 800 nm, the forward transmitted light through an Ag CAH consists of diffraction part, guided mode part, and that from LSPs. Among these three parts, the light from LSPs is weak, while the other two parts are the main contributors for transmission over the visible region. In contrast, the backward transmitted light through the Ag CAH includes almost the diffracted light. Therefore, the Ag CAH with appropriate structural parameters can achieve nonreciprocal optical transmission over a broad wave band. For instance, for the radii of two ends of hole being 1500 nm and 600 nm, with a film thickness of 1500 nm, its average ratio of forward transmission to backward one over the visible range is about 3.7, as shown in Fig. 5(g), being a broad band nonreciprocal transmission, which is very different from the above case.

3. Conclusion

In conclusion, we have theoretically investigated the nonreciprocal optical zero-order transmission through the Ag CAH. The results show that the ratio of forward transmission to backward one can be larger than 9 at a proper wavelength in visible range under optimized conditions. This novel micro-nano structure, due to its simplicity, small size, and stable performance, could have the potential served as all-optical diode and have an important application in micro photonic devices.