1. Introduction

Young’s double-slit interference is a fundamental optical phenomenon in wave optics. However, a subwavelength metallic double slit plays an important role in surface plasmon subwavelength optics [1], and it was proved to be an excellent basic element with which to investigate the interaction between subwavelength objects [2,3], to reveal the physical mechanism of extraordinary optical transmission through subwavelength holes arrays [4–6], and to analyze the near-field characteristics of surface plasmon polaritons (SPPs) [7–9]. Furthermore, single slits or multislits were also widely investigated in order to construct the basic theory of surface plasmon optics [10–12] and to explore practical applications such as surface plasmon photolithography [13,14]. So far, these studies have given us a profound understanding of surface plasmons and correlative phenomena. Recently we found that for a metallic double nanoslit with different widths, the Young’s interference order could be shifted readily via changing the width of the two slits. We could even get a dark fringe at the perpendicular bisector of the two slits, which is contrary to the classic Young’s interference pattern. To reveal the physical mechanism of the phenomena, we designed a series of double slits with variant slit width differences, film thicknesses, and slit distances, and then performed finite-difference time-domain calculations to analyze the relationship between these parameters and the shift of interference fringes.

2. Simulations and results

The schematic of a double nanoslit with different widths is illustrated in Fig. 1. Two slits are formed on a silver film with a thickness of h; the widths of the first and the second slits are a₁ and a₂, separately; and the distance between the two slits is d. When a TM-polarized light is incident from the left side, it excites surface plasmon modes at the slit entrance, while at the slit exit the propagated surface plasmon modes are scattered into free space light as well as converted into surface plasmons that propagate on the metal–air interface. The interference of the diffracted lights from the two slit exits creates the near-field pattern and the corresponding far-field fringe. In the finite-difference time-domain calculations, a 632.8 nm TM-polarized light is utilized, and the corresponding dielectric constant of silver is fixed to ε_m=-16.24+i0.66.

 

Fig. 1. Sketch of metallic double nanoslit with different widths.

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In the classic Young’s double-slit experiment, the two slits with dimensions comparable to wavelengths are utilized. If the incident light is normal to the surface, a bright fringe will appear at the perpendicular bisector of the two slits, for the phase difference of light emerging from the two slits is equal to zero at this line. In Fig. 2(a), a calculated near-field |Hy| distribution pattern of the classic Young’s interference is given, and the parameters in the calculation are set as follows: the film thickness h=700 nm, the slit width a₁=a₂=100 nm and the slit distance d=2.6 µm. The figure shows seven bright fringes in the near field, and the bright region is at the perpendicular bisector. The far-field pattern shown in Fig. 2(b) is determined by the near-field information, which also has seven bright fringes (not shown), and a peak appears at the center of the far-field divergence distribution. Here, only the two lowest interference orders are shown to reveal the order shift in what follows.

Suppose a₂ is reduced to 25 nm and other parameters are fixed as in Fig. 2(a); the near-field |Hy| distribution and the corresponding far-field divergence can be obtained as shown in Fig. 2(c) and Fig. 2(d), respectively. The results suggest that the dark fringe appears at the perpendicular bisector and, thus, far-field fringes shift. At the center of the far-field divergence pattern the energy becomes the lowest, which is contrary to the classic Young’s interference pattern.

 

Fig. 2. (a) Near-field |Hy| distribution pattern, a ₁=a ₂=100 nm. (b) Far-field divergence angle, a ₁=a ₂=100 nm. (c) Near-field |Hy| distribution pattern, a ₁=100 nm and a ₂=25 nm. (d) Far-field divergence angle, a ₁=100 nm and a ₂=25 nm.

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For analyzing the effect of slit width differences on the shift of interference orders, a series of numerical calculations were performed with the geometrical parameters designed as follows: the film thickness h=700 nm, the distance between slits d=2.6 µm, the width of the first slit a₁=100 nm, and the width of the second slit a₂ varies from 15 to 95 nm. From the far-field calculation results shown in Fig. 3(a), we know that the interference orders shift to the second slit along with the decreasing width of the second slit. When the a₂ width is reduced to 25 nm, the original dark fringe moves to the center of the two slits and a reverse pattern compared to the original one is formed. If a₂ continuously decreases, the interference orders will keep shifting to the second slit. It is found from Fig. 3(b) that the larger slit width difference corresponds to the rapid shift of the interference orders. In our earlier work, we demonstrated that the narrower slit gives a faster change in the propagation constant of the surface plasmon mode in a metallic nanoslit [15]. Based on the similar rule, the phenomena can be interpreted as follows: the relative phase retardation between the two SPPs launched by the slits varies when one slit width is decreased, which thus leads to the shift of interference orders.

 

Fig. 3. (a) Far-field divergence angle with different a₂. (b) Fringe positions versus slit width differences.

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Fig. 4. (a) Far-field divergence angle of variant film thicknesses. (b) Fringe positions versus film thicknesses. (Color symbols and lines: simulated results. Black lines: theoretical results).

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Further investigation focuses on the effect of film thickness on the shift of interference fringes. Assume the slit widths are a ₁=100 nm and a ₂=25 nm, the distance between two slits is d=2.6 µm, and the film thickness varies from 400 to 1300 nm with a step of 50 nm. From the calculated far-field results shown in Fig. 4(a), we learned that when the film thickness increases steadily from 400 to 1300 nm, the interference orders gradually shift to the second slit. The far-field pattern experienced two particular states during the process: one state with dark fringe at the center when the film thickness h=700 nm, and the other state with bright fringe at the center when the film thickness h=1300 nm. Figure 4(b) presents the shift of the two lowest dark and bright fringes on the film thickness, and the fringes shift to the second slit at a steady pace and a periodical vibration. From the idea illustrated in the previous section, we can find that the difference of the surface plasmon mode remains constant when the slit widths are fixed; the longer the propagation length, the larger the phase retardation difference between the two slits. When the phase retardation difference reaches 2π, the bright fringe appears at the center again, which agrees well with the calculation results of the film thickness of 1300 nm. However, the periodical vibration characters are not included in the model and will be further discussed in the next section.

Assume the distance between the two slits varies from 2.0 to 2.8 µm, while the slit widths a ₁=100 nm and a ₂=25 nm and the film thickness h=700 nm are fixed. From the near-field |Hy| distribution shown in Figs. 5(a) and 5(b) and far-field results shown in Fig. 5(c), we can see that the number of near-field fringes increases from 6 to 8 when the slit distance increases from 2.0 to 2.8 µm, and the far-field patterns have the same performance, i.e., the width of the bright and dark fringes are all compressed. Furthermore, it is worth noting that the fringe at the center region is dark for all slit distances, namely, the phase retardation difference between the two slit exits is independent of the slit distances.

 

Fig. 5. (a) Near-field |Hy| distribution pattern, d=2.0µm. (b) Near-field |Hy| distribution pattern, d=2.8µm. (c) Far-field divergence angle of variant slit distances.

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3. Discussions

From the previous results, we can fully confirm that the difference of the surface plasmon mode in two slits is the decisive factor in the shift of the interference orders. As we demonstrated in the previous work, the narrower the slit waveguide, the stronger the coupling between the SPPs of the two walls. This results in the increase of the SPP constant [15], i.e., the increase of the waveguide effective refractive index (k_sp/k₀). An illustrative effective refractive index versus metallic slit width is shown in Fig. 6. For an incident light with a wavelength of 632.8 nm, the effective refractive index in a 100 nm width slit is 1.24 and the effective index increases rapidly with the decrease of slit width; it reaches to 1.72 when the slit width decreases to 25 nm. Thus, the phase retardation difference between the two slits of Δφ=2π(n₂-n₁)h/λ varies with the slit width difference and film thickness. As in the previous section about changing the slit width difference, the film thickness and the first slit width are fixed to 700 nm and 100 nm, respectively, and the slit distance is 2.6 µm. Hence, the phase retardation difference Δϕ=2π(n₂-1.24)0.7/0.6328, n₂ increases with the decrease of the second slit width a₂ and reaches to 1.72 when a₂=25 nm, which leads to Δϕ≈π and thus the dark fringe appears in the center. We achieve extremely similar performances by varying the film thickness, where the slit width a₁=100 nm, a₂=25 nm, and slit distance d=2.6 µm are fixed, giving a phase retardation difference of Δϕ=2π(1.72-1.24)h/0.6328. We can obtain Δϕ≈π and Δϕ≈2π when h=700 nm and h=1300 nm, respectively. Thus, the two particular phase retardation differences lead to the reverse far-field patterns.

We also theoretically calculated more details of the far-field fringe position dependence of slit width differences and film thicknesses, as shown in Fig. 3(b) and Fig. 4(b), utilizing the model of Δϕ=2π(n₂-n₁)h/λ. The results show that the trend of the simulation curves and the theoretical curves agrees well, but the simulation results have an oscillation nature, which has a period of about 187 nm [Fig. 4(b)]. As the effective refractive index of the second waveguide is n₂=1.72, i.e., the effective wavelength in it is about 368 nm, we can obtain that the oscillation period of 187 nm is half an effective wavelength in the second waveguide, which indicates the F–P-like multi-interference process in the waveguide. At the film thickness of λ_eff/4+ m*λ_eff/2 (m is an integer), the resonant condition is fulfilled, which can affect the phase effectively at the slit output. Actually, the resonant effect in the first slit waveguide also works, and the superposition of the resonant effect of the two slits generates the oscillation shown in Fig. 4(b), whose period is approximately equal to that of half an effective wavelength in the second waveguide. The time–average results of Fig. 2(a) and Fig. 2(c) reveal the standing wave pattern in the waveguide, which also validates the F–P resonant in the waveguides.

 

Fig. 6. Effective refractive index of surface plasmon mode in metallic nanoslit with variant widths and wavelengths.

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In addition, a slight change in wavelength also results in the shift of interference orders for the same structure. For example, when the two slits have widths a₁=100 nm and a₂=25 nm, the effective refractive index of n₁=1.51 and n₂=2.65 can be calculated to correspond to the incident wavelength of 365 nm and Δn=1.14; if the incident wavelength is 360 nm, the effective index becomes n₁=1.54 and n₂=2.75, and the corresponding Δn=1.21. Therefore, the interference pattern of the two cases could be the inverse for the film with a thickness of several wavelengths. This sensitivity of the effective refractive index on the wavelength is extremely high in the ultraviolet region, where Δn/Δλ reaches the order of 10^-2. Further exploration of this dispersion character associated with the action of film thickness is expected to open up a new way of ultrahigh resolution spectral analysis.

4. Conclusion

In this paper we proposed a type of metallic double nanoslit with different widths. The position of Young’s interference fringes could be controlled readily through modulating the slit width differences and film thicknesses. We demonstrated that the shift of the interference orders is mainly due to the additional phase retardation produced by the surface plasmon mode in slits with different widths. Further research expanded to polychromatic light is expected to have potential applications in ultrahigh resolution spectral analysis.