1. Introduction

Recently, subwavelength slot waveguides have been intensively investigated due to its unique properties and potential applications. Generally speaking, there are two types of slot structures, i.e., plasmonic slot waveguide (PSWG) [1, 2] and dielectric slot waveguide (DSWG) [3, 4]. For PSWG, the photonic energy is confined near the interface of a metal and a dielectric media. Due to the very small skin depth of metal, the confinement dimension can be as small as tens of nanometers. However, the disadvantage of PSWG is the loss of metal, which is absent in DSWG.

Due to the low loss and subwavelength confinement merits of DSWG, many investigations related to which have been intensively carried out. To make full use of DSWG, the coupling of DSWG with PSWG [5], dielectric slab waveguide [6, 7], and photonic crystal (PC) waveguide [8] have been investigated. Introducing SWG in PCs, one can achieve slow light and rich dispersion of PC simultaneously [9]. DSWG are also widely used in biological and chemical sensing for the deep subwavelength confinement [10, 11]. Ref. [12] reported that the half-wave voltage of optical-electric modulation in a slotted resonant PC heterostructure is very low. What’s more, DSWG is also very useful in all-optical switching [13], and polarization splitter [14]. For a wave guiding structure, two basic functional components are very important, i.e., the sharp bends and high efficient crosses. For PC waveguide [15, 16], dielectric waveguide [17–20], and metallic waveguide [21, 22], the bending and crossing have been intensively investigated. Up to now, the bends of DSWG are also reported [23], and 90° sharp bend have been achieved [24, 25]. However, high efficient transmission of crossing DSWG is still an open problem.

In this paper, we investigate the transmission characteristics of two crossing DSWGs using the finite difference in time domain (FDTD) method. For the direct crossing of DSWG’s, the transmission is only about 40%, and the crosstalk is about 10%, while the reflection and radiation losses are about 12% and 37%, respectively. By filling up the slots locally around the cross, a high transmission of 97.2% is achieved, and the crosstalk, reflection and radiation losses are decreased to 0.3%, 0.1% and 2%, respectively.

2. Method, results and discussion

The schematic structure investigated in this paper is formed by two crossing DSWGs along the x (DSWG-x) and y (DSWG-y) axes, respectively, as shown in Fig.1. The width and refractive index (RI) of the slot regions are w_s = 70nm and n_s = 1.0 (vacuum), respectively. The width and RI of the high refractive layers are w_h and n_h = 3.2, respectively. T, R, C and RL represent the normalized transmission, reflection, crosstalk and radiation loss coefficients, respectively. The slot waveguide is proposed to operate around the communication wavelength of 1.5μm. In order to decrease R, C and RL, the two segments of A_xB_x and A_yB_y with a length of l are filled up using high RI medium.

 

Fig. 1 Schematic structure and parameters of the crossing dielectric slot waveguide (DSWG). The width and refractive index (RI) of slot (yellow regions) and high RI media (light blue regions) are w_s and n_s, w_h and n_h, respectively. Around the cross center, the segments of A_xB_x and A_yB_y (dark blue regions, with a length of l) of the slots are filled up using high RI medium (n_h). In, R, T, C and RL represent the incident, reflection, transmission, crosstalk and radiation loss, respectively. The two red arrows show the diffraction of the slot modes around the cross center.

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Using the finite difference in time domain (FDTD) method, we analyze the normalized transmission (T), reflection (R), crosstalk (C) and radiation loss (RL) of the structure. In order to obtain accurate spectra, a short Gaussian pulse is excited at the left port of the individual DSWG-x (without the disturbance of DSWG-y), and the spectrum of T₀(λ) at the right port of the DSWG-x is calculated. Then, the crossing DSWG-y is introduced, and the spectra of T₁(λ), C₁(λ), and R₁(λ) are calculated at corresponding ports (with the same excitation source). Finally, the normalized coefficients are obtained by X(λ) = X₁(λ)/T₀(λ) (X to be T, R or C). According to the conservation of energy, the radiation loss is given by RL = 1 – T – R – 2C.

Using the method mentioned above, we investigate the performances of the crossing DSWG with the parameters of w_s = 70nm and w_h = 160nm, and the results are shown in Fig.2. Fig.2(a) shows the normalized transmission T (line with circles, left y axis), radiation loss RL (line with triangles, right y axis), reflection R (line with squares, right y axis), and crosstalk (line with asterisks, right y axis), respectively. At the wavelength of 1.55μm, T is only about 35%, while the crosstalk is about 8%. The low transmission mainly results from high reflection of R = 12% and radiation loss of RL = 37%. Fig.2(b), (c) and (d) respectively show the steady field patterns of E_x, E_y and H_z at λ₀ = 1550nm. Large radiation loss from the crossing center can be observed clearly.

 

Fig. 2 Spectra and field distributions of direct crossing with w_s = 70nm, w_h = 160nm, and n_h = 3.2. (a) Spectra of transmission T (line with circles), radiation loss RL (line with triangles), reflection R (line with squares), and crosstalk C (line with asterisks). (b), (c) and (d) are the field distributions of E_x, E_y and H_z at the wavelength of λ₀ = 1.55μm, respectively. Red and blue colors represent positive and negative values, respectively (All the color maps of field pattern figures in this paper are the same).

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When the two segments of A_x,yB_x,y are filled up by using high RI medium, we investigate the performances, and show the results in Fig.3. From Fig.3(a), one can find that T increases to 97.2%, while C, R, and RL decrease to 0.3%, 0.12% and 2.1%, respectively. Compare with Fig.2(a), the transmission increases from 35% to 97.2%, while the crosstalk C and reflection R both decrease to negligible small. The radiation loss RL decreases sharply from 37% to 2.1%. Fig.3(b), (c) and (d) shows the field patterns of E_x, E_y and H_z at λ₀ = 1.55μm, respectively. Obviously, the radiation loss RL is efficiently suppressed comparing with Fig.2.

 

Fig. 3 Same as in Fig.2, except that the two segments of A_xB_x and A_yB_y with a length of l = 0.51μm are filled up using the high RI medium.

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When the operation wavelength and the slot width w_s are determined, the performances of the modified crossing are sensitively affected by the length l and width w_h of the high RI layers. Fig.4(a) shows the changes of T (line with circles), RL (line with triangles), R (line with squares) and crosstalk C (line with asterisks) with l, while the width w_h = 160nm is fixed. One can see that R, C and RL tend to minimums when l changes around 0.51μm, and T reaches to a peak value of 97.2%. Fig.4(b) shows the changes of T, RL, R and C with w_h, and l = 0.50μm is fixed. Around the value of w_h ∼ 0.155μm, T reaches the maximum of 96.0%, and R and RL both are about 2%, and the cross talk C is as small as 0.1%.

 

Fig. 4 Changes of T, RL, R, and C with l and w_h at the wavelength of 1.55μm. (a) w_h = 160nm and l ∈ (260, 650)nm. (b) l = 0.5μm and w_h ∈ (110, 180)nm.

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Now, let’s see the physical origins for the improvement of transmission in the modified crosses. We know that the slot waveguide only support TM modes (nonzero components of E_x, E_y and H_z). For a slot mode, the field of E_y is confined in a subwavelength slot region(w_s = 70nm ∼ λ₀/22 in this paper), and undergoes large reflection and strong diffraction when it transmits through the cross formed by DSWG-y, as denoted by the red arrows in Fig.1. Most part of the diffracted energy, however, can not be converted back into the guiding mode of DSWG-x due to the large mismatch of wave vector. One part of the diffraction energy results in crosstalk, and the other part radiates from the structure. This is the main reason of the radiation loss and crosstalk of the cross. This process can also be observed clearly from Fig.2.

When the segments of A_xB_x and A_yB_y are filled up with high RI (n_h) material, the slot mode localized in the width of w_s is convert into guiding mode in the slab with a width of w₁ = w_s + 2w_h. In the current situation, w₁ = 390nm is comparable with the wavelength λ_h (= 484nm) of light in the high RI medium. Therefore, the diffraction angle is greatly suppressed (compared with the diffraction of slot mode) when light propagates through the crossing region. Due to this process, the radiation loss and crosstalk are both greatly suppressed.

Although the filled segments of A_x,yB_x,y can suppress the C and RL effectively as analyzed above, the suppression of R may be somewhat surprising because additional reflection would appear at the interfaces of slot waveguide and dielectric waveguide. The physical origins of this surprising point result from the Fabry-Perot-like (FP) behavior of the filled segments of A_x,yB_x,y, which ensures maximum transmissions around the desired wavelength (1.55um in this paper) by the optimization of l and w_h. From this point of view, the condition for maximum transmission is

Using a mode analysis method and three-layer-planar-waveguide approximation of the filled segments, one can derive the value of n_e for a group of parameters of (w,n₁, n₂, n₃) for the 3-layer-plannar-waveguide. Here w and n₂ are the thickness and RI of the core layer, respectively. n_1,3 are the RI of the substrate and covering layers, respectively. In the crossing regions, however, the thickness of w changes from w₁ = 2w_h + w_s to w₂ → ∞ and w₃ = l (as shown in Fig.1). The corresponding values of n_e for w_1,2,3 respectively are (with n₁ = 3.2, n₂ = n₃ = 1.0)

When the parameters of w_h, l are optimized for a given wavelength at the case of θ_c = 90° crossing, results show that the structure is valid for θ_c in (80° ∼ 100°), as shown in Fig.5(a). When θ_c decreases from 90° to 83°, the transmission T is still above 95%, R and C [the average of C₁ and C₂ as shown in Fig5(b)] are both less than 0.6%, and the radiation loss RL reaches 4.1%. The performance can also be observed directly in Fig.5(b), which shows the field pattern of H_z at the case of θ_c = 85° and λ₀ = 1.55μm. These results shows that the crossing of DSWG is insensitive to θ_c, which makes the experimental verification of these results easier.

 

Fig. 5 (a) Changes of T, RL, R, and C with the crossing angle θ_c at λ₀ = 1.55μm. Here C is the average of C₁ and C₂ shown in (b). (b) Field pattern of H_z at the case of θ_c = 85°.

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3. Conclusions

In conclusion, we have investigated the crosses formed by two deep subwavelength dielectric slot waveguides (with a slot width < λ₀/20) in a silicon-air-silicon system. Results show that the transmission performance is very poor at the case of direct crossing. Using a simple method of filling up a section of the slot region around the cross, the transmission efficiency increases from 35% to about 97.2% in a large bandwidth around 1.55μm, and a wide crossing angle of θ_c ∈ (80°,100°). The physical origins of improvement in transmission are analyzed in detail. What’s more, the improved cross is compact, and occupies an area of less than (λ₀/2)². The results and method presented in this paper are helpful for the applications of slot waveguide in nano-photonic systems.