## Abstract

Transmission properties of two crossing dielectric slot waveguides (Si-Air-Si) are investigated using the finite difference in time domain method. Results show that the low transmission of this system mainly results from the reflection and radiation loss rather than the crosstalk. Using a simple method of filling up the crossing slots locally, the reflection, radiation losses and crosstalk are all greatly suppressed. With moderate parameters in this paper, the transmission efficient increases from 35.0% to more than 97% in a wide range of wavelength around 1.55*μm*. The results and method presented in this paper may be very useful in the application of slot waveguide in micro and nano photonics.

© 2011 OSA

## 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*= 1.0 (vacuum), respectively. The width and RI of the high refractive layers are

_{s}*w*and

_{h}*n*= 3.2, respectively.

_{h}*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 1.5

*μm*. In order to decrease

*R*,

*C*and

*RL*, the two segments of

*A*and

_{x}B_{x}*A*with a length of

_{y}B_{y}*l*are filled up using high RI medium.

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 *T*_{0}(*λ*) at the right port of the DSWG-*x* is calculated. Then, the crossing DSWG-*y* is introduced, and the spectra of *T*_{1}(*λ*), *C*_{1}(*λ*), and *R*_{1}(*λ*) are calculated at corresponding ports (with the same excitation source). Finally, the normalized coefficients are obtained by *X*(*λ*) = *X*_{1}(*λ*)/*T*_{0}(*λ*) (*X* to be *T*, *R* or *C*). According to the conservation of energy, the radiation loss is given by *RL* = 1 – *T* – *R* – 2*C*.

Using the method mentioned above, we investigate the performances of the crossing DSWG with the parameters of *w _{s}* = 70nm and

*w*= 160nm, and the results are shown in Fig.2. Fig.2(a) shows the normalized transmission

_{h}*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 1.55

*μm*,

*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*and

_{y}*H*at

_{z}*λ*

_{0}= 1550nm. Large radiation loss from the crossing center can be observed clearly.

When the two segments of *A*_{x,y}*B*_{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*and

_{y}*H*at

_{z}*λ*

_{0}= 1.55

*μm*, respectively. Obviously, the radiation loss

*RL*is efficiently suppressed comparing with Fig.2.

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*of the high RI layers. Fig.4(a) shows the changes of

_{h}*T*(line with circles),

*RL*(line with triangles),

*R*(line with squares) and crosstalk

*C*(line with asterisks) with

*l*, while the width

*w*= 160nm is fixed. One can see that

_{h}*R*,

*C*and

*RL*tend to minimums when

*l*changes around 0.51

*μm*, and

*T*reaches to a peak value of 97.2%. Fig.4(b) shows the changes of

*T*,

*RL*,

*R*and

*C*with

*w*, and

_{h}*l*= 0.50

*μm*is fixed. Around the value of

*w*∼ 0.155

_{h}*μm*,

*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%.

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*and

_{y}*H*). For a slot mode, the field of

_{z}*E*is confined in a subwavelength slot region(

_{y}*w*= 70nm ∼

_{s}*λ*

_{0}/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 _{x}B_{x}* and

*A*are filled up with high RI (

_{y}B_{y}*n*) material, the slot mode localized in the width of

_{h}*w*is convert into guiding mode in the slab with a width of

_{s}*w*

_{1}=

*w*+ 2

_{s}*w*. In the current situation,

_{h}*w*

_{1}= 390nm is comparable with the wavelength

*λ*(= 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.

_{h}Although the filled segments of *A*_{x,y}*B*_{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,y}*B*_{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

*n*is the effective index of refraction in the filled segments,

_{e}*l*is the length, and

*λ*

_{0}= 1.55

*μm*is the operation wavelength in vacuum.

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*,

*n*

_{1},

*n*

_{2},

*n*

_{3}) for the 3-layer-plannar-waveguide. Here

*w*and

*n*

_{2}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

*w*

_{1}= 2

*w*+

_{h}*w*to

_{s}*w*

_{2}→ ∞ and

*w*

_{3}=

*l*(as shown in Fig.1). The corresponding values of

*n*for

_{e}*w*

_{1,2,3}respectively are (with

*n*

_{1}= 3.2,

*n*

_{2}=

*n*

_{3}= 1.0)

*n̄*= [

_{e}*n*(

_{e}*h*

_{1}) +

*n*(

_{e}*h*

_{2}) +

*n*(

_{e}*h*

_{3})]/3 = 3.024 in Eq.(1). Substituting the values of

*l*= 0.51

*μm*,

*n̄*= 3.024, and

_{e}*λ*

_{0}= 1.55

*μm*into Eq.(1), one can obtain that

*m*= 2

*n*×

_{e}*l*/

*λ*

_{0}= 1.99 ≈ 2. This result shows that the suppression of reflection loss is result from the FP-like resonance of filled segments.

When the parameters of *w _{h}*,

*l*are optimized for a given wavelength at the case of

*θ*= 90° crossing, results show that the structure is valid for

_{c}*θ*in (80° ∼ 100°), as shown in Fig.5(a). When

_{c}*θ*decreases from 90° to 83°, the transmission

_{c}*T*is still above 95%,

*R*and

*C*[the average of

*C*

_{1}and

*C*

_{2}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*at the case of

_{z}*θ*= 85° and

_{c}*λ*

_{0}= 1.55

*μm*. These results shows that the crossing of DSWG is insensitive to

*θ*, which makes the experimental verification of these results easier.

_{c}## 3. Conclusions

In conclusion, we have investigated the crosses formed by two deep subwavelength dielectric slot waveguides (with a slot width < *λ*_{0}/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 1.55*μm*, 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 (

*λ*

_{0}/2)

^{2}. The results and method presented in this paper are helpful for the applications of slot waveguide in nano-photonic systems.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (NNSFC) under grants 11004041 and 10874036.

## References and links

**1. **J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B **73**, 035407 (2006). [CrossRef]

**2. **G. Veronis and S. Fan, “Modes of subwavelength plasmonic slot waveguides,” J. Lightwave Technol. **25**, 2511–2521 (2007). [CrossRef]

**3. **V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. **29**, 1209–1211 (2004). [CrossRef] [PubMed]

**4. **Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. **29**, 1626–1628 (2004). [CrossRef] [PubMed]

**5. **R. Yang, R. A. Wahsheh, Z. Lu, and M. A. G. Abushagur, “Efficient light coupling between dielectric slot waveguide and plasmonic slot waveguide,” Opt. Lett. **35**, 649–651 (2010). [CrossRef] [PubMed]

**6. **G. Veronis and S. Fan, “Theoretical investigation of compact couplers between dielectric slab waveguides and two-dimensional metal-dielectric-metal plasmonic waveguides,” Opt. Express **15**, 1211–1221 (2007). [CrossRef] [PubMed]

**7. **C. Ma, Q. Zhang, and E. Van Keuren, “Direct integration of nanoscale conventional and slot waveguides,” J. Nanosci. Nanotechnol. **11**, 2524–2527 (2011). [CrossRef] [PubMed]

**8. **X. Chen, W. Jiang, J. Chen, L. Gu, and R. T. Chenb, “20 dB-enhanced coupling to slot photonic crystal waveguide using multimode interference coupler,” Appl. Phys. Lett. **91**, 091111 (2007). [CrossRef]

**9. **A. D. Falco, L. O’Faolain, and T. F. Krauss, “Dispersion control and slow light in slotted photonic crystal waveguides,” Appl. Phys. Lett. **92**, 083501 (2008). [CrossRef]

**10. **C. A. Barrios, K. B. Gylfason, B. Sánchez, A. Griol, H. Sohlström, M. Holgado, and R. Casquel, “Slot-waveguide biochemical sensor,” Opt. Lett. **32**, 3080–3082 (2007). [CrossRef] [PubMed]

**11. **A. D. Falco, L. O’Faolain, and T. F. Krauss, “Chemical sensing in slotted photonic crystal heterostructure cavities,” Appl. Phys. Lett. **94**, 063503 (2009). [CrossRef]

**12. **J. H. Wülbern, J. Hampe, A. Petrov, M. Eich, J. Luo, A. K.-Y. Jen, A. D. Falco, T. F. Krauss, and J. Bruns, “Electro-optic modulation in slotted resonant photonic crystal heterostructures,” Appl. Phys. Lett. **94**, 241107 (2009). [CrossRef]

**13. **A. Martínez, J. Blasco, P. Sanchis, J. V. Galan, J. García-Rupérez, E. Jordana, P. Gautier, Y. Lebour, S. Hernandez, R. Spano, R. Guider, N. Daldosso, B. Garrido, J. M. Fedeli, L. Pavesi, and J. Martí, “Ultrafast all-optical switching in a silicon-nanocrystal-based silicon slot waveguide at telecom wavelengths,” Nano Lett. **10**, 1506–1511 (2010). [CrossRef] [PubMed]

**14. **M. Komatsu, K. Saitoh, and M. Koshiba, “Design of miniaturized silicon wire and slot waveguide polarization splitter based on a resonant tunneling,” Opt. Express **17**, 19225–19234 (2009). [CrossRef]

**15. **A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystals waveguides,” Phys. Rev. Lett. **77**, 3787–3790 (1996). [CrossRef] [PubMed]

**16. **S. Johnson, C. Manolatou, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. **23**, 1855–1857 (1998). [CrossRef]

**17. **C. Manolatou, S. Johnson, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “High-density integrated optics”, J. Lightwave Technol. **17**, 1682–1692 (1999). [CrossRef]

**18. **W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Low-loss, low-cross-talk crossings for silicon-on-insulator nanophotonic waveguides,” Opt. Lett. **32**, 2801–2803 (2007). [CrossRef] [PubMed]

**19. **P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. **34**, 2760–2762 (2009). [CrossRef] [PubMed]

**20. **J. Feng, Q. Li, and S. Fan, “Compact and low cross-talk silicon-on-insulator crossing using a periodic dielectric waveguide,” Opt. Lett. **35**, 3904–3906 (2010). [CrossRef] [PubMed]

**21. **W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Compact and low crosstalk waveguide crossing using impedance matched metamaterial,” Appl. Phys. Lett. **96**, 111114 (2010). [CrossRef]

**22. **W. Ding, D. Tang, Y. Liu, L. Chen, and X. Sun, “Arbitrary waveguide bends using isotropic and homogeneous metamaterial,” Appl. Phys. Lett. **96**, 041102 (2010). [CrossRef]

**23. **P. A. Anderson, B. S. Schmidt, and M. Lipson, “High confinement in silicon slot waveguides with sharp bends,” Opt. Express **14**, 9197–9202 (2006). [CrossRef] [PubMed]

**24. **C. Ma, Q. Zhang, and E. V. Keuren, “Right-angle slot waveguide bends with high bending efficiency,” Opt. Express **16**, 14330–14334 (2008). [CrossRef] [PubMed]

**25. **C. Ma, S. Qi, Q. Zhang, and E. Van Keuren, “High efficiency right-angle bending structures in continuous slot waveguides,” J. Opt. A, Pure Appl. Opt. **11**, 105702 (2009). [CrossRef]