Analysis of symmetric and asymmetric nanoscale slab slot waveguides

Optics Communications 282 (2009) 324–328

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Analysis of symmetric and asymmetric nanoscale slab slot waveguides C. Ma a, Q. Zhang b, E. Van Keuren a,* a b

Department of Physics, Georgetown University, Washington, DC 20057, USA Department of Electrical and Computer Engineering, Minnesota State University, Mankato, MN 56001, USA

a r t i c l e

i n f o

Article history: Received 11 June 2008 Received in revised form 10 September 2008 Accepted 20 September 2008

PACS: 42.82.m Keywords: Slot waveguides Asymmetric Confinement analysis Modes Designing rules

a b s t r a c t Nanoscale slab slot waveguides provide for high optical confinement and have found abundant applications in silicon photonics. After developing an analytical mode solver for general asymmetric slot waveguides, the confinement performance of symmetric as well as asymmetric geometries was systematically analyzed and compared. For symmetric structures, 2D confinement optimization by varying both lowindex slot and high-index slab width revealed a detailed saturation trend of the confinement factor with the increase of the studied width. Furthermore, simple design rules on how to choose the slot and slab width for achieving optimal confinement was obtained. For asymmetric structures, we demonstrated that the confinement performance was always lower than the 2D optimized confinement of the symmetric structures providing the two high-index slab layers and the two cladding layers have same refractive indices, respectively. In addition, the sensitivity of the confinement to the degree of asymmetry was studied, and we found that the fabrication tolerance on the material and structural parameters may be reasonably large for symmetric structures designed at optimal confinement. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Since the discovery of the slot waveguide structure [1], which takes advantage of the discontinuity of the electric field at a high-index-contrast interface to strongly enhance and confine light in a nanoscale region of low-index material, it has been researched for its fundamentals such as horizontal slot waveguides, multi-slot waveguides [2,3] and slot waveguide right-angle bends [4], etc., and many applications such as microring resonators [5], directional couplers [6], modulators [7], sensors [8], light enhancement [9], all-optical logic gate [10], multimode interference waveguides [11] etc. Although many of these applications utilize symmetric slot waveguide structures, asymmetric slot waveguide structures have also been used to improve the efficiency of slot waveguide bends [12,13] and design functional devices [5,8,14]. Furthermore, when symmetric slot waveguides are fabricated, variations in both refractive index and geometry can occur using current micro-/ nano-fabrication technology, thus introducing asymmetry into the waveguide devices. As a result, there is a need to study in more detail the confinement performance of asymmetric slab slot waveguides. With an explicit and easy to implement analytical TM fundamental modal solution of a general asymmetric slot waveguide that includes the solution for the symmetric slot waveguide [1] * Corresponding author. Tel.: +1 202 687 5982; fax: +1 202 687 2087. E-mail address: [email protected] (E. Van Keuren). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.09.085

as a special case, we systematically study and compare the confinement performance of nanoscale symmetric vs. asymmetric slab slot waveguides. A general asymmetric slot waveguide is shown in Fig. 1, in which a low-index slot is embedded between two high-index slabs with different refractive indices and thicknesses, and then the high-low-high-index structure is sandwiched by two different low-index claddings. It is equivalent to laying two conventional asymmetric slab waveguide close to each other in parallel, so the TM fundamental Eigenmodes of the two conventional waveguides overlap or couple with each other. While the analytical TM fundamental modal field expression and the transcendental dispersion relation for symmetric slot waveguide have been given in [1], there is a need for an explicit analytical TM fundamental modal solution of a general asymmetric slot waveguide in order to facilitate the investigation of asymmetric devices. The refractive index profile in Fig. 1 is equivalent to an asymmetric five-layer slab waveguide, of which the TE and TM field expressions have been formulated [15,16], although the middle layer was not nanoscale and thus the phenomenon of high enhancement of a TM fundamental mode in the middle low-index layer was not demonstrated. The TE and TM modal field expression was given in [15] for an asymmetric five-layer slab slot waveguide, with the middle layer being a low-index layer. However, the spatial coordinate system was displaced compared to that in [1]. General formulas describing the TE and TM modes in multilayer asymmetric slab waveguides was

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n H4

nS

n C5

y x z

-a

-b

2

1

0

a

3

4

5

Fig. 1. Asymmetric slab slot waveguide.

developed in [16] with similar spatial coordinate system set-up as that in [1]. But the middle layer indexed with subscript 0 is a highindex layer so a one-layer shift is needed for representing the TM E-field in the slot. For convenience purpose, and also to be consistent with the formulation given in [1] for symmetric slab slot waveguide, we reformulated the analytical TM fundamental modal field expression, the transcendental dispersion relation and their analytical solutions for an asymmetric five-layer slab waveguide (shown in the Appendix A). Note that practical slot waveguides have high-index layers with finite height, which reduces the confinement in the slot compared to the 2D slot waveguide discussed here, which has infinite high high-index layers. Note also that although Fig. 1 is a vertical slab slot waveguide, the formulas are also applicable to a horizontal slab slot waveguide [2], which may be easier to fabricate. Finally, we used exponential functions to represent the Eigenmodes inside the low-index layers 1, 3 and

a

1

b

We first use the formulae given in the Appendix A to calculate the field profiles for a symmetric and two asymmetric slab slot waveguides. For the first asymmetric case we randomly set a = 25 nm, b = 205 nm, c = 155 nm, nH2 = nH4 = 3.48, nC1 = nC5 = nS = 1.44, while we set the parameters of the second asymmetric case using a real material system: a = 25 nm, b = 205 nm, c = 155 nm, nH2 = 3.48 (silicon), nH4 = 2.0 (SiN), nC1 = nC5 = 1.0 (air), nS = 1.44 (silica). For the symmetric case we set a = 25 nm, b = c = 205 nm, nH2 = nH4 = 3.48, nC1 = nC5 = nS = 1.44. For comparison purpose, there are some common parameters for all the three cases. The vacuum wavelength is k0 = 1550 nm, which is used throughout this paper. Fig. 2 shows the normalized electric field Ex profile of the fundamental TM Eigenmode for the three studied cases. As is seen, the symmetricity of the field profiles shows the symmetricity of the waveguide structures. The calculated effective indices of the two asymmetric examples are 1.9824 and 1.6412, respectively, and for the symmetric example the effective index is 2.2735. As is well known, the most significant feature of a slot waveguide is its high field confinement in the slot layer. Hereafter we examine the effect of geometric parameters and refractive indices on field confinement. We start with symmetric structures. The refractive indices are nH2 = nH4 = 3.48 (silicon), nC1 = nC5 = nS = 1.0 (air). The effect of geometric parameters on the confinement factor, which is defined as the ratio of the power inside the slot to the total

1

c

0.8

Ex (a.u.)

Ex (a.u.)

0.8

0.6

0.4

0.2

0 -1

2. Calculations and analysis

c

0.6

0.4

0.2

0

X (um)

1

0 -1

1

0.8

Ex (a.u.)

n H2

n C1

5, in contrast to the cosh (sinh) functions used in [1,15,16]. We verified that our formulae reduce to these in [1] when geometric and index parameters are set the same way as in [1] i.e., the structure becomes a symmetric structure.

0.6

0.4

0.2

0

X (um)

1

0 -1

0

1

X (um)

Fig. 2. Normalized electric field Ex profile of the fundamental TM Eigenmode: (a) for the asymmetric waveguide with the random parameters; (b) for the asymmetric waveguide with the real material system; (c) for the symmetric waveguide.

Fig. 3. Confinement factor (a) and (c) and effective index (b) vs. a and b = c. Note that (a) is a 3-D mesh surface plot, and (c) is a contour plot with illustrated gradient.

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Ra

R power carried by the mode, i.e., a Ex ðxÞ  Hy ðxÞdx= total Ex ðxÞ Hy ðxÞdx, was studied by varying slot width a and (b = c). The results are summarized in Fig. 3. Only the modes with the highest confinement are selected whenever there are multiple valid modes. The confinement of Ex, the corresponding neff values, and the confinement contours and gradients, respectively, are presented in Fig. 3a–c. Note that for reference purpose the confinement of corresponding Ez was also included in Fig. 3a. From Fig. 3a and especially Fig. 3c, we found that when a increases, (b = c) needs to be increased by although not exactly the same, but a similar amount to optimize the confinement factor. For example, with a = 11 nm, b (= c) needs to be 148 nm to maximize the confinement (the resulting confinement is 50.29%). When a increases to 47 nm, the optimal b (= c) becomes 188 nm to have an optimal confinement (65.17%). The required increase in (b = c) is thus similar to the increase in a. Furthermore, with a = 91 nm, b (= c) must be set to 238 nm to achieve the maximum confinement (67.55%). With the above quantitative results, our study not only reveals design rules regarding the widths of the layers, but also indicates that when a increases, the optimized confinement factor tends to saturate. For example, when a increase from 36 nm to 91 nm, the optimized confinement factor only increases by a mere 67.55%  65.17% = 2.38%. Note that with the similar confinement factor, light power density decreases as a increases from 36 nm to 91 nm. Fig. 3c can be readily used to obtain the parameter sensitivity and manufacturing tolerance around a design point. For example, b = c = 168 nm gives an optimal confinement factor of 59.87% when a is 25 nm. Around this design point when a is fixed at 25 nm, the confinement factor is within the 55% contour even if b (= c) varies from 140 nm to 210 nm. On the other hand, when b (= c) is fixed at 168 nm, the confinement factor sustains above

Confinement factor

0.60 0.55 0.50 0.45 0.40 0.35 60

80

100

120

140

160

180

200

c-a (nm) Fig. 4. Dependence of the confinement factor on (c  a).

220

the level of 55% even if a varies from 10 nm to 50 nm. This shows around the design point the parameter sensitivity is low and thus the manufacturing tolerance is high. We next study the asymmetric case by fixing all the obtained parameter values except allowing the value of c to vary. We chose 2a = 50 nm. The optimized confinement factor with 2a = 50 nm is 59.87%, and the corresponding optimizing parameters values are b = c = 168 nm. Fig. 4 shows the dependence of the calculated confinement factor on (c  a), i.e., the thickness of the high-index layer on the right side in the range from 55 nm to 205 nm. We found that at (c  a) = 143 nm, or c = 168 nm, the confinement factor reaches its maximum 59.87% with neff = 1.5157. This result shows that with a = 25 nm and b = 168 nm, changing c away from 168 nm, which corresponds to a symmetric case, will reduce the confinement. Note that the plot of the dependence of the confinement vs. (c  a), though, is slightly asymmetric, meaning that for example, the case of (b = 168 nm, c = 148 nm) gives a slightly different confinement factor compared to the case of (b = 168 nm, c = 188 nm). In addition, the sensitivity around the optimal point is not very high, evidenced by the fact that the confinement factor only drops up to 2.37% of the optimal value within a wide range of (c  a) from 120 nm to 163 nm. This means that the studied optimized symmetric slot waveguide structure has a good manufacturing tolerance. We extend our results to a 2-D optimization case i.e., we allow both b and c to vary independently and optimize the confinement factor. The results are summarized in Fig. 5. Fig. 5a and b shows, respectively, the calculated confinement factor and its corresponding gradient as well as its contours vs. b and c. We found that b = c = 168 nm is actually the global optimal in terms of confinement. However, away from the global optimal point, the asymmetric configuration may perform better than the symmetric one. For example, when c = 178 nm, the best performance is achieved for b = 118 nm: the confinement factor is 54.4%. In contrast, the symmetric case of (b = c = 118 nm) only leads to a confinement of 35.31%. In addition, we observe the large confinement tolerance to the variations of b and c around the global optimal confinement point, represented by a large size contour at the confinement factor level of 57.5%. Recall that the global optimal confinement factor is 59.87%. Although not presented here, we also studied the cases in which a has different values other than 25 nm, and we obtained similar conclusions and trends as the results presented here. We finally study the effect of the refractive index for the previously optimized geometry i.e., 2a = 50 nm and b = c = 168 nm. We keep nH2 = 3.48 (silicon), nC1 = nC5 = nS = 1.0 (air) but allow nH4 to vary. Fig. 6 shows the dependence of the confinement factor on the variation of nH4 in a range from 1.48 to 5.48 for the studied structure with those parameter values that give the global optimal

Fig. 5. Confinement factor vs. b and c in (a) 3-D mesh surface plot, and (b) contour plot. Note that in (b) we also plotted the gradient of the confinement factor field.

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On the other hand, many devices that have been developed involve asymmetric slot waveguides. In [12,13], an asymmetric slot waveguide was used to improve the bending efficiency of a slot waveguide bend. Barrios et al. [8] demonstrated an ultrasensitive nanomechanical photonic sensor based on an asymmetric slot waveguide. Baehr-Jones et al. [5] and Barrios et al. [14] reported optical resonators based on slot waveguides with asymmetric cladding, and Xu et al. [17] have shown coupling between two slot waveguides where the cladding is also asymmetric. The results presented here provide a convenient analytical confinement calculation tool and design guidelines for these devices.

Confinement factor

0.6 0.5 0.4 0.3 0.2 0.1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

nH4 Fig. 6. Dependence of the confinement factor on nH4.

confinement factor. The confinement factor reaches its maximum value of 62.09% at nH4 = 4.12, and the corresponding effective index is neff = 1.6327. In our studied case, the optimal refractive index profile is slightly asymmetric i.e., nH4 = 4.12 vs. nH2 = 3.48, although the confinement enhancement is not very significant compared to the optimized symmetric case with nH2 = nH4 = 3.48.

Acknowledgement This material is based upon work supported by the National Science Foundation under Grant No. 0348955. Appendix A. Formulation For the general asymmetric structure shown in Fig. 1, the rigorous analytical solution for the transverse E-field profile Ex of the fundamental TM Eigenmode can be found by overlapping the TM fundamental Eigenmodes of the two conventional waveguides, and has the following form

3. Conclusions and discussion Using reformulated field expression for a general asymmetric slab slot waveguide we systematically studied the confinement performance and sensitivity vs. the waveguide geometry and index for both asymmetric and symmetric structures. For symmetric structures, 2D confinement optimization by varying both low-index slot and high-index slab width revealed a detailed saturation trend of the confinement factor with the increase of the studied width. Furthermore, simple design rules for choosing the slot and slab width were obtained for achieving optimal confinement. We found that for a given slot width 2a, and with the constraints of nH2 = nH4 and nC1 = nC5 = nS, the optimal geometric structure for light confinement is a symmetric structure. However, away from this optimal point, an asymmetric structure could perform better than a symmetric structure. When the constraint of nH2 = nH4 is relaxed and one of nH2 and nH4 is allowed to vary without changing the optimized symmetric geometric structure, we found that the optimal refractive index profile will be one that is slightly asymmetric with a higher index in one of the high-index layers, although the corresponding confinement enhancement is not significant in our studied case. These results indicate the effect of structural and material variations on the propagation and the precision needed in order to obtain desired values of confinement. For the specific case studied here, we conclude that the manufacturing tolerance on the material and structural parameters is reasonably large for a structure close to the global optimum. The analytical solution of the TM fundamental mode of an asymmetric slab slot waveguide will facilitate the research and design of both symmetric and asymmetric slot waveguide devices. A number of devices employing symmetric slot waveguides for nanophotonics have been demonstrated. For example, Fujisawa and Koshiba have developed an optical directional coupler [5] and a 1  2 multimode interference device [11], both based on symmetric slot waveguides. Xu and Lipson proposed all-optical logic devices using micro-ring resonators which employ slot waveguides [10]. For these types of devices to be commercially viable, their properties must be robust with respect to variations in material properties and layer thicknesses that occur in manufacture. Our results yield the dependence of the efficacy of the waveguides as a function of these variations from which manufacturing tolerances can be determined.

Ex ¼

8 p exp½cC1 ðx þ bÞ > > > 1 > > > < p21 cos½kH2 ðx þ aÞ þ p22 sin½kH2 ðx þ aÞ p31 exp½cS x þ p32 exp½cS x

> > > p41 cos½kH4 ðx  aÞ þ p42 sin½kH4 ðx  aÞ > > > : p5 exp½cC5 ðx  cÞ

x < b b 6 x < a a 6 x 6 a ac ð1Þ

where kH2 and kH4 are the transverse wave number in the high-index layers 2 and 4, respectively, cC1 and cC5 are the field decay coefficients in the cladding layers 1 and 5, respectively, cS is the field decay coefficient in the slot layer 3, p1, p21, p22, p31, p32, p41, p42 and p5 are field scaling coefficients and a, b, c are positive numbers specifying the width of the low-index slot and the two high-index layers as shown in Fig. 1. The transverse parameters kH2, kH4, cC1, cC5 and cS simultaneously satisfy the following relations 2

2

2

2

2

2

2

k0 n2H2  kH2 ¼ k0 n2H4  kH4 ¼ k0 n2C1 þ c2C1 ¼ k0 n2C5 þ c2C5 ¼ k0 n2S þ c2S ¼ b2 ð2Þ where k0 = 2p/k0 is the vacuum wave vector with k0 being the wavelength in vacuum, nH2, nH4, nC1, nC5, nS are the refractive indices of the five layers, respectively, as shown in Fig. 1, b = neff k0 is the propagation constant of the Eigenmode with neff being the effective index. The mode is a guided mode, which is theoretically lossless. The tangential electric field Ez ¼ ði=bÞ@Ex =@x and the transverse electric flux density Dx are continuous at the boundaries between layers 1 and 2, layers 2 and 3, layers 3 and 4, and layers 4 and 5. Matching boundary conditions at the four interfaces, the following transcendental dispersion equation can be obtained

uI uII expð4cS aÞ ¼ uIII uIV

ð3Þ

where

8 2  n c c n2 n2 K > > u ¼ ðn2C1 cS þ n2S cC1 Þ  H2K H2C1 S  C1n2S H2 tan½K H2 ðb  aÞ I > > H2 > > 2  > n c c n2 n2 K > > < uII ¼ ðn2C5 cS þ n2S cC5 Þ þ H4K H4C5 S  C5n2S H4 tan½K H4 ðc  aÞ H4 2  nH2 cC1 cS n2C1 n2S K H2 > 2 2 > u ¼ ðn c þ n c Þ  þ tan½K H2 ðb  aÞ > 2 III C1 S S C1 K > n H2 > H2 >   > 2 2 2 > > : uIV ¼ ðn2C5 cS  n2S cC5 Þ þ nH4 cC5 cS þ nC5 n2S K H4 tan½K H4 ðc  aÞ K H4 n H4

ð4Þ

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Solving this transcendental dispersion equation, the effective index neff or the propagation constant b can be obtained, then further the transverse parameters kH2, kH4, cC1, cC5 and cS can be obtained using Eq. (2). In order to calculate the mode profile of the field, it is practical to set one of the coefficients p31 and p32 to be 1, say set p31 = 1, although it can be precisely determined based on a normalization condition. The other coefficients p1, p21, p22, p32, p41, p42 and p5 can then be calculated in the following form and order:

8 p32 ¼ p31 e2cS a uII =uIV > > > > > p21 ¼ ðp31 ecS a þ p32 ecS a Þn2S =n2H2 > > > > cS a >  p32 ecS a ÞcS =kH2 > < p22 ¼ ðp31 e p1 ¼ fp21 cos½kH2 ðb  aÞ  p22 sin½kH2 ðb  aÞgn2H2 =n2C1 > > > > p41 ¼ ðp31 ecS a þ p32 ecS a Þn2S =n2H4 > > > > > p22 ¼ ðp31 ecS a  p32 ecS a ÞcS =kH4 > > : p5 ¼ fp41 cos½kH4 ðc  aÞ þ p42 sin½kH4 ðc  aÞgn2H4 =n2C5

ð5Þ

Note that p31 and p32 indicate the symmetricity of the field profile. For an asymmetric structure, p31 – p32 but if the structure is symmetric, i.e., b = c, nH2 = nH4 and nC1 = nC5 = nS, and these formulae for an asymmetric slab slot waveguide reduce to the formulae for a symmetric slot waveguide, and then p31 = p32 can be derived. Because it is the electric flux density D = n2E that is continuous at boundaries 23 and 34, the electric field at the boundary 23 in the

slot side (layer 3) is (nH2/nS)2 times of that in the high-index layer 2, and the electric field at the boundary 34 in the slot side (layer 3) is (nH4/nS)2 times of that in the high-index layer 4. When the width of the slot is small enough, usually in nanoscale, the coupled field will remain high across the whole slot and the coupled field appears asymmetric in general for asymmetric configurations. References [1] V.R. Almeida, Q. Xu, C.A. Barrios, M. Lipson, Opt. Lett. 29 (2004) 1209. [2] R. Sun, P. Dong, N.-n. Feng, C.-y. Hong, J. Michel, M. Lipson, L. Kimerling, Opt. Express 15 (2007) 17967. [3] N.-N. Feng, J. Michel, L.C. Kimerling, IEEE J. Quantum Electron. 42 (2006) 885. [4] C. Ma, Q. Zhang, E. Van Keuren, Opt. Express 16 (2008) 14330. [5] T. Baehr-Jones, M. Hochberg, C. Walker, A. Scherer, Appl. Phys. Lett. 86 (2005) 081101. [6] T. Fujisawa, M. Koshiba, Opt. Lett. 31 (2006) 56. [7] T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. Sullivan, L. Dalton, A. Jen, A. Scherer, Opt. Express 13 (2005) 5216. [8] C.A. Barrios, IEEE Photon. Technol. Lett. 18 (2006) 2419. [9] M. Galli, D. Gerace, A. Politi, M. Liscidini, M. Patrini, L.C. Andreani, A. Canino, M. Miritello, R. Lo Savio, A. Irrera, F. Priolo, Appl. Phys. Lett. 89 (2006) 241114. [10] Q. Xu, M. Lipson, Opt. Express 15 (2007) 924. [11] T. Fujisawa, M. Koshiba, IEEE Photon. Technol. Lett. 18 (2006) 1246. [12] P.A. Anderson, B.S. Schmidt, M. Lipson, Opt. Express 14 (2006) 9197. [13] C.-Y. Chao, J. Opt. Soc. Am. B 24 (2007) 2373. [14] C.A. Barrios, M. Lipson, Opt. Express 13 (2005) 10092. [15] K.-Y. Liou, U. Koren, E.C. Burrows, M. Young, M.J.R. Martyak, M. Oron, G. Raybon, J. Quantum. Electron. 26 (1990) 1376. [16] Y.-F. Li, J.W.Y. Lit, J. Opt. Soc. Am. A 4 (1987) 671. [17] Q. Xu, V.R. Almeida, R.R. Panepucci, M. Lipson, Opt. Lett. 29 (2004) 1626.