Tuning of zero group velocity dispersion in infiltrated vertical silicon slot waveguides

doi:10.1364/OE.21.001741

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Tuning of zero group velocity dispersion in infiltrated vertical silicon slot waveguides

Peter W. Nolte, Christian Bohley, and Jörg Schilling »View Author Affiliations

Optics Express, Vol. 21, Issue 2, pp. 1741-1750 (2013)
http://dx.doi.org/10.1364/OE.21.001741

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▸ Article Outline
– Abstract
1. Introduction
2. Optimized waveguide cross section
3. Manipulation of the group velocity dispersion
4. Conclusion
– References and links

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Abstract

In this work the design of Si / hybrid waveguides which contain a vertical infiltrated slot is studied. The case of slots infiltrated with a χ⁽³⁾ nonlinear material of relatively high refractive index (e.g. chalcogenide glasses) is specifically discussed. An optimized waveguide geometry with periodic refractive index modulation, a nonlinear figure of merit > 1 and minimum effective mode cross section is presented. Introducing a periodic refractive index variation along the waveguide allows the adjustment of the group velocity dispersion (GVD). Choosing the period accordingly, the phase matching condition for degenerate four wave mixing (GVD = 0) can be fulfilled at virtually any desired frequency and independently from the fixed optimized waveguide cross section.

1. Introduction

In the last years great efforts led to a strong miniaturization of optical components, as several devices were realized on the silicon-on-insulator (SOI) platform, which is compatible to CMOS technology. Mature processing techniques can be used to fabricate wave guiding structures. The very high refractive index contrast between the Si core (n_Si = 3.5) and the oxide cladding (n_SiO₂ ≈ 1.45) and air (n_Air = 1), respectively, enables a high confinement of light inside such waveguides. This enables the fabrication of small footprint devices. However, for many applications active devices exhibiting a nonlinear optical behavior are needed. Although Si shows large n₂-values (n₂ = 6 · 10⁻¹⁸m²/W [1

M. R. Lamont, C. M. de Sterke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As₂S₃ waveguides for parametric gain and wavelength conversion,” Opt. Express 15, 9458–9463 (2007). [CrossRef] [PubMed]

], approx. 100 times stronger than SiO₂[2

T. Liang and H. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 10, 1149 – 1153 (2004). [CrossRef]

]) it suffers from strong two photon absorption (β = 6.7 · 10⁻¹² m/W [2

T. Liang and H. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 10, 1149 – 1153 (2004). [CrossRef]

, 3

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2008).

]), leading to a small figure of merit (FOM = n₂/λβ). For Silicon the FOM is approximately 0.6 at λ = 1550 nm. Generally a FOM > 1 is necessary for efficient nonlinear optical processes [4

V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14, 1140–1142 (1989). [CrossRef] [PubMed]

One way to enhance the performance of SOI based devices is to combine Si with a material showing a higher FOM such as organic materials [5

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]

], chalcogenide glasses, e.g. As₂S₃[6

M. Asobe, T. Kanamori, K. Naganuma, H. Itoh, and T. Kaino, “Third order nonlinear spectroscopy in As₂S₃ chalcogenide glass fibers,” J. Appl. Phys. 77, 5518–5523 (1995). [CrossRef]

, 7

L. Zhang, Y. Yue, Y. Xiao-Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express 18, 13187–13193 (2010). [CrossRef] [PubMed]

] or silicon nanocrystals [8

Q. Liu, S. Gao, Z. Li, Y. Xie, and S. He, “Dispersion engineering of a silicon-nanocrystal-based slot waveguide for broadband wavelength conversion,” Appl. Opt. 50, 1260–1265 (2011). [CrossRef] [PubMed]

L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18, 20529–20534 (2010). [CrossRef] [PubMed]

–10

S. Mas, J. Caraquitena, J. V. Galn, P. Sanchis, and J. Mart, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express 18, 20839–20844 (2010). [CrossRef] [PubMed]

]. For this combination several geometries such as horizontal [11

P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17, 9282–9287 (2009). [CrossRef] [PubMed]

] and vertical [5

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]

] slot waveguides filled with nonlinear material, have been suggested. However, for nonlinear optical frequency transformation (e.g. second harmonic generation, four wave mixing) the phase matching is crucial, as well. Controlling the waveguide dispersion is, therefore, essential in designing devices using these processes. For instance for maximum performance of degenerate four wave mixing processes a group velocity dispersion (GVD) of zero is required [12

G. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic Press, 2006).

This publication focuses on infiltrated vertical slot waveguides. Here the horizontal electric field of the waveguide mode is highly concentrated in the vertical slot, leading to an enhanced nonlinear optical performance. The suggested waveguides might be built by fabricating slot waveguide with e-beam lithography and covering those waveguides consecutively with solution processed As₂S₃[13

C. Tsay, E. Mujagi, C. K. Madsen, C. F. Gmachl, and C. B. Arnold, “Mid-infrared characterization of solution-processed As₂S₃ chalcogenide glass waveguides,” Opt. Express 18, 15523–15530 (2010). [CrossRef] [PubMed]

C. Tsay, Y. Zha, and C. B. Arnold, “Solution-processed chalcogenide glass for integrated single-mode mid-infrared waveguides,” Opt. Express 18, 26744–26753 (2010). [CrossRef] [PubMed]

–15

G. C. Chern, “Spin-coated amorphous chalcogenide films,” J. Appl. Phys. 53, 6979 (1982). [CrossRef]

Up to now the linear and nonlinear optical properties of waveguides were mainly tailored applying specific designs of the waveguide cross section [4

V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14, 1140–1142 (1989). [CrossRef] [PubMed]

, 11

P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17, 9282–9287 (2009). [CrossRef] [PubMed]

, 16

Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express 18, 22061 (2010). [CrossRef] [PubMed]

, 17

P. Muellner, “Fundamental characteristics of the soi slot waveguide structure,” Ph.D. thesis, Faculty of Physics, University of Vienna (2010).

]. This generally resulted in a trade off between the aim of a a large FOM with strong field confinement and the often contradicting goal of a zero or near zero GVD. In this paper we discuss a design route which allows waveguide geometries with FOM > 1 and a maximum field concentration within the infiltrated material and a vanishing GVD at the same time. The independent adjustment of the GVD and the mode profile is achieved by varying the refractive index along the waveguide periodically. This introduces a photonic band gap for the waveguide mode leading to additional band bending in the vicinity of this band gap. Therefore the periodicity gives another degree of freedom to tailor the GVD of such hybrid waveguide systems.

2. Optimized waveguide cross section

2.1. Basic waveguide geometry and numerical model

In Fig. 1(a) a sketch of the investigated waveguide geometry is shown. The structure consists of a silicon on insulator (SOI) strip waveguide of width w_wg with a vertical centered slot. A layer of height h_wg of an optical nonlinear material is situated on top of the substrate and is infiltrated into the slot. The half space above the waveguide is filled with air. Throughout the paper we consider the chalcogenide glass As₂S₃ as the infiltrated/coating nonlinear material. It is one of the most common chalcogenide glasses. Its refractive index of approx 2.34 is still quite low compared to other chalcogenide glasses and the linear absorption nearly vanishes around a wavelength of 1500 nm - our spectral range of interest. To calculate the TE-like mode frequencies and field distributions of the waveguide modes, the commercial finite element solver COMSOL was used [18

“www.comsol.com”.

]. The dispersion of the refractive indices of the materials was taken into account in these simulations using the Sellmeier Eq. (11) with the material specific coefficients presented in Table 1.

Fig. 1 a) Schematic drawing of the waveguides investigated in this paper. The SOI slot waveguide is infiltrated and surrounded by a layer of the chalcogenide glass As₂S₃ with a refractive index of 2.34. b) Cross section of the TE-like mode profile of the infiltrated slotted Si waveguide. The inset shows the field intensity along the depicted line

Table 1 Material parameters used in this work

	Si [2 T. Liang and H. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 10, 1149 – 1153 (2004). [CrossRef] , 32 M. Bass, C. DeCusatis, J. Enoch, G. Li, V. N. Mahajan, E. V. Stryland, and C. MacDonald, Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics (McGraw-Hill Prof Med/Tech, 2009). ]	SiO₂[3 G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2008). , 32 M. Bass, C. DeCusatis, J. Enoch, G. Li, V. N. Mahajan, E. V. Stryland, and C. MacDonald, Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics (McGraw-Hill Prof Med/Tech, 2009). ]	As₂S₃ unexposed [6 M. Asobe, T. Kanamori, K. Naganuma, H. Itoh, and T. Kaino, “Third order nonlinear spectroscopy in As₂S₃ chalcogenide glass fibers,” J. Appl. Phys. 77, 5518–5523 (1995). [CrossRef] , 27 R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As₂S₃ In₂S₃ films,” J. Phys. D: Appl. Phys. 44, 305401 (2011). [CrossRef] ]	As₂S₃ exposed [27 R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As₂S₃ In₂S₃ films,” J. Phys. D: Appl. Phys. 44, 305401 (2011). [CrossRef] ]

B₁	10.668429	0.6961663	4.24	4.68
B₂	0.0030435	0.4079426
B₃	1.5413341	0.8974794
C₁ [μm²]	0.3015165	0.0684043	2.5758	2.6481
C₂ [μm²]	1.13475115	0.1162414
C₃ [μm²]	1104.0	9.896161

n₂ [m²/W]	6 · 10⁻¹⁸	2.6 · 10⁻²⁰	2 · 10⁻¹⁸
β [m/W]	6.7 · 10⁻¹²	0	6.2 · 10⁻¹⁵

Figure 1(b) shows the field distribution of such a waveguide, as can be seen in the inset the field is enhanced in the slot region by a factor of two. This relatively moderate enhancement, compared to what is achievable with air slotted waveguides, is caused by the smaller index contrast between the silicon core and the infiltrated material.

From the distribution of the electric and magnetic fields (e, h) the complex third order nonlinear parameter γ of the waveguide is calculated [19

S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part i: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009). [CrossRef] [PubMed]

γ = \frac{ε_{0}}{μ_{0}} \frac{\int n^{2} (x, y) (ω / c n_{2} (x, y) + i / 2 β (x, y)) {| e |}^{4} d A}{{| \int {(e \times h^{*})}_{z} d A |}^{2}} .

(1)

The material specific Kerr coefficients n₂ and nonlinear absorption coefficients β are in general anisotropic [20

J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered si photonic wires,” Opt. Express 16, 1280–1299 (2008). [CrossRef] [PubMed]

, 21

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

]. Nevertheless as the direction of Si-photonic waveguides is in practice not restricted to a specific crystallographic direction, we assume average isotropic values for n₂ and β which are listed in Table 1. γ measures the strength of the 3rd order nonlinear interactions in wave guiding structures corresponding to χ⁽³⁾ in bulk materials [21

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

]. The FOM is calculated from γ with FOM = Re{γ}/4π Im{γ}.

Closely related to the nonlinear parameter γ, an effective mode area A_eff can be defined. It describes the confinement of the electric field, which is responsible for the nonlinear optical response. Following the suggestion of Koos et al. [5

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]

] and others [19

, 20

, 22

I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett. 37, 2295–2297 (2012). [CrossRef]

] the corresponding effective mode Area A_eff can be defined as

A_{eff} = \frac{1}{n_{A s_{2} S_{3}}^{2}} \frac{μ_{0}}{ε_{0}} \frac{{| \int {(e \times h^{*})}_{z} d A |}^{2}}{\int_{D_{A s_{2} S_{3}}} {| e |}^{4} d A} .

(2)

In this definition of A_eff, |e|⁴ is integrated over the areas filled with As₂S₃ (D_As₂S₃). Thus, A_eff becomes minimal if the concentration of the field is maximized inside the strongly nonlinear optical material As₂S₃ leading to waveguide geometries with compact mode volumes.

To find the waveguide cross section with minimum A_eff the width and the height of the waveguides were varied while a fixed slot width of 100 nm was assumed. It has been reported that a smaller slot width gives rise to stronger field enhancement and thus a higher nonlinear optical FOM [5

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]

,11

P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17, 9282–9287 (2009). [CrossRef] [PubMed]

,17

P. Muellner, “Fundamental characteristics of the soi slot waveguide structure,” Ph.D. thesis, Faculty of Physics, University of Vienna (2010).

]. Nevertheless, there is certainly a technological limit of the minimum slot width of vertical slots as were investigated in this paper. Therefore, we limit our slot width to a rather conservative value of 100 nm to obtain structures, which can be easily realized with standard e-beam lithography and consecutively filled with As₂S₃.

2.2. Results

In Fig. 2(a) the FOM of the waveguides and its dependence on the waveguide width and height are shown. The blank space in the lower left corner indicates the parameter values, where no guided modes could be found. For wide waveguides the FOM is quite low, decreasing below unity for waveguide width w_wg > 550 nm. For these wide waveguides a large part of the mode is propagating within the two silicon rails at both sides of the slot, so that the low FOM of silicon starts to govern the FOM of the whole waveguide. On the other hand for waveguide widths approaching 300 nm a large part of the mode propagates inside the slot or leaks out beyond the silicon rails into the adjacent As₂S₃ slab regions. For these geometries the highest FOMs of about 3 to 4 are expected (the FOM of bulk Silicon and bulk As₂S₃ differ by a factor of 100). Finally, for ultra-narrow waveguide widths below 200 nm the waveguide modes become unbound since the radiation can couple to propagating modes inside the neighboring As₂S₃ slab regions. The modes within the infinitely extended As₂S₃ slab regions left and right of the waveguide therefore determine the ”light cone” for the waveguide modes.

Fig. 2 a) Figure of merit of such a waveguide. The circle indicates the maximum point of the 1/A_eff plot. At this point the value of the FOM is1.3. b) 1/A_eff as defined in Ref. [5

C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]

] for constant slot width w_slot = 100 nm and varying height h_wg and width w_wg of a chalcogenide coated slot waveguide. The optimum parameters were found to be h_wg = 210 nm and w_wg = 420 nm.

According to this analysis waveguide widths in the range of 250 – 300 nm with modes close to the light cone appear most favorable. However, a closer look at the field distributions for these waveguides already reveals considerable mode spreading that could lead to severe cross talk between neighboring waveguides. Furthermore, the operation of such narrow waveguides close to the light cone requires a highly accurate fabrication to avoid leakage radiation. Finally, for our dispersion engineering in order to work, we need waveguide modes whose dispersion curves are well below the light line. In conclusion we do not optimize our waveguide cross section towards a maximum value of the FOM, but try to maximize the field concentration in the slot region. This still results in FOMs between 1 to 2 as discussed in the following.

To find an optimum waveguide geometry with large field concentration inside the slot, the waveguide with minimal effective mode area is determined. Figure 2(b) shows A_eff for a varying waveguide width w_wg and height h_wg. There is a maximum of 1/A_eff (corresponds to a minimum of A_eff) at w_wg = 420 nm and h_wg = 210 nm. The mode field distribution of the geometry with minimized A_eff is shown in Fig. 1(b). Although most of the mode field intensity is confined in the slot a significant part of the mode field is already propagating in silicon giving rise to two photon absorption. Therefore the overall FOM for this geometry has a moderate value of 1.3.

3. Manipulation of the group velocity dispersion

For efficient nonlinear frequency transformation processes, a large FOM is not sufficient. In addition, phase matching conditions have to be fulfilled to yield efficient nonlinear amplification and generation. Since the infiltrated nonlinear material (chalcogenide glass) is amorphous only third order nonlinear optical processes, which are based on χ⁽³⁾, are of interest here. Next we will specifically investigate the phase matching conditions for the degenerate four wave mixing process and how these can be fulfilled for the mentioned infiltrated slot waveguides.

In the case of degenerate four wave mixing two waves at ω_p and ω_s interact to form a new wave ω_i. For an efficient frequency conversion the frequencies and phases of the three interacting waves have to be matched according to:

ω_{i} = 2 ω_{p} - ω_{s}

(3)

k_{i} = 2 k_{p} - k_{s}

(4)

For simplicity we will consider only the linear dispersion in equation (4) and neglect the nonlinear phase shift due to the Kerr effect for the moment. The consequences of a more comprehensive treatment incorporating the nonlinear change of the refractive index is shortly discussed at the end.

Condition (3) can be recast as:

ω_{i} = ω_{p} - Δ ω

(5)

ω_{s} = ω_{p} + Δ ω,

(6)

while k_i(ω_i) and k_s(ω_s) in (4) can be developed into Taylor series around ω_p.

k_{i} = k_{p} - {\frac{d k}{d ω} |}_{ω_{p}} Δ ω + \frac{1}{2} {\frac{d^{2} k}{d ω^{2}} |}_{ω_{p}} Δ ω^{2} \dots

(7)

k_{s} = k_{p} + {\frac{d k}{d ω} |}_{ω_{p}} Δ ω + \frac{1}{2} {\frac{d^{2} k}{d ω^{2}} |}_{ω_{p}} Δ ω^{2} \dots

(8)

Combining Eqs. (7) and (8) with Eq. (4) and disregarding higher order terms, leads to

\frac{d^{2} k}{d ω^{2}} = 0 .

(9)

This means that the group velocity dispersion (GVD), which is described by the term d²k/dω², vanishes if the phase matching condition is fulfilled. Silicon as well as As₂S₃ and standard waveguides show normal dispersion, i.e., the phase index is larger for shorter wavelengths. One way to compensate this behavior is to use the anomalous dispersion of silica in the wavelength region around 1500 nm. Waveguides are then designed in such way that for longer wavelength (when the wave leaks out into the surrounding material, i.e. silica) its anomalous dispersion compensates the waveguide dispersion [23

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006). [CrossRef] [PubMed]

]. Another way is to find an advanced waveguide design, which shows anomalous dispersion [7

, 9

L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18, 20529–20534 (2010). [CrossRef] [PubMed]

, 24

L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012). [CrossRef] [PubMed]

, 25

L. Zhang, Y. Yue, Y. Xiao-Li, R. G. Beausoleil, and A. E. Willner, “Highly dispersive slot waveguides,” Opt. Express 17, 7095–7101 (2009). [CrossRef] [PubMed]

]. Yet another way is to use a slow (a ≫ λ₀/n) periodic modulation of the effective mode index along the propagation direction, with a periodicity a. This adds a negative contribution of 2πm/a (m = 1, 2, 3,...) to the phase matching condition (4). If properly designed, this allows the fulfillment of Eq. (4) in many conditions [26

J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express 20, 9227–9242 (2012). [CrossRef] [PubMed]

We propose a concept, which uses a small scale periodic modulation of the refractive index with a periodicity of the cladding material in propagation direction a < λ₀/n. This refractive index modulations causes a 1-D photonic bandgap, which leads to additional band bending in its vicinity. This band bending yields an anomalous contribution to the GVD that can be used to achieve zero GVD. In practice, such a structure could be achieved by exploiting the photorefractive effect of As₂S₃. It is widely reported that the refractive index of chalcogenide glasses can be changed by illumination with above bandgap radiation. In the case of As₂S₃ the refractive index can be altered up to 0.1 by exposure to visible or UV light [27

R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As₂S₃ In₂S₃ films,” J. Phys. D: Appl. Phys. 44, 305401 (2011). [CrossRef]

]. Therefore, we again focus on this material in our model system to proof our concept.

3.1. Numerical Model

The model used to calculate the GVD of a waveguide infiltrated with a material with alternating refractive indices n₁ ≈ 2.3 and n₂ = n₁ + 0.05 is depicted in Fig. 3(a). The index n1 corresponds to that of unexposed As₂S₃ calculated by Sellmeier eq. with parameters given in Table 1. The offset of 0.05 was arbitrarily chosen as half of the maximum refractive index change reported in [27

R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As₂S₃ In₂S₃ films,” J. Phys. D: Appl. Phys. 44, 305401 (2011). [CrossRef]

]. Again the thickness of As₂S₃-layer is the same as the height of the waveguide and the upper half space above the waveguide is assumed to be air. Such a structure might be fabricated by periodic exposure of an already deposited As₂S₃ film. The eigenfrequencys ω(k) of such a periodic system for given k = 2π/λ-vectors were calculated in a 3D geometry using periodic boundary conditions in the propagation direction z[28

A. von Rhein, S. Greulich-Weber, and R. B. Wehrspohn, “Multiphysics software gazes into photonic crystals,” Physics Best pp. 38–39 (2007).

]. The finite element software COMSOL [18

“www.comsol.com”.

] allows to take the varying refractive indices n₁ of the materials for different eigenfrequencies into account. The cross section of the waveguide geometry corresponds to the optimum condition with minimum A_eff as described above. For a one-dimensional periodic index modulation a band gap around λ₀/2n = a is expected [29

J. D. Joannopoulos and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

]. For a design wavelength of λ₀ = 1550 nm this leads to a ≈ 330 nm. To test the flexibility of our strategy we calculated the eigenfrequencies for different lattice constants a.

Fig. 3 a) Schematic drawing of the investigated waveguide structure with index modulation along the waveguide. In the simulations w_slot is set to 100 nm. w_wg and h_wg are set to the optimized parameters h_wg = 210 nm and w_wg = 420 nm. The refractive indices of the cladding are varied with periodicity a between n₁ and n₂ = n₁ + 0.05. n₁ corresponds to the refractive index of unexposed As₂S₃ (compare Table 1) and n₂ to the index of the exposed As₂S₃. b) Dispersion of a periodic slot waveguide with lattice constant a = 380 nm.

3.2. Results

The result of such a eigenfrequency calculation for a = 380 nm is shown in Fig. 3(b). It shows the waveguide mode which is folded back, forming a bandgap at the Brillouin zone edge. Above the gap the guided modes of the second band extend up to the shaded area where the waveguide modes are able to couple to the As₂S₃-slab modes left and right of the waveguide and become lossy (light cone). To compensate the usual normal dispersion of the material and the waveguide mode the band bending of the second band around the upper band edge will be exploited. Only there the GVD vanishes.

From ω(k) we are able to calculate the GVD parameter as

- \frac{2 π c}{λ^{2}} \frac{d^{2} k}{d ω^{2}} = \frac{2 π c}{λ^{2}} {(\frac{d ω}{d k})}^{- 3} \cdot \frac{d^{2} ω}{d k^{2}} .

(10)

The results of the calculations for different lattice constants a = 300 nm, 320 nm,...,400 nm are shown on Fig. 4(a). There a clear zero crossing is observed, which shifts to longer wavelengths for larger lattice constants. A |GVD| < 0.5 ps/(nm m) is achieved within a wavelength range of Δλ ≈ 10 nm. This graph also represents the huge flexibility for tuning the GVD = 0 frequency to the desired spectral region. By simply changing the period of the index modulation the condition GVD = 0 can be shifted to nearly any wavelength within the near infra-red. The cross section of the waveguide can be left unchanged. In this way the method used here allows to choose the cross section of the waveguide in such a way to optimize parameters like effective mode area or FOM and then adjust the period of the index modulation to achieve the phase matching condition (GVD = 0) at the desired wavelength.

Fig. 4 a) GVD parameters for optimized transverse waveguide profile and a lattice constant a varying between a = 300 nm and a = 400 nm. The inset shows the linear dependence of the GVD = 0 wavelength and the period of the index modulation. b) Bandwidth (|ΔβL| < π/2) of the wavelength conversion for different waveguide lengths L. c) GVD parameters for fixed lattice constant a = 340 nm and thickness of the cladding layer Δh_wg

To estimate the useful bandwidth for degenerate four wave mixing around the GVD=0 frequency we apply the condition |ΔβL| < π/2 from [21

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

]. With this the tolerable wave-number mismatch Δβ = 2k_p(ω_p) − k_i(ω_i) − k_s(ω_i) depends on the length of the waveguide L. Assuming waveguide lengths between 1 and 5 mm the corresponding bandwidths were determined from the original dispersion curves and are presented in Fig. 4(b). As can be seen the bandwidth is well above 40 nm in any case and decreases more slowly with increasing waveguide length allowing even longer waveguides for narrow band four wave mixing.

As it might be difficult to prepare a chalcogenide glass layer whose thickness exactly matches the slot waveguide height, we investigated the properties of structures with different thickness mismatch Δh_wg where the As₂S₃ layer extends by 20 nm to 100 nm above the waveguide. This is a realistic scenario when a fabricated silicon slot waveguide is infiltrated afterward with the chalcogenide glass and no further leveling or polishing is performed. The results of these investigations are shown in Fig. 4(c). The zero point of the GVD parameter shifts to longer wavelengths for thicker cladding layer. This is expected as the field of the mode extends a bit upward out of the slot and the mode ”sees” now still the relatively high refractive index of the added As₂S₃ with Δh_wg. As expected the impact of Δh_wg on the shift of the GVD = 0 frequency levels off for larger Δh_wg as the mode decays quickly above the slot. However, care has to be taken that the As₂S₃-layer does not become too thick, as then the frequency of the As₂S₃-slab modes decreases considerably and the shaded area of the “radiative modes” in Fig. 3(b) extends close to the bandgap. Ultimately, the GVD = 0 frequency would be shifted into the leaky mode regime. This places a limit to the overall allowed As₂S₃ layer thickness in the area between 500 nm to 1 micron depending on waveguide cross section parameters and contrast of index modulation.

Finally we revisit Eq. (4) where the impact of the nonlinear phase shift Δβ_NL = 2P_pRe {γ} induced by the Kerr effect was neglected. To estimate the influence of this phase shift we calculated the phase mismatch Δβ = Δβ_NL − (2k_p(ω_p) − k_i(ω_p − Δω) − k_s(ω_p + Δω)) for Δω = 2π1.2 · 10¹²Hz ≙ Δλ ≈ 10nm and Re{γ} = 90(Wm)⁻¹ at a pump power P_p = 20dBm [21

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]

, 26

]. As a result the wavelength for which Δβ = 0 shifts by 2.5 nm towards the red from the original GVD = 0 wavelength. This demonstrates that the introduced band bending due to the photonic band gap is strong enough to compensate for linear and nonlinear dispersion characteristics. Even smaller index contrasts below Δn = 0.1 are sufficient to compensate for the material and waveguide dispersion. Furthermore the idea to use a periodic index modulation to achieve the phase matching for the degenerated four wave mixing process is not limited to photo-refractive materials. A periodic modulation of waveguide width or height or slot width will have a similar effect and will create a photonic bandgap with the required band bending as it was already reported for ”nanobeams” [30

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lonar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94, 121106–121106–3 (2009). [CrossRef]

]. Our strategy clearly aims to fulfill the phase matching condition in the spectral range around the telecom wavelengths (1550 nm wavelength.) Since we use the strong band bending around a photonic band edge our bandwidth is limited to about Δλ = 50 nm. In this respect our work has a certain similarity to the work on flat band slow light reported by Li et al. [31

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008). [CrossRef] [PubMed]

]. However, if four wave mixing should be achieved over a much wider spectral range (e.g.(λ_i − λ_p) > 200 nm) the quasi phase matching strategy investigated by Driscoll et al. [26

] is favored. With this approach basically any wave-vector mismatch can be overcome by adding a reciprocal lattice vector of a long period grating. However, this technique becomes ineffective for four wave mixing processes of narrow bandwidth as the resulting smaller wave-vector mismatches would require gratings with periods in the cm range. This would require undesirably large optical chip sizes.

4. Conclusion

A design route for Si/As₂S₃ waveguides with optimized waveguide cross section and tailored group velocity dispersion is presented. This method is based on a refractive index modulation along the waveguide and allows to independently optimize the effective mode area A_eff and the GVD. An FOM of about 1.3 and GVD parameter < 0.5 ps/(nm m) was achieved for a wavelength range of Δλ = 10 nm. The bandwidth for degenerate four wave mixing processes amounts to Δλ ≈ 50 nm for waveguide lengths of a few mm.

Appendices

5. Appendix

In this work, the Sellmeier Eq.

n_{0}^{2} (λ) = 1 + \sum_{i} \frac{B_{i} λ^{2}}{λ^{2} - C_{i}}

(11)

with B_i and C_i as in Table 1 has been used to calculate the refractive indices of the materials.

Acknowledgment

We gratefully acknowledge the funding by the Federal Ministry of Education and Research (BMBF) within the Centre for Innovation Competence SiLi-nano^® project 03Z2HN12.

References and links

1.	M. R. Lamont, C. M. de Sterke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As₂S₃ waveguides for parametric gain and wavelength conversion,” Opt. Express 15, 9458–9463 (2007). [CrossRef] [PubMed]
2.	T. Liang and H. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron. 10, 1149 – 1153 (2004). [CrossRef]
3.	G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2008).
4.	V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14, 1140–1142 (1989). [CrossRef] [PubMed]
5.	C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express 15, 5976–5990 (2007). [CrossRef] [PubMed]
6.	M. Asobe, T. Kanamori, K. Naganuma, H. Itoh, and T. Kaino, “Third order nonlinear spectroscopy in As₂S₃ chalcogenide glass fibers,” J. Appl. Phys. 77, 5518–5523 (1995). [CrossRef]
7.	L. Zhang, Y. Yue, Y. Xiao-Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express 18, 13187–13193 (2010). [CrossRef] [PubMed]
8.	Q. Liu, S. Gao, Z. Li, Y. Xie, and S. He, “Dispersion engineering of a silicon-nanocrystal-based slot waveguide for broadband wavelength conversion,” Appl. Opt. 50, 1260–1265 (2011). [CrossRef] [PubMed]
9.	L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express 18, 20529–20534 (2010). [CrossRef] [PubMed]
10.	S. Mas, J. Caraquitena, J. V. Galn, P. Sanchis, and J. Mart, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express 18, 20839–20844 (2010). [CrossRef] [PubMed]
11.	P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express 17, 9282–9287 (2009). [CrossRef] [PubMed]
12.	G. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic Press, 2006).
13.	C. Tsay, E. Mujagi, C. K. Madsen, C. F. Gmachl, and C. B. Arnold, “Mid-infrared characterization of solution-processed As₂S₃ chalcogenide glass waveguides,” Opt. Express 18, 15523–15530 (2010). [CrossRef] [PubMed]
14.	C. Tsay, Y. Zha, and C. B. Arnold, “Solution-processed chalcogenide glass for integrated single-mode mid-infrared waveguides,” Opt. Express 18, 26744–26753 (2010). [CrossRef] [PubMed]
15.	G. C. Chern, “Spin-coated amorphous chalcogenide films,” J. Appl. Phys. 53, 6979 (1982). [CrossRef]
16.	Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express 18, 22061 (2010). [CrossRef] [PubMed]
17.	P. Muellner, “Fundamental characteristics of the soi slot waveguide structure,” Ph.D. thesis, Faculty of Physics, University of Vienna (2010).
18.	“www.comsol.com”.
19.	S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part i: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009). [CrossRef] [PubMed]
20.	J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. Osgood Jr., “Nonlinear-optical phase modification in dispersion-engineered si photonic wires,” Opt. Express 16, 1280–1299 (2008). [CrossRef] [PubMed]
21.	Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007). [CrossRef] [PubMed]
22.	I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett. 37, 2295–2297 (2012). [CrossRef]
23.	A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14, 4357–4362 (2006). [CrossRef] [PubMed]
24.	L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express 20, 1685–1690 (2012). [CrossRef] [PubMed]
25.	L. Zhang, Y. Yue, Y. Xiao-Li, R. G. Beausoleil, and A. E. Willner, “Highly dispersive slot waveguides,” Opt. Express 17, 7095–7101 (2009). [CrossRef] [PubMed]
26.	J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express 20, 9227–9242 (2012). [CrossRef] [PubMed]
27.	R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As₂S₃ In₂S₃ films,” J. Phys. D: Appl. Phys. 44, 305401 (2011). [CrossRef]
28.	A. von Rhein, S. Greulich-Weber, and R. B. Wehrspohn, “Multiphysics software gazes into photonic crystals,” Physics Best pp. 38–39 (2007).
29.	J. D. Joannopoulos and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).
30.	P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lonar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94, 121106–121106–3 (2009). [CrossRef]
31.	J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008). [CrossRef] [PubMed]
32.	M. Bass, C. DeCusatis, J. Enoch, G. Li, V. N. Mahajan, E. V. Stryland, and C. MacDonald, Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics (McGraw-Hill Prof Med/Tech, 2009).

OCIS Codes
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: October 12, 2012
Revised Manuscript: November 22, 2012
Manuscript Accepted: November 24, 2012
Published: January 16, 2013

Citation
Peter W. Nolte, Christian Bohley, and Jörg Schilling, "Tuning of zero group velocity dispersion in infiltrated vertical silicon slot waveguides," Opt. Express 21, 1741-1750 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1741

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References

M. R. Lamont, C. M. de Sterke, and B. J. Eggleton, “Dispersion engineering of highly nonlinear As2S3 waveguides for parametric gain and wavelength conversion,” Opt. Express15, 9458–9463 (2007). [CrossRef] [PubMed]
T. Liang and H. Tsang, “Nonlinear absorption and Raman scattering in silicon-on-insulator optical waveguides,” IEEE J. Sel. Top. Quantum Electron.10, 1149 – 1153 (2004). [CrossRef]
G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2008).
V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett.14, 1140–1142 (1989). [CrossRef] [PubMed]
C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, “Nonlinear silicon-on-insulator waveguides for all-optical signal processing,” Opt. Express15, 5976–5990 (2007). [CrossRef] [PubMed]
M. Asobe, T. Kanamori, K. Naganuma, H. Itoh, and T. Kaino, “Third order nonlinear spectroscopy in As2S3 chalcogenide glass fibers,” J. Appl. Phys.77, 5518–5523 (1995). [CrossRef]
L. Zhang, Y. Yue, Y. Xiao-Li, J. Wang, R. G. Beausoleil, and A. E. Willner, “Flat and low dispersion in highly nonlinear slot waveguides,” Opt. Express18, 13187–13193 (2010). [CrossRef] [PubMed]
Q. Liu, S. Gao, Z. Li, Y. Xie, and S. He, “Dispersion engineering of a silicon-nanocrystal-based slot waveguide for broadband wavelength conversion,” Appl. Opt.50, 1260–1265 (2011). [CrossRef] [PubMed]
L. Zhang, Y. Yue, R. G. Beausoleil, and A. E. Willner, “Flattened dispersion in silicon slot waveguides,” Opt. Express18, 20529–20534 (2010). [CrossRef] [PubMed]
S. Mas, J. Caraquitena, J. V. Galn, P. Sanchis, and J. Mart, “Tailoring the dispersion behavior of silicon nanophotonic slot waveguides,” Opt. Express18, 20839–20844 (2010). [CrossRef] [PubMed]
P. Muellner, M. Wellenzohn, and R. Hainberger, “Nonlinearity of optimized silicon photonic slot waveguides,” Opt. Express17, 9282–9287 (2009). [CrossRef] [PubMed]
G. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2006).
C. Tsay, E. Mujagi, C. K. Madsen, C. F. Gmachl, and C. B. Arnold, “Mid-infrared characterization of solution-processed As2S3 chalcogenide glass waveguides,” Opt. Express18, 15523–15530 (2010). [CrossRef] [PubMed]
C. Tsay, Y. Zha, and C. B. Arnold, “Solution-processed chalcogenide glass for integrated single-mode mid-infrared waveguides,” Opt. Express18, 26744–26753 (2010). [CrossRef] [PubMed]
G. C. Chern, “Spin-coated amorphous chalcogenide films,” J. Appl. Phys.53, 6979 (1982). [CrossRef]
Y. Yue, L. Zhang, J. Wang, R. G. Beausoleil, and A. E. Willner, “Highly efficient nonlinearity reduction in silicon-on-insulator waveguides using vertical slots,” Opt. Express18, 22061 (2010). [CrossRef] [PubMed]
P. Muellner, “Fundamental characteristics of the soi slot waveguide structure,” Ph.D. thesis, Faculty of Physics, University of Vienna (2010).
“ www.comsol.com ”.
S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part i: Kerr nonlinearity,” Opt. Express17, 2298–2318 (2009). [CrossRef] [PubMed]
J. I. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. Osgood, “Nonlinear-optical phase modification in dispersion-engineered si photonic wires,” Opt. Express16, 1280–1299 (2008). [CrossRef] [PubMed]
Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express15, 16604–16644 (2007). [CrossRef] [PubMed]
I. D. Rukhlenko, M. Premaratne, and G. P. Agrawal, “Effective mode area and its optimization in silicon-nanocrystal waveguides,” Opt. Lett.37, 2295–2297 (2012). [CrossRef]
A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express14, 4357–4362 (2006). [CrossRef] [PubMed]
L. Zhang, Q. Lin, Y. Yue, Y. Yan, R. G. Beausoleil, and A. E. Willner, “Silicon waveguide with four zero-dispersion wavelengths and its application in on-chip octave-spanning supercontinuum generation,” Opt. Express20, 1685–1690 (2012). [CrossRef] [PubMed]
L. Zhang, Y. Yue, Y. Xiao-Li, R. G. Beausoleil, and A. E. Willner, “Highly dispersive slot waveguides,” Opt. Express17, 7095–7101 (2009). [CrossRef] [PubMed]
J. B. Driscoll, N. Ophir, R. R. Grote, J. I. Dadap, N. C. Panoiu, K. Bergman, and R. M. Osgood, “Width-modulation of Si photonic wires for quasi-phase-matching of four-wave-mixing: experimental and theoretical demonstration,” Opt. Express20, 9227–9242 (2012). [CrossRef] [PubMed]
R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, “Structure and optical properties of thin As2S3 In2S3 films,” J. Phys. D: Appl. Phys.44, 305401 (2011). [CrossRef]
A. von Rhein, S. Greulich-Weber, and R. B. Wehrspohn, “Multiphysics software gazes into photonic crystals,” Physics Best pp. 38–39 (2007).
J. D. Joannopoulos and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).
P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, and M. Lonar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett.94, 121106–121106–3 (2009). [CrossRef]
J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express16, 6227–6232 (2008). [CrossRef] [PubMed]
M. Bass, C. DeCusatis, J. Enoch, G. Li, V. N. Mahajan, E. V. Stryland, and C. MacDonald, Handbook of Optics: Optical Properties of Materials, Nonlinear Optics, Quantum Optics (McGraw-Hill Prof Med/Tech, 2009).

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