Slow light miniature devices with ultra-flattened dispersion in silicon-on-insulator photonic crystal

doi:10.1364/OE.17.013315

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Slow light miniature devices with ultra-flattened dispersion in silicon-on-insulator photonic crystal

Swati Rawal, Ravindra Sinha, and Richard M. De La Rue »View Author Affiliations

Optics Express, Vol. 17, Issue 16, pp. 13315-13325 (2009)
http://dx.doi.org/10.1364/OE.17.013315

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Abstract

We propose a silicon-on-insulator (SOI) photonic crystal waveguide within a hexagonal lattice of elliptical air holes for slow light propagation with group velocity in the range 0.0028c to 0.044c and ultra-flattened group velocity dispersion (GVD). The proposed structure is also investigated for its application as an optical buffer with a large value of normalized delay bandwidth product (DBP), equal to 0.778. Furthermore it is shown that the proposed structure can also be used for time or wavelength-division demultiplexing to separate two telecom wavelengths, 1.31µm and 1.55µm, on a useful time-scale and with minimal distortion.

1. Introduction

Slow light refers to reduction of the group velocity of light - and is a promising technology for future all-optical communication networks. It can be used for the enhancement of light-matter interaction and for the miniaturization of optoelectronic integrated circuits (OEICs) [1

T. F. Krauss, “Why do we need slow light?,” Nat. Photon. 2, 448–450 (2008). [CrossRef]

–5

R. S. Jacobsen, k. Andersen, P. I. Borel, J. F. Pedersen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441, 199–202 (2006). [CrossRef] [PubMed]

]. Photonic crystal (PhC) line-defect waveguides that are created within the photonic bandgap (PBG) are extensively used for achieving slow light in the flat band regions of guided modes supported by these waveguides [6

M. Soljacic, S. G. Jhonson, S. Fan, M. I. Baneseu, E. Ippen, and J. D. Joannopolous, “Photonic crystal slow light enhancement of non linear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002). [CrossRef]

–9

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006). [CrossRef] [PubMed]

]. Therefore they have strong application possibilities in optical buffers and other optical storage devices [10

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow light optical buffers: Capabilities and fundamental limitations,” J. Lightwave Technol. 23, 4046–4066 (2005). [CrossRef]

]. The high refractive index contrast achievable in SOI structures provides strong confinement of light in the vertical direction, while the PBG of the PhC lattice provides confinement of light in the horizontal direction. Slow light in PhCs has already been observed by several authors [11

M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulous, “Slow light, band edge waveguides for tunable time delays,” Opt. Express 13, 7145–7159 (2005). [CrossRef] [PubMed]

–13

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys. 40, 2666–2670 (2007). [CrossRef]

] near the edge of the Brillouin zone. Recently Mori and Baba have experimentally demonstrated a slow light device based on a chirped PhC coupled waveguide [14

D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005). [CrossRef] [PubMed]

, 15

D. Mori and T. Baba, “Dispersion controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004). [CrossRef]

] and have achieved the low group velocity of 0.017c. However fabrication of such complex devices is challenging and may limit their practical utilization in large scale integration.

In this paper, we report the design of SOI based PhC structures having elliptical air holes within a silicon core. For symmetric structures in the vertical direction, like air-bridge structures, light can be strictly classified into Transverse Electric (TE)-and Transverse Magnetic (TM) modes – and as even and odd modes. In such structures the PBG for TE even modes can be used selectively. However for asymmetrical structures, such as SOI structures, the modes cannot be classified selectively- and each mode has even as well as odd components. Modes with H_y as the predominant component are referred to as TE-like modes - and those with E_y as the predominant component is referred to as TM-like modes. (See Fig.1) Thus in asymmetrical structures, TE-like modes have both TE and TM components. If the amount of the TM component is negligible for the TE-like mode, there is a quasi PBG in the PhC properties, where propagation of the TE-like mode is almost completely stopped. The thickness of the PhC waveguide core layer should then satisfy the single-mode condition, in order to exploit the quasi PBG effectively [16

A. Shinya, M. Notomi, I. Yokohama, C. Takahashi, and J. Takahashi “Two-dimensional Si photonic crystals on oxide using SOI substrate,” Opt. Quantum. Electron. 34 113–121 (2002) [CrossRef]

]. The SOI waveguide structure is made up of a silicon core layer sandwiched between a silica cladding layer below and an air cladding above. The major and minor axes of the elliptical air holes are tuned to obtain a flat section of dispersion curve below the silica lightline for slow light propagation.

Fig. 1. (a) Schematic of the proposed design for the W1 line defect channel waveguide configuration with super-cell dimensions – and with length equal to 15a. (b) Diagram defining the coordinate system.

Figure 1 shows a schematic of the proposed design that has elliptical holes in a hexagonal arrangement - with a W1 line-defect channel waveguide configuration. The inset in Fig. 1 shows the super-cell used in computational modelling of the designed structure. For the device applications of such waveguides, the group velocity dispersion (GVD) and other higher-order dispersion parameters should be very low [17

A. Di 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]

-,18

F. Wang, J. Ma, and C. Jiang, “Dispersionless slow wave in Novel 2-D photonic crystal line defect waveguides,” J. Lightwave Technol. 26, 1381–1386 (2008). [CrossRef]

] thereby enabling the optical signal to propagate with reduced distortion - as we have demonstrated using Finite-Difference Time-Domain (FDTD) simulation. The band structure was obtained using a 3D Plane Wave Expansion (PWE) method [19

M. Plihal and A. A. Maradudin, “Photonic band structures of two dimensional systems- The Triangular lattice,” Phy. Rev. B 44, 1865–8571 (1991). [CrossRef]

,20

A. Taflove “Advances in Computational Electrodynamics- The Finite Difference Time Domain Method,” Artech House (1998).

]. Using these simulation tools we have been able to design an SOI based PhC channel waveguide with slow light behaviour - having a group velocity in the range from 0.0028c to 0.044c – and with vanishing GVD, third order dispersion (TOD) and fourth order dispersion (FOD) parameters. The structure designed has also been investigated for its possible application in the design of (i) an optical buffer with a value of normalized DBP that is equal to 0.778 - a value that is higher than previously reported values [14

D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005). [CrossRef] [PubMed]

, 21

M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, “Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth,” Opt. Express 15, 219 (2007). [CrossRef] [PubMed]

, 22

J. Ma and C. Jiang, “Demonstration of ultra slow modes in asymmetric line defect photonic crystal waveguides,” IEEE Phot. Technol. Lett. 20, 1237–1239 (2008). [CrossRef]

] - and (ii) a time and wavelength division demultiplexing device. The proposed device is designed to de-multiplex the two telecommunication wavelengths, 1.31 µm and 1.55 µm as well as other wavelengths around 1.55 µm, using time discrimination and with minimal distortion, while exhibiting an ultra-flattened dispersion curve for both wavelengths.

2. Design aspects and device description

For the design, we have firstly assumed a hexagonal arrangement of air holes in an SOI based PhC structure. The refractive index of silicon has been taken to be 3.5 and its thickness has been taken to be 450 nm. Computed results for the band-structure of a W1 channel guide aligned along the Γ-K direction. They indicate that only a single guided mode exists below the silica light-line - and the calculated value of the effective index of the guided mode for the fundamental TE-like guided mode is calculated to be 3.25. This core guiding layer of silicon is typically bonded on to a silica (SiO₂) lower cladding layer that may be as much as 3 µm thick, in practice. The lattice constant for the air holes has been chosen to be a=0.42 µm, with a basic air hole radius of 0.33a.

As shown in Fig. 2(a), the bandgap of the designed structure lies in the frequency range 0.2130(2πc/a) to 0.3058(2πc/a), for the TE like polarization. A W1 PhC channel waveguide was then obtained by creating a single line defect in the photonic crystal. By examining the dispersion diagram in Fig. 2(b), it is possible to identify a flat section for the even-symmetry transverse mode of the W1 waveguide located at frequencies around 0.24(2πc/a). The inset in the figure shows the super-cell used in the 3D PWE calculation. It is observed that the group velocity becomes very low because of the flat (i.e. near horizontal) dispersion curve, but the waveform of the optical signal is likely to be severely distorted by the large GVD parameter, on the order of 10⁸ ps²/km for this situation, thus limiting the achievable bandwidth for slow light transmission.

Fig. 2. (a) TE Bandgap map and (b) dispersion diagram having flat section of guided modes in the hexagonal lattice with lattice constant a=0.35 µm and basic air hole radius r=0.33a. The inset in the figure shows the super-cell used in the 3D PWE method. The black solid line is the silica lightline, while the red line is the dispersion curve.

Our main goal in this work is to obtain slow guided-light modes that feature the combination of low group velocity and vanishing GVD parameter. The main requirements for such modes to be achieved are: (a) operation below the silica light-line, because the modes which lie above the light-line are intrinsically lossy (i.e. leaky) in the vertical direction - and (b) a flat section of dispersion curve should be obtained i.e. the slope of the dispersion curve should not only be small but it should also be close to constant for a given range. Otherwise the higher order derivatives of the dispersion curve will lead to GVD, third-order dispersion (TOD) and fourth-order dispersion (FOD).

Fig. 3. Movement of the dispersion curves when (a) the semi-minor axis A and (b) the semi-major axis B changes gradually.

To satisfy the above requirements we change the embedded circular air holes into an array of identical elliptical air holes with semi minor axis (A) and semi major axis (B), while retaining a hexagonal lattice arrangement. The dependence of the dispersion diagram on the semi-minor axis (A) and semi-major axis (B) of the air holes is shown in Fig. 3(a) and 3(b). As indicated earlier, these are the normalized dispersion curves for propagation for a W1 channel guide oriented along the ΓK axis of the PhC lattice, for different values of A and B. Figure 3(a) shows that the dispersion curves move up in frequency in the slow light regime when the magnitude of the semi-minor axis A of the elliptical air holes is varied progressively from 0.20a to 0.32a -and the effective indices of the modes of the structure increase. In obtaining these curves, the value of the semi-major axis B has been fixed at 0.45a. In the same way, when the semi-major axis B is varied progressively from 0.39a to 0.48a, the dispersion curve again moves up in frequency, as shown in Fig. 3(b) - where A is fixed at 0.28a. The tuning of the two structural parameters, A and B has been performed in order to tailor the dispersion properties of the waveguide in the slow light regime below the silica light line and for the desired frequency range.

Finally we obtain an optimal waveguide with elliptical air holes that have a semi-minor axis A=0.286a and semi-major axis B=0.457a. Fig. 4(a) shows the band structure for TE-like modes of the designed structure with lattice constant a=0.42 µm, semi-minor axis A=0.286a and semi-major axis B=0.457a. For these parameter values, the bandgap ranges from 0.2352(2πc/a) to 0.3277(2πc/a). The dispersion curve obtained for the fundamental mode of the W1 waveguide is shown in Fig. 4(b), together with the super-cell used in the 3D PWE method. The region 0.41<(2π/a)<0.5 is chosen for slow light transmission - because in this region the flattest section of the dispersion curve is obtained that lies below silica light-line and is within the PBG region. i.e. the slow modes are confined vertically by total internal reflection and horizontally by the photonic bandgap of the PhC regions.

Fig. 4. (a) TE band diagram of finally designed structure. (b) Dispersion diagram for the proposed SOI based single line defect photonic crystal having elliptical air holes with semi-minor axis A=0.286a and semi-major axis B=0.457a. Flat section of dispersion curve corresponds to slow light region.

3. Numerical results and discussion

3.1. Group velocity and Group Velocity Dispersion (GVD):

In the previous section we have designed a W1 channel waveguide in an SOI based photonic crystal structure with a hexagonal lattice of elliptical air holes in silicon. The structure shows a flat dispersion curve below the silica light line. The key velocity of such modes is their group velocity, defined as the velocity with which the envelope of a short pulse propagates through space. The standard definition for group velocity is [23

G. P. Agarwal, Fiber Optic Communication systems, Hoboken, NJ: Wiley-Interscience (1997).

v_{g} = \frac{d ω}{d k}

(1)

where ω is the angular frequency and k is the wave-vector along the waveguide.

3.2. Group Velocity Dispersion (GVD):

The GVD is defined as the derivative of the inverse group velocity w.r.t. angular frequency.

β = \frac{d^{2} k}{d ω^{2}}

= \frac{d}{d ω} (\frac{1}{d ω ⁄ d k})

= - \frac{1}{{(d ω ⁄ d k)}^{3}} \frac{d^{2} ω}{d k^{2}}

β = - \frac{1}{v_{g}^{3}} \frac{d^{2} ω}{d k^{2}}

(2)

From Eq. (2), it can be observed that if the group velocity converges to zero, the GVD parameter goes to infinity - which causes the spreading of an optical pulse in time and as a result of different frequency components of the pulse travelling at different velocities may merge together. Keeping the above fact in view, we have designed our structure to obtain a flat-band situation that is characterized by small and nearly constant slope, with relatively low group velocity and GVD parameter.

Fig. 5. Variation of group velocity and Group Velocity Dispersion (GVD) parameter for flat section of dispersion curve plotted in Fig. 4.

In Fig. 5 the frequency dependence of the group velocity - and the GVD parameter - on normalized frequency are plotted for the slow wave region shown in Fig. 4(b). We have considered the flat band of the dispersion curve lying below silica lightline i.e. the frequency range between 0.2700(2πc/a) and 0.2733(2πc/a). It can be observed that, for this range of frequencies, the group velocity remains in the range 0.0028c to 0.044c - and the GVD parameter lies in the range of 10² ps²/km. The GVD has positive as well as negative values and, for a particular spectral region, it becomes flat - i.e., for a bandwidth of

Δ υ = \frac{Δ ω}{2 π} = 2.1 THz

, the GVD parameter lies below 10 ps²/km.The low group velocity obtained at the extreme points (~0.005c) is at the cost of large GVD, on the order of 10² ps²/km - while near the mid point, around ω=0.2718 (2πc/a), vg~0.044c and the GVD parameter vanishes. Hence a very low GVD value is obtained over the wide spectral bandwidth of 2.1 THz.

3.3 Third Order Dispersion (TOD) and Fourth Order Dispersion (FOD):

The TOD parameter was deduced by calculating:

TOD = \frac{1}{v_{g}} \frac{\partial (GVD)}{\partial k}

(3)

i.e. the first derivative of the GVD. and theFOD parameter was deduced by calculating:

FOD = \frac{1}{v_{g}} \frac{\partial (TOD)}{\partial k}

(4)

i.e. first derivative of the TOD.

Figure 6(a) shows that the upper limit of the TOD is on the order of 10⁴ ps³/km for the slow light regime below the silica light-line i.e. in the frequency range from 0.2700 (2πc/a) to 0.2733 (2πc/a) However we obtain a flat curve near the centre of frequency range, over a bandwidth of 2.1 THz, where TOD varies from 10¹ to 10² ps³/km. Figure 6(b) shows that the upper limit of the FOD is on the order of 10⁶ ps⁴/km for the slow light regime, but again a flat curve is obtained for a bandwidth of 2.1 THz near the centre. These values of TOD and FOD are less than those reported in previous work [17

A. Di 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]

, 24

S. Assefa and Y. A. Vlasov, “High order dispersion in photonic crystal waveguides,” Opt. Express 15, 17562 (2007). [CrossRef] [PubMed]

]. Thus both TOD and FOD vanish for a particular frequency range in the proposed SOI based structure having elliptical air holes PhC.

Fig. 6. Variation of higher order dispersion parameters (a) TOD and (b) FOD with frequency.

The results obtained for the TOD and FOD are important for evaluating the pulse broadening that is due to higher order dispersion, as well as for evaluating nonlinear effects that are dependent on high order dispersion in the slow light regime. Their low values, tending to zero in a bandwidth of 2.1 THz, show that the proposed waveguide can be used for device applications such as multiplexing and demultiplexing, as well as in realizing delay lines for optical buffers.

4. Photonic crystal waveguide as an optical buffer

The optical Buffer is a device that temporarily stores and adjusts the timing of optical packets. Application of PhC waveguides with slow light in optical buffers has recently been attracting wide attention [10

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow light optical buffers: Capabilities and fundamental limitations,” J. Lightwave Technol. 23, 4046–4066 (2005). [CrossRef]

]. We shall now investigate the properties of the SOI-based elliptical air hole W1 channel waveguide for use as an optical buffer - and determine the limitations of slow light propagation in the proposed structure. The number of bits that can be stored in a slow light device is given by its Delay Bandwidth Product (DBP). The expression for the DBP is:

DBP = T_{d} . B

(5)

where T_d is the time of propagation of a pulse in the waveguide and B is the bandwidth. If L is the length of the structure and the pulse is propagating with minimum distortion, the upper limitation on the DBP is given by [25

J. Ma and C. Jiang, “Flat band slow light in asymmetric line defect photonic crystal waveguide featuring low group velocity and dispersion,” IEEE J. Quantum Electron. 44, 763–769 (2008). [CrossRef]

]

D B P = T_{d} . B \leq \frac{L}{{\tilde{v}}_{g}} {(\frac{1}{4 π L ∣ β ∣})}^{1 ⁄ 2}

\Rightarrow D B P \leq {(\frac{L}{4 π})}^{1 ⁄ 2} \frac{1}{{\tilde{v}}_{g} {∣ β ∣}^{1 ⁄ 2}}

(6)

Thus the DBP depends upon the length of the structure, the average group velocity and the GVD parameter. If the length of the structure is fixed, the DBP is inversely proportional to

{\tilde{v}}_{g} {∣ β ∣}^{1 ⁄ 2}

. i.e., in order to increase the value of DBP, we need to decrease the value of the average group velocity ṽg and the GVD parameter, β. We shall next calculate the upper limit of the DBP in the proposed SOI-based PhC having a hexagonal arrangement of elliptical air holes, in the slow light regime below the silica light-line. At frequencies ω₀=0.2709 (2πc/a), i.e. for an incoming wave at λ₀=1.55 µm, the average group velocity is calculated to be 0.035c - and the average GVD parameter is β=0.5 ps²/km (from Fig. 5). Therefore,

{\tilde{v}}_{g} {∣ β ∣}^{1 ⁄ 2} ≃ 0.0247 {c . pskm}^{1 ⁄ 2}

. This value is highly reduced as compared to the value of 4.4c ps.km^1/2 in reference [25

]. Even for the slowest velocity achieved at a frequency of 0.27009(2πc/a) i.e. λ₀=1.555 µm, the average group velocity is 0.0028c and β=376 ps²/km - and therefore,

{\tilde{v}}_{g} {∣ β ∣}^{1 ⁄ 2} ≃ 0.0543 {c . pskm}^{1 ⁄ 2}

. Hence we observe that the upper limitation on the delay bandwidth product is strongly enhanced in our structure.

Fig. 7. Variation of group velocity and group index with frequency in the slow light region below the silica light-line.

The DBP, defined by T_d.B, provides a measure of the buffering capacity that a slow light device potentially provides. However its normalized form can become more useful if devices that have different lengths and different frequencies are compared. We shall now calculate the value of the normalized DBP for the proposed SOI based photonic-crystal channel-waveguide. Figure 7 shows the variation of the group index and group velocity with frequency. The average group index in the frequency range Δω is calculated as:

{\tilde{n}}_{g} = \int_{ω_{0} - Δ ω ⁄ 2}^{ω_{0} + Δ ω ⁄ 2} n_{g} (ω) \frac{d ω}{Δ ω}

(7)

From Fig. 7, the average value of group index is- ñ _g =66.60, in the normalized frequency bandwidth of

\frac{Δω}{ω} = 0.0117

. The normalized DBP is calculated to be-

{\tilde{n}}_{g} \frac{Δω}{ω} = 0.778

, which is substantially greater than that reported in references [14

D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005). [CrossRef] [PubMed]

and 22

J. Ma and C. Jiang, “Demonstration of ultra slow modes in asymmetric line defect photonic crystal waveguides,” IEEE Phot. Technol. Lett. 20, 1237–1239 (2008). [CrossRef]

]. The normalized DBP calculated in [21

] is equal to 0.359, which is also less than the value of 0.778 that we have calculated for the proposed structure. The proposed structure can therefore be used as an optical buffer with a high DBP value.

5. FDTD simulation

In this section an investigation is described for the broadening of pulses propagating through the waveguide of the proposed structure, using FDTD simulations. FDTD simulations are carried out using the RSOFT FULLWAVE package with the mesh size taken to be Period/3 and device dimensions- taken to be 6.72µm×5.04µm in XZ plane. If we set the central frequency at ω₀=0.2709(2πc/a) i.e. at λ₀=1.55 µm, the group velocity and GVD parameter at this wavelength are 0.035c and 0.5 ps²/km. As shown in Figure 8(a), for wavelength of 1.55 µm, time delay between the input peak and output peak is nearly 0.57 ps for a waveguide length of 15a. Hence the corresponding group velocity is 0.036c which is slightly larger than that observed using PWE method (i.e. 0.035c). The slowdown factor, which is defined as the ratio of the phase velocity to the group velocity (S=v_p/v_g) is calculated to be 17 at λ₀=1.55 µm. Thus the width of the Gaussian pulse at the output end is approximately equal to that of the pulse at the input end, with very little pulse expansion, as shown in Fig. 8(a). Here we have considered the length of waveguide to be 15a, as shown in Fig. 1. The incident pulse with a full width at half-maximum (FWHM) of 0.07 ps expands to 0.11 ps at the output. We observe the modal field distribution for an incident wave at a wavelength of 1.55 µm in Fig. 8(b) - which shows that the light spreads substantially into the first and second nearest rows of elliptical air holes of the PhC, due to the slow light propagation in the waveguide. Figure 8(c) shows the spreading of the wave for λ₀=1.555 µm - for which the slowest group velocity value of 0.0028c is achieved in the proposed SOI based photonic crystal waveguide.

Fig. 8. (a) Field amplitude of the Gaussian pulse recorded at the input end and output end of the waveguide as a function of time for λ₀=1.55µm. Modal field distribution in the PhC waveguide for (b) 1.550 µm and (c) 1.555 µm.

6. Time and Wavelength Division Demultiplexing (TDM and WDM)

The proposed SOI based PhC, with elliptical air holes, is next investigated for use in time and wavelength division demultiplexing (TDM and WDM). It is found that the proposed structure has a bandgap in the frequency range of 0.2352 (2πc/a) to 0.3277 (2πc/a) as shown in Figure 4(a). Hence the bandgap exist for both the wavelengths, 1.31 µm, which corresponds to a frequency of 0.3206 (2πc/a) and 1.55µm, which corresponds to a frequency of 0.2709 (2πc/a).

Fig. 9. (a)Variation of group velocity and Group Velocity Dispersion (GVD) parameter for region having central wavelength near 1.31µm (b) Modal field distribution in PhC channel waveguide for a wavelength of 1.31µm.

Figure 9(a) shows the frequency dependence of the group velocity and GVD parameter for a section of dispersion curve below the silica light-line. For an incoming pulse of light at a wavelength of 1.31 µm, the group velocity is calculated to be 0.178c and the GVD parameter is 0.0083 ps²/km, which implies low signal distortion for a wavelength of 1.31 µm. Thus a signal at a wavelength of 1.31 µm is more than five times faster than one at 1.55 µm, where the velocity is equal to 0.035c. Therefore the channel guides a signal at a wavelength of 1.31µm at faster velocity -and hence it reaches the output end of the W1 waveguide earlier than a signal at the wavelength of 1.55 µm. The FDTD simulation for 1.31 µm is shown in Fig. 9(b), indicating that the spread of light wave in theW1 waveguide at 1.31 µm is less than that for 1.55 µm, as observed in Fig. 8(b).

Fig. 10. Schematic for separation of pulses of light at telecom wavelength 1.31 µm, 1.5500 µm, 1.5534 µm and 1.5550 µm for time and wavelength division de-multiplexing.

If we now assume the device length to be 600 µm, the delay time for a wavelength of 1.55 µm is calculated to be 57 ps - and for a wavelength of 1.5534 µm it is calculated to be 89 ps. For a wavelength of 1.5550 µm, it is calculated to be 704 ps, while for a wavelength of 1.31 µm; it is calculated to be 11 ps (Fig. 10). Here we observe that even for the slowest velocity achieved i.e. 0.0028c, the GVD parameter is relatively small (on the order of 10² ps²/km) and hence this waveguide PhC structure can be used in both time and wavelength division de-multiplexing.

7. Conclusions

In this paper, we have reported on the design of an SOI based PhC channel waveguide with elliptical air holes for use in slow light propagation with group velocities in the range from 0.0028c to 0.044c – and with extremely low GVD, TOD and FOD parameters. This combination of small values for key parameters causes an input signal to pass through the PhC waveguide with much less distortion. The proposed structure has considerable potential for use in photonic device applications such as optical buffers, TDM and WDM processing -as we have demonstrated.

Acknowledgements

The authors gratefully acknowledge the initiatives and support towards establishment of the “TIFAC Centre of Relevance and Excellence in Fiber Optics and Optical Communication at the Delhi College of Engineering, Delhi” through the “Mission REACH” program of Technology Vision-2020 of the Government of India. Two of the authors (R.K.Sinha and Richard M De La Rue) are also grateful to the Royal Academy of Engineering (UK) for providing financial support to carry out research and development work in the area of Photonic Crystal Devices.

References and links

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7.	H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. Van Hulst, T. F. Krauss, and L. Kuipers, “Real space observation of ultra slow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005). [CrossRef] [PubMed]
8.	T. Baba and D. Mori, “Slow light engineering in photonic crystals,” J. Phys. D: Appl. Phys. 40, 2659–2665 (2007) [CrossRef]
9.	L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006). [CrossRef] [PubMed]
10.	R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow light optical buffers: Capabilities and fundamental limitations,” J. Lightwave Technol. 23, 4046–4066 (2005). [CrossRef]
11.	M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulous, “Slow light, band edge waveguides for tunable time delays,” Opt. Express 13, 7145–7159 (2005). [CrossRef] [PubMed]
12.	T. Baba, D. Mori, K. Inoshita, and Y. Kuroki, “Light localization in line defect photonic waveguides,” IEEE J. Quantum Electron. 10, 484–491 (2004). [CrossRef]
13.	T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys. 40, 2666–2670 (2007). [CrossRef]
14.	D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005). [CrossRef] [PubMed]
15.	D. Mori and T. Baba, “Dispersion controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85, 1101–1103 (2004). [CrossRef]
16.	A. Shinya, M. Notomi, I. Yokohama, C. Takahashi, and J. Takahashi “Two-dimensional Si photonic crystals on oxide using SOI substrate,” Opt. Quantum. Electron. 34 113–121 (2002) [CrossRef]
17.	A. Di 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]
18.	F. Wang, J. Ma, and C. Jiang, “Dispersionless slow wave in Novel 2-D photonic crystal line defect waveguides,” J. Lightwave Technol. 26, 1381–1386 (2008). [CrossRef]
19.	M. Plihal and A. A. Maradudin, “Photonic band structures of two dimensional systems- The Triangular lattice,” Phy. Rev. B 44, 1865–8571 (1991). [CrossRef]
20.	A. Taflove “Advances in Computational Electrodynamics- The Finite Difference Time Domain Method,” Artech House (1998).
21.	M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, “Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth,” Opt. Express 15, 219 (2007). [CrossRef] [PubMed]
22.	J. Ma and C. Jiang, “Demonstration of ultra slow modes in asymmetric line defect photonic crystal waveguides,” IEEE Phot. Technol. Lett. 20, 1237–1239 (2008). [CrossRef]
23.	G. P. Agarwal, Fiber Optic Communication systems, Hoboken, NJ: Wiley-Interscience (1997).
24.	S. Assefa and Y. A. Vlasov, “High order dispersion in photonic crystal waveguides,” Opt. Express 15, 17562 (2007). [CrossRef] [PubMed]
25.	J. Ma and C. Jiang, “Flat band slow light in asymmetric line defect photonic crystal waveguide featuring low group velocity and dispersion,” IEEE J. Quantum Electron. 44, 763–769 (2008). [CrossRef]

OCIS Codes
(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers
(130.2790) Integrated optics : Guided waves
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Slow Light

History
Original Manuscript: May 6, 2009
Revised Manuscript: June 18, 2009
Manuscript Accepted: June 18, 2009
Published: July 20, 2009

Citation
Swati Rawal, Ravindra Sinha, and Richard M. De La Rue, "Slow light miniature devices with ultra-flattened dispersion in silicon-on-insulator photonic crystal," Opt. Express 17, 13315-13325 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13315

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References

T. F. Krauss, "Why do we need slow light?," Nat. Photon. 2, 448-450 (2008). [CrossRef]
Richard M. De La Rue, "Slower for longer," Nat. Photon. 2(12), 715-716 (2008). [CrossRef]
Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, "Active control of slow light on a chip with photonic crystal waveguides," Nature 438, 65-69 (2005). [CrossRef] [PubMed]
V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All optical control of light on a silicon chip," Nature 481, 1081-1084 (2004). [CrossRef]
R. S. Jacobsen, K. Andersen, P. I. Borel, J. F. Pedersen, O. Hansen, M. Kristensen, A. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, "Strained silicon as a new electro-optic material," Nature 441, 199-202 (2006). [CrossRef] [PubMed]
M. Soljacic, S. G. Jhonson, S. Fan, M. I. Baneseu, E. Ippen, and J. D. Joannopolous, "Photonic crystal slow light enhancement of non linear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002). [CrossRef]
H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. Van Hulst, T. F. Krauss, and L. Kuipers, " Real space observation of ultra slow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903 (2005). [CrossRef] [PubMed]
T. Baba and D. Mori, "Slow light engineering in photonic crystals," J. Phys. D: Appl. Phys. 40, 2659-2665 (2007) [CrossRef]
L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, "Photonic crystal waveguides with semi-slow light and tailored dispersion properties," Opt. Express 14, 9444-9450 (2006). [CrossRef] [PubMed]
R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, "Slow light optical buffers: Capabilities and fundamental limitations," J. Lightwave Technol. 23, 4046-4066 (2005). [CrossRef]
M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulous, "Slow light, band edge waveguides for tunable time delays," Opt. Express 13, 7145-7159 (2005). [CrossRef] [PubMed]
T. Baba, D. Mori, K. Inoshita, and Y. Kuroki, "Light localization in line defect photonic waveguides," IEEE J. Quantum Electron. 10, 484-491 (2004). [CrossRef]
T. F. Krauss, "Slow light in photonic crystal waveguides," J. Phys. D: Appl. Phys. 40,2666-2670 (2007). [CrossRef]
D. Mori and T. Baba, "Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide," Opt. Express 13, 9398-9408 (2005). [CrossRef] [PubMed]
D. Mori and T. Baba, "Dispersion controlled optical group delay device by chirped photonic crystal waveguides," Appl. Phys. Lett. 85, 1101-1103 (2004). [CrossRef]
A. Shinya, M. Notomi, I. Yokohama, C. Takahashi, and J. Takahashi "Two-dimensional Si photonic crystals on oxide using SOI substrate," Opt. Quantum. Electron. 34113-121 (2002) [CrossRef]
A. Di 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]
F. Wang, J. Ma, and C. Jiang, "Dispersionless slow wave in Novel 2-D photonic crystal line defect waveguides," J. Lightwave Technol. 26, 1381-1386 (2008). [CrossRef]
M. Plihal and A. A. Maradudin, "Photonic band structures of two dimensional systems- The Triangular lattice," Phy. Rev. B 44, 1865-8571 (1991). [CrossRef]
A. Taflove "Advances in Computational Electrodynamics- The Finite Difference Time Domain Method," Artech House (1998).
M. D. Settle, R. J. P. Engelen, M. Salib, A. Michaeli, L. Kuipers, and T. F. Krauss, "Flatband slow light in photonic crystals featuring spatial pulse compression and terahertz bandwidth," Opt. Express 15, 219 (2007). [CrossRef] [PubMed]
J. Ma and C. Jiang, "Demonstration of ultra slow modes in asymmetric line defect photonic crystal waveguides," IEEE Phot. Technol. Lett. 20, 1237-1239 (2008). [CrossRef]
G. P. Agarwal, Fiber Optic Communication systems, Hoboken, NJ: Wiley-Interscience (1997).
S. Assefa and Y. A. Vlasov, "High order dispersion in photonic crystal waveguides," Opt. Express 15, 17562 (2007). [CrossRef] [PubMed]
J. Ma and C. Jiang, "Flat band slow light in asymmetric line defect photonic crystal waveguide featuring low group velocity and dispersion," IEEE J. Quantum Electron. 44, 763-769 (2008). [CrossRef]

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