the serpentine optical waveguide: engineering the dispersion relations and the stopped light points

The Serpentine Optical Waveguide: engineering

the dispersion relations and the stopped light

points

Jacob Scheuer* and Ori Weiss

School of Electrical Engineering Tel Aviv University, Ramat Aviv, Tel-Aviv 69978, Israel *[email protected]

www.eng.tau.ac.il/~kobys/

Abstract: We present a study a new type of optical slow-light structure

comprising a serpentine shaped waveguide were the loops are coupled. The

dispersion relation, group velocity and GVD are studied analytically using a

transfer matrix method and numerically using finite difference time domain

simulations. The structure exhibits zero group velocity points at the ends of

the Brillouin zone, but also within the zone. The position of mid-zone zero

group velocity point can be tuned by modifying the coupling coefficient

between adjacent loops. Closed-form analytic expressions for the dispersion

relations, group velocity and the mid-zone zero vg points are found and

presented.

©2011 Optical Society of America

OCIS codes: (130.2790) Guided waves; (260.2030) Dispersion.

References and links

1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).

2. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,”

Opt. Lett. 24(11), 711–713 (1999). 3. J. E. Heebner and R. W. Boyd, “'Slow’ and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14),

2629–2636 (2002).

4. A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35(4/5), 365–379 (2003).

5. J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing

and storing of light,” Opt. Photon. News 16(2), 36–40 (2005). 6. J. Heebner, P. Chak, S. Pereira, J. Sipe, and R. Boyd, “Distributed and localized feedback in microresonator

sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004).

7. M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92(8), 083901 (2004). 8. B. Z. Steinberg, “Rotating photonic crystals: a medium for compact optical gyroscopes,” Phys. Rev. E Stat.

Nonlin. Soft Matter Phys. 71(5), 056621 (2005).

9. J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006).

10. B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-

degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24(5), 1216–1224 (2007). 11. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J.

Lightwave Technol. 15(6), 998–1005 (1997).

12. J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).

13. T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally

coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003). 14. R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett.

76(26), 3858–3860 (2000).

15. X. Fengnian, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006).

16. J. Heebner, R. Boyd, and Q. Park, “SCISSOR solitons and other novel propagation effects in microresonator-

modified waveguides,” J. Opt. Soc. Am. B 19(4), 722–731 (2002).

#144634 - $15.00 USD Received 22 Mar 2011; revised 2 May 2011; accepted 24 May 2011; published 31 May 2011(C) 2011 OSA 6 June 2011 / Vol. 19, No. 12 / OPTICS EXPRESS 11517

17. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Y. S. Kivshar,

“Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express 16(2), 1104–1114 (2008).

18. A. A. Sukhorukov, A. V. Lavrinenko, D. N. Chigrin, D. E. Pelinovsky, and Y. S. Kivshar, “Slow-light dispersion

in coupled periodic waveguides,” J. Opt. Soc. Am. B 25(12), C65–C74 (2008). 19. P. Chak, J. K. Poon, and A. Yariv, “Optical bright and dark states in side-coupled resonator structures,” Opt.

Lett. 32(13), 1785–1787 (2007).

20. O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW--formation of mid-band zero group velocity,” Opt. Express 17(17), 14817–14824 (2009).

21. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and

dispersion relation,” Opt. Express 13(11), 4331–4340 (2005). 22. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to

gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).

23. A. G. Yamilov, M. R. Herrera, and M. F. Bertino, “Slow-light effect in dual-periodic photonic lattice,” J. Opt. Soc. Am. B 25(4), 599–608 (2008).

24. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional

photonic crystals,” Opt. Express 4(5), 167–176 (1999). 25. S. Nojima, “Enhancement of optical gain in two dimensional photonic crystal with active lattice points,” Jpn. J.

Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998).

26. S. Mookherjea, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. 84(17), 3265–3267 (2004).

27. J. B. Khurgin, “Dispersion and loss limitations on the performance of optical delay lines based on coupled

resonant structures,” Opt. Lett. 32(2), 133–135 (2007).

28. D. Miller, “Fundamental limit for optical components,” J. Opt. Soc. Am. B 24(10), A1–A18 (2007).

1. Introduction

During the past decade, substantial research efforts were focused on slow light photonic

structures and their applications. One of the most attractive approaches to realize such

structures is to let the light propagate through a chain of coupled cavities. Significant

reduction of the group velocity in such coupled cavity waveguides (CCWs) have been

predicted and demonstrated experimentally. Because CCWs can be realized in chip-scales, the

slowing effect opens up new avenues for numerous integrated optics applications e.g. delay

lines, rotation sensors, optical memories, optical filters, wavelength converters and more [1–

10]. In particular, micro-resonators based slow-light structures received substantial attention

due to their interesting properties and potential applications [11–14]. Much focus was devoted

to two different types of slow-light structures: the CROW (Coupled Resonator Optical

Waveguide) [2,5,15], and the SCISSOR (Side Coupled Integrated Spaced Sequence of

Resonators) [6,16]. Additional structures, exhibiting a combination of the properties of the

SCISSOR and the CROW have been proposed and demonstrated as well [17–20].

One of the most important characteristics of periodic slow-light photonic structures is the

formation of forbidden and allowed frequency bands in the dispersion relations of the

structure. The edges of the forbidden gaps are characterized by very slow group velocity

regimes and stopped-light (vg = 0) at the band-edge. In most of the more extensively studied

slow-light structures (SLSs) such as the CROW and the SCISSOR, the zero group-velocity

points are formed at the edges of the Brillouin zone (i.e. at K = 0 and K = π where K is the

Bloch wavenumber) due to the Bragg resonances and the cavity resonances of the structure.

Consequently, the dispersion relations of these structure exhibit cosine-like shapes and the

ability to engineer and modify these relations is quite limited. Formation of zero group

velocity points inside the Brillouin zone has also been shown to exist in structures allowing

for two distinct, counter-propagating, light-paths such as the side-coupled adjacent resonators

CROW (SC-CROW) [19,20] and coupled photonic crystal waveguides [18]. In these cases the

mid-zone zero group velocity points were found to emerge in the vicinity of the cavity [19,20]

and Bragg [18] resonances. Mid Brillouin zone zero vg points were also found theoretically in

the dispersion relations of optical microcoil resonator structures [21] for inter-winds coupling

levels which exceed a certain value.

In this paper, we introduce a new type of slow-light structure consisting of a snake-like

structure where optical coupling is introduced between adjacent curved sections (see Fig. 1).


We find that unlike the CROW and the SCISSOR structures but similar to the SC-CROW, the

serpentine optical waveguide (SOW) exhibits stopped-light points within the Brillouin zone.

This is despite the fact that, unlike the SC-CROW, the SOW structure does not explicitly

include resonators (although resonating optical paths do exist) and that the bands in the

structure are formed by the Bragg condition (as in a conventional Bragg stack) stemming

from the optical length of each unit cell. As discussed in Section 3 below (see also Fig. 3), the

spectral response of the SOW is periodical with a free spectral range (FSR) which is

determined by the phase accumulated by propagation along the optical length of a unit cell –

k0neff(2πR + 2LB). The couplers in the structure serve as partial reflectors due to the geometry

of the SOW, similar to the role of the interfaces between adjacent layers in a Bragg stack.

The position of the mid-zone stopped light point in the structure is determined by the

coupling coefficients between the loops of the SOW (see Fig. 1), and therefore can be

relatively easily modified and tuned, thus allowing much flexibility in the engineering and

tailoring of the dispersion relations to specific applications. Similar to the optical microcoil

resonator structure [21], the SOW provides two alternative paths for the light propagating

through it – conventional propagation through the waveguide, and “tunneled” propagation

through the couplers connecting adjacent unit cells. These two alternative routes are the key

for the formations of the mid Brillouin zone zero group velocity regimes.

The existence of a tunable slow group velocity region inside the Brillouin zone is highly

attractive for enhancing gain in lasers [22–26] and light-matter interactions. In Bragg or

CROW based lasers [22–26], lasing is attained only at the band edges while in the structures

such as the SC-CROW and SOW, the lasing point in the dispersion relation can be modified.

Fig. 1. The SOW structure: Unit cell and parameter definitions.

2. Structure parameters and transfer matrix

Figure 1 depicts a schematic of the SOW structure as well as the unit-cell and fields

definitions. The structure is defined by three parameters: 1) The phase accumulation in a

quarter-circle of the loop (section A); 2) The phase accumulation in the curved waveguide

connecting the upper and lower loops (section B); 3) the coupling coefficient, κ, between the

loops. Regarding the third parameter, we note that the structure can be generalized by

allowing the coupling coefficients between the upper and lower loops to be different.

However, such generalization does not modify substantially the properties of the structure and

in order to simplify the analysis we strict ourselves to identical coupling.


A single unit cell of the SOW is marked in Fig. 1. The field in the cell can be represented

by six amplitude E1..E6 as defined in Fig. 1. The electric field amplitudes at adjacent unit cells

can be connected by a transfer matrix M:

T T

1 1 6 1 61... ...n n n

E E E M E E (1)

where M is a 6 × 6 unitary matrix given by:

01

011

0

10

100

1

01

001

0011

0

00001

0001

0

2)(

2)(

2)(

2)(

)(

)(

ABA

ABA

ABA

ABA

BA

BA

ii

ii

ii

ii

i

i

eie

eie

ei

e

ei

e

ei

i

ei

i

M

(2)

where A and B represent the phase accumulation through sections A and B respectively.

Invoking the Bloch theorem, Eq. (1) can be used to derive the spectral properties of the

structure as well as the field distribution for any set of the parameters.

3. Band structure, dispersion relations and group velocity

According to Bloch theorem the field at each unit cell is related to the field at its neighbors by

a phase shift, i.e. nn EiKE )exp(1

, where K is the Bloch wavenumber and Λ is the length

of a unit-cell. Substitution of the Bloch condition into Eq. (1) yields an eigenvalue problem

which eiganvalues and eigenvectors are the Bloch wavenumber and field solutions:

0 I

iKeM (3)

where I is an 6x6 unit matrix. Introducing Eq. (2) into Eq. (3) yields the characteristic

equation which constitute the dispersion relation of the structure, i.e. the relation between the

frequency and the Bloch wavenumber. Extracting the dispersion relation from Eq. (3) is

laborious but straight-forward:

2 3 2cos(4 2 ) 4 cos ( ) 4 (1 )cos ( ) (1 4 )cos( )A B K K K (4)

Note, that the phase term in the left-hand side of Eq. (4) is the overall phase accumulated

by a wave which propagates along the whole waveguide length of a unit cell – 4A + 2B =

k0neff(2πR + 2LB), where k0 is the wavenumber, neff is the effective index of the waveguide

mode, R is the radius of section A and LB is the overall length of section B (see Fig. 1) given

by LB = 2αR.

As a concrete example we consider an SOW structure consisting of Silicon over insulator

structure where the Si layer is 230nm thick and the waveguide width is 430nm. The radius of

section A is R = 5μm and section B consists of two circular sections with radius R = 5μm and

an angle of α = 58.5°. Note, that the choice of α not only affects the overall length of section

B but also the width of the gap between the waveguide loops and, hence, the coupling

coefficient.


Figure 2 illustrates the mode profile of the waveguide super-imposed on the index

structure. At λ = 1.55mm the refractive indices of the Si and the silica are respectively nSi =

3.477 and nsilica = 1.4455. The resulting effective index is neff = 2.362 for the lowest TE mode.

Figure 3(a) depicts the dispersion relation of an SOW structure with the above-mentioned

cross-section and coupling coefficient of κ = 0.24. Figure 3(b) depicts the corresponding

normalized group velocity (vg/Λ) for the band which is marked in Fig. 3(a). For simplicity,

the dispersion of the waveguide cross-section and the material was neglected. Nevertheless,

this dispersion can be readily introduced by modifying neff according to the wavelength.

Fig. 2. The mode profile of the waveguide.

As can be expected, the dispersion relations consist of a series of bandgaps and passbands.

The FSR, given by ΔfFSR = c/neffR(2π + 4α), is 2.45THz. Each passband consists of a pair of

mirror image dispersion curves where the symmetry lines are located at frequencies which are

“anti-resonant” with the optical length of the waveguide structure in each unit cell i.e.

frequencies satisfying k0neff(2πR + 2LB) = 2π(m + 1/2) where m is an integer. This can be

easily obtained from Eq. (4) by substituting KΛ = ± π which yields cos(4A + 2B) = 1. The

term “anti-resonant” is in quotation marks because there is no actual resonance or anti-

resonance associated with this optical length.

Fig. 3. Dispersion relation (a) and normalized group velocity (b) for the marked band. The

structure parameters are defined in the text.

Another interesting feature of the dispersion relations shown in Fig. 3 is the emergence of

a zero group velocity point located within the Brillouin zone (at KΛ ± 0.491π for the

parameters chosen in Fig. 3). It should be noted that in the vicinity of the mid-zone zero group

velocity point there are two possible (positive) group velocities for every frequency. For

example, at f = 190.4THz there are two possible group velocities – vg/π = 1.06ps1

and vg/π =


2.38ps1

. The reason for the two possible group velocities stem for the two paths through

which the light can propagate along the structure – along the waveguide or by tunneling

through the directional couplers. As a result, for a given frequency there are two possible

Bloch modes (with different direct/tunneled propagation combination) which possesses

different Bloch wavenumbers and, correspondingly, group velocities.

Figure 4 shows the dispersion relations of the SOW structure for various representative

coupling coefficients. There are several interesting features which should be noted: 1) Once

the coupling coefficient, κ, exceeds a certain value, a zero group velocity point is formed

within the Brillouin zone; 2) As the coupling coefficient is increased, the zero group velocity

point shifts to larger Bloch wavenumbers, K, in the dispersion diagram, approaching KΛ =

2π/3 for κ1; 3) At κ = 0.5, the two curves forming each transmission band collide at K = 0.

Increasing the coupling coefficient further results in a formation of a crossing of the two

curves.

Fig. 4. The dependence of the dispersion curve on the coupling coefficient.

The features of the dispersion relations can be easily obtained from Eq. (4). Calculating

the derivative of Eq. (4) with respect to ω and substituting vg = (dK/dω)1

yields the group

velocity at each point:

2 3

g

0

sin( ) 12 cos ( ) 8 (1 )cos( ) (1 4 )v

(4 2 ) sin[ (4 2 )]

eff

eff A B eff A B

cn K K K

n L L k n L L

(5)

Substituting vg = 0 to Eq. (5) yields a quadratic equation for cos(KΛ) which solutions

provide the zero vg point:

gv 0

cos( ) (1 4 ) 6K

(6)

For meaningful solution, the absolute value of the right-hand side of Eq. (6) must be

smaller than 1 thus posing a limit on the coupling coefficients for which a zero group velocity

point is formed within the Brillouin zone is formed:

1.0 (7)


Figure 5 illustrates the position of the mid Brillouin-zone zero group-velocity point as a

function of the coupling coefficient. Substituting κ = 1 in Eq. (6) yields the maximal Bloch

number for which such point is formed – KΛ = cos1

(1/2).

Fig. 5. The dependence of the mid Brillouin zone zero group velocity point on the coupling

coefficient.

It is interesting explore the group velocity vg = dω/dK for κ = 0.1, i.e. at the coupling level

where the mid-zone zero group velocity is formed. Figure 6 depicts the group velocity of the

structure defined in Section 3 for κ = 0.1. As shown in the figure, an inflection point

accompanied by a flat region is formed around K = 0.

Fig. 6. The group velocity of SOW structure with κ = 0.1 vs. the wavenumber (a) and the

frequency (b). Note the formation of the inflection point at K = 0.

4. Finite difference time domain simulations

To verify the properties of the analytic model we studied the structure using finite difference

time domain (FDTD) simulation tool. In order to calculate the dispersion relation of the

structure we simulated a single unit cell with Bloch boundary conditions at x (corresponding

to the horizontal coordinate in Fig. 1) and perfectly matched layers in all other boundaries.

The dimensions and parameters of the structure were as detailed in Section 3.

Figure 7 depicts the dispersion relation of the structure calculated by the FDTD tool. The

stars indicates the results of the simulations while the solid line indicate the corresponding

analytic result based on Eq. (4), where the coupling coefficient and the effective index where

calculated by an FDTD simulation and a TE semi-vectorial finite difference mode solver and


were found to be neff = 2.362 and κ = 0.245 respectively. The excellent agreement between the

simulations and the analytic model is clear, thus validating the theoretical analysis of Section

3.

Fig. 7. Comparison between the numerically (blue stars) and the theoretically (solid green)

dispersion relations. The structure parameters are defined in the text.

5. Practical considerations – finite SOW structure and propagation losses

The transfer matrix method can also be used for exploring the properties of a realistic SOW

structure, incorporating propagation losses and finite length. Given a finite SOW structure

consisting of N unit cells, one can relate the fields at the output to the fields at the input by a

product of three matrices: 1) an input matrix, connecting the fields at the input to the fields at

the first unit cell; 2) N-unit-cell transfer matrix which is given by MN = MN where M is given

by Eq. (2); and 3) an output matrix, connecting the fields at the last unit cell to the fields at the

output. Note that assuming the SOW is excited from one of its sides, the outputs of the

structures are the transmitted field and the reflected field (see Fig. 1).

When analyzing the properties of a finite structure, the important parameters are the

number of unit cells and the coupling coefficient. Figure 8 depicts a color representation of

the transmission function (|t|2 in Fig. 1) of a lossless SOW structure consisting of five unit

cells for coupling coefficients ranging from 0.01 to 0.99. The dependence of the transfer

function on the coupling coefficient is quite complex. It seems that the SOW response can be

divided into several sub-sections depending on the coupling level. At low coupling levels

(κ<0.3) the transmission function consists of several transmission peaks (see inset 1).

Although similar property is exhibited by several, finite, slow-light structures (e.g. CROW) it

should be noted that the number of the transmission peaks decreases as the coupling

coefficient is increased. This is in contrast to CROWs, for example, in which the number of

transmission peaks is determined solely by the number of unit cells.

At larger coupling coefficients, asymmetric and narrow transmission peaks appear (see

inset 2). These peaks grow wider as the coupling is further increased until they merge with the

central peak and form a relatively flat, “box”-shaped spectral transmission (see inset 3). At

larger coupling coefficients, a rather interesting transmission function emerges, consisting of

three narrow peaks at the center and edges of the transmission band where the central peaks

exhibits an EIT-like transmission profile (see Fig. 9).


Fig. 8. Transmission spectral response as a function of the coupling coefficient for a 5 unit-

cells long SOW. The rest of the parameters are as in Fig. 4. Insets – transmission function for various coupling levels.

Fig. 9. Transmission spectral response of a 5 unit-cell SOW with κ = 0.9. The rest of the parameters are as in Fig. 4.

In addition to the interesting details of the finite SOW transmission function, the global

dependence of the transmission function on the coupling differs substantially from that of a

CROW. Referring to Fig. 8, note that for very low and very high coupling levels the

transmission function of the SOW approaches almost unity. In addition, the over-all

transmission band becomes narrow as the coupling is increased, reaching a minimum at

κ~0.35 and then broadening again as the coupling is further increased. The reason for this

behavior is that for low coupling levels the SOW approaches a simple curved waveguide

(which does not exhibit structure dependent spectral properties) while for κ1 the strong

coupling provide a shorter route for the light, with basically no back reflections, thus yielding


an almost unity and wavelength independent spectral transmission. This is in contrast to

CROW where the transmission band monotonically increases when the coupling in increased.

SOW consisting of more unit cells exhibit similar properties although the fine details vary.

Figure 10(a) depicts the transmission function of a lossless SOW structure consisting of seven

unit cells for coupling coefficients ranging from 0.01 to 0.99. Figure 10(b) illustrates a similar

plot for a similar structure with waveguide propagation losses of 1dB/cm.

Fig. 10. Transmission spectral response as a function of the coupling coefficient for a 7 unit-

cells long SOW with no loss (a) and 1dB/cm propagation loss (b).

Referring to Fig. 10(a), the longer SOW structure exhibits spectral properties which are

very similar to those of the five unit-cells SOW. There are some changes in the fine details

though, e.g. in the number of transmission peaks at weak coupling, the coupling levels a

which the “box”-shaped transmission function emerges, etc. but the global properties are

similar. In addition, referring to Fig. 10(b), it should be noted that the introduction of practical

loss levels does not substantially modifies the spectral properties of the SOW. In particular,

the impact of loss can be observed only close to the edges of the transmission band (see also

Section 6 below).

6. Pulse propagation

The results of the band structure analysis provide the necessary information for understanding

the propagation of a pulse through the structure. Understanding the dynamics of pulse

propagation is important for non linear optics applications (the intensity of the light inside the

resonator can be substantially larger than that of the injected signal), and for optical

communication systems, where the information is being transmitted in pulses.

In order to model the propagation of the injected pulse, it is separated into its harmonic

components. The output spectrum is obtained by multiplying the injected spectrum by the

SOW frequency response which is directly extracted from the dispersion diagram (Fig. 4).

The time domain dynamics is then calculated by inverse Fourier transforming the spectrum at

each point in the structure. Figure 11 illustrate the propagation of a 30ps wide pulse through

an SOW with coupling coefficient of κ = 0.15. The central frequency is set to be 191.72THz

(1564.8nm) which is approximately at the center of the slow-light band (the shallow and

linear slope regime of the red curve in Fig. 4) which is formed around KΛ = 0.17π. The linked

avi file (Media 1) shows the temporal evolution of the field in the structure. As seen in the

figure the pulse maintains its shape as it propagates through the structure although some

broadening is also observed. The distortion is formed by the GVD of the structure at the

wavelength of the pulse.

The group velocity of the pulse propagating along the SOW can be extracted directly from

the dispersion relations or from the temporal evolution of the pulse. For the case shown in


http://www.opticsinfobase.org/oe/viewmedia.cfm?uri=oe-19-12-11517-1

Fig. 11 (κ = 0.15) the group velocity is found to be 0.35 Unit-cells/ps. Taking the length of the

unit-cell to be the over-all length of the waveguide comprising it (i.e. its length for κ = 0 – a

simple curved waveguide), the corresponding group velocity is found to be 0.14 × c.

Reducing the coupling coefficient towards 0.1 would yield slower group velocities.

Fig. 11. Propagation of a 30 ps pulse along a SOW comprising 300 unit-cells. The pulse profile is shown at 100ps snapshots. The group velocity is approximately 0.35 unit-cells per ps

corresponding to vg0.14 × c. media 1 – animation of the pulse propagation (Media 1).

Fig. 12. Phase (a) and amplitude (b) responses of a 7 unit-cell long SOW. The average time

delay is ~40ps yielding group velocity of approximately 0.2 unit-cells per ps.

When the pulse is propagating through a finite structure, the spectral features of the finite

device must be also considered. Figure 12 depicts the amplitude and the phase response of a

SOW consisting of 7 unit-cells, identical in parameters to the infinite structure used in Fig. 11,

for a lossless case (blue) and for 1dB/cm propagation loss (red). The figure is zoomed on the

slow-light band spectral range as discussed above. The important points to note are that even

for non-negligible losses, the structure is quite transparent (only 1.5dB transmission loss)

and that the phase responses are almost identical.

The phase response allows for the extraction of the group delay which is in average over

the pulse bandwidth approximately 40ps. This corresponds to a group velocity of ~0.2 unit-

cells per ps which differs from that of an infinite SOW but within the same order of

magnitude. The differences in the group velocities between the finite and the infinite


http://www.opticsinfobase.org/oe/viewmedia.cfm?uri=oe-19-12-11517-1

structures is not surprising, in particular because the propagation takes place close to the edge

of the pass band and the finite structure is rather short.

7. Conclusions

We proposed and analyzed a new slow-light structure consisting of coupled loops that form a

serpentine-like optical structure. Unlike the most commonly studied slow-light structure, the

band structure of the SOW exhibits a zero group velocity point located within the Brillouin

zone. The frequency and, in particular, the Bloch wavenumber at which this point is formed is

determined by the coupling coefficient between adjacent loops and is insensitive to the

waveguide cross-section and length of the optical loops.

Transfer matrix formalism (TTF) was developed and was employed for the calculation of

the dispersion relations of the structure. Very good agreement was found between the TTF

and a FDTD analysis of the structure. An almost distortion-less propagation of a 30ps pulse in

the structure is shown at the relatively linear part of the dispersion relation with group

velocity in the order of 0.1 the speed of light in vacuum. Although some distortion is visible,

most of the power is concentrated in the main (leading) lobe and can be used for data

transmission.

An important point which must yet be addressed is the interfacing of the SOW with

external I/O ports. In order to avoid back reflections at the interfaces between the I/O

waveguides and the periodic structure, impedance matching sections must be placed at these

interfaces (like in any finite periodic structure). The design of such sections deserves a

comprehensive study and is, therefore, beyond the scope of this paper.

From the performances aspect, when compared with the CROW and the SCISSOR, all

three devices are linear periodic photonic structures exploiting resonating optical paths to

achieve low group velocity, there are both subjected to similar limitations (see e.g. [27,28]).

In all structures the reduction of the group velocity is attained by forcing the light to circulate

in closed loop, thus effectively extending the optical path. In practice the limitations on the

achievable delays, bandwidth GVD, etc. stem primarily from the technology used for the

realization of the structure (i.e. index contrast, intrinsic losses, etc.) and less from the

specifically employed geometry of the device. Thus one can expect that CROWs, SCISSORs

and SOWs employing similar waveguide structure exhibit similar performances from these

aspects. The uniqueness and advantage of the SOW is that it can exhibit a low GVD zero

group velocity region which can be tuned by the coupling coefficient, thus allowing for much

flexibility and engineering of the dispersion relation and light propagation properties.

Acknowledgments

The authors thank the Israel Ministry of Science and Technology for partially supporting this

research.


the serpentine optical waveguide: engineering the dispersion relations and the stopped light points

Documents