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Degenerate band edges in optical

fiber with multiple

grating: efficient coupling to slow light

Nadav Gutman,1,* Lindsay C. Botten,2Andrey A. Sukhorukov,3and C. Martijn de Sterke1

1Institute of Photonics and Optical Science (IPOS) and Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS),

School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia

2CUDOS, School of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007, Australia

3CUDOS, School of Physics, Australian National University, Canberra, Australian Capital Territory 0200, Australia

*Corresponding author: nadav@physics.usyd.edu.au

Received June 10, 2011; accepted July 8, 2011;

posted July 18, 2011 (Doc. ID 149063); published August 15, 2011

Degenerate band edges (DBEs) of a photonic bandgap have the form ðω − ωDÞ ∝ k2mfor integers m > 1, with ωD

the frequency at the band edge. We show theoretically that DBEs lead to efficient coupling into slow-light modes

without a transition region, and that the field strength in the slow mode can far exceed that in the incoming medium.

A method is proposedto create a DBE of arbitrary order m by coupling m optical modes with multiple superimposed

gratings. The enhanced coupling near a DBE occurs because of the presence of one or more evanescent modes,

which are absent at conventional quadratic band edges. We furthermore show that the coupling can be increased

or suppressed by varying the number of excited evanescent waves.

OCIS codes:050.2770, 060.2310.

© 2011 Optical Society of America

Slow-light propagation in photonic band gap (PBG) ma-

terials, such as photoniccrystals (PCs) and fiber gratings,

leads to the strengthening of light–matter interactions,

including an enhancement of nonlinear optical effects

[1]. In lossless PBG materials the speed of light is deter-

mined by the group velocity vg¼ ∂ω=∂k, with ω the op-

tical frequency and k the wavenumber. The magnitude

of the group velocity is reduced, in particular, close to

photonic band edges. For a regular photonic band edge

the dispersion is quadratic, i.e., ðω − ωDÞ ∝ k2, where ωD

is the band edge frequency [2]; then vg∝ ðω − ωDÞ1=2.

Despite the ubiquitous nature of band-edge slow-light

states in various PBG structures, it is difficult to couple

light into these slow propagating modes [3]. The reason

follows from the requirement to preserve the parallel

field components at the boundary between media with

different group velocities. The ratio between the forward

energy fluxes S of the two media defines the coupling

efficiency η ¼ S2=S1. For PBG materials with a quadratic

band edge, which only have a single propagating mode,

the boundary conditions impose a linear scaling between

the coupling and the group velocity η ∝ vg2. Thus cou-

pling into slow light in these materials is inefficient,

and thus the field strength in the slow-light medium does

not exceed that in the incoming medium (however, see

[4]). To overcome this, various configurations have been

used, which involve an additional layer [5] or a modifica-

tion of the interface to match the impedance [6].

In this Letter,we demonstrate that much more efficient

coupling to slow-light modes can be achieved near

nonquadratic degenerate band edges (DBEs) of the form

ðω − ωDÞ ∝ k2m, and we examine specifically quartic (k4),

sextic (k6), and octic (k8) cases. First we present a con-

ceptual method to create DBEs with arbitrary m. Then,

we show that weakly evanescent modes, decaying eva-

nescently away from the interface, are excited for m > 1,

allowing efficient coupling to slow light, so that the field

strength in the slow mode can far exceed that in the in-

coming medium. Moreover, the coupling strength can be

adjusted by varying the input mode amplitudes to vary

the number of excited evanescent modes.

It was previously shown that a quartic band edge

(m ¼ 2) can be achieved in an optical fiber with two

superimposed gratings that couple two different optical

modes [7]. Here we generalize this and show that DBEs

of arbitrary order 2m can be generated in a multimode

fiber with m modes with different wavenumbers K

We introduce m superimposed Bragg gratings with per-

iods such that they resonantly couple one forward mode

to all m backward modes and vice versa, as shown sche-

matically in Fig. 1. The dielectric modulation of the

superimposed gratings can be written as Δϵðx;y;zÞ ¼

Σm

the grating wavenumbers, Δϵjare the grating amplitudes,

Rðx;yÞ defines the transverse cross section, and δjare

small detunings. While considering the Bragg grating

configuration, care must be taken in order to prevent

the grating from scattering light into radiative modes.

For fiber gratings, typically Δϵj< 10−3, and so time-

independent coupled mode theory [8] can be used. It

describes this system by a set of 2m coupled ordinary

differential equations, one for each forward and back-

ward mode. We determine the dispersion by taking the

modes to vary as eikz, which reduces the differential

∼

j.

j¼1Δϵjcos½ðκjþ δjÞz?Rðx;yÞ, where κj¼ K

∼

1þ K

∼

j are

Fig. 1.

the multimode fiber has m incident and m reflected fiber

modes. For z > 0 multiple gratings with different periodicities

couple a single forward mode of the uniform fiber to all back-

ward modes and vice versa.

Schematic of the approach to create DBEs. For z < 0

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0146-9592/11/163257-03$15.00/0© 2011 Optical Society of America

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equations to a set of linear algebraic equations that, in

matrix form [7], read ½ΘðωDÞ þ ΔωC?Uj¼ kjUj, where

Θ and C are 2m × 2m matrices, Δω ≡ ω − ωDis the fre-

quency detuning from the band edge ωD. For a given fre-

quency this is an eigenvalue equation in k, with the Ujthe

associated eigenvectors. For lossless dielectric struc-

tures [6], the k are either real or occur in complex con-

jugate pairs, representing evanescently growing and

decaying modes.

We now consider the complex band structure close to

a DBE. To obtain a DBE of order 2m, we choose through

numerical optimization the values of the detunings δjthat

provide a dispersion relation satisfying the conditions for

derivatives ∂lω=∂kl¼ 0 for l ¼ 1;…;ð2m − 1Þ. For defi-

niteness we chose, without loss of generality, to consider

an upper PBG edge, so Δω > 0, since it is used to control

the dispersion of high energy pulses [9]. At the DBEs Θ is

defective, and by using perturbation theory of defective

matrixes [10], it can be shown that the splitting of kjfrom

zero follows

k ¼

ffiffiffiffiffiffiffiffiffiffi

ξΔω

2mp

eið2π=2mÞj

j ¼ 0;…;ð2m − 1Þ;

ð1Þ

where ξ is a system-dependent positive constant. Thus

near a DBE, Δω ∝ k2m, as illustrated in Fig. 2.

Close to ωDeach complex band structure has a single

forward propagating mode, indicated by the dotted black

curvein Fig. 2, which has a realand positive wavenumber

k, together with a single backward propagating mode

(dotted gray). The additional 2m − 2 eigenvalues k of the

2m × 2m matrix are complex, corresponding to m − 1

pairs offorward and backward evanescently decaying so-

lutions. These evanescent modes only occur near DBEs

and are the key difference between DBEs and conven-

tional, quadratic band edges [Fig. 2(a)]. The presence of

evanescent modes near a DBE is illustrated in Figs. 2(b)

and 2(c). The quartic DBE in Fig. 2(b) show one pair of

evanescent modes (solid gray and black curves),

whereas the sextic DBE in Fig. 2(c) has two pairs of such

modes. In all cases, at the band-edge frequency ωD, the

wavenumbers of the fiber modes converge, kj¼ 0, and

have a single eigenvector Uj¼ Uo[11,12].

We now consider the semi-infinite coupling problem

between uniform fiber for z < 0 and the fiber with multi-

ple gratings for z > 0 (see Fig. 1). For z < 0 there exist m

fiber modes, which can be used as distinct input chan-

nels. For z > 0, only grating modes with positive group

velocity or those that decay away from the interface

[i.e., with ImðkÞ > 0] are excited. Dispersion branches

corresponding to such modes are shown with solid and

dotted black curves in Fig. 2.

We now show that by adjusting the phase and ampli-

tude in each input channel, ij, different sets of modes

inside the grating region can be excited or suppressed,

which can be used to tailor the coupling efficiency. Con-

sider first the quartic case, m ¼ 2, when one propagating

and one evanescent mode can be excited at the interface

with the uniform fiber. We parameterize a complete set of

input conditions by two variables α and β:

i1¼ cosðαÞeiβ=V1;i2¼ sinðαÞ=V2;

ð2Þ

where the Vj are the group velocities of the homo-

geneous fiber modes. Figure 3(a) shows how the

overall reflection depends on the input parameters α

and β for frequency ω1¼ ωDþ 0:01ðω=2K

vg¼ 0:045c. Other physical parameters are as in [7].

We find that by special choices of α and β, it is possible

either to excite only the propagating or evanescent

modes or to minimize the reflectivity. The conditions

for each case are shown in Fig. 3(a) as dashed black,

solid black, and solid white curves, respectively, for fre-

quencies between ω1and ωD. We show in Fig. 3(b) the

coupling efficiency for each case versus the group velo-

city. We see that optimum coupling is achieved when

both the propagating and evanescent modes are excited.

The origin of the enhanced coupling is related to an in-

crease of the mode amplitudes inside the grating region.

We perform analytical analysis for a quartic DBE and

determine that the amplitudes of propagating and eva-

nescent modes diverge close to the DBE as aj∝ k−1. To

explain the divergence of aj, we turn to the eigenmodes

Uj. Close to the DBE the evanescent and propagating

grating modes have independent eigenvectors similar

to Uo, so the boundary condition is fulfilled only if the

propagating and evanescent modes have large and oppo-

site amplitudes. Close to the interface the evanescent

modes cancel the propagating mode, so the total field

satisfies the boundary condition. While the evanescent

modes decay away from the interface, the propagating

mode persists, so the total field increases. We find the

energy coupling efficiency η by calculating the energy

∼

jcÞ for which

(a)

Re(k)

Freq (ω)

(b)

Im(k)

(c)

Fig. 2.

(ω > ωD): (a) quadratic, (b) quartic, and (c) sextic. Each has

one forward (dotted black) and one backward (dotted gray)

propagating mode. (b), (c) DBEs also posses forward (black)

and backward (gray) decaying modes.

Complex band structure near an upper band edge

Fig. 3.

different incident fields [Eq. (3)] for vg¼ 0:045c. (b) Coupling

efficiency versus vg. White line in (a) and gray line in (b) repre-

sent maximum coupling. In both (a) and (b), solid and dashed

black lines represent coupling only to the propagating and eva-

nescent modes, respectively. In (b), dashed gray line represents

average coupling over all α and β values.

(a) Reflection from a fiber with quartic band edge for

3258 OPTICS LETTERS / Vol. 36, No. 16 / August 15, 2011

Page 3

flux inside the grating S2, given a fixed incident flux S1.

Only the propagating mode carries energy; thus the total

flux is η ∝ S2=S1∝ japropj2vg. For the quartic case,

vg∝ k3, and we find η ∝ v1=3

represents a key improvement compared to cases in

which only the propagating mode is excited, such as near

a conventional quadratic band edge, for which η ∝ vg.

The gray dashed curve in Fig. 3(b) shows that the aver-

age coupling scales in the same way as the maximum

coupling efficiency. Thus, for a generic incident field and

for sufficiently low vg, the coupling efficiency near a

quartic DBE exceeds the coupling near a quadratic band

edge at the same vg. If the band edge is not completely

degenerate, the 2m bands [Fig. 2] do not cross at the

band edge. This does not alter the high coupling effi-

ciency at frequencies detuned from the band edge, since

the band structure and the associated eigenmodes do

not change.

Next, we consider the sextic case where more than one

evanescent mode exists, so the propagating mode can be

excited on its own or together with either one or two eva-

nescent modes. We find that the amplitudes of the pro-

pagating and evanescent modes close to a sextic DBE

scale as aj∝ k−1if one evanescent mode is excited and

aj∝ k−2if two such modes are excited. If no evanescent

modes are excited, the propagating mode amplitude

aj∝ 1. As illustrated in Fig. 4(a), when only the propagat-

ing mode is excited, the coupling efficiency scales with

group velocity, exactly as for a quadratic band edge

[Fig. 4(a) dashed black]. If one or two evanescent modes

are excited, η ∝ v3=5

g

solid gray and black].

The generalization of results for an arbitrary DBE

is aj∝ k−p, where p is the number of evanescent

modes excited (together with the propagating one) and

g . At low group velocities this

and η ∝ v1=5

g , respectively [Fig. 4(b),

0 ≤ p ≤ ðm − 1Þ. This scaling can be shown by represent-

ing the eigenmodes Ujfor Δω ≠ 0 in Jordan normal form

[11] and solving the boundary conditions, which are writ-

ten as a set of linear equations. The resulting coupling

efficiency using the generalization of the modes ampli-

tudes in an arbitrary DBE is η ∝ vð2m−2p−1Þ=ð2m−1Þ

highest coupling efficiency is achieved when all the eva-

nescent modes are excited, which leads η to scale as

g

. The

η ∝

ffiffiffiffiffi

vg

2m−1p

:

ð3Þ

This is the main result of this article and gives the achiev-

able scaling between the energy coupling efficiency to

slow light for a DBE of arbitrary order, as confirmed by

results of numerical simulations presented in Fig. 4(b) for

quadratic, quartic, sextic, and octic cases.

In conclusion, we have described a procedure for gen-

erating DBEs of arbitrary order and show that the pre-

sence of evanescent modes can profoundly enhance

the coupling efficiency into slow-light modes. We further

demonstrate that the presence of multiple evanescent

modes further improves the coupling compared to the

case of a single evanescent mode considered previously

[4]. The high coupling efficiency can be thought of as

being created by a gradient in field intensity between the

fast- and slow-light modes, while the evanescent waves

compensate for the field gradient.

This work was supported by the Australian Research

Council. The authors thank R. McPhedran, C. Poulton,

and K. Dossou for fruitful discussions.

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PropertiesofPhotonic Crystal

00.01

Group velocity (vg/c)

0.020.030.04 0.05

0

0.2

0.4

0.6

0.8

1

02468

x 10-3

10

-0.1

0

0.1

0.2

0.3

0.4

Coupling (ω)

Group velocity (vg/c)

(b)(a)

Fig. 4.

sextic DBE when exciting only: evanescent modes (dashed

gray), propagating mode (dashed black) multiplied by 5 to fit

the scale, propagating mode and one evanescent mode (gray),

and propagating mode with both evanescent modes (black);

(b) coupling efficiency for quadratic (dashed gray), quartic

(dashed black), sextic (gray), and octic (black) DBE.

(a) Coupling efficiency versus group velocity close to a

August 15, 2011 / Vol. 36, No. 16 / OPTICS LETTERS3259