Degenerate band edges in optical fiber with multiple grating: efficient coupling to slow light.
ABSTRACT Degenerate band edges (DBEs) of a photonic bandgap have the form (ω-ω(D)) ∝k(2m) for integers m>1, with ω(D) 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 proposed to 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.
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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: email@example.com
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
. 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 ; 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 . 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
). To overcome this, various configurations have been
used, which involve an additional layer  or a modifica-
tion of the interface to match the impedance .
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 . 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
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  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
j¼1Δϵjcos½ðκjþ δjÞz?Rðx;yÞ, where κj¼ K
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
August 15, 2011 / Vol. 36, No. 16 / OPTICS LETTERS3257
0146-9592/11/163257-03$15.00/0 © 2011 Optical Society of America
equations to a set of linear algebraic equations that, in
matrix form , 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 , the k are either real or occur in complex con-
jugate pairs, representing evanescently growing and
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 . At the DBEs Θ is
defective, and by using perturbation theory of defective
matrixes , it can be shown that the splitting of kjfrom
j ¼ 0;…;ð2m − 1Þ;
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;
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 .
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
(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
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
3258OPTICS LETTERS / Vol. 36, No. 16 / August 15, 2011
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
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
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
 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
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
. 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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Group velocity (vg/c)
Group velocity (vg/c)
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 LETTERS 3259