Efficient light coupling into a photonic crystal waveguide with flatband slow mode

0.25 MB·English

by  A. Säynätjoki

E(cid:30)
ient light
oupling into a photoni

rystal
waveguide with (cid:29)atband slow mode
08 a,∗ b a,c b
0 A. Säynätjoki K. Vyn
kc M. Mulot a D. Cassagne
2
J. Ahopelto H. Lipsanen
n
a
a
J
Helsinki University of Te
hnology (TKK), Department of Mi
ro and
1
2 Nanos
ien
es, Mi
ronova, Tietotie 3, FIN-02015 Espoo, FINLAND
b

Groupe d'Etude des Semi
ondu
teurs, UMR 5650 CNRS - Université Montpellier
]
s II, CC074 Pla
e Eugène Bataillon, 34095 Montpellier Cedex 05, FRANCE
c c
i VTT Information Te
hnology, Mi
ronova, P.O. Box 1208, FIN-02044 VTT,
t
p
FINLAND
o
.
s
c
i
s
y
h
p Abstra
t
[

We design an e(cid:30)
ient
oupler to transmit light from a strip waveguide into the
1
v (cid:29)atband slow mode of a photoni

rystal waveguide with ring-shaped holes. The
9

oupler is a se
tion of a photoni

rystal waveguide with a higher group velo
ity,
5
2 obtained by di(cid:27)erent ring dimensions. We demonstrate
oupling e(cid:30)
ien
y in ex
ess
3 of95%overthe8nmwavelengthrvan≈gecw/h3e7rethephotoni

rystalwaveguideexhibits
. g
1 a quasi
onstant group velo
ity . An analysis based on the small Fabry-
0
Pérot resonan
es in the simulated transmission spe
tra is introdu
ed and used for
8
studying the e(cid:27)e
t of the
oupler length and for evaluating the
oupling e(cid:30)
ien
y
0
: in di(cid:27)erent parts of the
oupler. The mode
onversion e(cid:30)
ien
y within the
oupler
v
i is more than 99.7% over the wavelength range of interest. The parasiti
re(cid:29)e
tan
e
X
in the
oupler, whi
h depends on the propagation
onstant mismat
h between the
r
a slow mode and the
oupler mode, is lower than 0.6% within this wavelength range.
Key words: Guided waves, Integrated opti
s materials, Opti
al systems design,
Waveguides, Photoni
Integrated Cir
uits, Dispersion, Photoni

rystals.
PACS: 42.15.Eq, 42.70.Qs, 42.82.-m, 42.82.Cr, 42.82.Et, 42.82.Gw
∗
Email address: antti.saynatjokitkk.fi (A. Säynätjoki).
PreprintsubmittedtoPhotoni
sandNanostru
tures-FundamentalsandAppli
ations2February2008
1 Introdu
tion
Photoni

rystal waveguides (PhCWs) exhibit slow opti
al modes with group
velo
ities at least one order of magnitude smaller than in
onventional waveg-
uides [1,2,3,4,5℄. Enhan
ed nonlinearity e(cid:27)e
ts have been demonstrated for
slow modes, whi
h may allow s
aling down the size of a
tive integrated opti
s
devi
es [6,7℄. Slow modes
an be used in tele
ommuni
ations systems pro-
vided that they have su(cid:30)
iently low group velo
ity dispersion (GVD), and
PhCWs with su
h (cid:29)atband slow modes have re
ently been designed [8,9,10℄.
We have designed a waveguide based on a photoni

rystal with ring-shaped
v ≈ c/37
g
holes(RPhCW)withlowGVDandgroupvelo
ity overawavelength
range of several nanometers [11℄. The feasibility of the RPhCW was shown in
[12℄, where we observed slow light propagation in an RPhCW fabri
ated on a
sili
on-on-insulator substrate.
E(cid:30)
ient
oupling between waveguides with di(cid:27)erent group velo
ities is not
trivial [13,14,15,16℄. Transmission from strip waveguides (SWs) to slow modes
in PhCWs has been improved by optimizing the termination of the PhCW
[13℄, tapering the PhCW [15℄ and by using high group velo
ity PhCWs at
both ends of the slow light PhCW [16℄. In these examples, e(cid:30)
ient
oupling
into slow modes in PhCWs was a
hieved with
ouplers signi(cid:28)
antly shorter
than required for adiabati
mode
onversion. De Sterke et al. [17℄ and Velha
et al. [18℄ showed that perfe
t
oupling
ould be a
hieved by utilizing the
interferen
e between the forward and ba
kward propagating modes in the

oupler. Hugonin et al. [19℄ noti
ed the appearan
e of a transient zone a few
latti
e periods long at the interfa
e between PhCW se
tions of di(cid:27)erent group
velo
ities, where light is smoothly slowed down as it penetrates the slow-
light waveguide, and they demonstrated e(cid:30)
ient mode
onversion in su
h a
stru
ture. This approa
h therefore makes it possible to optimize the di(cid:27)erent
interfa
es of the slow-light devi
e independently.
In this paperwe designan e(cid:30)
ient
oupler intotheslow,dispersionengineered
v
g
mode with a nearly
onstant in the RPhCW presented in [11℄. We present
a simple way of studying
oupling e(cid:30)
ien
ies in di(cid:27)erent parts of the
oupler,
based on the small F-P resonan
es in the transmission spe
trum.
2
R
out
R
in
w d a
Fig. 1. S
hemati
of a W1 RPhC w ith an input strip waveguide.
0.260
R =0.385a
luancy ( = / ) 000...222555468 outR=0.315aouRt=0.360a out Rout=0.375a
e
Normalized frequ 000...222455802 Decrgev=ca/3.8sing vg vg=c/37
0.246
0.25 0.30 0.35 0.40 0.45 0.50
Propagation constant b [2p/a]
FRig. 2. Dispersion relaRtion o−f Rthe e=ve0n.1m5aode in RPhCWs with di(cid:27)erent outer radius
out out in
. The ring width for all waveguides. In the normvali=zedc/f3r.e8-
g
quen
yRrang=e0w.3it1h5ian thevfra=mce/,3t7heaveraRge gr=ou0p.3v8e5loa
ity de
reasesfrom
out g out
when to when .
2 Coupler design
2.1 Group velo
ity in RPhC waveguides
Figure 1 shows a s
hemati
of an RPhCW de(cid:28)ned by one missing row of holes
in an otherwise perfe
t RPhC. The RPhC is a triangular latti
e of rings with
a
a latti
e
onstant . The ring is de(cid:28)ned by two parameters, ring outer and
R R
out in
inner radii and , respe
tively. The RPhCW is
oupled to SWs with
w d
width at the interfa
e. The parameter de(cid:28)nes the position at whi
h the
RPhCW is terminated.
Band stru
tures of the RPhCWs are
al
ulated using the plane wave expan-
sion (PWE) method des
ribed in [20℄. The PWE simulations yield dispersion
u(β) u u = a/λ β
relations , where is the normalized frequen
y and is the
v
g
propagation
onstant. The group velo
ity is de(cid:28)ned as
ω 2πc u
v = d = d .
g
β a β (1)
d d
v
g
Therefore, is dire
tly proportional to the slope of the dispersion
urve in
3
Rout<0.36a Rout=0.385a
Inputdete
tor Outputdete
tor
Sour
e
Fig. 3. S
hemati
image of the
oupler design. A slow-light RPhCW is
oupled
R v
out g
to SWs through RPhCWs with smaller , i.e. with higher . The inset shows
the beginning of the RPhCW. The sour
e and dete
tor
on(cid:28)guration in FDTD
simulations is also shown in the s
hemati
.
Fig. 2, where we plot the dispersion relation of the even TE-like mode in
R
out
RPhCWs with di(cid:27)erent values of . The ring width is kept
onstant at
R −R = 0.15a
out in
. The simulations were
arried out in 2D for the TE polar-
n = 3.178
eff
ization. An e(cid:27)e
tive index of the diele
tri
material equal to was
used,
orresponding to the e(cid:27)e
tive refra
tive index of a 400 nm thi
k sili
on
slab on sili
a at the wavelength of 1550 nm.
R =
out
We have demonstrated in our previous work that an RPhCW with
0.385a R = 0.235a
in
and exhibits a (cid:29)atband mode with a quasi
onstant and
v ≈ c/37
g
relatively low group velo
ity over a wavelength range of 8 nm [11℄.
v
g
The frequen
y range
orresponding to the nearly
onstant is highlighted in
v
g
Fig. 2. The values in this paper are given for this frequen
y range, unless
the frequen
y or propagation
onstant range is expli
itly spe
i(cid:28)ed.
For all waveguides in Fig. 2, the even mode is in the index-guided regime
v ≈ c/3.8 β < 0.32(2π)
with g when a . When approa
hing the Brillouin zone
β = 0.5(2π) v
edge at a , the mode be
omes di(cid:27)ra
tion-guided: g de
reases and
β = 0.5(2π) R
eventually vanishes when a . If out is de
reased, the air-(cid:28)ll fa
tor
of the RPhCW de
reases, therefore the dispersion
urve of the guided mode
is shifted to smaller frequen
ies. Consequently, the average group velo
ity
v = c/3.8
g
within the frequen
y range of interest
an be
hanged from with
R = 0.315a v = c/37 R = 0.385a
out g out
to with .
2.2 Optimization of the
oupler
The stru
ture we
onsider is shown s
hemati
ally in Fig. 3. The slow-light
waveguide is sandwi
hed between two
ouplers, whi
h
onsist of RPhCWs
R < 0.36a
out
with and are
oupled with strip waveguides. As the (cid:28)rst step in
the
oupler design, we optimize transmission from SWs to the
ouplers.
T
Thetransmission ofthe
ouplersissimulatedusingthe(cid:28)nite-di(cid:27)eren
etime-
domain (FDTD) method [21℄. The 2D FDTD simulations were
arried out for
4
0.30 1.0
0.25 0.9
0.20 0.8 Tn
o
vc / g0.15 0 .7 nsmissi
a
0.10 0.6 Tr
T
0.05 v / c 0.5
g
0.30 0.32 0.34 0.36 0.38 0.40
R / a
out
10a
Fig. 4. Group velo
ity in the RPhCW (solid line) and the transmission of a long
RRPhCW sandwi
hed between twous=tri0p.2w50aveguides (dashed line) as a fun
tion of
out
at the normalized frequen
y .
n = 3.178
eff
the TE polarization using an e(cid:27)e
tive refra
tive index , as in the

ase of the PWE simulations. The sour
e and dete
tors are pla
ed into the
SWs, as shown in Fig. 3. First we optimize the termination of the RPhCW
w = 1.6a
and (cid:28)nd that optimal
oupling is obtained with parameters and
d = 0.625a
.
u = 0.250 R
out
In Figure 4, group velo
ity at is plotted as a fun
tion of , to-
10a
gether withthe transmissionthrough a longRPhCW sandwi
hed between
R = 0.385a
out
two SWs. For the RPhCW with , transmission is only 52% at
u = 0.250 T
. in
reases with in
reasing group velo
ity, being more than 98%
R ≤ 0.315a v > c/4 R = 0.315a
out g out
when and . Therefore we
hoose for the

oupler.
n < 40
g
With in our RPhCW, e(cid:30)
ient
oupling is expe
ted with an abrupt
transition between the RPhCW se
tions [19℄. Therefore, we study a simple

oupler, where the interfa
e between the
oupler and the slow-light waveguide
R
out
is realized with an abrupt
hange of . In our initial stru
ture, we use
5a

oupler length of . We will dis
uss the e(cid:27)e
t of the length of the
ouplers
later in this paper.
3 Results and dis
ussion
3.1 Transmission properties
a
Figure 5 shows the transmission spe
tra of a 50 long slow-light RPhCW with
T T
opt ref
and without
ouplers ( and , respe
tively). Transmission through the
R = 0.315a T
out cpl
RPhCW with sandwi
hed between SWs ( ) is also plotted
intoFig.5inordertoshowthetransmissionbetweentheSWsandthe
ouplers.
n = c/v R =
g g out
The PWE
al
ulated group index in the RPhCW with
5
1.00 100
0.95 TTTorepft T > 95% 80
TTransmission 00..8950 ngcpl Quasi constant ng 6400 nGroup index g
0.80 20
1550 1555 1560 1565 1570 1575
Wavelength (nm)
50a
Fig. 5R. FD=TD0.3si8m5aulatedatr=an3s9m2ission spe
tra through a long RPhC waveguTide
out opt
withT andT nm, with and without
oupl1e0ras at both ends (
ref cpl
aRnd = 0,.r3e1s5pae
tively). isthetransmission spe
trum of a long RPhCWwith
out
n(i=.e.,c/avwaveguide similar to the
ouplers) sandwi
hed between SWs.
g g
Group index is dedu
ed from the PWE simulations [11℄.
0.385a
, presented in [11℄, is plotted into the same graph. All the
urves are
λ a = 392
plotted as a fun
tion of wavelength with a latti
e
onstant nm,
whi
h yields a
ut-o(cid:27) wavelength of 1575 nm.
T v
opt g
is higher than 95% over the wavelength range of nearly
onstant be-
tween 1560 nm and 1568 nm,
orresponding to a bandwidth of about 1 THz.
Our devi
e provides both high transmission e(cid:30)
ien
y and low group velo
-
ity dispersion for a broadband opti
al signal, whi
h is important if slow-light
waveguides are
onsidered to be used in data transmission systems.
1567 T T
opt cpl
Uptothewavelength nm, isvery
loseto .Onlysmallos
illations
are seen in the spe
trum, with their period varying as a fun
tion of the group
index of the RPhCW. These os
illations are F-P resonan
es due to re(cid:29)e
tions
between the two
oupler-RPhCW interfa
es. In the next se
tion we will show
that these F-P os
illations are a sensitive indi
ator of mode
onversion e(cid:30)-

ien
y and re(cid:29)e
tions in the
oupler and therefore
onstitute an e(cid:30)
ient way
of studying the various
oupling me
hanisms in su
h stru
tures.
3.2 Fabry-Pérot analysis
PhCW modes withinthe photoni
band gap are ne
essarily lossless in 2D sim-
ulations, as out-of-plane losses do not exist and in-plane losses are prohibited
by the PhC. Therefore, non-ideal transmission in our simulation is indu
ed
by in-plane s
attering and/or by parasiti
ba
k re(cid:29)e
tion at ea
h waveguide
interfa
e.
6
1.00
0.95
Tcpl
T / R0.90
Quasi constant n
0.85 g
m = 10
m = 5
0.80 m = 3
m = 2
1550 1555 1560 1565 1570 1575
Wavelength (nm)
T /T
opt,R cpl
Fig. 6. A measure for mode
onversion e(cid:30)
ien
y, , with di(cid:27)erent
oupler
T
opt,R
lengths. is the linear interpolation between the points
orresponding to F-P
m
resonan
es in the transmission spe
tra with a
oupler with a length of latti
e
T
cpl
periods, and is the transmission spe
tra of the
oupler sandwi
hed between
strip waveguides.
3.2.1 Mode
onversion e(cid:30)
i
ien
y
The transmission of a lossless F-P
avity is equal to unity at resonan
e [23℄.
T T
opt cpl
should therefore be equal to at ea
h resonan
e, provided that the

oupler
onverts the slow mode properly into the
oupler mode; this is the

ase in the transmission spe
trum shown in Fig. 5 for wavelengths up to 1572
T < T
opt cpl
nm. For resonan
es at longer wavelengths, , suggesting that mode

onversion is not perfe
t.
The e(cid:30)
ient mode
onversion experien
ed between PhCW se
tions of sig-
ni(cid:28)
antly di(cid:27)erent group velo
ities has been explained by the
reation of a
transient zone within a few latti
e periods of the slow-light waveguide result-
ing from interferen
es between propagating and evanes
ent Blo
h modes. To
get more insight on the mode
onversion along the
oupler, we repeat the
T m m
opt
FDTD simulation of with di(cid:27)erent
oupler lengths , where is the
a

oupler length in latti
e periods . The results are shown in Figure 6, where
T /T T
opt,R cpl opt,R
we plot , where is the linear interpolation between the points
T
opt

orresponding to F-P resonan
es in . Perfe
t mode
onversion should yield
T /T = 1
opt,R cpl
.
2a
The
oupler with a length of just is enough to improve transmission
om-
T
ref
pared to the waveguide with no
ouplers (
ompare to in Fig. 5). A
oupler
m = 3
with provides nearly perfe
t
oupling in the quasi
onstant group in-
m = 5 λ = 1566
dex range. With , transmission is enhan
ed between nm and
λ = 1574
nm.
Further in
rease in the
oupler length maintains the ex
ellent transmission,
whi
h shows that e(cid:30)
ient
oupling o

urs when the evanes
ent modes ex
ited
at the two
oupler interfa
es do not overlap and interfere destru
tively [19℄.
5a
The
oupler length needed for this is . The weak transmission variations
that are observed for longer
ouplers
orrespond to F-P os
illations in the
7
0.02
Quasi constant n
g
R 0.01

0.00
0.06 0.08 0.10 0.12
D b
R
F∆iβg. 7. Para5saiti
ba
k-re(cid:29)e
tan
e as a fun
tion of propagation
onstant mismat
h
witha long
oupler.Thewavelengthrangeofthe(cid:28)gureisfromabout1555nm
n
g
to about 1570 nm, the quasi
onstant region is drawn between 1560 nm and
1568 nm and the s
atter symbols have an interval of 1 nm.

oupler [17℄. However, these os
illations are not dominant in our
ase. For a
5a

oupler length of , we see that the mode
onversion losses are lower than
n T
g opt,R
0.3% in the wavelength range of quasi
onstant . drops near
ut-o(cid:27)
for all
oupler lengths, indi
ating that a more
ompli
ated
oupler might be
n
g
needed for higher [15,19℄.
3.2.2 Ba
k-re(cid:29)e
tan
e
The signature of the ba
k re(cid:29)e
tions are the F-P os
illations in the transmis-
T
opt
sion spe
trum, whi
h
an be seen in . In this
ase, the RPhCW
an be
R
regarded as a symmetri
, lossless F-P
avity. The re(cid:29)e
tan
e at the ends
of su
h a
avity
an be dedu
ed from the F-P os
illations in the transmission
spe
trum as [22℄
T /T −1
q R A
R = ,
T /T +1 (2)
q R A
T T
R A
where and are the transmission at resonan
e and antiresonan
e, respe
-
tively.
The amplitudeof the F-P os
illationsinFig. 5 in
reases at higher wavelengths
n
g
already within the nearly
onstant regime. The os
illation amplitude is
indeed a fun
tion of the propagation
onstant mismat
h between the slow
∆β
mode and the
oupler mode, . Figure 7 shows the ba
k-re(cid:29)e
tan
e as a
∆β n R
g
fun
tion of . In the wavelength range with
onstant , is lower than
0.6%. Although the ba
k-re(cid:29)e
tan
e is very low, phase mat
h between two
stru
tures is
ru
ial in the design of
ouplers for slow-light devi
es.
8
4 Con
lusion
We present a highly e(cid:30)
ient
oupler into a slow mode with low group ve-
lo
ity dispersion in a photoni

rystal waveguide with ring-shaped holes. The

oupler is a se
tion of a waveguide with a di(cid:27)erent ring parameter and with
an optimized interfa
e with the strip waveguide. The slow mode of interest
37 ± 3
exhibits a quasi
onstant group index of on a bandwidth of 8 nm at
tele
ommuni
ations wavelengths.
We introdu
e simple and e(cid:30)
ient methods based on the study of transmission
spe
tra to evaluate the
oupling e(cid:30)
ien
ies in di(cid:27)erent parts of the
oupler.
The
oupling e(cid:30)
ien
y at the interfa
e between the strip waveguide and the

oupler is higher than 98%. We show that the mode
onversion e(cid:30)
ien
y in a
5a
long
oupler is more than 99.7% over the wavelength range of interest. The
parasiti
ba
k-re(cid:29)e
tan
e in the
oupler, whi
h depends on the propagation

onstant mismat
h between the slow mode and the
oupler mode, is smaller
than 0.6% in the same wavelength range.
On the te
hnologi
al point of view, the
oupler is very
ompa
t and it is
manufa
turable with a pro
ess des
ribed in our earlier work. The total trans-
mission of the slow-light waveguide
oupled to strip waveguides is higher than
95% over the 1 THz bandwidth of the dispersion tailored slow mode. While
slow-lightphotoni

rystal waveguides with lowgroup velo
itydispersionhave
re
ently been presented, e(cid:30)
ient
oupling with strip waveguides is the next,
ne
essary step towards integrating them into photoni

ir
uits.
A
knowledgments
This work was (cid:28)nan
ed by the A
ademy of Finland under the PHC-OPTICS
proje
t,graduates
hoolGETA andtheEuropeanUnionunder proje
t PHAT.
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10