Table Of ContentExtraction of optical Bloch modes in a photonic-crystal waveguide
S.R. Huisman,1,∗ G. Ctistis,1 S. Stobbe,2 J.L. Herek,1 P. Lodahl,2 W.L. Vos,1 and P.W.H. Pinkse1,†
1MESA+ Institute for Nanotechnology, University of Twente,
PO Box 217, 7500 AE Enschede, The Netherlands
2Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, DK-2100, Copenhagen, Denmark
(Dated: January 5, 2012)
Weperformphase-sensitivenear-fieldscanningopticalmicroscopyonphotonic-crystalwaveguides.
TheobservedintricatefieldpatternsareanalyzedbyspatialFouriertransformations,revealingsev-
eral guided TE- and TM-like modes. Using the reconstruction algorithm proposed by Ha, et al.
(Opt. Lett. 34 (2009)), we decompose the measured two-dimensional field pattern in a superposi-
tionof propagatingBloch modes. Thisopensnew possibilities tostudy specificmodesin near-field
2 measurements. We apply the method to study the transverse behavior of a guided TE-like mode,
1 where the mode extends deeper in the surrounding photonic crystal when the band edge is ap-
0 proached.
2
n
INTRODUCTION (a) (b)
a LL
J 1 2 3 4 N
Light TM
4 Near-field scanning optical microscopy (NSOM) is a
cone
s] pthoewedrifffurlactotioolntolimstitud[1y].obAjecutnsiqwuiethfeaatruerseolouftiNonSObMelowis B C A
c the ability to tap light from structures that are designed
i TE
t to confine light, such as integrated optical waveguides
p y
[2,3]andcavities[4–7]. Usingtheeffectoffrustratedto- x
o a
. tal internal reflection, light that is invisible to other mi-
s
c croscopytechniquescanbedetected. Itismainlyforthis Figure 1: (color online) (a) Top view of a photonic-crystal
i reason that NSOM is so useful in the study of photonic- waveguide that is divided in N unit cells (rectangles) for the
s
y crystalwaveguides. Photonic-crystalwaveguidesaretwo- reconstruction algorithm. (b) Calculated bandstructure for a
h dimensional (2D) photonic-crystal slabs with a line de- GaAsphotonic-crystalwaveguideshowingbothTE-like(red)
p fect wherein light is guided [8]. They possess unique dis- and TM-like (blue) guided modes. The gray area marks the
[ lightcone,LListhelightline,theblue(pink)areamarksTM-
persion relations, supporting slow-light propagation and (TE)-like slab modes, the purple area marks both TE- and
2 enhanced light-matter interactions [9, 10]. TM-like slab modes. The labeled arrows mark the different
v With NSOM, one can measure the dispersion relation modes considered here.
9
inthesewaveguides,maplightpulsesspatiallyandstudy
3
slow-light propagation [3, 11, 12]. It is also possible to
5
5 measurethefieldpatterns,whichcanbecomplicatedbe- Blochmodesarereconstructed. Weapplythisalgorithm
. cause of the multimodal nature of the structures. Spa- to study the behavior of the spatial width of the lowest
0
1 tial Fourier transforms are especially useful to analyze frequency TE-like guided mode in the 2D band gap for
1 e.g., the dispersion [3] or individual mode contributions TE-polarized light as a function of the wavevector.
1 [13]. For ballistic light propagation in photonic-crystal
:
v waveguidesthedetectedfieldpatternisasuperpositionof
i Bloch modes determined by the symmetry of the waveg- SAMPLES AND METHODS
X
uide. Ha et al. [14] recently proposed an algorithm that
r
a uses these symmetry conditions to extract Bloch modes Figure 1(a) illustrates the top view of the waveguide
fromarbitrarymeasuredfieldpatterns[15]. Intheoptical studied here. It consists of a GaAs photonic-crystal slab
domainthealgorithmhassofarbeenusedtoidentifydis- withholesformingatriangularlatticewithpitchsizea=
persion relations in photonic-crystal waveguides [16, 17]. 240±10 nm, normalized hole radius r = 0.309±0.002,
a
To date, however, 2D spatial patterns of Bloch modes waveguidelengthofapproximately1mmandslabthick-
at optical frequencies have not been obtained with this ness h = 160 ± 10 nm. A single row of missing holes
method. formsaW1waveguide. Lightisguidedinthexˆ-direction.
Here, we show the power of the Bloch mode recon- Each numbered rectangle represents a unit cell. Fig.
struction algorithm by extracting individual 2D mode 1(b) shows the calculated bandstructure along xˆ for the
patterns from phase-sensitive NSOM measurements on photonic-crystal waveguide surrounded by air using a
a GaAs photonic-crystal waveguide. We discuss in de- plane wave expansion [18] (assuming a constant refrac-
tail a specifically measured field pattern for which the tive index of n =3.56 and thickness h =0.67). The
GaAs a
2
black diagonal line represents the light line, correspond-
ing to light propagation in air. Modes above the light
line couple to modes outside the waveguide, are there- (a) measured
fore lossy [8] and are not considered here. The blue
and pink areas mark a continuum of modes propagat-
inginthesurroundingphotoniccrystalforTM-andTE-
1 (cid:109)m
A
polarized light, respectively. The blue and red bands (b) reconstructed m
describe modes for, respectively, TM- and TE-polarized
p
light that are guided by the line defect. Here we concen-
l
i
trate on mode A that is TE-polarized, and modes B and t
u
C that are TM-polarized. d
(c)
Acontinuous-wavediodelaser(TopticaDLpro940)is residuals e
used with a linewidth of 100kHz and an emission wave-
lengthbetween907−990nm,correspondingtoareduced
frequency in the range 0.24...0.26a/λ. Light is side-
coupled on a cleaved end-facet of the 1 mm long waveg- A LL B C A
(d)
uide with a high-NA glass objective (NA=0.55). The C
A B A
incidentlightislinearlypolarizedwithanangleofabout
45o with respect to the normal of the waveguide to ex-
cite both TE- and TM-like modes. The field pattern is C C
collected approximately 100 µm away from the coupling
facet using an aluminum coated fiber tip with an aper-
ture of 160±10 nm. We perform phase-sensitive NSOM
using heterodyne detection. Detailed descriptions of a Figure 2: (color online) (a) Measured amplitude for a
similar setup are presented in Ref. 11. photonic-crystal waveguide at 931.6 ± 0.1 nm. (b) Fitted
The propagating modes considered here represent amplitude using 7 reconstructed Bloch modes over 30 unit
cells. (c) Residual amplitude. (d) Amplitude coefficients of
eigenmodes of the crystal and can therefore be repre-
thespatialFouriertransformsofthemeasurednear-fieldpat-
sented by Bloch modes. A 2D Bloch mode propagating
tern. Labels correspond to those in Fig. 1(b).
in the xˆ-direction at position r = [x,y] and frequency ω
is described by Ψ (r,ω) = ψ (r,ω)exp(cid:0)ik x(cid:1). Here
m m ma
ψ (r,ω) is an envelope that is periodic with the lat-
m sultsinthefieldpatternA =a ψ (r,ω)andtheBloch
tice and satisfies ψ (r,ω) = ψ (r + axˆ,ω), m labels m m m
m m vectork form=1...M. Wecomparetheextractedk
theBlochmode,andk isthecorrespondingnormalized m m
m with wavevectors determined from spatial Fourier trans-
Bloch vector (we consider normalized wavevectors only).
forms to confirm the accuracy of the algorithm.
Werestrictourselvestopropagatingmodes(k ∈R). Itis
assumed that the measured field pattern Φ(r,ω) can be
describedbyasuperpositionofM Bloch-modesΨ (r,ω)
m
with amplitude a and one overall residual ε(r,ω): RESULTS
m
Figure 2(a) shows the measured field amplitude
M
(cid:88) |Φ(r,ω)|forawaveguidesectionof30unitcells(7.2±0.5
Φ(r,ω)= a Ψ (r,ω)+ε(r,ω) (1)
m m
µm) at a wavelength of λ = 931.6±0.1nm. A beating
m=1
pattern with a period of 5.7±0.2 unit cells reveals that
The residual ε(r,ω) describes measured field patterns multiple modes are involved in the spatial pattern. To
that cannot be described by the M Bloch modes, such ensure that each U (r’,ω) describes precisely one unit
n
asnon-guidedmodes, butalsoaccountsforexperimental cell,theoriginalmeasurementwasresampledonadiffer-
artifacts and noise. ent grid. Figure 2(b) shows the reconstructed amplitude
Inthereconstructionalgorithm[14,17]asectionofthe (cid:80)M
| A (r,ω)| with M = 7 Bloch modes, which is in
waveguide is separated into N unit cells, see Fig. 1(a). m
m=1
The algorithm uses the property that each ψ (r,ω) is excellent agreement with the measured amplitude. We
m
periodic in the photonic-crystal lattice, and requires us have chosen M = 7 because we expect from Fig. 1(b)
toanalyzethemeasuredfieldforeachunitcellU (r’,ω), a forward propagating mode corresponding to the light
n
withr’thecoordinatewithinoneunitcell. Themeasured line and for modes A, B and C both forward and back-
Φ(r,ω)canbefittedwithaseriesofBlochmodesusinga ward propagation, hence M = 1+(2×3) = 7. The low
least squares optimization that minimizes the functional values for the functional W = 0.0133 demonstrates that
W = (cid:82) |ε(r,ω)|2dr/(cid:82) |Φ(r,ω)|2dr. This procedure re- indeedΦ(r,ω)iswelldescribedbythesuperpositionof7
3
(a)
Table I: Comparison between fitted wavevectors from Bloch-
reconstructed
mode reconstruction (k ), and obtained wavevectors from
m
spatialFouriertransforms(k ). Thefirstcolumnlabelsthe
SFT
Bloch modes. The second column gives the fitted k . The
m
third column gives a measure how strongly present a mode
is. The fourth column gives the k , where the superscript
SFT
F marksthefundamentalwavevector. Inthefifthcolumnwe
(b) k=0.668 (mode A)(c) k=0.487 (mode C)
identify the modes from the calculated bandstructure in Fig.
1(b), the propagation direction, where +(−) corresponds to
|A|
the positive(negative) xˆ-direction, and polarization.
m k (2π) c k (2π) Label
m a m SFT a
1 0.251(1) 0.0073 0.251(3) +LL Im(A)
2 0.330(3) 0.0117 −0.668(3)F, 0.331(3) -A, TE
3 0.379(2) 0.0086 0.370(3) +B, TM
Re(A)
4 0.488(2) 0.6714 −0.516(3), 0.489(3)F +C, TM
5 −0.331(2) 0.2202 −0.332(3), 0.668(3)F +A, TE
(d) |E| |(cid:65)| (e) |E| |(cid:65)|
6 −0.367(6) 0.0014 −0.370(3) -B, TM
7 −0.486(2) 0.0681 −0.489(3)F, 0.516(3) -C, TM
Bloch modes. This conclusion is confirmed by the abso-
luteresiduals|ε(r,ω)|plottedinFig. 2(c). Thefittedk Figure3: (coloronline) (a)ReconstructedamplitudeforFig.
m
2(a) using Bloch modes m=4 and m=5 only. (b−c) Am-
are presented in the second column of Tab. I. The third
plitude, imaginary part and real part for both Bloch modes.
column describes the relative contribution of each mode
(d−e) Comparison between calculated |E| (left) and recon-
asc =|(cid:82) A∗ (r,ω)Φ(r,ω)dr|/(cid:82) |Φ(r,ω)|2dr. Notethat
m m structed |A| (right).
the 7 contributions plus that of the residual add up to
unity. The fifth column describes which modes of Fig.
1(b)correspondtok ,thepropagationdirectionandpo-
m whenonlythesetwomodesaretakenintoaccountforthe
larization. We observe mainly the forward propagating
reconstruction. A very good agreement is observed with
TE-like mode A (m = 5) and the forward propagating
Φ(r,ω) of Fig. 2(a). Especially the diagonal beats are
TM-like mode C (m = 4). The errors in k are esti-
m well reproduced. The difference wavevector of the two
mated by varying the grid element size and allowing for
modes corresponds to a beating period of (5.5±0.2)a.
a relative increase of ∆W by maximum 10%; within this
The beating pattern of two orthogonal modes is the re-
range the mode patterns A (r,ω) and A (r,ω) do not
4 5 sultofquasi-interference;theNSOMtiptherebyprojects
change noticeably.
bothorthogonalpolarizationsonadetectionbasiswhere
Next, the fitted km are compared with wavevectors these modes interfere [19]. Figure 3(b) shows the ampli-
determinedfromthespatialFouriertransformsshownin tude, the real part and the imaginary part of the recon-
Fig. 2(d) (kSFT). A Fourier transform in the xˆ-direction structedTE-likeBlochmodeAwithk5 =−0.331±0.002.
was made for each line parallel to the waveguide over a The mode profile is symmetric in the yˆ-direction about
range of 35.4±0.9 µm, which includes the range shown the center of the waveguide. Figure 3(d) shows the cal-
in Fig. 2(a). For comparison km and kSFT are listed culated[18]time-averagedamplitude(cid:104)|E|(cid:105)(left)andthe
in Tab. I, showing an excellent agreement. The spa- measured amplitude |A| for approximately 3 unit cells.
tial Fourier transforms show for modes A and C higher Both show a similar pattern. Figure 3(c) shows the am-
Bloch harmonics. Both the fundamental kSFT and the plitude,therealpartandtheimaginarypartoftherecon-
observed higher Bloch harmonics are listed in the table. structedTM-likeBlochmodeC withk =0.487±0.003.
4
InFig. 2(d)themodesfromFig. 1(b)areidentified. The Figure3(e)showsthecalculated(cid:104)|E|(cid:105)(left)andthemea-
amplitude coefficients confirm that we detect mainly the suredamplitude|A|forapproximately3unitcells(right).
forward propagating TE-like mode A (kSFT = −0.332± For mode C the agreement is poor, likely because the
0.003,0.668±0.003) and the forward propagating TM- near-field tip has a low response to E and a non-trivial
z
like mode C (kSFT =−0.516±0.003,0.489±0.003). response to Ex and Ey, see Ref. 13, in addition to its
We have demonstrated that the forward propagating finite resolution. The reconstructed field patterns are an
TE-like mode A and the forward propagating TM-like approximation of the Bloch modes propagating in the
mode C are the most prominent Bloch modes present system and moreover, they are not necessarily orthogo-
in the data of Fig. 2. Figure 3(a) shows the amplitude nal because of quasi-interference. Although the effect of
4
(a) measured (b) calculated a transverse mode profile is apparent that can be mainly
described by one prominent maximum at y = 0 that is
ofile ofile slightly asymmetric, describing light guidead in the line
de pr kx=0.375 de pr defect. Additional side lobes are observed at ay = −0.9
nsverse mo nsverse mo kx=0.330 kx=0.380 wraoneudnoadbtisneaygrv=peh1ao.4tco,enrneitcprarcelrsymesntataxilni.mgAulimtghktaxtex=ayte0n=.d33i0n2,ga(irnnedtdotdthhaeeshcseoudnr)--
Tra kx=0.332 kx=0.285 Tra kx=0.280 ttrhiebunteiownspeoafktsheobssiedreveldobaets byec=om−e3.b2i,g−ge2r..2,−A1ls.5o,naontde
a
(c) 2.8. When the wavevector is increased, the relative
contributions of these additional peaks increase. At
k = 0.375 the central peak is still present, and the
x
surrounding peaks have grown. Qualitatively, the mea-
suredtransversemodeprofilescorrespondwiththecalcu-
lated ones shown in Fig. 4(b), which were obtained from
the time-averaged amplitude of the total electric field
a
(cid:82)
|E(r,ω)|dx. Themaximaandminimaoccuratapprox-
0
imatelythesamelocationsandthewidthwofthecentral
maximumisgrowingwithincreasingk . Notallfeatures
x
are resolved of the calculated transverse mode profile in
our measurements. For example, the measured relative
amplitude of the central maximum compared with the
additional maxima differs from the calculations.
Thecentralmaximaofthetransversemodeprofilesare
Figure 4: (color online) (a) Measured normalized transverse
fitted with a Gaussian with width w. Figure 4(c) shows
modeprofilefork =0.285(black),k =0.332(reddashed),
x x the reduced width w versus wavevector (black symbols,
and kx =0.375 (blue). (b) Calculated normalized transverse a
modeprofilefork =0.280(black),k =0.330(reddashed), barsrepresent95%convergenceintervals). Theredsym-
x x
andk =0.380(blue). (c)Determined(black)andcalculated bols interpolated by the dashed line represent w deter-
x a
width w (red) versus longitudinal wavevector kx for the TE- mined from the calculated transverse mode profile. The
like guided mode. Inset: measured (symbols) and calculated measured w increases with k . For k < 0.34 the mea-
x x
dispersion (red).
sured w matches the calculated w well. For k > 0.34
x
the measured w becomes larger than the calculated w.
Weattributethistothefiniteresolutionofthenear-field
the tip is far from straightforward ([5, 20, 21]), we antic-
tip and to its response function. For most considered
ipate that the comparison between calculated modes of
k -values, however, the measured w/a matches the cal-
x
anopticalsystemandreconstructedmodescouldleadto
culated with a 10% accuracy.
methods to deduce the response function of a near-field
tip.
Next, we demonstrate the power of the reconstruction
algorithmbystudyingthetransversebehaviorofTE-like
SUMMARY
mode (A) versus wavevector. We consider the forward
propagatingTE-likemode(k =−0.331,0.669inFig. 3)
5
and take its reduced wavevector (0 < kx < 0.5) to com- In conclusion, we have implemented an algorithm pro-
pare directly with the folded bandstructure of Fig. 1(b). posedbyHaetal. toextractBlochmodesfromnear-field
We have measured field patterns between λ=907−944 measurements on a photonic-crystal waveguide. The ex-
nm,andapplythereconstructionalgorithmtodetermine tractedwavevectorsareinverygoodagreementwiththe
A(r,ω)andkx forthismodeateachλ. Wehaveselected wavevectors determined from spatial Fourier transforms.
Φ(r,ω) where this TE-like mode is prominently present We have studied two extracted Bloch modes to explain
in spatial Fourier transforms. The inset in Fig. 4(c) the observed near-field pattern. We also have studied
shows the fitted kx versus reduced frequency λa. how the width of a selected mode changes with wavevec-
In order to concentrate on the transverse behavior, we torandfindgoodagreementwithcalculations. Weantic-
(cid:82)a ipatethatthisalgorithmcanbeusedtofilterstatesthat
define the transverse mode profile |A(r,ω)|dx. Fig-
cannot be described by propagating Bloch modes, such
0
ure4(a)showsthemeasurednormalizedtransversemode as Anderson-localized states observed in the slow-light
profile for 3 different wavevectors. At k =0.285 (black) regime [22–26].
x
5
Acknowledgments [11] R.J.P. Engelen, T.J. Karle, H. Gersen, J.P. Korterik,
T.F. Krauss, L. Kuipers, and N.F. van Hulst, Opt. Exp.
13, 4457 (2005).
We kindly thank Dirk Jan Dikken, Herman Offer-
[12] V.S.Volkov,S.I.Bozhevolnyi,P.I.Borel,L.H.Frandsen,
haus, Kobus Kuipers and Allard Mosk for stimulating
and M. Kristensen, Phys. Rev. B 72, 035118 (2005).
discussions, Cock Harteveld, Jeroen Korterik, and Frans
[13] M. Burresi, R.J.P. Engelen, A. Opheij, D. van Oosten,
Segerinkfortechnicalsupport. Thisworkwassupported D.Mori,T.Baba,andL.Kuipers,Phys.Rev.Lett.102,
by FOM and NWO-Nano. 033902 (2009).
[14] S.Ha,A.A.Sukhorukov,K.B.Dossou,L.C.Botten,C.M.
de Sterke, and Y.S. Kivshar, Opt. Lett. 34, 3776-3778
(2009).
[15] S. Ha, A.A. Sukhorukov, A.V. Lavrinenko, I.V.
∗ Electronic address: [email protected] Shadrivov, D.A. Powell, and Y.S. Kivshar, Appl. Phys.
† Electronic address: [email protected], Lett. 98, 061909 (2011).
[16] M. Spasenovic, T.P. White, S. Ha, A.A. Sukhorukov, T.
www.utwente.nl/mesaplus/anp
Kampfrath, Y.S. Kivshar, C.M. de Sterke, T.F. Krauss,
[1] L. Novotny and B. Hecht, Principles of Nano-Optics,
and L. Kuipers, Opt. Lett. 36, 1170-1172 (2011).
Cambridge University press, Cambridge, first edition
[17] S. Ha, M. Spasenovic, A.A. Sukhorukov, T.P. White,
(2006).
C.M. de Sterke, L. Kuipers, T.F. Krauss, and Y.S.
[2] M.L.M Balistreri, H. Gersen, J.P. Korterik, L. Kuipers,
Kivshar, J. Opt. Soc. Am. B 28, 955-963 (2011).
and N.F. van Hulst, Science 294, 1080-1082 (2001).
[3] H.Gersen,T.J.Karle,R.J.P.Engelen,W.Bogaerts,J.P. [18] S.G. Johnson, and J.D. Joannopoulos, Opt. Express 8,
Korterik, N.F. van Hulst, T.F. Krauss, and L. Kuipers, 173-190 (2001)
Phys. Rev. Lett. 94, 073903 (2005). MIT photonic-bands (mpb) package, http://ab-
[4] N. Louvion, A. Rahmani, C. Seassal, S. Callard, D. initio.mit.edu/wiki/index.php/MIT Photonic Bands.
G´erard, and F. de Fornel, Opt. Lett. 31, 2160-2162 [19] M.L.M.Balistreri,A.Driessen,J.P.Korterik,L.Kuipers,
(2006). and N.F. van Hulst, Opt. Lett. 25, 637-639 (2000).
[5] S.Mujumdar,A.F.Koenderink,T.Su¨nner,B.C.Buchler, [20] W.C.L.Hopman,K.O.vanderWerf,A.J.F.Hollink,W.
M.Kamp,A.Forchel,andV.Sandoghdar,Opt.Express Bogaerts, V. Subramaniam, and R.M. de Ridder, Opt.
15, 17214-17220 (2007). Express 14, 8745-8752 (2006).
[6] L.Lalouat,B.Cluzel,F.deFornel,P.Velha,P.Lalanne, [21] L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade,
D.Peyrade,E.Picard,T.Charvolin,andEHadji,Appl. J.P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel,
Phys. Lett. 92, 111111 (2008). Phys. Rev. B 76, 041102 (2007).
[7] L. Lalouat, B. Cluzel, C. Dumas, L. Salomon, and F. de [22] J. Topolancik, B. Illic, and F. Vollmer, Phys. Rev. Lett.
Fornel, Phys. Rev. B 83, 115326 (2011). 99, 253901 (2007).
[8] J.D. Joannopoulos, S.G. Johnson, J.N. Winn, and R.D. [23] L.Sapienza, H.Thyrrestrup, S.Stobbe, P.D.Garc´ıa, S.
Meade, Photonic crystals, Molding the flow of light, Smolka, P. Lodahl, Science 327, 1352 (2010).
Princeton University Press, Princeton, second edition [24] P.D.Garc´ıa,S.Smolka,S.Stobbe,andP.Lodahl,Phys.
(2008). Rev. B 82, 165103 (2010).
[9] T.F. Krauss, J. Phys. D: Appl. Phys. 40, 2666-2670 [25] J.Yang,H.Noh,M.J.Rooks,G.S.Solomon,F.Vollmer,
(2007). and H. Cao, Appl. Phys. Lett. 98, 241107 (2011).
[10] T. Lund-Hansen, S. Stobbe, B. Julsgaard, H. [26] S.R. Huisman, et al., in preparation.
Thyrrestrup, T. Su¨nner, M. Kamp, A. Forchel, and P.
Lodahl, Phys. Rev. Lett. 101, 113903 (2008).
