Fano resonances in prism-coupled square micropillars

January 1, 2004 / Vol. 29, No. 1 / OPTICS LETTERS 5
Fano resonances in prism-coupled square micropillars
Ho-Tong Lee and Andrew W. Poon
Department of Electrical and Electronic Engineering, The Hong Kong University of Science and Technology,
Clear Water Bay, Hong Kong SAR, China
Received July 24, 2003
We report Fano resonances in the frustrated total internal ref lection (TIR) spectra of a prism-coupled square
micropillar. Our angle-selective frustrated TIR technique reveals characteristically asymmetric resonance
line shapes, which evolve spectrally over approximately a 2p phase in the far field within a subdegree range
of ref lection angles. We theoretically model the asymmetric line shapes by the interference between a high-Q
resonance that is evanescently coupled and partially confined by TIR, and a coherent background that is total
internally ref lected at the prism surface without coupling to the micropillar. © 2004 Optical Society of
America
OCIS codes: 230.5750, 260.5740.
Optical resonances circulating in dielectric micropillar
(m-pillar) cavities in the form of circular disks, 1 rings, 2
and squares 3–5 have attracted considerable interest for
wavelength-division multiplexing channel add–drop
applications. It is well known that a light wave can
be partially confined by total internal ref lection (TIR)
at the m-pillar sidewalls. When the m-pillar is input
side coupled by either a dielectric coupler 1–3,5 or a
Gaussian beam, 4 optical resonances are excited when
the cavity round-trip wave is wave-front matched
with the input coupled wave. The excited modes,
without interfering with a coherent background, are
generally considered to have a symmetric Lorentzian
line shape. 6 However, recent theoretical and numerical
investigations revealed pronounced asymmetric
resonance line shapes in waveguide-coupled optical
microresonators. 7 Such characteristically asymmetric
line shapes, known as Fano resonances, result
from the interference between a single resonance and
a continuum of background modes. Recently, Fano
resonances have also been found in two-dimensional
nonlinear photonic crystal slabs. 8
In this Letter we report, for the first time to
our knowledge, Fano resonances in frustrated TIR
spectra of a prism-coupled square m-pillar. Our
angle-selective frustrated TIR technique reveals
characteristically asymmetric line shapes that evolve
spectrally over an approximately 2p phase change
in the far field within a subdegree range of ref lection
angles. We theoretically model the asymmetric
line shapes by the interference between a high-Q
resonance that is evanescently coupled and partially
confined by TIR, and a coherent background that is
total internally ref lected at the prism surface without
coupling to the m-pillar.
Figure 1(a) shows a schematic of the experimental
setup. A commercially available 200-mm-sized
fused-silica square m-pillar (with rounded corners) in
the form of an optical fiber was evanescently coupled
with a hemicylindrical UV-grade fused-silica bulk
prism. The m-pillar was positioned perpendicular
to the plane of incidence and near the hemicylindrical
prism center. To maintain a fixed air gap
separation between the m-pillar and the prism we
coated the m-pillar with two stripes of 0.1-mm-thick
gold spacers (to set the minimum gap separation)
that were 1 cm apart and glued onto the prism. A
Gaussian beam from a wavelength-tunable diode laser
at a 1550-nm wavelength (with a laser linewidth of
300 kHz) was focused with an f 25 cylinder lens
onto the prism (with cone angle 2.3 ± and spot size
50 mm) at an incident angle u 45 ± 6 0.5 ± , where
u is near TIR critical angle u c 44 ± for refractiveindex
contrast n 1.44. Such values of u favor
input coupling to the four-bounce ray orbits in square
m-pillars, 4,5 as shown in Fig. 1(a). The focused line
beam illuminated only the m-pillar between the
spacers. The laser was 45 ± linearly polarized to
measure either the TM- (E-field k the m-pillar axis) or
the TE- (E-field the m-pillar axis) polarized mode.
TM measurement is emphasized in the following
discussion. The ref lected light after coupling to
the m-pillar was collimated by an f 25 cylinder lens
at 90 ± relative to the laser beam and selectively
collected with a 62.5-mm-core multimode fiber at the
far field after an analyzer. The multimode fiber
was mounted upon a translation stage to facilitate
angle-selective ref lection measurement along the Z
axis (with an angle resolution of 0.03 ± ) while the
prism–pillar coupling remained fixed. The light
transmitted tangentially to the m-pillar sidewall was
imaged with an objective lens onto another multimode
fiber after an analyzer and was monitored with a
near-IR CCD camera. Both the ref lected and the
Fig. 1. (a) Schematic of the experimental setup for measuring
Fano resonances in a prism-coupled square m-pillar:
MM1, MM2, multimode fibers; PD1, PD2, photodiodes;
P, polarizer; A, analyzer. Z, translation. (b) Measured
sidewall-transmitted spectrum, (c) measured ref lected
spectrum at u 45 ± ; FSR, free spectral range.
0146-9592/04/010005-03$15.00/0 © 2004 Optical Society of America

6 OPTICS LETTERS / Vol. 29, No. 1 / January 1, 2004
transmitted spectra were lock-in detected with similar
photodiodes. The spectral resolution was 0.03 nm,
which was limited only by our data-taking rate.
Figures 1(b) and 1(c) show the simultaneously
measured TM-polarized sidewall-transmitted and
-ref lected spectrum of the prism-coupled square
m-pillar. The free spectral range of both the ref lected
(u 45 ± ) and the transmitted spectrum is 2.86 nm,
consistent with the calculated free spectral range of
wave-front-matched four-bounce orbits. 4 Only one
set of modes is preferentially coupled. The ref lected
spectrum [Fig. 1(c)] reveals approximately symmetric
dips with a maximum Q exceeding 10 4 , whereas the
transmitted spectrum [Fig. 1(b)] shows approximately
symmetric peaks with a higher Q (2 3 10 4 ). We attribute
the difference in Q to the fact that the ref lected
and transmitted spectra can originate from different
heights along the illuminated m-pillar, which can have
a nonuniform surface roughness or nonuniform gap
spacing. Figure 1(c) shows a coupling efficiency that
exceeds 50%, which can be improved by narrowing
the air gap separation between the prism and the
m-pillar. 5 The ripplelike modulation in Fig. 1(c)
is due to residual Fabry–Perot interference in the
external-cavity diode laser.
Figure 2 shows the measured TM-polarized angleselective
spectra at ref lection angles from u 44.94 ±
to u 45.71 ± . Characteristically asymmetric line
shapes (Fano line shapes) can be clearly discerned
in Figs. 2(b) and 2(d). The Fano line shapes evolve
continually over an approximately 2p phase change.
The Q value measured in Fig. 2(a) exceeds 5000, and
the coupling efficiency exceeds 68%. We observed
that the Fano line-shape evolution repeats for another
approximately 2p-phase change from u 44.45 ± to
u 44.94 ± (not shown here).
Figure 3 is a schematic of our theoretical model.
We consider the far-field interference between ray R,
which is evanescently coupled to a high-Q square
cavity four-bounce mode, and ray B, which is a coherent
background that is due to TIR at the prism
surface without coupling to the m-pillar. The interference
between an optical resonance and the coherent
background is analogous to the interference between
a single resonance and a continuum of background
modes according to the theory of Fano resonances.
The square cavity has plane-to-plane distance a and
a refractive index n m that is the same as that of the
prism. The cavity is separated from the prism with
air gap spacing x and refractive index n a 1. Ray R
is evanescently input coupled to the square cavity
at u u R , which satisfies the square cavity TIR
confinement condition u c , u R , 90 ± 2 u c . 4 We
express the resultant ref lected electric field amplitude
E R (normalized to incident field amplitude E 0 ) as
follows 9 :
where r FTIR and t FTIR are the frustrated TIR complex
ref lection and transmission coefficients between
semi-infinite dielectric media separated by an air
gap, 9,10 f is the cavity round-trip phase shift, a is
the cavity round-trip loss, and F R is the phase of
E R relative to E 0 at the far field. We adopt a 2p2
phase difference between r FTIR and t FTIR . 9,10 The
wavelength-dependent value of f is given as
fl 2pnLl 1g TIR , where L 2acos u1sin u
is the wave-front-matched round-trip length 4 and g TIR
is the TIR phase shift at the four cavity sidewalls. 10
Ray B represents the coherent background that is
due to TIR at the prism surface without coupling to the
m-pillar. Such coherent background can be attributed
to the fact that part of the laser beam may not spatially
overlap the prism–pillar coupling region, as shown in
Fig. 2. Measured Fano resonances in TM-polarized ref lection
spectra (a) u 45.71 ± , (b) u 45.49 ± , (c) u 45.27 ± ,
(d) u 45.11 ± , (e) u 44.94 ± . Int., intensity.
E R
E 0
r FTIR 1 t FTIR 2 expifexp2a
1 2 r FTIR expifexp2a
Ç
ER
E 0
Ç
expiF R , (1)
Fig. 3. Schematic of the theoretical model. Ray R is
coupled to a four-bounce ray orbit. Ray B represents a
coherent background that is total internally reflected at
the prism surface without coupling to the cavity. The
dashed arrow represents the surface wave leakage.

January 1, 2004 / Vol. 29, No. 1 / OPTICS LETTERS 7
Fig. 4. Calculated resonance line shapes with a coherent
background. x 800 nm, u R 44 ± , a 0.19.
(a) jF R 2F B j 0, (b) jF R 2F B j 0.5p, (c) jF R 2F B j
p, (d) jF R 2F B j 1.5p.
Fig. 3. Moreover, it is conceivable that the m-pillar in
the experiment can have a lateral tilt angle of the order
of 1 ± with the prism’s f lat surface, and thus even
part of the laser beam illuminating the prism–pillar
coupling region can be total internally ref lected without
coupling to the m-pillar. We express the coherent
TIR background electric field E B (normalized to E 0 )as
E B E 0 expiF B , where F B is the phase of E B relative
to E 0 at the far field.
The evanescently coupled high-Q resonances (ray R)
and the coherent TIR background (ray B) can spatially
overlap and interfere in the far field. The resultant
ref lected intensity I (normalized to incident intensity
I 0 ) is given as follows:
µÇ Ç
I
2
∂ Ç Ç
ER
ER
11 1 2 cosF R 2F B . (2)
I 0 E 0 E 0
Phase jF R 2F B j at resonance wavelength l 0 determines
the asymmetric line shapes.
Figure 4 shows the interference intensity calculated
with Eq. (2). TM polarization (or s polarization 10 )is
assumed. We chose a 200 mm, n m 1.44, u R 44 ± ,
x 800 nm, and a 0.19 and found good agreement
between the measured Fano line shapes (Fig. 2) and
the line shapes calculated for jF R 2F B j 0, 0.5p, p,
1.5p. The calculated resonances in Fig. 4(a) have a
coupling efficiency of 69% and a Q value of 5000,
which match those of Fig. 2(a).
We remark that the evanescently coupled high-Q
resonances can output couple tangentially as a surface
wave along the cavity sidewall, shown as a dashed
arrow in Fig. 3. Inasmuch as the square m-pillar is
multimode, 4,5 it is conceivable that the high-Q resonance
field and a continuum of multimodes (evanescently
coupled by the Gaussian beam) can interfere in
the far field and thus display Fano resonances.
In conclusion, we have experimentally demonstrated
Fano resonances by using angle-selective
prism coupling to the high-Q optical resonances of a
square m-pillar. Our frustrated TIR spectra revealed
asymmetric line shapes, which evolve spectrally over
approximately a 2p phase in the far field within a
subdegree range of angles. We theoretically modeled
the measured Fano resonances by the interference
between a high-Q resonance that is evanescently
coupled and partially TIR confined, and a coherent
background that is due to TIR at the prism surface
without coupling to the m-pillar. We found good
agreement between the measured and the calculated
characteristically asymmetric line shapes. Fano
resonances in microresonators should be of particular
interest in relation to integrated optical switches and
filters, for which the optical resonance line shapes can
be tuned by application of a relative phase between
the high-Q resonances and the coherent background.
This study was substantially supported by grants
from the Research Grants Council and the University
Grants Council of the Hong Kong Special Administrative
Region, China (project HKUST6166/02E & HIA01/
02.EG05). We thank C. H. Lam, K. C. Ho, and D. H. T.
Chan for their helpful contributions. We thank Y. L.
Pan and R. K. Chang of Yale University for providing
the square fiber. A. W. Poon’s e-mail address is
eeawpoon@ust.hk.
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