Power transfer between neighboring planar waveguides

Power transfer between neighboring
planar waveguides
˜ ∗ and F. J. Garc´ıa de Abajo
X. M. Bendana
´
Instituto de Optica
- CSIC and Unidad Asociada CSIC-Universidade de Vigo, Serrano 121,
28006 Madrid, Spain
*[email protected]
Abstract:
The ability to control light over very small distances is a
problem of fundamental importance for a vast range of applications in
communications, nanophotonics, and quantum information technologies.
For this purpose, several methods have been proposed and demonstrated
to confine and guide light, for example in dielectric and surface plasmon
polariton (SPP) waveguides. Here, we study the interaction between
different kinds of planar waveguides, which produces dramatic changes
in the dispersion relation of the waveguide pair and even leads to mode
suppression at small separations. This interaction also produces a transfer of
power between the waveguides, which depends on the gap and propagation
distances, thus providing a mechanism for optical signal transfer. We
analytically study the properties of this interaction and the power transfer
in different structures of interest including plasmonic and particle-array
waveguides, for which we propose an experimental realization of these
ideas.
© 2012 Optical Society of America
OCIS codes: (130.2790) Guided waves; (230.7370) Waveguides; (230.7390) Waveguides, planar; (230.7400) Waveguides, slab; (240.6680) Surface plasmons; (250.5403) Plasmonics.
References and links
1. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater “Plasmonics—a
route to nanoscale optical devices,” Adv. Mater. 13, 1501 (2001).
2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).
3. H. Raether, Surface Plasmons (Springer-Verlag, 1988).
4. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric
structures,” Phys. Rev. B 61, 10484–10503 (2000).
5. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of asymmetric
structures,” Phys. Rev. B 63, 125417 (2001).
6. X. M. Bendana and F. J. Garcia de Abajo, “Confined collective excitations of self-standing and supported planar
periodic particle arrays,” Opt. Express 17, 18826–18835 (2009).
7. T. W. Ebbesen, C. Genet, and S. I. Bozhevolnyi, “Surface-plasmon circuitry,” Phys. Today 61, 44 (2008).
8. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics”, Nature (London) 424,
824–830 (2003).
9. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in
nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, 16356–16359 (2000).
10. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of
silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998).
11. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic
crystal slabs,” Phys. Rev. B 60, 5751 (1999).
12. E. Verhagen, M. Spasenovic, A. Polman, and L. Kuipers, “Nanowire plasmon excitation by adiabatic mode
transformation,” Phys. Rev. Lett. 102, 203904 (2009).
#155664 - $15.00 USD
(C) 2012 OSA
Received 30 Sep 2011; revised 6 Jan 2012; accepted 10 Jan 2012; published 26 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 3152
13.
14.
15.
16.
J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
E. D. Palik, Handbook of Optical Constants and Solids (Academic, 1985).
M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1970).
P. Nordlander, C. Oubre, E. Prodan, K. Li and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,”
Nano Lett. 4, 899–903 (2004).
17. A. Manjavacas and F. J. Garcia de Abajo, “Robust plasmon waveguides in strongly interacting nanowire arrays,”
Nano Lett. 9, 1285–1289 (2009).
18. Z. Chen, T. Holmgaard, S. I. Bozhevolnyi, A. V. Krasavin, A. V. Zayats, L. Markey, and A. Dereux, “Wavelengthselective directional coupling with dielectric-loaded plasmonic waveguides,” Opt. Lett. 34, 310–312 (2009).
19. A. V. Krasavin and A. V. Zayats, “Passive photonic elements based on dielectric-loaded surface plasmon polariton
waveguides,” Appl. Phys. Lett. 90, 211101 (2007).
Optical excitations propagating in planar waveguides find application to optical-signal processing [1], and they pose interesting questions from a fundamental viewpoint regarding the
behavior of light over small distances. Different realizations of planar waveguiding include
light confinement in dielectric films [2], surface plasmon polaritons (SPPs) [3–5], and planar
particle arrays [6]. Despite ohmic losses in metals, SPPs have the advantage of concentrating
light energy over small regions compared to the wavelength [7]. In contrast, dielectric films
can propagate light over much longer distances and are versatile elements used in integrated
optics [8]. Particle arrays have been demonstrated in 1D chains [9, 10], and they are intimately
related to 2D photonic crystal waveguides [11]. Guided modes in the latter depend strongly on
the environment and their existence can undergo a sudden transition as the degree of dielectric
asymmetry is varied [6].
Here, we study the interaction between planar waveguides placed in close proximity. The
guided modes of these systems are a combination of the modes in the individual waveguides,
which are hybridized due to their mutual interaction. We focus on the transfer of optical energy
between two parallel waveguides as a major in-out problem of signal processing and communications, and we consider the particular case of two gently curved waveguides, so that a parallel
waveguide configuration can be assumed at each position along the guiding direction as an
adiabatic approximation [12]. We show the energy transfer to strongly depend on the bending
radius of the waveguides and their minimum gap separation. For small gaps, only one mode is
supported, thus producing a sharp transition in the output power.
Planar waveguides are structures that confine light in one spatial direction and allow free
propagation along the remaining two dimensions. The normal wave vector iκz is imaginary outside the waveguide, thus producing evanescent fields that cannot propagate away from it. These
evanescent tails spread out of the guiding structure with a characteristic 1/e−field-amplitudedecay penetration distance Lz = 1/κz [13].
In Fig. 1(a-c) we show characteristic dispersion relations for several kinds of waveguides as
a function of wave vector kk parallel to the plane of the structure. Particle array waveguides
(a) have a confining resonance within the First Brillouin Zone (FBZ). This resonance defines
a dispersion relation with vanishing group velocity when the parallel wave vector approaches
π /a and degenerate TE and TM modes when the particles are spherical [6]. Dielectric slabs
(b) show a variety of TM and TE modes, with the fundamental TM mode having no cutoff
wavelength. These modes lie in between the light line in the surrounding medium and the light
line in the dielectric slab material. Finally, metal-dielectric interfaces (c) and metal films can
sustain propagating surface-plasmon polaritons (SPPs) [3].
The confining planar waveguides in a parallel-waveguide configuration interact with each
other when placed in close proximity via the evanescent tails of the modes. The resonances of
the resulting structure are then determined by the Fabry-Perot condition [15]
1 − r1σ r2σ exp (−2κz d) = 0,
#155664 - $15.00 USD
(C) 2012 OSA
(1)
Received 30 Sep 2011; revised 6 Jan 2012; accepted 10 Jan 2012; published 26 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 3153
(a) Particle array
(b)
Dielectric slab
(c) Plasmonic surface
---
π
0
k
k
0
(d)
π
k||a
2π
k||
k||
(f)
(e)
40
20
d (µm)
d (µm)
20
d (µm)
ka
2π
0
1
k||/k
1.005
0
1
k||/k
1.005
0
0
1
k||/k
1.012
Fig. 1. (a-c) Schematic view and dispersion relations for different planar waveguide structures. The guided mode dispersion relations are shown in red in these frequency-momentum
plots. (a) Planar array of dielectric particles. The light cone and the diffracted light line are
shown as black lines. (b) Dielectric slab. Straight lines show the light cones in vacuum
(black) and the dielectric medium (green). (c) SPP supporting metal-dielectric interface.
Straight lines indicate the light cone (solid line) and the electrostatic surface-plasmon frequency (dashed line). (d-f) Interaction between different combinations of confining structures placed at varying distance d. A similar behavior is observed in all cases, with a strong
repulsion between modes at small distances. (d) Two planar arrays of silicon (εSi = 12)
spheres of radii 200 nm and 175 nm, respectively, embedded in silica (εSiO2 = 2) and arranged in square lattices of period 1 µ m. The light wavelength is λ = 4 µ m. (e) Planar array
of silicon spheres of radius 200 nm arrange in a square lattice of period 1 µ m placed near a
dielectric slab of thickness 70 nm and permittivity ε = 6. (f) Slot waveguide consisting of
a silver-air-silver structure [14] illuminated with λ = 1550 nm light.
where riσ is the reflectivity of guide i for σ -polarized light and d is the separation distance. This
model can be easily generalized to an arbitrary number of waveguides. The results of this model
are shown in Fig. 1(d-f) for different combinations of waveguides. The effect of interaction is
qualitatively the same in all of these cavities: for large distances, the modes are at the position
of the original non-interacting resonances; however, as the separation is decreased, mode repulsion takes place down to a critical distance below which one of the modes is pushed beyond the
light line of the surrounding medium, thus being broadened and effectively non contributing
to propagating optical signal due to coupling to radiation; in contrast, the remaining mode is
still pushed towards larger parallel wave vectors that increase its degree of confinement. This
#155664 - $15.00 USD
(C) 2012 OSA
Received 30 Sep 2011; revised 6 Jan 2012; accepted 10 Jan 2012; published 26 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 3154
repulsive interaction is a common effect in many optical and plasmonic hybridized systems
when similar modes are placed at interacting distances [16]. In the particular case of symmetric
structures (e.g., Fig. 1(f)), the interaction breaks the mode degeneracy and produces characteristic symmetric and antisymmetric field distributions. In the metal-insulator-metal (MIM) layer
structure, the metal prevents radiation losses, so that the lowest-kk mode in Fig. 1(f) (the symmetric mode) is well defined down to kk = 0. Nevertheless, there is a critical distance d below
which the mode is suppressed, which occurs at kk = 0. This point describes non-propagating
modes of vanishing group velocity.
As a result of the mismatch in parallel wave vector for a given separation d, the two modes
accumulate different phase as they propagate along the structure. Therefore, the power density
moving in each part (i.e., each individual waveguide) of the interacting structure (Pi ) varies
along the propagation direction when both modes are excited, thus producing power transfer
back and forth between both waveguides. For simplicity, we focus on symmetric geometries
(e.g., the MIM structure or a double layer of equal-size particles), in which the field amplitudes ci (x) in waveguide i = 1, 2 at the position x along the propagation direction satisfies the
equations
cs (x) + ca (x)
√
,
2
cs (x) − ca (x)
√
c2 (x) =
,
2
c1 (x) =
(2)
where cs and ca are the complex amplitudes of the symmetric and antisymmetric modes.
We consider two neighboring curved waveguides, in which the distance between them d
varies very smoothly along the propagation direction. In such a system, the interaction (and
therefore, also the wave vector of the modes) varies adiabatically along the propagation direction x. Within the adiabatic regime, we can neglect losses and propagation constant shifts if
the penetration distance of the modes outside the waveguide is much smaller than the bending radius. Treating as a perturbation all terms in the Maxwell equations that differ from
the straight waveguide, one can show that radiative losses in a curved waveguide of large
bending radius R scale as (Lz /R)2 , where Lz is the penetration distance into the surrounding
medium (i.e., the transversal extension of the mode). Small radiative losses ∼ 3% per optical
cycle have been already predicted for small rings of radius R twice smaller than the wavelength and Lz /R ∼ 0.01 [17]. An extrapolation to the structures considered here (e.g., in Fig. 2,
Lz ≈ 2.65 µ m ≪ R ≥ 100 µ m, leading to (Lz /R)2 ∼ 7 × 10−4 ) renders radiative losses negligible compared to ohmic losses. This produces a power transfer that depends on the nature of the
waveguides, as well as on the details of the variation of the separation distance with x. Starting
with a mode prepared in guide 1 at large distances between the waveguides at position x0 , the
net power in each guide at a subsequent position x reduces to [18, 19]
Z x
P1 (x) ∝ cos2
dx′ ∆kk (d(x′ ))/2 ,
x
Z 0x
(3)
P2 (x) ∝ sin2
dx′ ∆kk (d(x′ ))/2 ,
x0
where ∆kk (d) is the wave vector difference between the two modes, which is calculated from
the Fabry-Perot condition (Eq. (1)), rendering kk for each mode as a function of separation d
(see Fig. 1). In the derivation of these equations, we consider the incident wave to be equally
split between in-phase symmetric and antisymmetric modes at x0 (this places the weight of the
incident wave only in waveguide 1 according to Eqs. (2)), we follow the adiabatic evolution
#155664 - $15.00 USD
(C) 2012 OSA
Received 30 Sep 2011; revised 6 Jan 2012; accepted 10 Jan 2012; published 26 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 3155
(a)
A
R
d
x
z
y
(b)
R (µm)
100
A
d (µm)
0.5
d = 3 µm
200
1
300
1.5
400
R = 200 µm
3
10 µm
10 µm
500
6
−100 µm
−10 µm
100 µm
−100 µm
−10 µm
100 µm
Fig. 2. (a) SPP energy transfer between neighboring silver cylinders. A SPP is assumed
to be excited at line A (e.g., by external illumination over a grating parallel to the left
cylinder), so that it propagates along the surface polar direction, as shown by arrows. As
the SPP reaches the gap between both wires, their interaction produces power transfer to
the right cylinder. (b) Left: Power transfer under the configuration of (a) as a function of
cylinder radius R for a gap distance d = 3 µ m. Right: power transfer as a function of gap
distance d for cylinder radius R = 200 µ m. Black dots display the line A where the modes
are excited.
of the modes (i.e., they pick up a phase as they propagate), and we recombine the resulting
amplitudes using Eqs. (2) to obtain the power density Pi ∝ |ci |2 .
We have solved Eq. (3) for planar waveguides curved onto the surface of large-radius cylinders, with the guided modes propagating along the polar direction of the latter, for both particle
arrays and metal-dielectric SPP-supporting interfaces (Fig. 2(a)). The results are shown in Fig.
2(b) for different values of the cylinder radius (left) and the gap distance (right) in silver-air
SPP waveguides. For a fixed gap distance, the power is only partially transferred when the
bending radius is small. When the radius increases so does the interaction region, thus enhancing the power transfer, until 100% transfer is achieved at R ≈ 200 µ m for a d = 3 µ m gap. If
we continue increasing the radius, total transfer occurs at intermediate propagation distances,
from where the power would be transferred back to the first waveguide. The corresponding oscillations of the power back and forth between both curved waveguides along the propagation
direction in the interaction arc is clearly observed for a large value of the radius in Fig. 3(c).
The same pattern occurs when the gap is reduced and the radius is kept constant. For large
separations, the interaction is weak and only a fraction of the power is transferred between the
waveguides. When the gap is small enough (d ≈ 3 µ m for R = 200 µ m), the power is completely
transferred to the second waveguide. Smaller gap values produce several cycles of power transfer and, when the gap distance is smaller than a critical value, the suppression of one of the
hybrid modes leads to a featureless distribution of power on both sides of the system. In Fig.
3(a), we show that the power changes rapidly between both waveguides when we increase the
bending radius R or decrease the gap distance d. When the constitutive guiding structure is
#155664 - $15.00 USD
(C) 2012 OSA
Received 30 Sep 2011; revised 6 Jan 2012; accepted 10 Jan 2012; published 26 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 3156
5
20 (b)
(a)
1 (c)
d
d (µm)
d (µm)
d
0.75
7
10
P2(x)
R
R
R (µm)
400 10-2
0.5
P1(x)
R (mm)
100
0
−60 −40 −20 0 20
x (µm)
40
60
Fig. 3. (a) Transferred power at the exit of the system described in Fig. 2(a) as a function
of cylinder gap distance d and radius R for a wavelength of 1550 nm. (b) Same as (a)
for particle arrays (square lattice of period 1 µ m, particle radius 200 nm, and wavelength
4 µ m) arranged in a cylindrical geometry, as shown in the inset. (c) Normalized power in
each guide i, Pi , along the interaction length for a configuration of two silver cylinders with
radius R = 400 µ m and separation d = 2 µ m (see yellow point in (a)).
surrounded by a dielectric material, such as in Fig. 3(b), the modes become leaky below the
light line and the gap distance has larger critical values, and therefore the changes in the output
signal are slower and occur for larger values of the radius.
In summary, we have analyzed the dispersion relations of planar waveguides placed in close
proximity. The interaction produces repulsion between the modes of the individual waveguides.
The hybridized modes of the interacting structures propagate at different phase velocities, thus
producing a power transfer between both waveguides that depends on the traveled distance.
When the gap distance is varied adiabatically (e.g., by gently bending the waveguides so that
they form cylinders of large radius compared to the minimum gap separation between them),
a net power transfer is produced that exhibits a smooth dependence on bending radius and
minimum gap distance. Eventually, when the gap is below a critical value, only a single mode is
able to propagate inside the waveguide and the system sharply changes to an equally distributed
power output. The present study is relevant for advancing in the in/out coupling problem, which
is still a major issue in the design of integrated optics waveguides based upon plasmons and
particle arrays.
Acknowledgments
We acknowledge support from the European Union (NMP4-2006-016881-SPANS, FP7ICT-2009-4-248855-Nanophotonics4Energy, FP7-ICT-2009-4-248909-LIMA and NMP4-SL2008-213669-ENSEMBLE) and the Spanish MICINN (MAT2010-14885 and Consolider
NanoLight.es). X.M.B. acknowledges a JAE scholarship from CSIC.
#155664 - $15.00 USD
(C) 2012 OSA
Received 30 Sep 2011; revised 6 Jan 2012; accepted 10 Jan 2012; published 26 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 3157

Download Report