Coupled Mode Analysis in Nano-photonics
In physical systems composed of a number of electrons or photons, couplings between
eigenstates of particles make a critical contribution for many phenomena distinctive from an
isolated state. To investigate these coupling phenomena, the coupled mode theory (CMT) which defines the relation between states (or modes) with self-evolution and coupling terms
- has been studied intensively in many fields. For example in optics, by assigning each optical
component into ‘resonator’ or ‘waveguide’ and deriving equations between them, the
behavior of light in complex systems can be well-described without full solving of the
Maxwell’s equations, for general platforms such as dielectric waveguides / microdisk,
photonic crystals or plasmonics. Here we will study the CMT, which can be applied
successfully not only to the classical optics, but to nano-photonics, with two examples.
1. (Side-Coupled Resonator) As a simplest example, we will consider the optical system
composed of an ideal lossless resonator coupled with a single waveguide (Fig. 1). For the
simplicity, we assume that the wave in the waveguide is coupling into the resonator in a
specific point with a same rate for the left and right section of the waveguide. Here, ω0 / a
denotes the resonance frequency / energy amplitude of the resonator, while S+k (S-k) depicts
the amplitude of input (output) wave along the waveguide (k=1 for the left section and k=2
for the right section). Because of the coupling between the waveguide and the resonator,
the a of the resonator is decayed into the waveguide with a life time of τ0 for each section.
In this case, the time derivative of the resonator energy amplitude is described as,
d
1 1
a  (i 0   )a   ( S 1  S  2 ) .
(1)
dt
0 0
From the phase relation between waves, the two unknown waves (S-1, S-2) have the relations
as,
S1  S2  a ,
S2  S1  a .
(2)
(3)
Fig. 1
1-1. To obtain the solution of the equations, it is necessary to obtain the relation between α,
γ, and τ0. Find the relation between α and τ0, by considering the law of energy
conservation, and the relation between γ and τ0, by considering the time-reversal
symmetry. (See C. Manolatou, et al., IEEE Journal of Quantum Electron., 35, 1322 (1999) for
more details)
1-2. Plot the frequency dependent reflection (|S-1/S+1|2) and transmission (|S-2/S+1|2) when
the light is excited only through a left port (S+1=1, S+2=0) for the coupling Q-factor Qcoup= 103,
104, and 105 (Qcoup= ω0∙τ0 /4). The resonance frequency is ω0=2π∙(200THz).
1-3. In a realistic condition, the total quality factor (Qtotal) of the resonator is defined by the
contribution of intrinsic Q-factor (Qint) and the coupling Q-factor (Qcoup) as 1/Qtotal = 1/Qint +
1/Qcoup. Because the intrinsic Q-factor (Qint) of the resonator is not infinite due to the
imperfect fabrication and material absorption, it is necessary to consider in a real design.
Include the intrinsic loss into the above coupled mode equation (Eq. 1-3) and repeat 1-2
for Qint=104.
1-4. (Optical Nonlinearity) Based on the Kerr optical nonlinearity, the refractive index is
changed dependent on the intensity of the light. By changing the resonance frequency
dynamically, this effect can be utilized to realize the ‘all-optical switch’. If the resonance
frequency of the Fig. 1 is expressed as ω=ω0-ρ∙ω0∙|a|2 (ρ: Kerr coefficient, = 10-8), find the
transmitted power |S-2|2 with the variation of S+1=0~100 (the frequency of the incident wave
is ωin=1.001∙ω0, Qcoup= 103, S+2=0, Qint=∞).
2. (Dynamic Q) Dynamic control of the device bandwidth is an attractive topic in nanophotonics. For example, there was a work for stopping light all-optically in photonic crystal
platform by changing the bandwidth dynamically (M. F. Yanik, S. Fan, Phys. Rev. Lett., 92,
083901 (2004)). For the dynamic control of the bandwidth, here we provide the simple
example of controlling Q of the photonic crystal resonator (Y. Tanaka, et al., Nature
Materials, 6, 862 (2007)).
Fig. 3 shows the photonic crystal dynamic Q structure and its CMT model. As can be seen,
the resonator is side-coupled for the photonic crystal waveguide which is blocked by the
hetero-interface mirror (Across the mirror interface, the lattice constant is different realizing
the perfect mirror). By exerting the pump pulse on the waveguide section between the
resonator and the mirror which changes the refractive index of the right section of the
waveguide, now we can change the quality factor of the resonator.
2-1. The intrinsic Q of the resonator is Qv and the coupling Q (or in-plane Q) without mirror
is Qin0. By exciting the pump pulse, the phase difference θ between the backward wave from
the resonator and the forward wave from the resonator which is reflected by the mirror, is
changed. Find the Qtotal with Qv, Qin0, and θ, by applying CMT.
Fig. 2 (Left) Photonic crystal structure for dynamic Q (Right) CMT schematics for the left
structure
2-2. Plot the Qtotal dependent upon θ (0≤θ≤π) for Qv=104, 105, and 106 when Qin0=103 and
104.
2-3. For realizing the perfect mirror, the hetero-interface structure (with different lattice
constant) is utilized. Why didn’t they use the photonic bandgap mirror?