Revisiting the absorption and transmission
properties of coupled open waveguides
LEI CHEN AND KENG C. CHOU*
Department of Chemistry, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
*Corresponding author: kcchou@chem.ubc.ca
Received 26 April 2018; revised 20 August 2018; accepted 31 August 2018; posted 5 September 2018 (Doc. ID 330305);
published 12 October 2018
Open waveguides are widely used in modern photonic device s, such as microstructured fiber filters and sensors.
Their absorption and transmission spectra are the most important properties in determining the overall perfor-
mance of the photonic devices. The imaginary parts of their eigenvalues have been commonly used to calculate the
absorption and consequently the transmission spectra. Here we show that this formulism is generally incorrect
and not consistent with the simulation results obtained by the beam propagation method. We revisit the
fundamental theory for the absorption of open waveguides and present a general formulism. We found that
parity-time-symmetry transitions, which have been conventionally ignored, play a critical role in the properties
of the coupled waveguide. The absorption and transmission are highly dependent on the physical length of the
system. On the basis of our findings, optimization criteria for designing photonic sensors and filters are
presented.
© 2018 Chinese Laser Press
https://doi.org/10.1364/PRJ.6.001003
1. INTRODUCTION
Many photonic crystal fiber (PCF) devices, such as PCF filters
[1,2] and sensors [3–5], utilize the coupling between the
dominant core and cladding eigenmodes to produce the desired
transmission resonance. In these devices, the refractive indices
(RIs) of the eigenmodes and the dispersion can be precisely
controlled by manipulating their geometrical structures. As a
result, the resonance wavelength is tunable and has a narrow
bandwidth, which is desirable for designing an optical device.
Conventionally, the absorption spectrum of a PCF is
calculated by Beer’s Law using the imaginary part of its eigen-
mode’s RI as the absorption coefficient [6–9]. In this formu-
lism, the dominant core eigenmode and dominant cladding
eigenmodes are decoupled because of symmetry protection
[10]. The symmetry protection originates from the orthogonal-
ity of the eigenmodes of a Hermitian Hamiltonian. However,
the assumption that these two eigenmodes are decoupled is not
generally correct for a non-Hermitian Hamiltonian [11]. When
these two eigenmodes are coupled, energy transfer between
these two modes occurs. In this case, the absorption can no
longer be described by the simple exponential decay (Beer’s
Law) because of the interference between these two eigen-
modes. As described below, our study shows that Beer’s Law
is not consistent with the simulation results obtained by the
beam propagation method (BPM).
In the current study, we develop a formulism to calculate
the absorption and transmission spectra for a non-Hermitian
Hamiltonian of an open optical system. The parity-time
(PT) symmetry of the system plays a critical role in the
aforementioned interference between the eigenmodes.
Interestingly, we found that the PT phase transition depends
on both the waveguide absorption and the input wavelength.
Consequently, the absorption and transmission may exhibit
dramatic changes, instead of a mono-exponential decay or gain
when light propagates along the waveguide. The current study
shows that adjusting the length of a waveguide is also critical for
optimizing its absorption and transmission properties. On the
basis of our formulism, several optimization criteria for design-
ing photonic crystal sensors and filters are presented. In
addition, we show that an increase in the imaginary part of the
eigenmode is a sufficient condition, not a necessary condition,
for an increased absorption.
2. NON-HERMITIAN COUPLED WAVEGUIDES
For simplicity, a PCF [Fig. 1(a)] with coupled core and clad-
ding modes [Fig. 1(b)] can be mathematically represented by a
two-core fiber [Fig. 1(c)] with two coupled modes, as shown in
Fig. 1(d). The Hamiltonian of the system or any coupled
waveguides can be written as (ℏ 1)
ˆ
H ωn
1
a
†
a ωn
2
b
†
b ga
†
b b
†
a, (1)
where ω is the angular frequency, n
i
are the complex mode RIs,
a and b are the annihilation operators for the core and cladding
modes, respectively, a
†
and b
†
are the creation operators for the
Research Article
Vol. 6, No. 11 / November 2018 / Photonics Research 1003
2327-9125/18/111003-05 Journal © 2018 Chinese Laser Press