A highly efficient add-drop filter using a three-dimensional photonic crystal

The Internet is the driver for modern communication, transporting an
increasing density of data. Much of this is being carried over optical
fibers
using wavelength division multiplexing (WDM), in which multiple
wavelengths are transported along the same optical fiber. At different points on
the fiber it is necessary to pull off (drop) individual wavelength channels for
end-users. Simultaneously it is necessary to add data streams into empty
wavelength channels. Waveguides
in two-dimensional photonic crystals (PCs) and ring resonators have been
extensively investigated as all-optical add-drop filters. We show that
three-dimensional photonic band gap crystals with complete band gaps can be
novel add-drop filters.
We used a microwave-scale layer-by-layer PC that has been extensively developed
at Ames Laboratory and Iowa State University with a complete band gap from 11GHz
to 12.9GHz, for all directions of wave propagation. The add-drop filter has an
entrance waveguide and exit waveguide created by removing rod segments. These
waveguides were separated by a dielectric rod of length d. We created a cavity
of length L one unit cell above the waveguide layer. The cavity-waveguide
interaction and the crosstalk
between the waveguides is controlled by the separation d.
Since all modes carried by the waveguide are within the photonic band gap, there
is no leakage of modes to the outside—an inherent problem with both
two-dimensional PCs and add-drop filters using ring resonators.
Figure
3
shows the results of transmission measurements for a straight
waveguide, two waveguides separated by d=8a without a cavity, and two waveguides
with separation d=8a and a cavity with L=0.75a, where a is the rod spacing for
the PC. The straight waveguide has a strong transmission band from 11.8 to
12.8GHz. By separating entrance and exit waveguides, the transmission is reduced
by >20dB over all frequencies. With the cavity, a narrow transmission peak
appears at 12.22GHz (see Figure
3
). The peak transmission of this mode is nearly that of the straight
waveguide, suggesting excellent coupling. The 12.22GHz mode is the resonant
frequency of a single mode cavity of length L =0.75a. We studied the coupling
for different cavity sizes (L) and waveguide separations (d). Each cavity has a
different resonant frequency. Larger cavities support more than one mode. The
larger multimode cavity (shown in Figure
2
) of size L=5.5a (and separation d=9a) exhibits three cavity modes with
three strong transmission peaks.
Finite difference time domain (FDTD) simulations provide an appealing physical
picture. Simulations used a 20-layer PC similar with two waveguide sections
coupled by a cavity of length L. Computational constraints necessitated using
smaller L and d (d=6a; L=3a) than in the experiments. The frequency response was
obtained by exciting the input guide with a pulsed dipole source and simulating
the transmitted E fields in the exit guide. Three simulated transmission peaks
were obtained similar to those measured,1 indicating resonant cavity modes that
couple the input and output streams.
After identifying resonant modes, we obtained a visual understanding by exciting
the input guide with a constant frequency source tuned to a resonance (12.5GHz)
and simulating the temporal evolution of the fields. Initially (1000Δt,
where the time step Δt=2.06ps), large fields exist only in the input
waveguide (as shown in Figure
1
). As time progresses (3000Δt), the fields grow within the cavity,
indicating input waveguide-cavity coupling. At a later time (5000Δt), the
intensity gradually builds in the exit guide. Still later (9000Δt), a large
excitation of the output guide is accompanied by the excitation of the cavity
(see Figure
1
). The slow coupling requires >5000 time steps to excite the cavity
and then couple to the exit guide.
Waveguides in three-dimensional PCs can couple through defect cavities. The
resonant frequency of the cavity can be selected from the input guide and
dropped to the output guide. These designs can be scaled down to
telecommunications wavelengths (1.5μ). Controlling the geometry of defect
cavities can lead to realistic novel add-drop filters for telecommunications
applications.
Note for Wavelength-division multiplexing

In fiber-optic communications, wavelength-division multiplexing (WDM) is a
technology which multiplexes multiple optical carrier signals on a single
optical fiber by using different wavelengths (colours) of laser light to carry
different signals. This allows for a multiplication in capacity, in addition to
enabling bidirectional communications over one strand of fiber." This is a
form of frequency division multiplexing (FDM) but is commonly called wavelength
division multiplexing."
The term wavelength-division multiplexing is commonly applied to an optical
carrier (which is typically described by its wavelength), whereas
frequency-division multiplexing typically applies to a radio carrier (which is
more often described by frequency). However, since wavelength and frequency are
inversely proportional, and since radio and light are both forms of
electromagnetic radiation, the two terms are equivalent.

Note for Finite-difference time-domain

Finite-difference time-domain (FDTD) is a popular computational
electrodynamics modeling technique. It is considered easy to understand and easy
to implement in software. Since it is a time-domain method, solutions can cover
a wide frequency range with a single simulation run.
The FDTD method belongs in the general class of grid-based differential
time-domain numerical modeling methods. The time-dependent Maxwell’s equations
(in partial differential form) are discretized using central-difference
approximations to the space and time partial derivatives. The resulting
finite-difference equations are solved in either software or hardware in a
leapfrog manner: the electric field vector components in a volume of space are
solved at a given instant in time; then the magnetic field vector components in
the same spatial volume are solved at the next instant in time; and the process
is repeated over and over again until the desired transient or steady-state
electromagnetic field behavior is fully evolved.

Note for Photonic crystals

Photonic crystals are composed of periodic dielectric or metallo-dielectric (nano)structures
that affect the propagation of electromagnetic waves (EM) in the same way as the
periodic potential in a semiconductor crystal affects the electron motion by
defining allowed and forbidden electronic energy bands. Essentially, photonic
crystals contain regularly repeating internal regions of high and low dielectric
constant. Photons (behaving as waves) propagate through this structure – or not
– depending on their wavelength. Wavelengths of light (stream of photons) that
are allowed to travel are known as "modes". Disallowed bands of
wavelengths are called photonic band gaps. This gives rise to distinct optical
phenomena such as inhibition of spontaneous emission, high-reflecting
omni-directional mirrors and low-loss-waveguiding, amongst others.
Since the basic physical phenomenon is based on diffraction, the periodicity of
the photonic crystal structure has to be of the same length-scale as half the
wavelength of the EM waves i.e. ~200 (blue) to 350 (red) nm for photonic
crystals operating in the visible part of the spectrum – the repeating regions
of high and low dielectric constants have to be of this dimension. This makes
the fabrication of optical photonic crystals cumbersome and complex.

Pictures overview

Figure 1. E-field intensities from FDTD simulation in the plane
containing the input waveguide, the exit waveguide, and the cavity. Simulations
are for the resonant frequency of 12.5GHz at 1000, 3000, 5000, and 9000 time
steps. The intensity scale is logarithmic.

Figure 2. Measured transmission for a cavity of length L=5.5a (dashed)
displaying three resonant frequencies, compared to the straight waveguide and no
cavity (d=9a).

Figure 3. Measured transmission for the straight waveguide (solid)
compared to transmission for waveguides separated by 9a, with (dotted) and
without (dashed) a cavity of L=0.75a. The cavity-induced resonance (arrow) is
∼1dB below the straight guide. 

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