Figures from ch. 9 of Photonic Crystals
Steven G. Johnson, MIT course 18.369
May 3, 2010
(a)
(b)
(c)
a
a
R
Figure 1: Three examples of photonic-crystal fibers. (a) Bragg fiber, with a onedimensionally periodic cladding of concentric layers. (b) Two-dimensionally periodic
structure (a triangular lattice of air holes, or “holey fiber”), confining light in a hollow core
by a band gap. (c) Holey fiber that confines light in a solid core by index guiding.
1
1.6
Frequency ωa/2πc
Frequency (ωlight-line ω)a/2πc
0.045
0.04
0.035
0.03
photonic crystal
light-cone ω
1.2
light-line
0.8
guided mode
0.4
0
0
fundamental
guided mode
0.025
0.4
0.8
1.2
1.6
2
Wave vector kza/2π
0.02
0.015
0.01
higher-order
guided modes
0.005
0
0
2
4
6
8
10
12
14
16
18
20
Wave vector kza/2π
Figure 2: Band diagram of solid-core holey fiber as a function of axial wave vector k z . The
usual ω plot is inset, but for clarity we also plot the ∆ω between the guided bands and
the light line. The higher-order guided modes are three bands that are nearly on top of
one another.
y
x
neg
pos
Ex
neg
pos
Ey
Figure 3: Electric-field pattern for the doubly degenerate fundamental mode of figure 2.
Their polarizations are nearly orthogonal everywhere: the mode pictured at left is mostly
Ex , and the mode pictured at right is mostly Ey . The green circles show the locations of
the air holes.
1.45
Effective Index ckz/ω
1.43
1.41
1.39
photonic crystal
light-cone
1.37
1.35
1.33
1.31
1.29
0.1
1
10 13
Wavelength λ/a = 2πc/ωa
Figure 4: Bands of the solid-core holey fiber, from figure 2, plotted as “effective index”
ck z /ω versus wavelength, along with the light cone (gray region). In the limit of small λ,
the effective indices of both the modes and the light line approach the index 1.45 of the
bulk silica.
2
1.8
Frequency ωa/2πc ~ kt
1.6
1.4
1.2
1
0.8
0.6
M
K
0.4
Γ
0.2
0
Γ
M
K
Γ
Figure 5: Band diagram for TM-polarized modes of a two-dimensional triangular lattice
of perfect-metal cylinders. Two band gaps are shown (shaded yellow), where the lowest
band has a low-frequency cutoff characteristic of metallic structures. These bands are
equivalent to the modes of the holey fiber in the scalar (large k z ) limit.
3
photonic crystal
light cone
Frequency ωa/2πc
2.5
2
1.5
1
=
ck z
ω
0.5
0
0
0.5
1
1.5
2
2.5
3
Wave vector kza/2π
Figure 6: Projected band diagram, as a function of out-of-plane wave vector k z , for a
triangular lattice of air holes (inset: period a, radius 0.47a) in ε = 2.1. This forms the light
cone of the holey fiber from figure 1(b), with gaps appearing as open regions. The light
line of air, ω = ck z , is shown in red. (Dashed box indicates region plotted in figures 9 and 11
for the defect modes.)
2
Frequency ωa/2πc
1.9
1.8
1.7
1.6
M
1.5
K
Γ
1.4
Γ
M
Γ
K
Figure 7: Band diagram versus in-plane wave vector in the irreducible Brillouin zone (inset) for the triangular lattice of air holes from figure 6, at an out-of-plane wave vector
k z a/2π = 1.7. Gaps are shaded yellow: the lower gap corresponds to the index-guiding
region, and the upper gap corresponds to one of the band gaps inside the light cone
where guiding in an air core is possible.
silica glass
air
8.1microns
3.
ns
cro
i
2m
10 microns
Figure 8: Electron-microscope image of hollow-core holey-fiber cross section (black regions are air holes, and gray regions are silica glass). Central air defect, replacing 7 holes,
supports gap-guided modes around a wavelength of 1060 nm. (Image courtesy Karl Koch
and Corning, Inc.)
2
photonic crystal light cone
Frequency ωa/2πc
1.9
1.8
1.7
1.6
1.5
=
ck z
ω
1.4
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Wave vector kza/2π
Figure 9: Band diagram showing guided modes of hollow-core holey-fiber structure (inset,
similar to experimental structure of figure 8), corresponding to dashed region of figure 6.
(Air core is formed by a radius-1.202a air hole.) Three thick red lines indicate doubly degenerate bands that have the correct symmetry to couple to planewave input light. Thin
green lines indicate doubly degenerate bands with a different symmetry, and thin blue
lines indicate nondegenerate bands. Bands below the light line (thick black) are surface
states confined to the edge of the core. Three dots indicate the modes plotted in figure 10.
(a) ωa/2πc = 1.63
(b) ωa/2πc = 1.66
(c) ωa/2πc = 1.68
kza/2π = 1.6
kza/2π = 1.7
max
0
Intensity
Figure 10: Intensity patterns (ẑ · Re[E∗ × H]) of three doubly degenerate modes of a
hollow-core holey fiber (ε shaded green), corresponding to the dots on the thick red lines
in figure 9. (a) and (b) lie above the air light line at k z a/2π = 1.6, while (c) is a surface
state lying below the air light line at k z a/2π = 1.7.
2
photonic crystal
light cone
Frequency ωa/2πc
1.9
1.8
1.7
1.6
1.5
=
ck z
ω
1.4
1.3
max
0
Intensity
1.4
1.5
1.6
1.7
1.8
1.9
2
Wave vector kza/2π
Figure 11: Band diagram and mode intensity patterns (insets) of a hollow-core holey fiber
with a larger air core (radius 1.4a, left inset) corresponding to a crystal termination that
eliminates surface states. Thick red line (lower) indicates doubly degenerate mode with
correct symmetry to couple to input plane waves. Thin blue and green lines indicate
other symmetries (blue is nondegenerate and green is doubly degenerate); there are
actually two blue lines that are difficult to distinguish here. Intensity patterns are shown for
k z a/2π = 1.6, indicated by the dots. (The four higher-order modes all have similar intensity
patterns; a nondegenerate mode is shown.)
1 micron
100 microns
10 microns
Figure 12: Electron-microscope image of a hollow-core omnidirectional-mirror Bragg-fiber
cross section, with insets showing enlarged views of multilayer structures. Thin white layers
are a chalcogenide glass and gray regions are a polymer. This fiber was designed to
operate at a wavelength of 10.6 µm (?).
1
z
Frequency ωa/2πc
0.8
0.6
0.4
omnidirectional
TM
0.2
TE
TM & TE
0
0
0.2
0.4
0.6
0.8
1
Wave vector kza/2π
Figure 13: Projected band diagram for glancing-angle quarter-wave stack of indices n1 =
1.6 and n2 = 2.7, plotted versus wave vector k z parallel to layers (inset). Regions where
propagating modes exist in the stack are shaded: blue for TM (E out of the incidence
plane) and red for TE (E in the incidence plane), or purple for both. The frequency range
of omnidirectional reflection from an air medium (corresponding to the black light line
ω = ck z ) is shaded yellow. (Note that this “TM/TE” terminology is the reverse of the “te/tm”
terminology used later in the chapter.)
Frequency ωa/2πc
0.4
(b)
(a)
eh21
0.3
eh11te01
tm11te01
tm21
he31
0.2
he21
tm01
te31
he11
te21
tm01
0.1
te11
6a
6a
metal
0
0
0.1
0.2
0.3
Wave vector kza/2π
0.4
0
0.1
0.2
0.3
0.4
Wave vector kza/2π
Figure 14: (a) Band diagram of hollow-core ( R = 3a) Bragg fiber with layer indices of
2.7/1.6 (blue/green layers in inset); gray regions indicate extended modes that propagate
in the multilayer mirrors. (b) Band diagram showing the first few bands of a hollow perfectmetal waveguide of the same radius; light gray regions indicate extended modes of Bragg
mirrors from (a), for comparison. Modes are labelled by polarization and radial/angular
mode numbers as described in the text.
he11
te01
max
0
Intensity
Figure 15: Intensity patterns for two core-guided modes in hollow-core Bragg fiber of figure 14 at the quarter-wave frequency. Power intensity is shown by color scale, and transverse ( xy) electric-field pattern is shown by green arrows.
Absorption suppression factor α/α0
1
0.9
0.8
6a
0.7
0.6
0.5
tm01
0.4
eh11
0.3
te01
0.2
he11
0.1
0
0.22
0.24
0.26
0.28
0.3
0.32
0.34
Frequency ωa/2πc
Figure 16: Absorption suppression factor α/α0 of R = 3a Bragg fiber from figure 14(a)
versus frequency for four modes. α/α0 is the radio of the modes absorption losses to the
absorption losses for the bulk material, assuming that the absorption losses are dominated
by those of the low-index material (n = 1.6, shaded green in inset). Here, the he11 (fundamental) mode is the lowest loss, while for larger R the te01 mode becomes the lowest.
Absorption suppression factor α/α0
10
10
10
0
-1
eh11
-2
tm01
10
-3
1/R3
10
he11
-4
te01
6a
10
10
-5
-6
10
0
10
1
Core radius R/a
Figure 17: Scaling of the absorption suppression factor α/α0 versus core radius R, at
quarter-wave frequency, for several modes of the hollow-core Bragg fiber with indices
2.7/1.6 (blue/green in inset). The absorption loss is assumed to be dominated by that of
the low-index material. All modes asymptote to a 1/R3 scaling (shown as a black line, for
reference).