SUPPLEMENTARY INFORMATION
S. He, Y.R. He and Y. Jin
Table of Contents
1. Calculation of reflection by mode matching with a complete set of modes
2. The case of input port with a thicker core layer
3. Field pattern and wave packet propagation in a lossy case
4. Negligible reflection in a tapered slow light photonic crystal waveguide
5. Animation for the propagation of a wave packet in a tapered lossless or lossy metamaterial
waveguide
6. Reflection arising from intermodal coupling in a flat slab waveguide with varying
refractive index
7. The calculation of momentum flow inside a metamaterial waveguide
1
1. Calculation of reflection by mode matching with a complete set of modes
The mode matching technique requires orthogonality and completeness of modes to describe
an arbitrary electromagnetic field in terms of orthogonal modes. Therefore, both point modes and
continuous modes should be taken into consideration for a rigorous calculation. One can add
either a PML (perfectly matched layer) or perfect electric conductor (PEC) boundary (which
should be far away enough, e.g., 300 m, from the waveguide boundary) on each side of the
waveguide to discretize the continuous modes. When the PEC boundary is used, analytical
expressions for modes are available. Let DPEC1 or DPEC2 denote the distance between the PEC
boundary and the surface of the LHM core layer. For example, the modes for core thickness d2=55
m with DPEC=300 m are shown in Fig. S1. There are 6 leaky modes on the real axis, besides 2
guided modes (one forward mode and one backward mode). As DPEC becomes large enough, these
quasi-continuous modes become dense enough to mimic the modes in open space (without PEC).
Figure S1. The complete set of modes. The LHM core thickness of the metamaterial slab waveguide is d=55 m.
Here the PEC boundaries are set at DPEC1 = DPEC2=300 m.
In Fig. S2a, a three-step tapered metamaterial waveguide is shown as an example for mode
matching calculation. The incident optical power carried by a forward mode (with wave vector f)
in the input waveguide will be channeled to different modes after it enters the tapered waveguide.
From Fig. S2b one sees that the optical power ratios oscillate at first when DPEC<100 m.
However, as DPEC increases further, the fluctuation becomes small, reaching a stable value in the
end. One also sees here that about 80% of the incident energy is coupled to the backward mode
(with negative group velocity but positive wave vector) and finally flow out from the input port
(as power reflection R12). The rest of the incident energy is mainly reflected directly by the steps,
denoted as R11, which will decrease if we taper the waveguide more gradually with more steps (as
shown in Fig. 2b).
2
Figure S2. An example for the calculation of reflection by the mode matching technique. (a) A three-step
tapered metamaterial waveguide. (b) Calculation results for reflection as DPEC (=DPEC1 = DPEC2) increases. Here din
and dout are chosen to be 55 m and 52.4 m (< dc=52.451813091371 m), respectively. The length of the middle
waveguide is 6 m.
Note that PEC becomes inappropriate when some forward modes become leaky in a tapered
metamaterial waveguide whose core-thickness is large at the left entrance (see Fig. S3 below).
2. The case of input port with a thicker core layer
Now we consider what would happen if the LHM core is very thick at the input waveguide. The
modal structure of Fig. S3a shows that: (1) the metamaterial waveguide may support some higher
order oscillating TMi modes when the LHM core thickness is larger than the corresponding critical
(i )
(2)
core thickness denoted by d c ( d c
is exactly the aforementioned critical thickness dc). (2)
Forward TMi mode (on a lower branch in Fig. S3a) will be cut off and become leaky when the
(i )
(2)
thickness of the LHM core layer exceeds d L . For example, d L
(denoted by dL for short) is
60.8 m for TM2 mode.
If we excite at the left input port the fundamental backward TM2 mode, whose wave vector is in
–z direction and energy flux in +z direction, the right-going backward mode (after passing Point
Q1 in Fig. S3a and the position with leakage thickness dL) will be coupled to a left-going forward
mode (on the lower branch of TM2 mode) near critical core thickness dc, due to the
3
aforementioned matching of wave vectors and modal profiles. The energy of this left-going
forward mode (near the position with critical core thickness dc) will then propagate along the –z
direction until reaching the position with leakage thickness dL, where the energy starts to leak
away into the RHM cladding. From Fig. S3b we clearly see energy leakage toward the RHM
cladding when the core thickness exceeds dL. To show that the energy of a left-going forward
mode will indeed leak to the RHM cladding in the above situation, we perform an additional
simulation to study the case when a left-going forward mode is excited from the right side of the
tapered section at the narrow end. Our simulation result in Fig. S3c evidently shows the leakage
behavior around the position of leakage thickness dL. The combination of Fig. S3b and Fig. S3c
shows clearly that mode transition from a right-going backward mode to a left-going forward
mode, and finally to leaky mode, will occur if the LHM core is thick at the input waveguide.
4
Figure S3. Coupling from a backward mode to a forward mode, and the leakage in a tapered metamaterial
waveguide whose LHM core is thick at the left entrance.
a, Various oscillating modes at c/2=1 THz. Modes
become leaky below the horizontal line, of which location is determined by the index of the cladding. b, c, FEM
simulation results when the LHM core of the left entrance waveguide is thicker than dL = 60.8 m. The incident
energy fluxes are along the +z and –z directions for b and c, respectively. Since the index of the upper cladding is
larger than that of the lower cladding, more power leaks towards the upper direction. For b, we choose din=100 m,
dout=52 m (<dc = 52.45 m), and the region with field magnitude larger than 0.1 is suppressed in order to show
the leaky field more clearly. For c, the core thicknesses of the narrow end and wide end in the tapered waveguide
are 55 m (between dc = 52.45 m and dL = 60.8m) and 100 m, respectively.
3. Field pattern and wave packet propagation in a lossy case
Since an LHM is usually accompanied with loss, here we consider the effect of loss on the
reflection/coupling properties of a tapered metamaterial waveguide.
First we consider the case of single wavelength (300 m), assuming 2=2=-5+0.002i. The
distribution of |Hy| is shown in Fig. S4a, and its plot along line x=0 is shown in Fig. S4b. Since
the imaginary parts of 2 and 2 are quite small, the incident forward mode can still get close to
the critical thickness region and form some local field enhancement. The left-going backward
mode is still excited as the field distribution shows a clear interference between the incident
(right-going) forward mode and the excited left-going backward mode. The excited left-going
backward mode undergoes attenuation and finally flows out of the input port. At the input port, the
interference effect becomes very weak since the backward mode has been attenuated to almost
zero (see Fig. S4b). We obtain less than 1% reflection (R12) at the input port. In this case, the weak
reflection is due to the intrinsic energy loss. Again, the light cannot be trapped at the critical
thickness in this case.
5
Figure S4. Field pattern in a tapered slow light waveguide when the LHM is lossy. Here we choose
2=2=-5+0.002i, and the other parameters are the same as those used for Fig. 2f and g. (a) Distribution of field
magnitude |Hy|. (b) Plot of |Hy| along line x=0.
The backward coupling in a lossy case can be shown more clearly by simulating the evolution
of a guided wave packet. As a numerical example, we choose =10-4c for the loss of the
metamaterial. The results calculated with the mode matching method are shown in Fig. S5a-d and
Animation S2, from which one sees that the incident optical pulse dissipates as it propagates in the
tapered metamaterial waveguide, and in the mean time is coupled to some backward mode near
the critical thickness for each frequency component. The excited backward mode (with positive
wave vector) then propagates to the right with attenuation, and finally flow out of the input port
with much lower (as compared to the lossless case) optical energy. The reflection behavior can
still be observed (see Fig. S5a for the trajectory of the wave packet peak). Obviously, light could
not be trapped in a lossy case either.
6
Figure S5. Wave packet propagation and reflection in a tapered slow light waveguide when the metamaterial
is lossy with =10-4c. The length of the tapering section is 30c. The LHM core thickness of the input and output
waveguides are 55 m and 52.2 m, respectively. (a) The peak location of |Hy| at different times (in the unit of
wavelength after multiplying with light speed c in free space). (b) – (d) The distribution of field magnitude |Hy| at
three different times (corresponding to the three times marked in a) after the wave packet enter the tapered
waveguide.
4. Negligible reflection in a tapered slow light photonic crystal waveguide
Here we use a tapered slow light photonic crystal (PhC) waveguide to show that the large
reflection in the tapered slow light LHM waveguide we studied before is not due to the tapering,
but the coupling between the forward and backward modes near the critical thickness.
The slow light photonic crystal waveguide is formed in the following way. First we remove
one row of cylinder holes from a triangle-lattice PhC, which has a background (relative)
permittivity of 3.6 (GaAs) with cylinder holes (radius r=0.31a, a is the lattice constant). Then we
displace Row A with 0.1a along the basic vector direction (indicated by the arrow in Fig. S6a),
and increase the radius of the holes to 0.35a. Finally, we displace Row B with 0.3a along the basic
vector direction. Here we define dgap as the distance between the facing edges of the two cylinders
in Rows A and B at the position of interest. A zero-group-velocity point appears at a frequency of
a/=0.2496 when the radius of holes in Row B is adjusted to 0.3557a (corresponding to dgap =
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0.8531a), see Fig. S6b and d. To taper the slow light PhC waveguide, we change the radii of the
holes for Row B linearly from 0.3538a at the input port (corresponding to dgap=0.855a) to 0.3568a
(corresponding to dgap = 0.852a) at the output port. The overall length of the tapered waveguide is
100a. From the dispersion curve Fig. S6c one sees that backward coupling will not occur even at
the zero-group-velocity point, which separates the upper and lower branches of modes. The optical
power transmission and reflection at the zero-group-velocity frequency of a/=0.2496 for a
tapered PhC waveguide (the input port has dgap=0.855a) are shown in Fig. S6e as the ending dgap
varies. From this figure one sees that nearly all the optical power transmits through the tapered
photonic crystal waveguide, instead of reflecting back.
Figure S6. (a) A tapered slow light photonic crystal waveguide. (b) Band structure for a photonic crystal
waveguide when the radius of all holes in Row B is fixed to 0.3557a (corresponding to dgap=0.8531a). A
zero-group-velocity point appears at a frequency of a/=0.2496. (c) The propagation constant as dgap varies at
frequency a/=0.2496. (d) The group velocity as dgap varies at the frequency of a/=0.2496. (e) The reflection,
transmission, loss for a tapered slow light PhC waveguide as the ending dgap varies at the output port (the input
8
port has dgap =0.855a).
This example also shows one of the ways to eliminate the reflection in a tapered slow light
waveguide through engineering the dispersion of the waveguide.
5. Animation for the propagation of a wave packet in a tapered lossless or lossy metamaterial
waveguide
Animation S1. Animation for the propagation and reflection (due to the coupling between the
forward and backward modes) of a wave packet in a tapered lossless metamaterial waveguide.
Detailed parameters and analysis are the same as those used for Fig. 4.
Animation S2. Animation for the propagation and reflection (due to the coupling between the
forward and backward modes) of a wave packet in a tapered lossy metamaterial waveguide.
Detailed parameters and analysis are the same as those used for Fig. S5 in Supplementary
Information.
6. Reflection arising from intermodal coupling in a flat slab waveguide with varying
refractive index.
We claim that the reflection is due to the physical intermodal coupling, rather than the inexact
numerical modeling of the tapered boundary. To verify this, we perform additional simulation on a
flat slab waveguide with gradually varying material parameters [see FDTD analysis of slow light
propagation in negative-refractive-index metamaterial waveguides, J. Opt. A: Pure Appl. Opt. 11,
114027 (2009)]. The flat boundary can be exactly modeling in a numerical simulation. By varying
gradually the refractive index of core layer n from -1.4 to -1.37, (the group velocity becomes zero
when n = -1.38) we obtain again the standing-wave characteristics for field magnitude |Hy| due to
strong reflection of incident energy, as shown in Fig. S7a. Both the mode matching and FEM
calculations are performed, and the two results show excellent agreement (see Fig. S7b). The
small values of non-zero minima are due to the difference in the modal profiles of the incident
forward mode and the reflected backward mode.
Figure S7. Calculation results on slow light waveguide with slow-varying core layer index at =200 μm. The
cladding layer is air, while the index of the core layer varies from -1.4 to -1.37 along an index-grading length of
100. The thickness of the core layer is fixed to 190 μm during the calculation, and thus the interface can be
exactly modeled. a, Field distribution for the magnitude |Hy|. b, The normalized magnitude |Hy| calculated with
FEM and mode matching method (MMT).
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7. The calculation of momentum flow inside a metamaterial waveguide
a) Consistent formulas for momentum in a dispersive medium.
Recent research shows that the momentum density and momentum flow density for
electromagnetic wave inside a medium contains two parts, the electromagnetic part and
mechanical part. The electromagnetic part momentum flow density is,
pem  gemvg
(S1)
where g em   0 0 E  H .
To calculate the mechanical momentum flow, we should first construct a wave packet using
two spectrum components a and b , whose frequencies are close to each other (i.e.,
a  b ). Then the mechanical momentum flow density is obtained using the
wave-packet-integral method, namely,
p mech 
The expressions for f
mech
and
1

vg t
0
0
 
f mech dzdt
(S2)
 are
f mech  ( P) E  0 ( M ) H  0 ( P / t )  H  0 0 ( M / t )  E
(S3)
  2 / (a  b )
(S4)
The readers can find more detail on the theoretical analysis and calculation concerning the
momentum in medium in the following paper (and references therein): Consistent formalism for
the momentum of electromagnetic waves in lossless dispersive metamaterials and the conservation
of momentum, Progress In Electromagnetics Research, vol. 116, 81-106 (2011).
b) Application of the momentum calculation formula to the metamaterial waveguide.
The uniform slab waveguide we studied here has the following parameters: 1=2.25 (relative
permittivity for upper cladding), 1=1, 2=2=1−p2/(2+i) (for the LHM core) with
p  6c , c/2=1 THz (corresponding to c=300 m), 3=2.56 (relative permittivity for
lower cladding), 3=1.
The field components for TM mode in the metamaterial waveguide are
e k1x , x  0

hy  H 0 cos(k2 x)  B sin(k2 x), 0  x  d
  k3 x
C e , x  d
10
(S5)
  k1x
e , x0

k

0
1

0  
ex 
H 
[cos(k2 x)  B sin(k2 x)], 0  x  d
 0 0  k 0 2
 
C e k3 x , x  d

 k0 3
 k1
e k1x , x  0

 ik01
0  k2
ez 
H 
[sin(k2 x)  B cos(k2 x)], 0  x  d
 0 0  ik0 2
 k3
C e  k3 x , x  d

ik

 0 3
(S6)
(S7)
where k0 is the free space wave vector, β is the propagation wave vector along the waveguide and
k1   2  k0211 , k2  k02 2 2   2 , k3   2  k02 33 . The z dependence exp(iβz) is
omitted in the above expression.
Using the boundary condition of continuity of the tangential field components, we obtain,
B
k1 / 1
k2 /  2
(S8)
C  e [cos(k2 d )  B sin(k2 d )]
k3d
The propagation wave vector β is obtained by solving the characteristic equation
tan(k2 d ) 
(k2 /  2 )(k1 / 1  k3 /  3 )
(k2 /  2 ) 2  k1k3 / 1 3
(S9)
The EM field inside our waveguide with two spectrum components a and b can be
expressed as,
hy 2  hya sin(  a z  a t )  hyb sin( b z  bt )



ex 2  ex a sin(  a z  a t )  ex b sin( b z  bt )

a
b
ez 2  ez sin(  a z  a t )  ez sin( b z  bt )


(S10)

where hy i , ex i , ez i is the field expression hy , ex , ez at frequency ωi, i=a, b.
em
The z direction electromagnetic momentum flow across the lateral cross section, Pz , is
em
obtained by integrating the electromagnetic momentum flow density p z
Pzem  


pzem dx  


gem vg dx 
11
d 1
d  c2



over x,
ex 2 hy 2 dx
(S11)
The mechanical momentum flow density should be calculated using eq. (2).
Take the term f e  0 (Px / t ) H y for example,
f e  0 0 (a *( a  1)exa cos(  a z  a t )  b *( b  1)exb cos(  b z  bt ))
*( hya sin(  a z  a t )  hyb sin( b z  b t ))
p fe 

vg t
0
0
1
 
In the limit
f e dzdt  0 0 [a ( a  1)exa hyb  b ( b  1)exb hya ] / (  a   b )
(S12)
(S13)
a  b   we obtain the momentum flow density corresponding to the
term f e  0 (Px / t ) H y ,
p fe  0 0
d [ (  1)ex / hy ]
d (  a  b )
hy2 / 2
(S14)
Similarly, the momentum flow density concerning the term f m  0 0 (M y / t ) Ex is
p fm  0 0
d [ (   1)hy / ex ]
d (  a  b )
ex2 / 2
(S15)
In the calculation of momentum flow regarding to the term f p  (  P) Ez , we notice that
(  P) is zero everywhere except the boundaries between metamaterial core and conventional
material cladding. Then we obtain the momentum flow corresponding to this term as follows,


1

d [( 3  1) ]
 1) ]
2
k3 k3
2
k1 k1
Pf 
(
) 0 H 02 / 2 
(
C e  k d ) 0 H 02 / 2
d (  a  b ) k01
d (  a  b ) k0 3
d [(
3
p
(S16)
We should bear in mind that the momentum flow across the lateral cross section is obtained
after integrating the momentum flow over x. Thus, the total momentum flow across the lateral
cross section of this waveguide is
Pztot  Pzem  


tot
This expression [eq. (S17)] for Pz
p fe dx  


p fm dx  Pf p
(S17)
(the overall momentum flow along the z direction) is used to
obtain the data shown in Fig. 5.
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