Chapter 2
Multimode Interference Theory
§2.1 Wave theory of dielectric slab waveguide [37-40]
§2.1.1 Maxwell equations
An optical wave is an electromagnetic field, which can be described by
Maxwell’s equations as
r
r
∂B
∇× E = −
,
∂t
r
v r ∂D
∇× H = J +
,
∂t
r
∇⋅ B = 0,
r
∇⋅D = ρ .
(2.1)
(2.2)
(2.3)
(2.4)
The principle of conservation of charge can be deduced by (2.2), (2.4) and
r
∇ ⋅ (∇ × A) = 0 as
r ∂ρ
∇⋅J +
= 0.
∂t
Poynting vector S (W/m2) is
r r r
S = E×H .
The constitutive relations (or material equations) in isotropic media are
15
(2.5)
(2.6)
r
r
D = εE ,
r
r
B = µH ,
r
r
J = σE ,
(2.7)
(2.8)
(2.9)
where ε, µ and σ are permittivity, permeability and electric conductivity. The
refractive index is n = ε r µ r , where ε r = ε / ε 0 and µ r = µ / µ 0 are relative
permittivity and relative permeability, respectively. Usually,
µ = µ 0 for
dielectric media, so n = ε r .
In the isotropic media without considering electric charges, the wave
equation can be deduced from the Maxwell’s equations as
r
r
2
r
∂
E
∂
E
∇ 2 E − µ 0σ
− µ 0ε 2 = 0 ,
∂t
∂t
r
r
2
r
∂
H
∂
H
∇ 2 H − µ 0σ
− µ 0ε 2 = 0 .
∂t
∂t
For a plane wave E = Re[e( x, y, z ) exp( jωt )] , (2.10) can be written as
r
r
∇ 2 E + (k 2 − jωµ 0σ ) E = 0 ,
r
r
∇ 2 H + (k 2 − jωµ 0σ ) H = 0 .
(2.10a)
(2.10b)
(2.11a)
(2.11b)
r
r
The phase velocity and group velocity are ν p = ω / k and ν g = ∂ω / ∂k ,
respectively.
Border conditions between two media are:
1) Electric field in transversal direction: E1t=E2t
2) Magnetic field in transversal direction: H1t=H2t
3) Electric density in perpendicular direction: D1p=D2p
4) Magnetic density in perpendicular direction: B1p=B2p
§2.1.2 Reflection and refraction
We consider a light k1 propagating from medium1 to medium2 as shown in
Fig. 2.1. The transmission k2 and reflection k3 occur at the interface between
medium1 and medium2.
16
y
k2
Medium n2
Medium n1
θ2
θ1
z
θ3
k1
k3
Fig. 2.1 Optical beams at the interface of two different media
Considering δ/δx=0, we can write the Maxwell’s equations as
∂E x
 ∂E x
 ∂z = − jωµ 0 H y , ∂y = jωµ 0 H z

− j ∂H z ∂H y

−
Ex =
(
)

∂z
ωε 0 n 2 ∂y
(2.12a) For senkrecht wave,
∂H x
 ∂H x
2
 ∂z = jωε 0 n E y , ∂y = − jωε 0 E z

j ∂E z ∂E y

Hx =
−
(
)

ωµ 0 ∂y
∂z
(2.12b) For parallel wave.
The senkrecht wave in medium 1 can be written as
r r
r r
 E x = E s1 exp(− jk1 ⋅ r ) + E s 3 exp(− jk 3 ⋅ r )
r r
r r

=
−
⋅
+
−
θ
θ
.
[
sin
exp(
)
sin
exp(
H
E
j
k
r
E
j
k
 y
s1
1
1
s3
3
3 ⋅ r )] / Z 1
r
r
 H = [− E cos θ exp(− jk ⋅ rr ) + E cos θ exp(− jk ⋅ rr )] / Z
s1
1
1
s3
3
3
1
 z
(2.13)
The senkrecht wave in medium 2 can be written as
r r
 E x = E s 2 exp(− jk 2 ⋅ r )
r r

 H y = E s 2 sin θ 2 exp(− jk 2r⋅ r ) / Z 2 .
 H = − E cos θ exp(− jk ⋅ rr ) / Z
s2
2
2
2
 z
The parallel wave in medium 1 can be written as
17
(2.14)
r r
r r
 H x = [ E p1 exp(− jk1 ⋅ r ) + E p 3 exp(− jk 3 ⋅ r )] / Z 1
r r
r r

θ
θ
E
E
j
k
r
E
j
k
=
−
−
⋅
−
−
⋅r) .
sin
exp(
)
sin
exp(
 y
p1
1
p3
3
r1
r3
 E = E cos θ exp(− jk ⋅ rr ) − E cos θ exp(− jk ⋅ rr )
p1
1
1
p3
3
3
 z
(2.15)
The parallel wave in medium 2 can be written as
r r
 H x = E p 2 exp(− jk 2 ⋅ r ) / Z 2
r r

 E y = − E p 2 sin θ 2 exp(− rjk 2 ⋅ r ) .
 E = E cos θ exp(− jk ⋅ rr )
p2
2
2
 z
(2.16)
where surge impedance (or characteristic impedance) is expressed as
Z i = E / H = µ 0 / ε 0 / ni , (i = 1,2) .
(2.17)
By the border conditions of E1t=E2t and H1t=H2t at y=0, we know
E s1 exp(− jk1z z ) + E s 3 exp(− jk 3 z z ) = E s 2 exp(− jk 2 z z ) ,
(2.18a)
− E s1 n1 cos θ1 exp(− jk1z z ) + E s 3 n1 cos θ 3 exp(− jk 3 z z ) = − E s 2 n2 cos θ 2 exp(− jk 2 z z )
.
(2.18b)
For arbitrary z, (2.18) exist. So the following relation should exist
k1z = k 2 z = k 3 z or n1 k 0 sin θ1 = n2 k 0 sin θ 2 = n1 k 0 sin θ 3 .
The law of reflection and Snell’s law can be obtained as
θ1 = θ 3
n1 sin θ1 = n2 sin θ 2 .
and
Fresnel’s relations can be obtained from (2.18) as
E s 3 n1 cos θ 1 − n 2 cos θ 2

=
=
r
s

E s1 n1 cos θ 1 + n 2 cos θ 2

E
2n1 cos θ 1
t s = s 2 =
E s1 n1 cos θ 1 + n 2 cos θ 2

(2.19)
for senkrecht wave.
By the same method, the reflection coefficient and transmission coefficient
can be written as
E p 3 n2 cos θ1 − n1 cos θ 2

=
r p =
E p1 n2 cos θ1 + n1 cos θ 2


E
2n1 cos θ1
t p = p 2 =

E p1 n2 cos θ1 + n1 cos θ 2
(2.20)
for parallel wave.
From (2.19) or (2.20), we can see the refection (rsrs* or rprp*) is zero at an
angle θB. The angle is called Brewster angle. For example, the Brewster angle is
18
560 for the incident ray from air to glass.
§2.1.3 Goos-Hanshen shifts
From Snell’s law, we know when a light input to a medium with a small
refractive index of n2 from a medium with a large refractive index of n1, and the
incident angle is larger than an angle θc, the total reflection occurs, as shown in
Fig. 2.2. The angle θc is critical angle defined as a sin( n 2 / n1 ) .
y
n2
ZGH
z
θ1 θ1
n1
θ1
Fig. 2.2 Goos-Hanshen shifts.
From Snell’s law, we know
cos θ 2 = ±
n2 − n1 sin 2 θ1
2
2
n2
2
,
(2.21)
where θ2 is the refractive angle in medium with the refractive index of n2.
For the case of total reflection with the condition of n1 sin θ1 > n2 , (2.21) can
be written as
cos θ 2 = − j
n1 sin 2 θ1 − n2
2
n2
2
2
.
(2.22)
It is worthy to note that the sign in (2.22) has to be minus. The transmission
in perpendicular direction shown in (2.16) is unlimited, if the sign is plus.
The reflection coefficient in (2.19) can be written as
n1 cos θ1 + j n1 sin 2 θ1 − n2
2
n1 cos θ1 − j n1 sin θ1 − n2
2
2
rs =
2
19
2
.
(2.23)
Obviously, the power of the reflected wave is 1. rs =1. So we can assume:
exp(−ϕ s ) = rs or exp(−ϕ s / 2) = n1 cos θ1 + j n1 sin 2 θ1 − n2 .
2
2
Then, the Goos-Hanshen shift for senkrecht wave can be written as
ϕ s = −2 tan (
−1
sin 2 θ 1 − sin 2 θ c
cos θ1
).
(2.24)
By the same method, the Goos-Hanshen shift for the parallel wave can be
written
ϕ p = −2 tan −1 (
sin 2 θ1 − sin 2 θ c
sin 2 θ c cos θ 1
).
(2.25)
The shift of reflection point can be written as
Z GH =
∂φ s , p
∂k z
2 tan θ 1


2
  k1 z 
2
  k  − n 2

0 
= 
2 tan θ1

  k   k    k
  1z  +  1z  − 1  1z
  k1   k 2    k 0
.
(2.26)
2

 − n 2 2

Note the Goos-Hanshen shift makes the phase faster, so the sign is minus,
although it looks that the Goos-Hanshen shift retards the phase in the Fig. 2.2.
§2.2 Dielectric optical waveguide [17-22]
The basic of multimode interference (MMI) is that multi modes exist in a
waveguide. The modes propagating in a waveguide can be described by Maxwell
equations or a ray optics method. Maxwell equations would give more information
about a mode behavior in an optical waveguide. However, the ray optics method
would give an explicit and direct image for understanding the modes in a
waveguide. Of course, many conclusions can be obtained by either of them.
§2.2.1 Mode conception
20
In the section, I show the modes concept by ray optics method. The light
propagating in a waveguide can be briefly understood that the light is confined in
the waveguide by total reflection as shown in Fig. 2.3. When a light with an
incident angle of ψ input the waveguide with a refractive index of nf from air, the
refractive angle θ1 can be written as
θ1 = a sin(sin(ψ ) / n f ) .
(2.27)
Clad nc
θ
Waveguide nf
θ1
T
ψ
Substrate ns
Fig. 2.3 Propagating ray in a waveguide.
The incident light in the waveguide has to be totally reflected at substrate
and clad interfaces. The critical angle of total reflection in clad and substrate
interfaces can be written as
θ c = sin −1 (nc / n f ) ,
(2.28)
θ s = sin −1 (n s / n f ) ,
(2.29)
respectively.
The condition to confine the light in the waveguide can be written as
θ1 < π / 2 − θ c , s .
(2.30)
When n s = nc , the maximum incident angle ψmax can be written as
ψ max = sin −1 (n f 2∆ ) ≡ N . A. (Numerical Aperture),
2
where ∆ =
n f − nc
2n f
2
2
.
21
(2.31)
The input lights can be classified into 3 kinds by incident angles. One is the
guided light (ψ < ψmax) confined within the waveguide without loss. The second is
the clad radiation light (ψ > ψmax) that radiating to the clad and being confined at
the substrate, given n s ≥ nc . The third is the substrate-clad radiation light
radiating to both of substrate and clad. Basically, my research is focused on the
guided modes propagating in a waveguide.
However, not all the lights matched the condition of ψ < ψmax can exist in a
waveguide, because they have to matched the stand-wave condition in transversal
direction of the waveguide as shown
Φ T = 2(m + 1)π , m = 0,1,K
(2.32)
ΦT is a transversal phase change of one circle trip. So, only the lights
matched (2.30) and (2.32) can exist in a waveguide. From (2.32) we can know the
lights are discrete, rather than continual. The discrete lights are called as modes.
Considering the Goos-Hanchen shifts in two interfaces of a waveguide, the
stand-wave condition (2.32) for the senkrecht wave can be written as
Φ T = 2n f Tk 0 cos θ + ϕ s + ϕ c = 2mπ
m = 0,1, K ,
(2.33)
or
n f sin 2 θ − n s
2
n f Tk 0 cos θ − tan (
−1
2
n f cos θ
n f sin 2 θ − nc
2
−1
) − tan (
2
n f cos θ
) = mπ , m = 0,1L .
(2. 34)
The discrete modes can be obtained by solving (2.34). For an SOI substrate
with the SiO2 layer of 1.0 µm-thick, the solutions of (2.34) are shown in Fig. 2.4
under various silicon layer thickness of 0.25 µm, 0.5 µm, and 1.0 µm, respectively.
The calculating parameters are shown in the following table.
Table.2.1 Waveguide parameters
Thickness, T(µm)
k0(1/µm)
nc
ns
nf
0.25/0.5/1.0
2π/1.55
1.46
1.46
3.46
From the calculated results we know there is a single mode (a fundamental
mode) for the thickness of 0.25 µm. The corresponding propagation angle is about
580. So the effective refractive index can be calculated by nfsin(580)=2.9342.
The calculated results can be summarized in table 2.2.
22
Table. 2.2 Calculated waveguide characteristics of a SOI substrate
Si thickness
Critical angle
0.25 µm
0.5 µm
250
1.0 µm
Mode numbers
Angles of mode 0
neff
1
580
2.9342
2
70.50
3.2615
4
79.50
3.4021
Usually the mode number in a MMI device is no less than 3 and no larger
than 8. The reasons are followings. When the mode number is 2, the waveguide
width is narrow, so the image spacing in transversal direction is narrow, which is
unexpected for some MMI devices such as splitters. When the mode number is
large, the complete phase matching between the modes is difficult as shown later.
Consequently, the image quality is poor. So, it is important to consider the mode
number in the designing process of an MMI device. The mode number can be
calculated by (2.34).
1.0µm
Mode 3
Phase (rad.)
0.5µm
Mode 2
Mode 1
0.25µm
Critical angle
Propagation angle of θ (degree)
Fig. 2.4 Discrete modes in SOI substrates under various thicknesses
§2.2.2 Mode propagation in optical slab waveguide
In the previous section, mode concept is introduced. In the section, I describe
a guided mode behavior in an optical slab waveguide in detail.
23
x
n0
n1
a
z
0
-a
ns
Fig. 2.5 Refractive index distribution in an optical slab waveguide
Considered a slab waveguide with the refractive index distribution as shown
as in Fig. 2.5, the electric field distribution of TE mode in the waveguide core and
claddings are
 A cos(ka − φ ) exp[−σ ( x − a)] ( x > a)

(−a ≤ x ≤ a )
E y =  A cos(kx − φ )
 A cos(ka + φ ) exp[ξ ( x + a)]
( x < −a)

,
(2.35)
where
k = k 0 n1 − β 2 = k 0 n1 − neff ,
2
2
2
2
σ = β 2 − k 0 2 n0 2 = k 0 neff 2 − n0 2 ,
ξ = β 2 − k 0 2 ns 2 = k 0 neff 2 − n s 2 .
Ey in (2.35) at x=a and –a is continual. And the difference of Ey
− σA cos(ka − φ ) exp[−σ ( x − a)] ( x > a)

= − kA sin(kx − φ )
(−a ≤ x ≤ a )
dx 
ξA cos(ka + φ ) exp[ξ ( x + a)] ( x < −a)
∂E y
(2.36)
is also continual at x=a and –a as
 kA sin(ka + φ ) = ξA cos(ka + φ )
.

σA cos(ka − φ ) = kA sin(ka − φ )
(2.37)
Removing A, the following equations can be obtained,
tan(u + φ ) =
24
ω
u
,
(2.38a)
tan(u − φ ) =
ω1
u
,
(2.38b)
where
 u = ka

 ω = ξa .
ω = σa
 1
(2.39)
So, the following relations can be gotten as
u=
ω
ω 1
mπ 1
+ tan −1 ( ) + tan −1 ( 1 ) (m = 0,1,2, L) ,
2
2
2
u
u
(2.40a)
φ=
ω
mπ 1
ω 1
+ tan −1 ( ) − tan −1 ( 1 ) (m = 0,1,2, L) ,
u
u
2
2
2
(2.40b)
u 2 + ω 2 = k 2 a 2 (n1 − ns ) ≡ v 2 ,
(2.41)
2
2
ω1 = γv 2 + ω 2 ,
γ =
b=
2
2
2
2
2
2
2
2
n s − n0
n1 − n s
ne − n s
n1 − ns
,
.
For propagating modes 0 < b < 1. The condition of cut off is b = 0.
2ν 1 − b = mπ + tan −1
b
b+γ
,
+ tan −1
1− b
1− b
(2.42)
 u = v 1− b

 ω=v b .
ω = v b + γ
 1
For a symmetric waveguide, (2.40) can be written as
mπ
ω
+ tan −1 ( )
2
u
mπ
φ=
2
u=
or
25
,
(2.43)
ω = u tan(u −
mπ
) .
2
(2.44)
Electric field distributions are followings
1
∞ 1
1 r r* r
*
*
( E × H ) ⋅ u z dx = ∫
( E x H y − E y H x )dx ,
−∞ 2
−∞ 2
P = ∫ dy ∫
0
∞
P=
Pcore =
βaA 2
2ωµ 0
β
2ωµ 0
∫
∞
2
−∞
E y dx ,
(2.45)
(2.46)
 sin 2 (u + φ ) sin 2 (u − φ ) 
+
 (−a ≤ x ≤ a) ,
1 +
2ω
2ω1


(2.47)
Psub
βaA 2 cos 2 (u + φ )
=
( x < −a) ,
2ωµ 0
2ω
(2.48)
Pclad
βaA 2 cos 2 (u − φ )
=
( x > a) ,
2ωµ 0
2ω1
(2.49)
P = P core + Psub + Pclad =
A=
1
1 
βaA 2 
+
1 +
,
2ωµ 0  2ω 2ω1 
2ωµ 0 P
.
βa(1 + 1 / 2ω + 1 / 2ω1 )
(2.50)
(2.51)
By the same method, the conclusions for TM modes can be written as
 A cos(ka − φ ) exp[−σ ( x − a)] ( x > a)

(−a ≤ x ≤ a ) ,
H y =  A cos(kx − φ )
 A cos(ka + φ ) exp[ξ ( x + a)]
( x < −a )

(2.52)
n ω
n ω
1
mπ 1
+ tan −1 ( 1 2 ) + tan −1 ( 1 2 1 ) (m = 0,1,2,L) ,
2
2
2
ns u
n0 u
(2.53)
2
u=
2ν 1 − b = mπ + tan (
−1
2
n1
2
ns
2
2
n
b
) + tan −1 ( 1 2
1− b
n0
b+γ
).
1− b
(2.54)
§2.2.3 Three-dimensions optical waveguide
The discussed waveguides above is of two dimensions (2-D) called as slab
optical waveguides. The analysis of 2-D waveguide is explicit for studying the mode
26
propagating characteristics in a multimode waveguide. However, the real world is a
three dimensions (3-D) space. Marcatili’s method is often used to analyze a 3-D
waveguide. The method is shown in Fig. 2.6. Considered the mode distribution in a
waveguide mentioned above, the power in shaded zones is small and neglected. The
refractive indexes in core and clad are n1 and n0, respectively. The cross section of
the waveguide is divided into three zones. The core is marked by I, the cladding in
y-direction is marked by II, and the cladding in x-direction is marked by III.
y
II
2d
III
I
n1
x
n0
2a
Fig. 2.6 Waveguide model of Marcatili’s method.
Considering the modes of Epqx, the wave equations in zones I, II and III can be
written as
 A cos(k x x − φ ) cos(k y y − ψ )
zoneI

H y =  A cos(k x a − φ ) exp[−γ x ( x − a )] cos(k y y − ψ ) zoneII
 A cos(k x − φ ) exp[−γ ( y − d )] cos(k d − ψ ) zoneIII
x
y
y

,
(2.55)
where
 − k x 2 − k y 2 + k 0 2 n1 2 − β 2 = 0 zoneI
 2
2
2
2
2
 γ x − k y + k 0 n0 − β = 0 zoneII ,
− k 2 + γ 2 + k 2 n 2 − β 2 = 0 zoneIII
0
0
y
 x
and
27
(2.56)
π

φ = ( p − 1) 2

π
ψ = (q − 1)
2

p = 1,2,L
.
(2.57)
q = 1,2,L
When x = a, Ez ∝ (1 / n 2 )∂H y / ∂x is continual. When y = d, Hz ∝ ∂H y / ∂y is
continual. Then the relations can be gotten as
k x a = ( p − 1)
k y d = (q − 1)
π
2
π
2
n1 γ x
2
+ tan −1 (
+ tan −1 (
2
n0 k x
γy
ky
),
) .
(2.58a)
(2.58b)
In order to improve the calculating preciseness, the improved Marcatili’s
method is also proposed. In the new method, the refractive index in shaded area is
presumed as
2
2
2n0 − n1 .
However, the Marcatili’s method would lead to more calculations for analyzing
a 3D multimode waveguide. In order to simplify the analysis of the 3D waveguide,
another method is often used, which is the effective index method (EIM). By the
method, a 3-D model can be simplified to a 2-D model. In EIM the electric field is
separated into two fields in x- and y-directions,
H y ( x, y ) = X ( x)Y ( y ) .
(2.59)
So the wave equation is written as
1 d 2 X 1 d 2Y
2
+
+ [k 0 n 2 ( x, y ) − β 2 ( x, y )] = 0 ,
2
2
Y dy
X dx
(2.60)
1 d 2Y
2
2
2
+ [k 0 n 2 ( x, y ) − k 0 neff ( x)] = 0 ,
2
Y dy
(2.61)
1 d2X
2
2
+ [k 0 neff ( x) − β 2 ( x, y )] = 0 .
2
X dx
(2.62)
By the continual conditions of fields at interfaces mentioned above, neff(x) can
be calculated, then the modes behaviors in the waveguide can be calculated by
(2.61) and (2.62).
28
§2.3 MMI principle
Self-imaging of periodic objects illuminated by coherent lights was first
described about 170 years ago [41]. Self-focusing (graded index) waveguides can
also produce periodic real images of an object [42]. However, the possibility of
achieving self-imaging in uniform index slab waveguide was first suggested by
Bryngdahl [15] and explained in more detail by Ulrich [16], The principle can be
stated as follows: Self-imaging is a property of multimode waveguides by which an
input field profile is reproduced in single or multiple images at periodic intervals
along the propagation direction of the waveguide.
§2.3.1 Multimode waveguide
Generally, the MMI devices comprise two parts. One is the center multimode
waveguide and the access waveguides as shown in Fig. 2.7. The center multimode
waveguide supports a large number of modes, where the MMI occurs. A number of
access waveguides are placed at its beginning and at its end. Such devices are
generally referred as M x N MMI couplers, where M and N are the number of
input and output waveguides, respectively. Usually, the access waveguides are
single-mode waveguides for the high-performance MMI device as discuss later.
Access
waveguides
Access
Multimode waveguide
waveguides
N
1
N-1
.
.
M-1
1
M
Fig.2.7 NxM multimode interference coupler
The 3-D multimode waveguide can be simplified to a 2-D model by EIM as
mentioned above. Fig. 2.8 shows a step-index multimode waveguide with an
effective refractive index nr and a clad index of nc. The multimode waveguide
width is WM and the propagating direction of modes is z-direction, as shown in Fig.
2.8. The supported modes number can be calculated by (2. 40). The power
29
distribution of these modes in the multimode waveguide is also shown in Fig. 2.8.
From the mode power distribution, we can see the guided modes penetrated into
clad, which is the Goos-Hahnchen shifts.
z
mode 5
Multimode
mode 3
waveguide
mode 2
y
mode 1
nr
WM
nc
y
mode 0
Fig. 2.8 Multimode waveguide, refractive index distribution and the
supported modes.
The lateral wavenumber kyv and the propagation constant βv are related to the
ridge index nr by the dispersion equation
k yv + β v = k 0 nr ,
2
2
2
2
(2.63)
where k 0 = 2π / λ0 .
The stand-wave condition in y-direction is
k yvWev = (v + 1)π ,
(2.64)
where ν = 0, 1, …(m-1), m is the mode number supported by the multimode
waveguide. The effective waveguide width can be written as
 λ  n
Wev ≈ WM +  0  c
 π  n r
σ = 0 for TE
.
σ = 1 for TM
where 
30
2σ

 (n r 2 − nc 2 ) − (1 / 2) ,

(2.65)
So the propagation constant of mode ν can be written as
β v = k 0 nr
2
2
 (v + 1)π
− 
 Wev



2
.
(2.66)
By Taylor expansion of (2.66), the propagation constant can be written as
β v = nr k 0 −
(v + 1) 2 πλ0
4n rWe
2
.
(2.67)
Then the propagation constant difference of mode ν and mode 1 can be
expressed as
β0 − βv =
v(v + 2)πλ0
4nrWe
2
.
(2.68)
By defining Lπ as the beat length of the two lowest-order modes
Lπ ≡
π
β 0 − β1
=
4n rWe
3λ0
2
,
(2.69)
the propagation constants spacing can be written as
(β 0− β v ) ≈
v(v + 2)π
.
3Lπ
(2.70)
Modes Power
Amplitude (arbitrary)
Coefficients
0
1
1
1
2
1
3
1
4
1
5
1
6
1
Waveguide width (µm)
Fig.2.9 Mode matching between input mode and the excited modes in a
multimode waveguide
31
§2.3.2 Mode matching technology
As mentioned above, the access waveguide is a single-mode waveguide, and
the center waveguide of MMI devices is a multimode waveguide. After a single
mode input the multimode waveguide, many modes supported by the waveguide
are excited. In the process, the power is reserved without considering the loss, it is
Pin = Pmod e 0 + Pmod e1 + ...Pmod ev ... + Pmod em .
(2.71)
Modes Power
Amplitude (arbitrary)
Coefficients
0
0.1
1
1.0
2
0.1
3
0.3
4
0.1
5
0.4
6
0.1
Waveguide width (µm)
Fig. 2.10 Mode matching between input mode and the excited modes in a
multimode waveguide
The power carrying by every mode can be calculated by the method of Fourier
conversion, as discussed in §2.3.3. However, there is a more simple method to
evaluate that, which is the method of mode matching technology. The idea of the
method is to adjust the power coefficient of every mode to match the power profile
of the input mode. The power distribution of every mode in the multimode
waveguide can be calculated, as shown in Fig. 2.8.
I gave three samples for the mode matching as shown in Fig. 2.9, Fig. 2.10
and Fig. 2.11. In Fig. 2.9, an input mode having this profile at the MMI input end
would be decomposed into many modes with equal amplitudes in the multimode
section. In fig. 2.10, the input mode is like an asymmetric profile, so the odd
modes have larges amplitude. In fig. 2.11 the input is like a Gauss profile, the
even modes have larger amplitude coefficients.
32
modes Power
Amplitude (arbitrary)
Coefficients
0
1.0
1
0.0
2
0.8
3
0.1
4
0.4
5
0.1
6
0.2
Waveguide width (µm)
Fig. 2.11 Mode matching between input mode and the excited modes in a
multimode waveguide.
§2.3.3 Guided-mode propagation analysis [17]
An input field profile Ψ(y,0) imposed at z = 0 and totally contained within the
multimode waveguide, will be decomposed into the modal field distributions ψν(y)
of all modes
Ψ ( y,0) = ∑ cvψ v ( y ) ,
(2.72)
v
where the summation should be understood as including guided modes as well as
radiative modes. The field excitation coefficients cν can be estimated using overlap
integrals
cν =
∫ Ψ ( y,0)ψ ( y)dy
∫ψ ( y )
v
2
,
(2.73)
v
based on the field-orthogonality relations.
If the “spatial spectrum” of the input field Ψ(y, 0) is narrow enough not to
excite unguided modes, it may be decomposed into the guided modes alone
m −1
Ψ ( y,0) = ∑ cvψ v ( y ) .
(2.74)
v =0
The field profile at a distance z can then be written as a superposition of all
the guided mode field distributions
33
m −1
Ψ ( y, z ) = ∑ cvψ v ( y ) exp( j (ωt − βν z )).
(2.75)
v =0
Taking the phase of the fundamental mode as a common factor out of the sum,
dropping it and assuming the time dependence exp(jωt) implicit hereafter, the
field profile Ψ( y , z ) becomes
m −1
Ψ ( y, z ) = ∑ cvψ v ( y ) exp( j ( β 0 − βν ) z ).
(2.76)
v =0
A useful expression for the field at a distance z = L is then found by
substituting (2.70) to (2.76)
m −1
Ψ ( y, L) = ∑ cvψ v ( y ) exp( j
v =0
v(v + 2)π
L).
3Lπ
(2.77)
The shape and the types of images formed will be determined by the modal
excitation cν, and the properties of the mode phase factor
exp( j
v(v + 2)π
L).
3Lπ
(2.78)
It will be seen that, under certain circumstances, the field Ψ(y, L) will be a
reproduction (self-imaging) of the input field Ψ(y, 0). It is called as general
interference to the self-imaging mechanisms, which are independent of the modal
excitation; and restricted interference to those which are obtained by exciting
certain modes alone.
The following properties will prove useful in later derivations
ψ ( y )
ψ v (− y ) =  v
ψ v (− y )
for v even
.
for v odd
(2.79)
§2.3.4 General interference
This
section
investigates
the
interference
mechanisms,
which
are
independent of the modal excitation, that is, I pose no restriction on the
coefficients cν, and explore the periodicity of (2.77).
A. Single Images
By inspecting (2.77), it can be seen that Ψ (y, L) will be an image of Ψ (y, 0) if
exp( j
v(v + 2)π
L) = 1 or (−1) v .
3Lπ
34
(2.80)
The first condition means that the phase changes of all the modes along L
must differ by integer multiples of 2π. In this case, all guided modes interfere with
the same relative phases as in z = 0; the image is thus a direct replica of the input
field. The second condition means that the phase changes must be alternatively
even and odd multiples of π. In this case, the even modes will be in phase and the
odd modes in antiphase. Because of the odd symmetry stated in (2.78), the
interference produces an image mirrored with respect to the plane y = 0.
z
y=0
Ψ(y,0)
(3Lπ)/2
z=0
(3Lπ)
3(3Lπ)/2
2(3Lπ)
y
Fig. 2.12 Multimode waveguide showing the input field Ψ(y,0), a mirrored single image
at (3Lπ), a direct single image at 2(3Lπ), and two-fold images at (3Lπ)/2 and 3(3Lπ)/2.
It is evident that the first and second conditions of (2.79) will be fulfilled at
L = p(3Lπ ) with
p = 0,1,L
,
(2.81)
for p even and p odd, respectively. The factor p denotes the periodic nature of the
imaging along the multimode waveguide. Direct and mirrored single images of the
input field Ψ( y , 0) will therefore be formed by general interference at distances z
that are, respectively, even and odd multiples of the length (3Lπ), as shown in Fig.
2.12. It should be clear at this point that the direct and mirrored single images
could be exploited in bar- and cross-couplers, respectively.
B. Multiple Images
In addition to the single images at distances given by (2.80), multiple images
can be found as well. Let us first consider the images obtained half-way between the
direct and mirrored image positions, i.e., at distances
L = p(3Lπ ) / 2 with
p = 1,3,5,L .
The total field at these lengths is found by substituting (2.81) into (2.76)
35
(2.82)
Ψ ( y,
m −1
p
π
3Lπ ) = ∑ c vψ v ( y ) exp( jv(v + 2)πp ( )) ,
2
2
v=0
(2.83)
with p an odd integer. The mode field symmetry conditions of (2.78), (2.82) can be
written as
Ψ ( y,
Ψ ( y,
+
p
3Lπ ) = ∑ cvψ v ( y ) + ∑ (− j ) p cvψ v ( y ).
2
v even
v odd
(2.84)
p
1
1
1
1
3Lπ ) = ∑ cvψ v ( y ) + ∑ cvψ v ( y ) + ∑ (− j ) p cvψ v ( y ) + ∑ (− j ) p cvψ v ( y )
2 ν odd
2 ν odd
2 ν even
2 ν even
2
1
1
1
1
cvψ v ( y ) − ∑ cvψ v ( y ) + ∑ (− j ) p c vψ v ( y ) − ∑ (− j ) p c vψ v ( y )
∑
2 ν even
2 ν even
2 ν odd
2 ν odd
Ψ ( y,
p
1
1
3Lπ ) = ( ∑ cvψ v ( y ) + ∑ cvψ v ( y )) + ( ∑ cvψ v ( y ) − ∑ c vψ v ( y ))
2 ν even
2 ν even
2
ν odd
ν odd
1
1
+ ( ∑ (− j ) p cvψ v ( y ) + ∑ (− j ) p cvψ v ( y )) − ( ∑ (− j ) p cvψ v ( y ) − ∑ (− j ) p cvψ v ( y ))
2 ν even
2 ν odd
ν even
ν odd
p
1
1
1
1
3Lπ ) = Ψ ( y,0) + Ψ (− y,0) + ( − j ) p Ψ ( y,0) − (− j ) p Ψ (− y,0)
2
2
2
2
2
p
p
1 + (− j )
1 − (− j )
=
Ψ ( y,0) +
Ψ (− y,0)
2
2
Ψ ( y,
(2.85)
The last equation represents a pair of images of Ψ(y, 0), in quadrature and with
amplitudes 1 / 2 , at distances z = (3Lπ)/2, 3(3Lπ)/2, . . . as shown in Fig. 2.12. This
two-fold imaging can be used to realize 2 x 2 3-dB couplers. Optical 2 x 2 MMI
couplers based on the single and two-fold imaging by general interference have been
realized in wafers with various materials [26, 36, 43,44].
In general, multi-fold images are formed at intermediate z-positions. Analytical
expressions for the positions and phases of the N-fold images have been obtained
[31] by using Fourier analysis and properties of generalized Gaussian sums. A very
brief summary of the bases and results of that derivation is given here. The starting
point is to introduce a field Ψin(y) as the periodic extension of the input field Ψ( y,0);
anti- symmetric with respect to the plane y = 0 (which, for this analysis, is chosen to
coincide with one guide’s lateral boundary), and with periodicity 2Wm
Ψin ( y ) =
∞
∑ (Ψ ( y − v 2W ) − Ψ (− y + v2W )) ,
e
e
(2.86)
v = −∞
and to approximate the mode field amplitudes by sin-like functions
ψ v ( y ) ≈ sin( k yv y ).
(2.87)
36
Based on these considerations, (2.73) can be interpreted as a (spatial) Fourier
expansion, and it is shown [31] that, at distance
L=
p
(3Lπ ) ,
N
(2.88)
where p ≥ 0 and N ≥ 1 are integers with no common divisor. The field will be of the form
Ψ ( y, L) =
1 N −1
∑ Ψin ( y − y q ) exp( jϕ q ) ,
C q=0
(2.89)
with
y q = p ( 2q − N )
We
,
N
(2.90)
ϕ q = p( N − q)
qπ
,
N
(2.91)
where C is a complex normalization constant with C =
N , p indicates the
imaging periodicity along z , and q refers to each of the N images along y.
The above equations show that, at distances z = L, N images are formed of the
extended field Ψin(y), located at the positions yq, each with amplitude 1 / N and
phase ϕ q . This leads to N images (generally not equally spaced) of the input
field Ψ(y, 0) being formed inside the physical guide (within the guide’s lateral
boundaries), as shown in Fig. 2.13. The multiple self-imaging mechanism allows for
the realization of N x N or N x M optical couplers. Shortest devices are obtained for
p = 1. In this case, the optical phases of the signals in a N x N MMI coupler are,
(apart from a constant phase), given by
ϕ rs =
π
4N
( s − 1)(2 N + r − s ) + π
for r + s even
,
(2.92)
for r + s odd
,
(2.93)
and
ϕ rs =
π
4N
(r + s − 1)(2 N − r − s + 1)
where r = 1,2,…,N is the (bottom-up) numbering of the input waveguides and s =
1,2,…, M is the (top-down) numbering of the output waveguides as shown in Fig.
2.7.
It is important to note that the phase relationships given by (2.92) and (2.93)
are inherent to the imaging properties of multimode waveguides. It appears that
the output phases of the 4 x 4 coupler satisfy the phase quadrature relationship,
and that this MMI device can be used as a 900-hybrid which is a key component in
37
phase-diversity or image rejection receivers and which can be used to avoid the
quadrature problem in interferometric sensors.
§2.3.5 Restricted interference
No restrictions have been placed on the modal excitation in the above
discussions. This section investigates the possibilities and realizations of MMI
couplers in which only some of the guided modes in the multimode waveguide are
excited by the input field(s). This selective excitation reveals interesting
multiplicities of ν(ν+ 2), which allow new interference mechanisms through shorter
periodicities of the mode phase factor of (2.77).
A. Paired Integerence
By noting that
mod 3 [ν (ν + 2)] = 0
for v ≠ 2, 5, 8,L ,
(2.94)
it is clear that the length periodicity of the mode phase factor of (2.87) will be
reduced three times if
for ν = 2,5,8,...
cv = 0
(2.95)
Therefore, as shown in [21], single (direct and inverted) images of the input
field Ψ(y,0) are now obtained at
L = p( Lπ )
with
p = 0,1,2,L ,
(2.96)
provided that the modes v = 2, 5, 8, . . . are not excited in the multimode waveguide.
By the same token, two-fold images are found at (p/2)Lπ, with p odd number.
One possible way of attaining the selective excitation of (2.94) is by launching
an even symmetric input field Ψ(y, 0) (for example, a Gaussian beam) at
y = ±We / 6 . At these positions, the modes ν = 2, 5, 8, . . . present a zero with odd
symmetry, as shown in Fig. 2.8. The overlap integrals of (2.72) between the
(symmetric) input field and the (antisymmetric) mode fields will vanish and
therefore cν = 0 for ν = 2, 5, 8, . . . Obviously, the number of input waveguides is in
this case limited to two.
When the selective excitation of (2.94) is fulfilled, the modes contributing to the
imaging are paired, i.e. the mode pairs 0-1,3-4, 6-7, . . . have similar relative
properties. (For example, each even mode leads its odd partner by a phase
difference of π/2 at z = Lπ/2 (the 3-dB length), by a phase difference of π at z = Lπ,(the
cross-coupler length), etc). This mechanism will be therefore called paired
interference. Two-mode interference (TMI) can be regarded in this context as a
38
particular case of paired interference.
2 x 2 MMI couplers based on the paired interference mechanism have been
demonstrated in silica-based dielectric rib-type waveguides with multimode section
lengths of 240 µm (cross state) and 150 µm (3-dB state) [45]. Insertion loss lower
than 0.4 dB, imbalance below 0.2 dB, extinction ratio of - 18 dB, and polarization
sensitivity loss penalty of 0.2 dB were reported for structures supporting 7-9 modes.
Calculations predict that power excitation coefficients as low as -40 dB for the
modes ν = 2 , 5 , 8 can be achieved through a correct positioning of the access
waveguides, remaining below -30 dB for a 0.1-pm misalignment [21].
B. Symmetric Interference
Optical N-way splitters can in principle be realized on the basis of the general
N-fold imaging at lengths given by (2.82). However, by exciting only the even
symmetric modes, l-to-N beam splitters can be realized with multimode waveguides
four times shorter [46].
In effect, by noting that
mod 4 [v(v + 2)] = 0
for v even
(2.97)
it is clear that the length periodicity of the mode phase of (2.79) will be reduced four
times if
cv = 0
for v = 1,3,5,L
(2.98)
Therefore, single images of the input field Ψ(y, 0) will now be obtained at
L = p(
3Lπ
)
4
with p = 0,1,2, L ,
(2.99)
if the odd modes are not excited in the multimode waveguide. This condition can be
achieved by center-feeding the multimode waveguide with a symmetric field profile.
The imaging is obtained by linear combinations of the (even) symmetric modes, and
the mechanism will be called symmetric interference.
In general, N-fold images are obtained [22,46] at distances
L=
p  3Lπ 


N 4 
(2.100)
with N images of the input field Ψ(y, 0), symmetrically located along the y-axis with
equal spacing We/N.
Fig. 2.13 shows the calculated intensity patterns inside the multimode waveguide
of single-input, symmetrically excited MMI couplers. At mid-way from the self-imaging
length, a two-fold image is formed. The number of images increases at even shorter
distances, according to (2.99), until they are no longer resolvable. A good rule of thumb
39
is that in order to obtain low-loss well-balanced 1 -to- N splitting of a Gaussian field, the
multimode waveguide is required to support at least m = N + 1 modes [47].
Fig. 2.13 Power distribution in a 1x1 rectangular MMI coupler simulated by
FD-BPM.
The 1 x 2 waveguide splitter combiner is perhaps the simplest MMI structure
ever realized, needing just two symmetric modes. Extremely short splitters (20-30
µm for silica-based and 50-70 µm for InP-based waveguides) have been fabricated
with excess losses of around 1 dB and imbalances below 0.15 dB [46], in agreement
with numerical predictions [48].
A number of 1 x N waveguide splitters/combiners covering a wide range of
different multimode guide widths (12-48 µm) and lengths (250-3800 µm) have been
demonstrated in GaAs-and InP-based rib waveguides which divide power with < 0.4
dB imbalances between N output guides, for values of N between 2 and 20 [22], [49].
These experiments permit to conclude that, setting 1 µm as an achievable
40
lithographic limit to the open gap, and 2 µm as a workable width of the access
waveguides, InP-based 1-to-N way splitters at X0 = 1.55 µm could be as short as N x
20 µm.
Linear tapered waveguide
Source
Single
Single
Multi
Single
image
image
images
image
Fig. 2.14 Linear tapered multimode waveguide.
§2.3.6
Linear tapered MMI
A. Concept of linear tapered MMI
After Bryngdahl suggestion, R. Ulrich and G. Ankele presented it is an inherent
property of multimode, parallel or weakly tapered guides, that to any interior object
point, there exist a number of real self-images further down the guide as shown in
Fig. 2.14. At intermediate positions, multiple self-images of P form.
The principle can be briefly explained as
u (Q) =
+J
∑ exp(i(2πnd
j
/ λ + j π )) .
(2.101)
j =− J
They gave three important conclusions for the MMI in waveguides.
1)
Magnified self-images can be obtained also with other than linear tapers. A
modal treatment, assuming adiabatic adaptation of all modes to the local cross
section of the guide, shows that any smooth slender taper can produce
self-images.
2)
Two-dimensional
self-images
should
be
formed
in
waveguides
of
rectangular cross section, provided an equivalent imaging condition (horizontal
direction) is satisfied also in the second traverse direction (vertical direction).
3)
Self-imaging must exist also in broad thin strip guides of integrated optics.
From the conclusions of R. Ulrich and G. Ankele, the followings can be known
41
as
a. Linear tapered MMI waveguides appear early as that of rectangular
waveguides
b. That the MMI images exist even in the linear tapered multimode
waveguide is theoretically confirmed.
c. The confirmation method is based on optical ray methods.
B. Linear tapered MMI devices with a variable splitting ratios
R. Ulrich and G. Ankele theoretically and experimentally confirmed that MMI
images even exist in a linear tapered waveguide. Pierre A. Besse et al. proposed a
new butterfly geometrical design for MMI devices with a variable splitting ratio by
the lineal tapered MMI waveguide. Also, the new MMI devices are compact,
polarization-insensitivity, and tolerant to fabrication error. The principle of
operation is explained in the followings. The coupler is cut in two or many sections,
and phase shifters are introduced between them. It results in multileg MZI’s. The
splitting ratios can be chosen by adapting the phase shifts. For a practical
realization, the phase shifters have to be accurate and ultra-compact. They
therefore introduce the butterfly geometrical configuration as shown in Fig. 2.15.
The two sections are transformed in a linearly down-tapered and a linearly
up-tapered section, respectively. By this transformation the self-imaging properties
remain [50].
Fig. 2.15 Butterfly geometrical configuration.
By the new design, they also provided a 1 x 3 splitter. The output ratios of the 1 x 3
splitters can be changed by changing the width at middle MMI position. The results are
shown in Fig. 2.16 [36].
42
Fig. 2.16 Simulated results of butterfly MMI couplers.
C. Linear tapered MMI splitters with a length reduction
MMI devices have become important components in integrated optical
circuits. Perhaps the most important MMI structure is the 2 x 2 coupler, due to its
applicability to most optical devices. As a result of interest in increased circuit
density, the size of these MMI structures has decreased to the “extremely small”
regime. Length scaling of MMI devices is most readily done via control of the
width of the multimode interference region. However, a practical limitation on
further reduction in device size is the proximity of the access waveguides, since
directional coupling among the access waveguides occurs
David S. Levy et al. propose a new MMI structure, which reduces the proximity
limitations by allowing the access waveguides to be well separated. This device
allows for a major reduction in the device length through a taper of the MMI
region width along the propagation axis. The shape of the MMI region is designed
to preserve the splitting ratio at the end of the MMI at the 3-dB point.
The schematic of the new 3-dB MMI coupler is shown in Fig. 2.17. The access
waveguides can be placed that their outside edges coincide with the edges of the
MMI region, and that the angle of the access waveguides is set to match the local
taper angle at the ends of the MMI region. Tilting these waveguide in this manner
keeps the phase tilt of the input image approximately along a coordinate system,
which is conformal with respect to the end walls of the tapered MMI region. This
combined with the fact that dWMMI / dz = 0 at LMMI/2 minimizes the phase
changes due to the discontinuous change in wavefront present in a linear taper
[54]. With these phase changes now negligible, the imaging properties of the
structure are preserved as the width is tapered, and the decrease in average
43
width leads to a reduction in the imaging/device length. Theoretical calculation
shows the length can be reduced by 60%[51]. However, the authors did not give a
MMI image analysis in detail.
Fig.2.17 Taper MMI couplers as splitters
§2.3.7 MMI reflection characteristics
Several applications such as lasers and coherent detection techniques are very
sensitive to reflections. Reflections in MMI devices may originate at the end of the
MMI-section in between the output guides. Non-negligible reflections may occur
when large refractive index differences are encountered such as the semiconductor
-air interface in deeply etched waveguides. For non-optimum lengths, some light
may be reflected off the end of the MMI section and may eventually reach the input
guides. However, even for optimum lengths, reflections in MMI devices can be
surprisingly effective, because the reflection mechanisms involve the very same
imaging property of multimode waveguides. Two different reflection mechanisms
have been identified, and they are summarized in the table 2.3:
Table 2.3: MMI reflection characteristics [33]
Device
MMI 3dB coupler
MMI-power splitter
Excitati
Single input
Single input
on
In-phase inputs
Out-of-phase
inputs
Transmission
Back-re
flection
44
Internal resonance
1) An “internal resonance” mechanism, which is caused by the presence of several
simultaneously occurring self-images. For example, the MMI 3-dB coupler is based
on the two-fold image occurring at a length of L = 3Lπ/2 as given by (2.81). This
length equals precisely twice the self-imaging length for symmetric excitation L = 3
Lπ/4 as given by (2.87). This symmetric self-imaging mechanism ensures efficient
imaging of both reflecting ends onto each other. In lasers employing such an MMI
3-dB coupler, this "internal resonance" may show as a separate contribution in the
laser spectrum, as shown in Fig. 2.18 [17]. Simultaneously occurring general and
symmetric self-imaging mechanisms can possibly be prevented by employing
couplers based on the paired interference mechanism.
Fig. 2.18 Measured MMI inner resonance.
2) A second type of reflection is encountered when an MMI power splitter is used in
reverse as a power combiner. Efficient combining operation requires inputs of equal
amplitude and phase. If, however, the two inputs are 1800 out of phase, power is
minimum in the output guide but maximum at the reflecting end of the MMI
section. This leads to perfect imaging of the input guides back onto themselves.
Back reflection can thus vary from a minimum for in-phase excitation to a
maximum for out-of-phase excitation for a single MMI combiner optimized for
maximum transmission. Note that this reflection mechanism can cause increased
45
back reflection during the off-state of a Mach-Zehnder modulator using a 2 x 1 MMI
combining element. For reflection-sensitive applications, several means can be used
to achieve an effective reduction of reflections, such as using low-contrast
waveguides or tapering the ends of the MMI section [52].
Back reflections exist in the conventional MMI couplers with a rectangular
shape [33-35], [53]. Figure 2 .19 shows optical power distributions in a 2 x 1 MMI
combiner with a rectangular shape, which was simulated by FD-BPM. The bold
lines in Fig. 2.19 are outlines of the simulated MMI combiner with the width and
length of 12 µm and 395 µm, respectively. There are two rib single-mode access
waveguides at the input and a rib single-mode access waveguide at the output.
Considering the single-mode conditions in the rib structure [54,55], I selected the
access waveguide with a width and refractive indices of a core and a clad of 2 µm,
3.35 and 3.34, respectively.
Fig.2.19 Power distributions in an MMI coupler with a rectangular shape in the
case of (a) in-phase inputs and (b) out-of-phase inputs.
In the case of in-phase inputs (no relative phase difference between two inputs),
even-modes are excited in the MMI section because of the symmetric inputs. As a
result of multimode interference, a single-mode interference image appears at a
position determined by the effective width and refractive index of the MMI
waveguide. The image can be extracted by forming an output waveguide at the
position, as shown in Fig. 19 (a). On the other hand, when there is a relative phase
difference between the two input lights, the MMI pattern is changed. In the case of
out-of-phase inputs (a relative phase difference of π between two inputs) only
46
odd-modes are excited. So the MMI pattern at the output is reverse, compared to
that of the in-phase inputs. No modes can be coupled into the output waveguide,
and most of the power is concentrated on the MMI shoulders, the facet wall, as
shown in Fig.19 (b).
Consequently, there is a large end-facet reflection. We
simulated the reflection backed into an input port to be − 16dB under the condition
of an end-facet reflectivity of 25%. The simulated result is larger than the
experimental result of –25 dB of a 2 x 2 MMI splitter [34], [35]. The reason is that
the output power of a 2 x 1 MMI combiner with out-of-phase inputs is mainly
concentrated on the facet wall, while the output power of a 2 x 2 MMI splitter is
mainly concentrated on the output ports. Especially when the MMI is deeply etched
or covered by a metal layer, the reflection is more severe. The reflection maybe is no
problem for some passive systems. However, when the combiner is connected with a
semiconductor optical amplifier (SOA) or a laser, the reflection would degenerate
the system performance.
§2.4 Non-linear tapered MMI combiners
In order to minimize the end-facet reflection mentioned above, several
proposals were reported [35], [52]. Previously we proposed another method using a
tapered MMI combiner to avoid the back reflection [51], [53], [56]. A tapered MMI
coupler with two symmetric end-facets has been reported as compact power
splitters [22-26], [57-59], but the tapered MMI with asymmetric end-facets for
power combiners has been little reported. Actually, the MMI coherent lightwave
combiners are different from the MMI splitters, although it seems that the MMI
splitters can be used as combiners by introducing lights propagation in the opposite
direction. In the case of MMI coherent lightwave combiners, relative phase changes
of input lights will change the power distribution in an MMI section. So, non-linear
tapered MMI combiners are new MMI-based components.
§2.4.1 MMI images existence in non-linear tapered multimode
waveguides
The tapered MMI combiner is schematically shown in Fig. 2.20, where the
47
combiner length, the initial MMI width, the access waveguide width and the access
waveguide spacing are denoted by L, W, ww and ws, respectively. The combiner
output end and the output waveguide have the same width. The tilted and curved
borders can be in any shapes such as an arc, an exponential curve and others.
Fig. 2.20 Schematic diagram of tapered MMI-based combiners.
The fundamental operation principle in the tapered waveguide is the same as
those of conventional MMI couplers [16]. By introducing an input light from a
single-mode access waveguide, there are several excited modes in the MMI section.
However, the excited mode number decreases with propagation in the combiner
because of the tapered structure, and there can be only a single mode at the output.
Since the supported mode number decreases, an interference image in the tapered
MMI is contributed only from those modes supported at the position with a local
waveguide width. I name these modes as effective modes of the image. High-order
modes disappear with propagation in the tapered MMI, however the power loss can
be small if the taper is not steep. As I discuss later, the low loss is mainly from the
power conversion from high-order modes to low-order modes because of the
interference of the hybrid modes.
In the conventional MMI coupler with a rectangular shape, the propagation
constant spacing is derived as
β 0 − βv ≈
v (v + 2)πλ0
,
4nW 2
(2.102)
where W is an effective width of the MMI waveguide with a refractive index of n, λ
0
is a free-space wavelength, and ν is a mode number. Since the propagation
constant spacing is independent of the propagation positions, the positions are
always exist, where the phase differences of all high-order modes and the
48
fundamental mode are the same, for example, 2lπ(self-image, l is an integer).
Therefore the stable and clear images periodically appear along the propagation in
the MMI waveguide. Stable and clear images mean that they are generated by the
contribution of all modes in the MMI section.
For the tapered MMI, the width W is not a constant, so the propagation
constant spacing given by (2.102) depends on the propagated position. In other
words, the border shape determines the propagation constant spacing. In order to
analyze the propagation characteristics of modes, I deduced the propagation
constant spacing in the tapered MMI waveguide [17], [57]. The dispersion relation
at any position z can be written as
(nK 0 ) 2 = K xν ( z ) 2 + βν ( z ) 2 ,
(2.103)
where Kxv(z) and βv(z) are a lateral wavenumber and a propagation constant of the
mode ν at a position of z, respectively. K0 is a free-space wavenumber. The lateral
wavenumber is described as
K xv ( z )W ( z ) = (v + 1)π
,
(2.104)
where W(z) is an effective MMI width at the position z, andνis a mode number,
such as ν= 0,1, …, (m-1). m is the number of effective modes.
From the analogy of
(2.102), the propagation constant spacing of theν-th
mode to the fundamental one with respective propagation constants of βv(z) and
β 0(z) can be expressed as
β 0 ( z ) − βν ( z ) ≈
where the effective width W(z)
ν (ν + 2 )πλ 0
4 nW ( z ) 2
,
(2.105)
depends on the tapered MMI position.
The relative phase difference ∆φv after the propagation from the position of z1 to
the one of z2 in the tapered section is give by [57]
v ( v + 2 )πλ 0 z 2 dz
∆ φ v ( z1 , z 2 ) = ∫ z 2 ( β 0 ( z ) − β v ( z ))dz =
∫z
2
z1
4n
1 W (z)
. (2.106)
(2.106) shows that the relative phase difference per unit length ∆φv(z1,z2)/(z2-z1)
depends on the propagation position, which is a large difference from that of the
conventional rectangular MMI waveguide. So the question is that if there exist the
positions in the tapered MMI, where the relative phase differences of all the modes
to the fundamental mode are the same, for example 2lπ (l is an integer). If these
49
positions do not exist, stable and clear interference images cannot exist in the
tapered MMI.
The ratios αν1 of relative phase differences of the ν-th and 1st modes to the
fundamental mode propagated from the position of z1 to the one of z2 can be written
as
αν 1 =
∆φv ( z1, z2 ) v(v + 2)
=
.
∆φ1 ( z1, z2 )
3
(2.107)
(2.107) describes an important characteristic of mode propagation in a tapered
MMI waveguide. Although every mode has different phase differences to the
fundamental mode in different propagated sections, the phase differences of all
modes proportionally change in a propagation section. I suppose there are m
effective modes with respective phases of
φ 0 (0), φ1 (0), K , φ m −1 (0) ,
(2.108)
at the initial position. I can slice the taper into many small segments. By (2.107),
the phases of the effective modes passed first slice can be written as
φ 0 (1), φ1 (1), L , φ ν (1) = αν 1 (φ1 (1) − φ 0 (1)) + φ 0 (1) ,
(2.109)
where ν = 2, 3, …m-1. Passed slice k, the phases of these effective modes can be
written as
Mode0 : ψ 0 = φ 0 (0) + φ 0 (1) + L + φ 0 ( k )
Mode1 : ψ 1 = φ1 (0) + φ1 (1) + L + φ1 (k )
L
.
(2.110)
Modeν : ψ ν = αν 1 (ψ 1 −ψ 0 ) + ψ 0
Since I only interest in the relative phase changes of higher modes to the
fundamental one, I can rewrite (2.110) as
0, (ψ 1 − ψ 0 ),8(ψ 1 − ψ 0 ) / 3, L ,ν (ν + 2)(ψ 1 − ψ 0 ) / 3 ,
(2.111)
for mode 0, mode 1, mode 2, …, modeν, respectively. Thus, the self-images appear
at the positions, where ψ1-ψ0 = 6lπ, l is an integer. So we can say that the positions
to give specific phase differences exist even in the tapered MMI. If the border shape
W(z) is known, the positions L of self-images can be calculated by
L
∫0
dz
8 nl
=
.
2
λ0
W (z)
(2.112)
(2.112) shows that the self-image spacing depended on the border shape is a
chirped period, which is different with that of conventional rectangular MMI
waveguides.
50
0, ∆ψ ,8∆ψ / 3, L ,ν (ν + 2) ∆ψ / 3
∆ψ = ψ 1 − ψ 0
or
Pk
P3
P2
P1
P0
φ 0 ( k ), φ1 ( k ), φ v ( k ) = α v1 (φ1 ( k ) − φ 0 ( k )) + φ 0 ( k )
v = 2,3, L m − 1
φ 0 (3), φ1 (3), φ v (3) = α v1 (φ1 (3) − φ 0 (3)) + φ 0 (3)
v = 2,3, L M − m1 − m 2 − m 3 , m 3 = 0,1,2, L M − m1 − m 2
φ 0 ( 2), φ1 ( 2), φ v ( 2) = α v1 (φ1 ( 2) − φ 0 ( 2)) + φ 0 ( 2)
v = 2,3, L M − m1 − m 2 , m 2 = 0,1,2, L M − m1
φ0 (1), φ1 (1), φv (1) = α v1 (φ1 (1) − φ0 (1)) + φ0 (1)
v = 2,3, L M − m1 , m1 = 0,1,2, L M
φ 0 = 0, φ1 = 0, L , φ m −1 = 0, φ m = 0, L , φ M = 0
Fig. 2.21 Modes propagation in a tapered MMI waveguide.
The mathematical descriptions can be easily understood by the Fig.2.21. At
input position P0, a single mode input is decomposed into M serious modes with
respective phases of φ0, φ1, …, φm-1, φm,…, φM. With the propagation, the excited mode
number decreases with propagation in the combiner because of the tapered
structure, and there can be only a single mode at the output. For example, arrived
at position P1, m1 high modes lost. The power of the lost high-order modes partially
is converted to low-order modes by the interference of the hybrid modes, while
others become radiation loss. From position p2 to position p3, m2 high-order modes
lost. The self images appear at position pk as shown above, where only m modes. In
other words, only m modes contribute to the images, although there are M modes at
51
the initial position. The m modes are effective modes for the images.
§2.4.2 Non-linear tapered MMI combiners
The non-linear tapered MMI combiners have many advantages over traditional
rectangular MMI devices such as no end-reflection, compactness, easy extension to
a multi-ports MMI combiner, and robustness to fabrication errors. In the section I
describe the non-linear tapered MMI combiners in detail.
A. Free of back reflection
In order to minimize the end-facet reflection, several proposals were reported
[35], [52]. In [35] authors gave a new design adding two dumpy ports in the both
sides of output waveguide of a 2 x 1 MMI coherent combiner. The method can
effectively reduce back reflection, however it has some shortages. First is that the
method cannot remove off the back reflection. It is difficult to judge if the device is
safe to be used in a system being sensitive to back reflection because of the
uncertain remained reflection. Second is the small waveguide spacing, since there
are three waveguides in the MMI ends that usually is narrow. The coupling between
these waveguide maybe leads to some problem such as loss or low extinction ratios.
Method reported in [52] can be used reduce the inner resonance in a 2 x 2 coupler,
however it is difficult to be used in the 2 x 1 combiners.
We proposed another method using a tapered MMI combiner to avoid the back
reflection [53], since the clear and stable MMI images even exist in the tapered
MMI waveguide as shown above. In Fig. 2.19 I showed that there is a severe
end-facet reflection for the rectangular MMI with out-of-phase inputs. Under the
same conditions, I simulated the characteristics of the tapered MMI with arc
borders, as shown by the bold lines in Fig. 2.22. In the case of in-phase inputs, only
even-modes are excited because of the symmetric inputs. Generated clear
single-mode image is coupled into the output waveguide, and the loss is small, as
shown in Fig. 2.22 (a).
52
Fig.2.22 Power distributions of a tapered MMI combiner in the cases of (a)
in-phase inputs and (b) out-of-phase inputs.
In the case of out-of-phase inputs, which means the asymmetric inputs, only
odd-modes were excited. So there is no power distribution on the centerline as
shown in Fig. 2.22 (b). No output is extracted from a centered single-mode
waveguide. High-order odd-modes leak out of the tapered structure. Therefore, the
device is free of the end-facet reflection problem mentioned above.
B. Device compactness
The compactness of the tapered MMI with symmetric ends has been discussed
[57]. Here I discuss the compactness of the one with asymmetric ends (horn-shaped
MMI). I take a single arc with a radius R as a border shape of the tapered MMI
combiner, and then the tapered MMI width W(z) can be written as
W ( z ) = ww + 2 y 0 − 2 R 2 − ( z − x0 ) 2
,
(113)
where x0 and y0 are the coordinates of the center of curvature of the arc, and ww is
an access waveguide width. Substituting W(z) in (2.106), I calculated the beat
53
length of the two lowest modes, which is an important parameter to characterize
the length of MMI waveguides.
Fig.2.23 Ratios of beat lengths of a tapered MMI coupler to that of a
rectangular one as a function of a curvature radius R
In order to show the compactness of the 2 x 1 tapered MMI, I compare the beat
length Lπ of the tapered coupler to that of the conventional rectangular one, which is
denoted by Lr, with the same initial width W of 5.5 µm and length L of 200 µm. The
results are shown in Fig. 2.23. When the curvature radius R is infinite, the border
shapes become straight tilted lines and the MMI has a shape of a trapezoid. The
simulated results show that the beat length of the tapered MMI coupler with a
trapezoidal shape is 77% of that of a rectangular MMI coupler and that of the
tapered MMI coupler with curved borders is shorter than that with a trapezoidal
shape. However, a very small curvature radius R is impractical because of the
severe scattering loss or no passing through the MMI section. So the curvature
radius R has to be no less than the minimum radius Rmin defined as
Rmin
(W − ww) 2 + 4 L2
=
,
4(W − ww)
(2.114)
where W, ww and L are an initial width, an access waveguide width and a length of
the tapered MMI. Rmin renders that the arc is tangential to the output waveguide, so
there seems no scattering loss at the joint between the MMI section and the output
waveguide. In this sense, Rmin is an optimal curvature radius, and the simulated
results confirmed it [53]. However, it is worthy to note the results in Fig. 2.23
depend on a device length, and the length reduction is limited by an expected
extinction ratio, which we discuss later.
54
Output imbalance (dB)
Port number
Fig. 2.24 Power imbalance of 8x1 coherent lightwave combiners.
C. Multi-port coherent lightwave combiners
Multi-port coherent lightwave combiners are widely applied in optical delay
line filters such as optical transversal filters. The implementation of the multi-port
combining of coherent inputs by a single MMI combiner is very attractive, because a
combiner comprised of several cascaded 2 x 1 ones becomes in a larger size,
sensitive to fabrication errors and more losses. The proposed tapered MMI combiner
is easy to be extended to multi-port combiners. The output power balance for the
inputs is necessary to keep effective interference of these coherent inputs.
Previously we discussed the output power balance for a horn-shaped MMI combiner,
and a power imbalance of 3dB was reported for an 8 x 1 combiner, as shown by the
line with squares in Fig. 2.24 [53]. After that, I reported a new structure, by which
the power imbalance is about 0.5 dB for an 8 x 1 tapered combiner with an
optimized structure. In the new structure, I adopt two arcs with equal curvature
radii Rn and they connect waveguides on both sides, as shown in the inset of Fig.
2.24. The curvature radius Rn in this structure can be written as
4 L2 + (W − ww) 2
Rn =
8(W − ww)
,
(2.115)
where W, L and ww are the an initial MMI width, an MMI length and an access
55
waveguide width. In order to attain a good power balance, I added two dummy input
ports, each of them located on the border sides. With the new structure I obtained a
power imbalance of about 0.5 dB by the optimized device structure of W = 26.5 µm, ww
= 2 µm, ws = 0.5 µm and L = 1000 µm.
As an application of multi-port tapered MMI combiners, I study the cases of an
optical transversal filter with four taps, as shown in Fig. 2.25 (a) [60, 61]. The input
light is tapped into four branched ones with sequential delays of τ0 by delay lines of
a length ∆L between the adjacent taps. Then the tapped lights enter the tapered
MMI combiner, interfere each other, and output.
Using FD-BPM I simulated the output spectrum of the optical transversal filter,
as shown in Fig. 2.25 (b), where the delay line length ΔL of 1 mm is assumed. The
used tapered combiner has a structure of W = 17.5 µm, L = 800 µm and ws = 0.5 µm.
In Fig. 2.25 (b), an evaluated FSR is 0.717 nm, which agrees with the analytical
value given by FSR=λ02/n/∆L, where n is a refractive index of the delay lines. The
simulated excess loss is less than 2 dB.
τ0
τ0
τ0
Tapered MMI
Combiner
(a)
Output power (dB)
0
-5
-10
-15
-20
-25
-30
1550.5
1551
1551.5
1552
Wavelength (nm)
(b)
Fig. 2.25 (a) An optical transversal filter with the tapered MMI combiner,
and (b) the simulated output spectrum.
56
The power distributions in the 4 x 1 tapered MMI combiner are simulated by
FD-BPM in the cases of in-phase and out-of-phase inputs as shown in Fig. 2.26. In
the case of in-phase inputs, the excited modes interfere each other in the taper
waveguide. As an interference result, a single mode image occurs in position P as
shown in Fig. 2.26 (a). Of course, the mode number at position P is less than that at
initial position because of the tapered structure. In the used structure, the mode
number at position P is about 2 modes. Then the single mode is decomposed into 2
modes. Then the 2 modes become one single mode again near the output port.
Finally the single mode image is output by a tapered waveguide. In the sense, the
tapered MMI combiner can be understood as a tapered MMI combiner and a spot
size converter (SSC). From the calculated results we can know the loss is small,
because most of the power of the lost modes is transferred to the low-order modes by
the mode interference because of existed hybrid modes.
P
(a). In-phase inputs
(b). Out-of-phase inputs
Fig. 2.26 Power distributions in a 4x1 tapered MMI combiner in the cases
of (a) in-phase inputs and (b) Out-of-phase inputs.
In the case of out-of-inputs, the MMI images are near to the both border sides.
The images lost easily as shown in Fig. 2.26 (b). In Fig. 2.26 (b) the phase
57
distribution of the inputs is 0, π, 0, π. The MMI patterns would be different for other
distributions such as π, π, 0, 0. My simulation shows that difference phase
distribution of inputs will affect the output extinction ratio, and the separate
distribution is better. So, it is important how to arrange the input waveguides in
designing a multi-ports MMI combiner.
It is worthy to note that the phase changes from every input port to the output
port are a little different. The phase differences will affect the MMI images
distribution in the tapered MMI combiner and shift the output spectrum, however
they don’t affect the output characteristics. The phase difference can be understood
an initial phase of the input signals, in order to simplify the tapered MMI analysis.
Mathematically, the largest phase difference between ports is the phase
difference of port1 and port (N/2), which can be written as
nπ ( ww + ws ) 2 ( N 2 − 2 N ) / 4 / λ / L ,
(2.116)
where n and λ are refractive index of MMI waveguide and wavelength at free-space,
ww , ws and L are waveguide width, access waveguide spacing and MMI combiner
length, respectively. N is the port number as shown in Fig. 2.20.
In fact, the phase differences are small, because of the long device length and
the narrow width. For example, when L = 1500 µm, ww = 2 µm, ws = 0.5 µm and N =
8, the largest phase difference is 0.1π. The initial phase problem can be solved by
properly designing the multi-port MMI combiner.
-2
15
-2.5
13
-3
11
9
Excess loss (dB)
Extinction ratio (dB)
17
-3.5
7
5
-4
-10
-5
0
5
10
15
Width error,δW/W (%)
Fig. 2.27 Robustness and excess loss to fabrication errors of a
device width.
58
D. Fabrication robustness
Robustness to fabrication errors is another advantage of the tapered MMI
combiner. For the rectangular MMI coupler, the fabrication error of a width or a
length will cause severe degradation of the performance. For example, for a
rectangular MMI coupler with a width of W = 18 µm, a wavelength of λ = 1.55 µm
and a refractive index of n = 3.3, the fabrication error of a width by δW/W = 5%, an
additional excess loss of over 2 dB [62]. But for the tapered MMI combiner, the
additional excess loss is less than 1 dB. This is because the tapered MMI waveguide
gradually changes its form to a single-mode waveguide. So a generated image can
be coupled into the output waveguide with a smaller loss, even when there is a
position error of the image. Therefore, the tapered device is more robust to
fabrication errors.
Fig. 2.27 shows the robustness to fabrication errors of a width of a 2 x 1 tapered
MMI combiner with the structure of L = 200 µm, W = 8 µm and ws = 2 µm. The
border of the combiner is comprised of two tangent arcs. Considering actual
fabrication errors, the width changes of the tapered MMI combiner and the output
waveguide are the same. The expected extinction ratios are 11.5 dB. When the
width error is 5%, the shifted extinction ratio is about 3.5 dB. Basically the change
of the extinction ratio is from the losses of high-order modes and the width change
of the output waveguide. However, the excess loss change with the width change is
not sensitive to be less than 1 dB for the width errors of 5% as shown in the figure.
In our experiments we found that a MZ modulator with a 2 x 1 tapered MMI
combiner has a less insertion loss than that with a 2 x 1 rectangular MMI combiner
by about 2 dB, which contradicted our simulated results. We believe the
contradiction is from that the length of the 2 x 1 rectangular MMI combiner is not
optimal. In fact, the optimal MMI length is difficult to be obtained in actual
fabrications, since it is sensitive to the waveguide dimension affected by fabrication
processes such as exposing, developing, etching and coating.
E. Multimode waveguide
All the discussions above are based on a single-mode output waveguide,
because a multimode output waveguide severely deteriorates the combiner
performance. In the case of out-of-phase inputs for a 2 x 1 tapered combiner, the
lowest-order mode in the MMI section is the first-order mode. When an access
waveguide at the output is a single mode waveguide, no modes can be coupled into
the output waveguide, so the extinction ratio of the combiner is high. However,
59
when the output waveguide is a multimode waveguide, high-order modes would
couple into it. So the extinction ratio of the tapered combiner will decrease.
Relative output power (dB)
0
-5
-10
-15
W=3µm
-20
W=2µm
-25
0
1
2
3
4
5
6
Relative phase difference (rad.)
Fig. 2.28 Relative output powers as a function of relative phase differences
of two inputs under different output waveguide widths.
In Fig. 2.28 we simulated relative output powers of a 2 x 1 tapered MMI
combiner with the parameters of W = 6 µm, ws = 2 µm and L = 300 µm. As
mentioned above, all the access waveguides at the input and the output with a
width of 2 µm are single-mode waveguides, and the refractive indices of core and
cladding regions are 3.35 and 3.34, respectively. The extinction ratio of the output
shown by the line with squares is about 18 dB. However, when the output
waveguide width changes from 2 µm to 3 µm, the output waveguide can hold two
modes, and the extinction ratio decreases to 7 dB, as shown by the line with
triangles. So a single-mode access waveguide is inevitable for a high-performance
tapered MMI combiner.
§2.4.3 Design of non-linear tapered MMI combiners
An output extinction ratio of a coherent lightwave combiner is a basic
parameter to characterize its performance. The extinction ratio of the tapered
combiner is affected by the device structure such as a curvature radius R of the
60
borders, an access waveguide spacing ws, an initial device width W and a device
length L, as shown in Fig. 2.29, which are simulated by FD-BPM [63].
61
Fig. 2.29 Relative output power as a function of relative phase difference of two
inputs under various parameters: a) border curvature radii, b) device length, c)
access waveguide spacing and d) device initial widths.
For a tapered MMI with a given initial width and a given length, the
curvature radius R of the borders gives a large effect on the extinction ratio. The
mechanisms are the followings.
In the case of in-phase inputs, all the excited
modes are even-modes, and basically the interference images tend to concentrate
along the taper centerline, as shown in Fig. 22 (a). So the high-order modes have
little losses. That is why the excess loss of a 2 x 1 MMI coupler with in-phase
62
inputs is less than that of the same 2 x 1 MMI coupler with one-port input. On
the other hand, in the case of out-of-phase inputs, only odd-modes are excited, and
the generated MMI images mainly concentrate on border sides, as shown in Fig.
2.22 (b), and almost all of the high-order modes leak out of the combiner because
of the taper structure. These phenomena lead to larger extinction ratio. These
mechanisms explain that smaller curvature radius R tends to lead to larger losses
for high-order modes and a large extinction ratio, as shown in Fig. 2.29 (a).
Diamonds, squares and triangles are the output powers of tapered combiners for
various curvature radii of Rmin = 22.5 mm, R = 40 mm and R = infinite, with
structure parameters of W = 6 µm, ws = 2 µm and L = 300 µm. The excess losses
defined as the output in the case of in-phase inputs are nearly the same for
different curvature radii R, so the Rmin given by (2.115) is an optimal curvature
radius. Fig. 2.29 (b) shows the simulated output powers of tapered combiners for
various device lengths L of 200 µm, 300 µm and 400 µm, with the structure
parameters of W = 6 µm, ws = 2 µm and R = Rmin. The longer device leads to a
higher extinction because of larger total losses of high-order modes. So the device
compactness is limited by an expected extinction ratio. For example, when the
expected extinction ratio is 15 dB, the shortest length is 120 µm with the
structure parameters of R = Rmin, ws = 1 µm and W = 5 µm.
The access
waveguide spacing ws affects the device performance largely as shown in Fig. 2.29
(c). The curves with diamonds, squares and triangles are the output powers of the
tapered combiners with L = 300 µm and R = Rmin. W = 5 µm, 6 µm and 7 µm are
adopted corresponding to various ws of 1 µm, 2 µm and 3 µm, respectively. The
smaller access waveguide spacing gives a larger extinction ratio and a small
excess loss. The large excess loss for the larger access waveguide spacing is due to
the following reasons. One is the spurious mode conversion by the tapered
structure; another one is that the excited modes are different depending on the
access waveguide spacing. However, very small access waveguide spacing is
limited by fabrication techniques and coupling between access waveguides. The
initial device width affects the device extinction ratio, too. Fig. 2.29 (d) is the
relative output powers of the tapered MMI combiners for different initial widths
W of 6 µm, 8 µm and 10 µm with structure parameters of L = 300 µm, ws = 2 µm
and R = Rmin. The narrower initial width leads to a larger extinction ratio, which
is from a larger loss of high-order modes.
63
0
35
-0.5
30
-1
25
20
-1.5
15
-2
10
Excess loss (dB)
Extinction ratios (dB)
40
-2.5
5
0
-3
0
100
200
300
400
500
600
Device length (µm)
Fig. 2.30 Dependence of extinction ratios and excess loss on tapered MMI length.
Fig. 2.29 (b) shows the simulated output powers of tapered combiners for
various device lengths L of 200 µm, 300 µm and 400 µm, with the structure
parameters of W = 6 µm, ws = 2 µm and R = Rmin. The longer device leads to a larger
extinction ratio because of more losses of high-order modes. So, a long tapered MMI
combiner is necessary for an expected extinction ratio, although compactness is one
of advantages of the tapered structure [57]. Fig. 2.30 shows a device length
dependence of extinction ratios and excess loss of a 2 x 1 tapered MMI combiner
with the structure parameters of R = Rmin, ww = 2 µm, ws = 1 µm and W = 5 µm.
Therefore, the device length is actually determined by balancing an expected
extinction ratio, excess loss and compactness. For example, when the expected
extinction ratio is larger than 15 dB, the device length should not be less than 140
µm.
The insertion loss is mainly from the scattering loss by the two MMI curved
borders in the case of in-phase inputs. The scattering loss depends on the border
shape and the distribution of excited modes in the taper MMI combiner. So, we can
improve the excess loss by forming the images of the excited modes as close to the
centerline of the tapered MMI as possible, which can be realized by reducing the
input access waveguide spacing, as shown in Fig. 2.29 c). There is an insertion loss
of about 3 dB in Fig. 2.29 a), b) and d) because of the large access waveguide spacing
of 2 µm. When the access spacing is less than 1 µm, the excess loss can be less than
2 dB.
64