chapter 2 multimode interference theory
TRANSCRIPT
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
tBE∂∂
−=×∇r
r, (2.1)
tDJH∂∂
+=×∇r
rv, (2.2)
r0=⋅∇ B , (2.3)
ρ=⋅∇ Dr
. (2.4)
The principle of conservation of charge can be deduced by (2.2), (2.4) and
as 0)( =×∇⋅∇ Ar
0=∂∂
+⋅∇t
J ρr. (2.5)
Poynting vector S (W/m2) is HESrrr
×= . (2.6) The constitutive relations (or material equations) in isotropic media are
15
EDrr
ε= , (2.7)
HBrr
µ= , (2.8)
EJrr
σ= , (2.9) where ε, µ and σ are permittivity, permeability and electric conductivity. The
refractive index is rrn µε= , where and are relative
permittivity and relative permeability, respectively. Usually, for
dielectric media, so
0/εεε =r 0/ µµµ =r
0µµ =
rn ε= .
In the isotropic media without considering electric charges, the wave equation can be deduced from the Maxwell’s equations as
02
2
002 =
∂∂
−∂∂
−∇tE
tEE
rrr
εµσµ , (2.10a)
02
2
002 =
∂∂
−∂∂
−∇tH
tHH
rrr
εµσµ . (2.10b)
For a plane wave )]exp(),,(Re[ tjzyxeE ω= , (2.10) can be written as
∇ 0)( 022 =−+ EjkE
rrσωµ , (2.11a)
0)( 022 =−+∇ HjkH
rrσωµ . (2.11b)
The phase velocity and group velocity are kp
r/ων = and kg
r∂∂= /ων ,
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.
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Medium n2
Medium n1
k1 k3
k2
θ1 θ3
θ2 z
y
Fig. 2.1 Optical beams at the interface of two different media Considering δ/δx=0, we can write the Maxwell’s equations as
∂
∂−
∂∂−
=
=∂∂
−=∂∂
)(
,
20
00
zH
yH
njE
HjyE
HjzE
yzx
zx
yx
ωε
ωµωµ (2.12a) For senkrecht wave,
∂
∂−
∂∂
=
−=∂∂
=∂∂
)(
,
0
02
0
zE
yEjH
EjyH
EnjzH
yzx
zx
yx
ωµ
ωεωε (2.12b) For parallel wave.
The senkrecht wave in medium 1 can be written as
⋅−+⋅−−=⋅−+⋅−=
⋅−+⋅−=
1333111
1333111
3311
/)]exp(cos)exp(cos[/)]exp(sin)exp(sin[
)exp()exp(
ZrkjErkjEHZrkjErkjEH
rkjErkjEE
ssz
ssy
ssx
rrrrrrrr
rrrr
θθθθ . (2.13)
The senkrecht wave in medium 2 can be written as
⋅−−=⋅−=
⋅−=
2222
2222
22
/)exp(cos/)exp(sin
)exp(
ZrkjEHZrkjEH
rkjEE
sz
sy
sx
rrrr
rr
θθ . (2.14)
The parallel wave in medium 1 can be written as
17
⋅−−⋅−=⋅−−⋅−−=
⋅−+⋅−=
)exp(cos)exp(cos)exp(sin)exp(sin
/)]exp()exp([
333111
333111
13311
rkjErkjEErkjErkjEE
ZrkjErkjEH
ppz
ppy
ppx
rrrrrrrr
rrrr
θθθθ . (2.15)
The parallel wave in medium 2 can be written as
⋅−=⋅−−=
⋅−=
)exp(cos)exp(sin
/)exp(
222
222
222
rkjEErkjEE
ZrkjEH
pz
py
px
rrrr
rr
θθ . (2.16)
where surge impedance (or characteristic impedance) is expressed as
)2,1(,/// 00 === inHEZ ii εµ . (2.17)
By the border conditions of E1t=E2t and H1t=H2t at y=0, we know
)exp()exp()exp( 223311 zjkEzjkEzjkE zszszs −=−+− , (2.18a)
)exp(cos)exp(cos)exp(cos 222233131111 zjknEzjknEzjknE zszszs −−=−+−− θθθ .
(2.18b) For arbitrary z, (2.18) exist. So the following relation should exist
zzz kkk 321 == or n . 301202101 sinsinsin θθθ knknk ==
The law of reflection and Snell’s law can be obtained as 31 θθ = and . 2211 sinsin θθ nn =
Fresnel’s relations can be obtained from (2.18) as
+==
+−
==
2211
11
1
2
2211
2211
1
3
coscoscos2
coscoscoscos
θθθ
θθθθ
nnn
EE
t
nnnn
EE
r
s
ss
s
ss
(2.19) for senkrecht wave.
By the same method, the reflection coefficient and transmission coefficient can be written as
+==
+−
==
2112
11
1
2
2112
2112
1
3
coscoscos2
coscoscoscos
θθθ
θθθθ
nnn
EE
t
nnnn
EE
r
p
pp
p
pp
(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 . )/sin( 12 nna
θ1
ZGH
n2
n1
y
z θ1 θ1
Fig. 2.2 Goos-Hanshen shifts.
From Snell’s law, we know
22
122
12
22
sincosnnn θ
θ−
±= , (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 , (2.21) can
be written as 211 sin nn >θ
22
221
221
2sincosn
nnj −−=
θθ . (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
221
22111
221
22111
sincos
sincos
nnjn
nnjnrs
−−
−+=
θθ
θθ. (2.23)
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Obviously, the power of the reflected wave is 1. sr =1. So we can assume:
ss r=− )exp( ϕ or 221
22111 sincos)2/exp( nnjns −+=− θθϕ .
Then, the Goos-Hanshen shift for senkrecht wave can be written as
)cos
sinsin(tan2
1
21
21
θθθ
ϕ cs
−−= − . (2.24)
By the same method, the Goos-Hanshen shift for the parallel wave can be written
)cossinsinsin
(tan21
2
21
21
θθθθ
ϕc
cp
−−= − . (2.25)
The shift of reflection point can be written as
−
−
+
−
=∂
∂=
22
2
0
1
2
1
1
1
1
22
2
0
1
1
,
1
tan2
tan2
nkk
kk
kk
nkk
kZ
zzz
z
z
psGH θ
θ
φ . (2.26)
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
. (2.27) )/)sin(sin(1 fna ψθ =
T Waveguide nf
Clad nc
Substrate ns
θ
θ1 ψ
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
)/(sin 1fcc nn−=θ , (2.28)
)/(sin 1fss nn−=θ , (2.29)
respectively. The condition to confine the light in the waveguide can be written as
. (2.30) sc ,1 2/ θπθ −<
When , the maximum incident angle ψcs nn = max can be written as
..)2(sin 1max ANn f ≡∆= −ψ (Numerical Aperture), (2.31)
where 2
22
2 f
cf
n
nn −=∆ .
21
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 . 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.
cs nn ≥
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
K,1,0,)1(2 =+=Φ mmT π (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
K,1,02cos2 0 ==++=Φ mmTkn csfT πϕϕθ , (2.33)
or
L1,0,)cos
sin(tan)
cos
sin(tancos
2221
2221
0 ==−
−−
− −− mmn
nn
n
nnTkn
f
cf
f
sff π
θ
θ
θ
θθ .
(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) c sk0(1/µm) n n 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 Mode numbers Angles of mode 0 neff
0.25 µm 1 580 2.9342 0.5 µm 2 70.50 3.2615 1.0 µm
250 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).
Phas
e (r
ad.)
0.25µm
0.5µm
1.0µm
Mode 3
Mode 2
Mode 1
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.
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z
x
a 0 -a ns
n1
n0
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
−<++≤≤−−
>−−−=
)()](exp[)cos()()cos(
)()](exp[)cos(
axaxkaAaxakxA
axaxkaAEy
ξφφ
σφ , (2.35)
where
2210
221
20 effnnknkk −=−= β ,
20
20
20
20
2 nnknk eff −=−= βσ ,
220
220
2seffs nnknk −=−= βξ .
Ey in (2.35) at x=a and –a is continual. And the difference of Ey
−<++≤≤−−−
>−−−−=
∂
)()](exp[)cos()()sin(
)()](exp[)cos(
axaxkaAaxakxkAaxaxkaA
dxEy
ξφξφ
σφσ (2.36)
is also continual at x=a and –a as
−=−+=+
)sin()cos()cos()sin(
φφσφξφ
kakAkaAkaAkakA
. (2.37)
Removing A, the following equations can be obtained,
uu ωφ =+ )tan( , (2.38a)
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uu 1)tan( ω
φ =− , (2.38b)
where
===
aakau
σωξω
1
. (2.39)
So, the following relations can be gotten as
),2,1,0()(tan21)(tan
21
2111 L=++= −− muu
mu ωωπ , (2.40a)
),2,1,0()(tan21)(tan
21
2111 L=−+= −− muu
m ωωπφ , (2.40b)
2221
2222 )( vnnaku s ≡−=+ω , (2.41)
221 ωγω += v ,
221
20
2
s
s
nnnn
−
−=γ ,
221
22
s
se
nnnn
b−
−= .
For propagating modes 0 < b < 1. The condition of cut off is b = 0.
bb
bbmb
−+
+−
+=− −−
1tan
1tan12 11 γπν , (2.42)
+==
−=
γωω
bvbvbvu
1
1.
For a symmetric waveguide, (2.40) can be written as
2
)(tan2
1
πφ
ωπ
mu
mu
=
+= −
, (2.43)
or
25
)2
tan( πω muu −= . (2.44)
Electric field distributions are followings
∫∫∫∞
∞−
∞
∞−−=⋅×= dxHEHEdxuHEdyP xyyxz )(
21)(
21 ***1
0
rrr , (2.45)
∫∞
∞−= dxEP y
2
02ωµβ , (2.46)
)(2
)(sin2
)(sin12 1
22
0
2
axauuaAPcore ≤≤−
−
++
+=ω
φω
φωµβ
, (2.47)
)(2
)(cos2
2
0
2
axuaAPsub −<+
=ω
φωµβ , (2.48)
)(2
)(cos2 1
2
0
2
axuaAPclad >−
=ω
φωµβ , (2.49)
++=++=10
2
21
211
2 ωωωµβaAPPPP cladsubcore , (2.50)
)2/12/11(2
1
0
ωωβωµ
++=
aP
A . (2.51)
By the same method, the conclusions for TM modes can be written as
−<++≤≤−−
>−−−=
)()](exp[)cos()()cos(
)()](exp[)cos(
axaxkaAaxakxA
axaxkaAH y
ξφφ
σφ, (2.52)
),2,1,0()(tan21)(tan
21
2 20
12
112
211 L=++= −− m
unn
unnmus
ωωπ, (2.53)
)1
(tan)1
(tan12 20
211
2
211
bb
nn
bb
nn
mbs −
++
−+=− −− γπν . (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.
n0
y
x III
II
In1
2a
2d
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
−−−−−−−−
−−=
zoneIIIdkdyxkAzoneIIykaxakAzoneIykxkA
H
yyx
yxx
yx
y
)cos()](exp[)cos()cos()](exp[)cos(
)cos()cos(
ψγφψγφ
ψφ , (2.55)
where
=−++−=−+−=−+−−
zoneIIInkkzoneIInkkzoneInkkk
yx
yx
yx
000
220
20
22
220
20
22
221
20
22
βγβγβ
, (2.56)
and
27
=−=
=−=
L
L
,2,12
)1(
,2,12
)1(
pp
πψ
πφ . (2.57)
When x = a, is continual. When y = d, is
continual. Then the relations can be gotten as
xHnEz y ∂∂∝ /)/1( 2 yHHz y ∂∂∝ /
)(tan2
)1( 20
211
x
xx kn
npak
γπ −+−= , (2.58a)
)(tan2
)1( 1
y
yy k
qdkγπ −+−= . (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 21
202 nn − .
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,
)()(),( yYxXyxH y = . (2.59)
So the wave equation is written as
0)],(),([11 22202
2
2
2
=−++ yxyxnkdyYd
YdxXd
Xβ , (2.60)
0)](),([1 220
2202
2
=−+ xnkyxnkdyYd
Y eff , (2.61)
0)],()([1 22202
2
=−+ yxxnkdxXd
X eff β . (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).
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§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.
N N-1 . 1
1 . M-1 M
Access waveguides
Access waveguides Multimode waveguide
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.
Multimode
waveguide
WM nr nc
mode 5
mode 2
mode 0
mode 1
mode 3
y
y
z
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
220
22rvyv nkk =+ β , (2.63)
where . 00 /2 λπ=kThe stand-wave condition in y-direction is
π)1( += vWk evyv , (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
)2/1(222
0 )( −−
+≈ cr
r
cMev nn
nn
WWσ
πλ
, (2.65)
where .
==
TMforTEfor
10
σσ
30
So the propagation constant of mode ν can be written as
222
0)1(
+−=
evrv W
vnk πβ . (2.66)
By Taylor expansion of (2.66), the propagation constant can be written as
20
2
0 4)1(
errv Wn
vkn
πλβ
+−= . (2.67)
Then the propagation constant difference of mode ν and mode 1 can be expressed as
20
0 4)2(
erv Wn
vv πλββ
+=− . (2.68)
By defining Lπ as the beat length of the two lowest-order modes
0
2
10 34
λββπ
πerWnL =
−≡ , (2.69)
the propagation constants spacing can be written as
π
πββLvv
v 3)2()( 0
+≈− . (2.70)
Modes Power Coefficients
0 1 1 1 2 1 3 1 4 1 5 1 6 1
Am
plitu
de (a
rbitr
ary)
Waveguide width (µm)
Fig.2.9 Mode matching between input mode and the excited modes in amultimode 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
emeveein PPPPP modmod1mod0mod ...... +++= . (2.71)
Modes Power
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)
A
mpl
itude
(arb
itrar
y)
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
Fig. 2.11 Mode matching between input mode and the excited modes in amultimode waveguide.
modes Power 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)
A
mpl
itude
(arb
itrar
y)
§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
∑=Ψv
vv ycy )()0,( ψ , (2.72)
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
∫∫Ψ=
)(
)()0,(2 y
dyyyc
v
v
ψ
ψν , (2.73)
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
∑−
=
=Ψ1
0)()0,(
m
vvv ycy ψ . (2.74)
The field profile at a distance z can then be written as a superposition of all the guided mode field distributions
33
.))(exp()(),(1
0∑−
=
−=Ψm
vvv ztjyczy νβωψ (2.75)
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
.))(exp()(),(1
00∑
−
=
−=Ψm
vvv zjyczy νββψ (2.76)
A useful expression for the field at a distance z = L is then found by substituting (2.70) to (2.76)
.)3
)2(exp()(),(1
0∑−
=
+=Ψm
vvv L
LvvjycLy
π
πψ (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
).3
)2(exp( LLvvj
π
π+ (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
−=−
oddvforyevenvfory
yv
vv )(
)()(
ψψ
ψ . (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
.)1(1)3
)2(exp( vorLLvvj −=+
π
π (2.80)
34
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.
y
z
2(3Lπ) 3(3Lπ)/2 (3Lπ) (3Lπ)/2 z=0
y=0
Ψ(y,0)
Fig. 2.12 Multimode waveguide showing the input field Ψ(y,0), a mirrored single imageat (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,1,0)3( == pwithLpL π , (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,5,3,12/)3( == pwithLpL π . (2.82) The total field at these lengths is found by substituting (2.81) into (2.76)
35
∑−
=
+=Ψ1
0))
2()2(exp()()3
2,(
m
vvv pvjvycLpy ππψπ , (2.83)
with p an odd integer. The mode field symmetry conditions of (2.78), (2.82) can be written as
.)()()()32
,( ∑∑ −+=oddv
vvp
evenvvv ycjycLpy ψψπΨ (2.84)
∑∑∑∑
∑∑∑∑
−−−+−+
−+−++=Ψ
evenvv
p
evenvv
p
oddvv
oddvv
oddvv
p
oddvv
p
evenvv
evenvv
ycjycjycyc
ycjycjycycLpy
νννν
ννννπ
ψψψψ
ψψψψ
)()(21)()(
21)(
21)(
21
)()(21)()(
21)(
21)(
21)3
2,(
∑∑∑∑
∑ ∑∑ ∑
−−−−−+−+
−++=Ψ
oddvv
p
evenvv
p
evenvv
p
oddvv
p
even oddvvvv
even oddvvvv
ycjycjycjycj
ycycycycLpy
νννν
ν νν νπ
ψψψψ
ψψψψ
))()()()((21))()()()((
21
))()((21))()((
21)3
2,(
)0,(2
)(1)0,(2
)(1
)0,()(21)0,()(
21)0,(
21)0,(
21)3
2,(
yjyj
yjyjyyLpy
pp
pp
−Ψ−−
+Ψ−+
=
−Ψ−−Ψ−+−Ψ+Ψ=Ψ π
(2.85)
The last equation represents a pair of images of Ψ(y, 0), in quadrature and with amplitudes 2/1 , 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
∑∞
−∞=
+−Ψ−−Ψ=Ψv
eein WvyWvyy ))2()2(()( , (2.86)
and to approximate the mode field amplitudes by sin-like functions
).sin()( yky yvv ≈ψ (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
)3( πLNpL = , (2.88)
where p ≥ 0 and N ≥ 1 are integers with no common divisor. The field will be of the form
∑−
=
−Ψ=Ψ1
0
)exp()(1),(N
qqqin jyy
CLy ϕ , (2.89)
with
NWNqpy e
q )2( −= , (2.90)
NqqNpqπϕ )( −= , (2.91)
where C is a complex normalization constant with NC = , 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 N/1 and
phase . This leads to N images (generally not equally spaced) of the input qϕ
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
evensrforsrNsNrs ++−+−= ππϕ )2)(1(
4 , (2.92)
and
oddsrforsrNsrNrs ++−−−+= )12)(1(
4πϕ , (2.93)
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 L,8,5,20)]2([mod3 ≠=+ vforνν , (2.94)
it is clear that the length periodicity of the mode phase factor of (2.87) will be reduced three times if
,...8,5,20 == νforcv (2.95) Therefore, as shown in [21], single (direct and inverted) images of the input
field Ψ(y,0) are now obtained at L,2,1,0)( == pwithLpL π , (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
. 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 c
6/eWy ±=
ν = 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. Symmetri Interference c
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 evenvforvv 0)]2([mod4 =+ (2.97)
it is clear that the length periodicity of the mode phase of (2.79) will be reduced four times if
L,5,3,10 == vforcv (2.98) Therefore, single images of the input field Ψ(y, 0) will now be obtained at
L,2,1,0)4
3( == pwithL
pL π , (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
=
43 πL
NpL (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 byFD-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
Singleimage
Multi images
Single image
Single image
Source
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
∑+
−=
+=J
Jjj jndiQu ))/2(exp()( πλπ . (2.101)
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 at L0/ =dzdWMMI MMI/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.
§2.3.7 MMI reflection characteristics
Fig.2.17 Taper MMI couplers as splitters
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 Excitation
Single input Single input In-phase inputs Out-of-phase inputs
Trans-mission
Back-re flection
44
Inter- nal re- sonance
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 thecase 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 dB16− 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
20
0 4)2(
nWvv
vπλββ +
≈− , (2.102)
where W is an effective width of the MMI waveguide with a refractive index of n, λ0 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
2220 )()()( zzKnK x νν β+= , (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
π)1()()( += vzWzKxv , (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 β0(z) can be expressed as
20
0)(4
)2()()(
znWzz
πλννββ ν
+≈− , (2.105)
where the effective width W(z) 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]
∫+
=∫ −=∆ 2
12
02
1021 )(4
)2())()((),( zz zW
dzn
vvdzzz zzzz vv
πλββφ . (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 z z1 to the one of 2 can be written as
3)2(
),(),(
211
211
+=
∆∆
=vv
zzzzv
φφαν . (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
)0(,),0(),0( 110 −mφφφ K , (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
)1())1()1(()1(,),1(),1( 001110 φφφαφφφ νν +−=L , (2.109)
where ν = 2, 3, …m-1. Passed slice k, the phases of these effective modes can be written as
0011
1111
0000
)(:
)()1()0(:1)()1()0(:0
ψψψαψν
φφφψφφφψ
νν +−=
+++=+++=
Mode
kModekMode
L
L
L
. (2.110)
Since I only interest in the relative phase changes of higher modes to the fundamental one, I can rewrite (2.110) as
3/))(2(,,3/)(8),(,0 010101 ψψννψψψψ −+−− L , (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
00 2
8)( λ
nlzWdzL =∫ . (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
P1
0,,0,0,,0,0 110 ===== − Mmm φφφφφ LL
MmmMvvv
LL ,2,1,0,,3,21())1()1(()1(),1(),1(
11
001110
=−=+−= φφφαφφφ
221
001110
,2,1,0,,3,2())2()2(()2(),2(),2(mMmmmMv
vv
−=−−=+−=
LL
φφφαφφφ
3321
001110
,2,1,0,,3,2)3())3()3(()3(),3(),3(MmmmmMv
vv
=−−−=+−=LL
φφφαφφφ
1,3,2))()(()(),(),( 01110
−=+−=
mvkkkkk vv
L
φφφαφφφ
01
3/)2(,,3/8,,0ψψψ
ψννψψ−=∆
∆+∆∆ L
or
)
1
)2
21 mm −−
)(0 k
P3
P2
Pk
P0
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.
c
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 compa tness 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
20
20 )(22)( xzRywwzW −−−+= , (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
)(44)( 22
min wwWLwwWR
−+−
= , (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
Out
put i
mba
lanc
e (d
B)
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
)(8)(4 22
wwWwwWLRn −
−+=
, (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.
Tapered MMI Combiner (a)
-30
-25
-20
-15
-10
-5
0
1550.5 1551 1551.5 1552
τ0 τ0 τ0
Out
put p
ower
(dB)
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
Fig. 2.26 Power distributions in a 4x1 tapered MMI combiner in the casesof (a) in-phase inputs and (b) Out-of-phase inputs.
(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
LNNwswwn //4/)2()( 22 λπ −+ , (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.
-4
-3.5
-3
-2
151050-5-105
7
9
11
15
13
17
Exce
ss lo
ss (d
B)
-2.5
Extin
ctio
n ra
tio (d
B)
Width error,δW/W (%)
Fig. 2.27 Robustness and excess loss to fabrication errors of adevice width.
58
D. Fabrication robus ness tRobustness 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.
0
-5
-10
-15
-20
-25
W=3µm W=2µm
Rela
tive
outp
ut p
ower
(dB)
0 1 2 3 4 5 6 Relative phase difference (rad.)
Fig. 2.28 Relative output powers as a function of relative phase differencesof 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, w = 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.
s
§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 twoinputs 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
-3
-2.5
-2
-1.5
-1
-0.5
0
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500 600
Extin
ctio
n ra
tios
(dB)
Exce
ss lo
ss (d
B)
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