C. L. SUI ET AL.

290

doesn’t equal to 0, the value of 2/RL

won’t be an

integer, which means we cannot evenly distribute every

period of orthogonal basis around the circle. But if L is

chosen to be a very large number, the approximate error

can be neglected. According (9), if L is given, we can get

a simple relationship between h and .

0

Phase shift due to rough surface should also be taken

into consideration when we investigate the mode splitting

in micro-disk. The phase shift is uniformly distributed in

the interval

l

[0,2 )

, so the orthogonal basis can be modi-

fied to be i

{() efz xp[ 2(0,1)j N]}

. Here is a

random variable that uniformly distributes in the interval

.

(0,1)N

[0,1)

If we have to analyze the effect of N modes with dif-

ferent azimuthal orders, we can find N-dimensional or-

thogonal basis from {()}

i

z to approximate the effect

of inhomogeneity, according to the representation theo-

rem in strictly normed linear space[13]. The weights for

each orthogonal basis can be obtained by matrix trans-

form.

Surface roughness effect caused by ()r

and '( )r

can be represented as an elementary function of ()r

and '( )r

, so we can easily obtain [12]:

[()]

['()]

ii

ii

kk

fx

fx

[()]

['()]

kk

Ffx F

FfxF

(10)

Thus, if 0

2

0|()|

l

k

xdx

keeps static when the value of

0

l changes, the value of [()

i

k]

x will be the same,

which means the ()r

nduced roughness effect will not

change. Similarly by keeping 0

2|'()|

l

i

0k

xdx

static, we can

keep the effect of r'()

unchanged even though the

value of 0

l changes. In this way, we can analyze the

effect of ()r

and r'()

respectively.

3. Simulation and Result Analysis

To confirm the theory above, we analysis the surface

roughness effect on Q-factor of micro-disk as an example.

Because we mainly concentrate on the power from the

in/out port, the standard operator norm defined in [14] is

carefully applied to estimate the transmittance of micro-

disk. The operator norm is defined as:

2

,0

(, )

sup (,)

xSx

Ex Ex

Exx

(11)

According to [6], the height of inhomogeneity ()r

induced scattering for 2D micro-disk resonator can be

written as

2

2

()

22

()

(1 )

hK

E

(12)

and if the surface is rather smooth, which means is

fairly small, it is the height of inhomogeneity )(r

,

rather than its first derivative '( )r

, that is the-

nant factor that leads to scattering loss.

To compare the deviation above wit

domi

orthogonal

is 3μm.

h the

ba

-disk

Th

sis method we proposed in Section 2, we conduct FDTD

simulations with on Ge whisper gallery mode (WGM)

micro-disk resonator. The simulation tool is FDTD solu-

tion release 8.0. Micro-resonators based on Ge are more

and more popular for the merit of especially high

Q-factor and graft ability on Silicon based device [11].

It’s of great value to have a thoroughly research on the

surface roughness effect of Ge micro-disk.

In our simulation, the diameter of micro

e waveguide coupled along the disk is made of SiO2

with 0.5μm width at 200nm distance from the disk. N =

1.42 is used as waveguide’s reflective index and a value

of n = 4.2 for Ge micro disk. Surface roughness is ma-

nipulated by orthogonal basis {()}

i

x and details of

orthogonal basis are carefully calculated following the

method described in Section 2. When we investigate the

propagation of fundament TE/TM mode, the mode num-

ber n in (9) is set to 1. Mean variance of roughness 2

is 1nm and correlation length c

L is 50 nm. Accord

to (8-9), if /4

ing

, we can obtain

2

0

2

/ (18

sin [0.01895(18)]0.0379(18)]

18 /52.8

0.4107

R

pl pl

p0

16 )L p

1 exp[hj

(13)

from (1e larger p is

3) that thIt’s easy to observe

m

, the

ore accurate we can calculate because we need to fix

the value of L to make 2/RL

to be an integer. In our

simulation, p is set to be 4, and the relationship between

0

,hl indicated in (17) can be further simplified.

s it is stated above, by keeping 0

2|()|

l

k

A0

xd

x static,

acss due to cording to (10), the scattering r'( )

lo

changes with 0

l increasing from 150nm to 400nm

the effect of ()r while

will be the same. Similarly, if the

value of 0

2|

l

0|'()

k

xdx

s static when 0

l s varied, the

scattering ()r

loss due to

will chan e accordingly g

and the effect of r'( )

kee static.

Figure 3 showlectromagnetic

p

e

s

s the (EM) field distri-

bution when the effect of ()r

varies with the in-

crease of 0

l. Obviously, Q-f micro-disk resona-

tor decreasramatically as the height of inhomogeneity

increases. On the contrary, if the value of 0

2|()|

l

k

actor of

e d

0

xdx

is a constant, the Q-factor almost keeps u nchanged with

the alternative of 0

l, as depicted in Figure 4. Figure 5

reveals different sensitivity of micro-resonator’s Q-factor

on the two scattering mechanism. The effect of ()r

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