Analysis of Microdisk/Microring’s Surface Roughness Effect by Orthogonal Decomposition

doi:10.4236/opj.2013.32B068

Paper Menu >>

Journal Menu >>

Optics and Photonics Journal, 2013, 3, 288-292

doi:10.4236/opj.2013.32B068 Published Online June 2013 (http://www.scirp.org/journal/opj)

Analysis of Microdisk/Microring’s Surface Roughness

Effect by Orthogonal Decomposition

Chengle Sui1, Qiangmin Wang2, Shilin Xiao1, Pingqing Li1

1State Key Lab of Advanced Optical Communication System and Networks, Shanghai Jiao Tong University, Shanghai, China

2School of Information Security Engineering Shanghai Jiao Tong University, Shanghai, China

Email: [email protected], [email protected]

Received 2013

ABSTRACT

Application of micro-resonator is limited by different types of surface inhomogeneity. The 1-th derivative of inho-

mogeneity (i.e. '( )r



) affects the wave transport as well as the height of inhomogeneity (i.e. ()r



). A method

based on orthogonal decomposition is proposed to analysis both scattering mechanism respectively. Then surface

roughness effect on Q-factor of micro-disk waveguide gallery mode (WGM) resonator is investigated with our method

and the analysis fits well with FDTD simulation results.

Keywords: Orthogonal Decomposition; Micro-resonator; Surface Roughness Effect

1. Introduction

Micro-ring/micro-disk resonators are very elemental build-

ing blocks of integrated photonic device for its compact

size and especially for its high quality factor (Q-factor)

[1]. High-Q micro resonators are more and more widely

applied in lasers, modulators and bio-sensors [2-4]. How-

ever, the Quality factor of the high-index-contrast system

is still mainly constrained by scattering loss and absorp-

tion loss due to surface roughness [5].

Surface roughness effect on Q-factor and mode split-

ting are widely investigated in [5-9]. According to [10],

wave transport along the waveguide can be affected by

two scattering mechanisms: one is induced by the height

of inhomogeneity (i.e. ()r



), the other is due to the

first-order derivative of inhomogeneity (i.e. '( )r



). In

related works, the surface roughness is modeled as a

random variable, and the scattering loss is calculated

stochastically. But all of these methods either fail to ob-

serve the effect of or are too complex for calculation,

because they cannot analyze independently the scattering

loss due to both scattering mechanism. Therefore, if a

simulation is needed to conduct for analysis, we cannot

calculate the result accurately. For example, in [9], sev-

eral Nano scale circular holes are carved evenly along the

microdisk to make intentional surface roughness. In [6],

random distributed inhomogeneity is approached by a

number of randomly spaced rectangle asperities. All these

approximation method is not very accurate, and cannot

help us to further our understanding on the underlying

scattering mechanism.

In this paper, a method based on orthogonal decompo-

sition is proposed to simplify the calculation of surface

roughness effect without losing accuracy. We try to ap-

proach the inhomogeneity infinitely by a linear combina-

tion of orthogonal basis we proposed, and then we can

analyze the effect of ()r





and '( )r



 independently

by carefully choosing proper orthogonal basis.

The paper is organized in the following way: in Sec-

tion 2, the model of orthogonal basis is introduced, and

then its correctness is proved by Fourier Transform. At

the end of Section II, the utilization of the proposed or-

thogonal basis is studied in detail. In Section 3, we ana-

lyze the surface roughness effect on Q-factor of micro-

disk resonator at first. Then we conduct the analysis and

FDTD simulation based on our method, the result of

which validates our derivation.

2. Model

2.1. Notations

The side boundary of micro resonator is rough, and is

shaped by closed contour S along the basis parallel to z

basis, as depicted in Figure 1. The distance from center

point to each side boundary is given by

()(), [, ]rRr



 



  (1)

where R is the average radius and is a random variable

with zero mean. The correlation function of is often fitted

to Gauss distribution, i.e.

*2 2

12 12

()( )exp[()/]

rr L

 

 2

(2)

C. L. SUI ET AL. 289

where Lc is the correlation length and σ is the root mean

square (rms) value of surface deviations. In addition, we

have

 expectation operator

 2-norm

()F

[(

elementary function

)]

x i-th square of ()

()

x i-th order derivative of ()

Figure 2 shows one period of a sequence of orthogo-

nal basis {()}

x, which is orthogonal with each other,

i.e.

() ()0

fxfxdx ij













 (3)

'' '

() ()() ()

ij ij

xfxdxf xfxdx



 











(4)

Finally, we define the period of orthogonal basis as L.

Figure 1. SNR degradation as function of laser line width.

Figure 2. Schematic of orthogonal basis in one period.

2.2. Orthogonal Decomposition

Now we’ll prove the fact by Fourier Transform that side

wall inhomogeneity can be approached by orthogonal basis











through orthogonal decomposition. The Fourier

Transform of fk(x) is given by

()(1)exp[2 ()/]

(/)

kkn

sinn lL

hnL







 



 nxilL

(5)

According to [12-13], there would always be a linear

combination of











that can approach ()r





infi-

nitely by meeting the condition

lim( )( )0

Af r



 



 (6)

Therefore, if the orthogonal basis and its weight are

carefully chosen, the orthogonal basis we proposed can

accurately approximate the transmittance of micro reso-

nator with rough surface.

For a specific resonator, several modes may be propa-

gated along the waveguide. As it is stated in [5],

3/2 2

()exp( )

2( )

exp 2

rajk

ar jnd

LkL

aa RR











 

 



















 (7)

Reference [5] shows that each micro resonator is

matched with a sequence of ak , and each ak corresponds

to the mode splitting for resonant mode with azimuthal

order k.

Now the way of getting the parameter of orthogonal

basis we proposed will be revealed. Comparing (5) with

(7), if we simply look into one specific mode splitting

with azimuthal order k, the surface roughness effect can

be accurately approached by one specific orthogonal ba-

sis







if only

22 /2

Lpp

 



 (8)

221

3/2 2

(/)

(1)exp(2/)

(/)

2( )

exp 2

sinn lL

LnL

















 











ilL

(9)

Here, 0

is normalized random variable whose mean

value is 1. If 2LR



, it’s reasonable to assume that

orthogonal basis spreads over micro resonator to infinite

period. From (8), we can get that if the azimuth



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 .

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

[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 {()}

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]:









[()]

['()]

[()]

['()]

Ffx F

FfxF



 (10)

Thus, if 0

0|()|

xdx

 keeps static when the value of

l changes, the value of [()

x will be the same,

which means the ()r



nduced roughness effect will not

change. Similarly by keeping 0

2|'()|

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:

(, )

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

()

(1 )



 (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

-disk



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 {()}

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



/ (18

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

18 /52.8

0.4107

pl pl

16 )L p

1 exp[hj













(13)

from (1e larger p is



3) that thIt’s easy to observe

, 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

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

s it is stated above, by keeping 0

2|()|

x static,

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





changes with 0

l increasing from 150nm to 400nm

the effect of ()r while





will be the same. Similarly, if the

value of 0

0|'()

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

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|()|

actor of

e d

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





C. L. SUI ET AL. 291

Figure 3. Electromagnetic (EM) field distribution profile at

the wavelength of 1548.39nm (a) flat surface (b) l0 is 200 nm

Figure 4. Transmission characteristics as a function o f wave-

length.

is the dominator factor that leads to the degrading of mi-

cro-disk resonator’s Q-factor, when side wall roughness

is not fairly obvious, i.e. c



. Results of the simula-

tion based on our method fit well with the analysis in [6].

4. Conclusions

In this paper, we proposed a novel method base on or-

thogonal decomposition to analyze the surface roughness

effect of micro-resonator. It becomes possible to calcu-

late the scattering loss of ()r



 and r'( )



 respec-

tively through our work, and so we can have a better un-

derstanding of the inhomogeneity’s effect on micro

resonator. Our theory is proved by Fourier Transform

and matrix analysis. Finally, we investigated the surface

roughness effect on Q-factor of micro-disk resonator, and

a FDTD simulation based on orthogonal basis is con-

ducted to validate our deviation.

REFERENCES

[1] R. Soref, “The of silicon photon-

ics,” IEEE Jo in Quantum Elec-

past, present, and future

urnal of Selected Topics

tronics, Vol.12, No.6, 2006, pp.1678- 1687.

doi:10.1109/JSTQE.2006.883151

[2] M. Soltani, S. Yegnanarayanan and A. Adibi,

Q Planar Silicon Microdisk Reson

“Ultra-High

ators for Chip-Scale

Silicon Photonics,” Optics Express, Vol.15, No.8, 2007,

pp.4694-4704.

doi:10.1364/OE.15.004694

[3] L.Q. Ren, X. Wu, M. Li, X.

“Ultrasensitive Label-Free C

Zhang, L. Liu, and L. Xu,

oupled Optofluidic Ring La-

ser Sensor,” Optics Letters, Vol.37, No.18, 2012,

pp.3873-3875.

doi:10.1364/OL.37.003873

[4] Y. Hu, X. Xiao, H. Xu, X. Li,

Yu, and J. Yu, “High-spee

K. Xiong, Z. Li, T. Chu, Y.

d silicon modulator based on

cascaded microring resonators,” Optics Ex-

press,Vol. 20,No.14, 2012, pp.15079-15085.

doi:10.1364/OE.20.015079

[5] Q. Li, A. A. Eftekhar, “Azimuthal-Order V

Surface-Roughness-Induced

ariations of

Mode Splitting and Scatter-

ingloss in High-Q Microdisk Resonators,” Optics Letters,

Vol. 37, No.9, 2012, pp. 1586-1588.

doi:10.1364/OL.37.001586

[6] E. Ganapolskii and Z. Eremenko,

Surface Inhomogeneities on

“Effect of Random

Spectral Properties of Di-

electric-Disk Microresonators: Theory and Modeling at

Millimeter Wave Range,” Physical Review E, Vol.79,

No.4, 2009, p. 041136.

doi:10.1103/PhysRevE.79.041136

[7] M. Borselli, T. J. Johnson and O. Painter

Rayleigh Scattering Limit in High-Q Silicon Mic

, “Beyond the

rodisks:

Theory and Experiment,” Optics Express, Vol.13, No.5,

2005, p.1515.

doi:10.1364/OPEX.13.001515

[8] X. Yi and Y. X

Figure 5. Q-factor as a function of the value of l0 in differ

scattering mechanism.

ent iao, “Multiple-Rayleigh-Scatterer Induced

C. L. SUI ET AL.

292

Whispering-Gallery-ModeMode Splitting in A High-Q

Microresonator,” Physical Review A, Vol.83, No.2, 2011,

p. 023803.

doi:10.1103/PhysRevA.83.023803

[9] S. Cho an

Q-Factor of Ge Whispering Gall

d S. Koo, “Surface Roughness Effect on

ery Mode Microdisk

Channels:

Resonator,” Optical Society of America, 2011.

[10] N. M. Makarov and Y. V. Tarasov, “Electron Localiza-

tion in Narrow Surface-Corrugated Conducting

Manifestation of Competing Scattering Mechanisms,”

Physical Review B, Vol. 64, No.23, 2001, p. 235306.

doi:10.1103/PhysRevB.64.235306

[11] H.-Y. Yu, M. Ishibashi, J. -H. Park, M. Kobayashi, and K.

C. Saraswat, "p-Channel Ge MOSFET by Selectively

Heteroepitaxially Grown Geon Si,” IEEE Electron De-

vice Letters, Vol. 30, No. 6, 2009, pp. 675-677.

doi:10.1109/LED.2009.2019847

[12] W. A. Gardener, “Introduction to Random Processes:

With Applications to Signals and Systems,” New York,

NY: McGraw-Hill 1990

[13] R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cam-

bridge University Press, Cambridge, 1990.

[14] E. Kreyszig, “Introductory functional analysis with ap-

plications,” John Wiley & Sons, New York, 1989.