**International Journal of Emerging Technology and Advanced Engineering **

**Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 9, September 2015)**

14

** Modeling of Ring Resonators for Optical Filters Applications **

## using MEEP

### I. M. Matere

1### , W. Ngetich

2### , D. W. Waswa

3*1,2,3 _{University of Eldoret, Department of Physics, Fibre Optics Research Group, P.O box 1125-30100 Eldoret, Kenya }*

**Abstract****—Ring Resonators are key component in modern **
**optical networks. Their size allows high density integration in **
**optical photonic circuits due to the use of high index contrast. **
**Ring Resonator based filters are an enabling technology in **
**wavelength division multiplexing optical networks. In this **
**study, we theoretically demonstrate ring resonator affecting **
**parameters for optical filter applications. This is achieved **
**through coupling a closed loop resonator with a straight **
**waveguide using evanescent. Coupling was necessary to **
**demonstrate ** **filter ** **new ** **structure ** **behavior. ** **A ** **Mit **
**Electromagnetic Equation Propagation (MEEP) was used to **
**simulate transfer functions, resonance frequency of 1.45 x 1015**
**rad/s, resonance wavelength of 1.2955 μm, factors affecting **
**the ring resonator; dielectric constant, frequency, perfect **
**matched layer and characteristics of a tunable optical filter **
**showed a good agreement between analytical and simulation **
**based results. This work is of great interest in designing **
**narrow bandwidth ring resonator based filters for DWDM **
**technology to provide high density spectral usage and **
**capacity. **

* Keywords*—

**Free Spectral Range (FSR), Finesse (F),**

**Quality factor (Q) and tunable filter**

### .

I. INTRODUCTION

The fundamental building blocks of Ring resonators based devices are a micro-ring with one or two waveguides [1]. In the former case, this leads to two-port devices, which act as all-pass filter. On the other hand, a Ring Resonator consisting of a ring with two straight waveguides represents a 4-port structure in photonic integrated circuits which are the next generation of optical networks where the optical components are of small size and high density. High refractive contrast waveguides represent good candidates for future optical circuits. Optical ring resonators are the main building blocks in the photonic circuits as all-pass filters , add/drop filters and biosensors[2] [3]. A waveguide with a perimeter of several micrometers is used to construct an optical resonator. This ring supports a number of circulating wavelengths that satisfy the resonant condition as illustrated in equation 1 [4].

(1)

Where is the phase of the coupler and *m* is an integer
representing the mode number and , *L *is the
circumference of the ring given by , *r* being the
radius of the ring measured from the center of the ring to
the center of the waveguide, c is the phase velocity**Error! **
**Reference source not found.** of the ring mode given by

. The fixed angular frequency, ,

refers to the speed of light in a vacuum. The vacuum

wavenumber *k * is related to the wavenumber λ

through: .

The free spectral range (FSR) is the difference between
two resonance wavelengths which is of interest in
wavelength division multiplexing WDM. The resonator is
then coupled to an external circuit to get the transfer of
stored energy this is achieved by using a single or double
straight waveguides close to the ring**Error! Reference **
**source not found.** **Error! Reference source not found.**.
The coupling between the evanescent modes of the ring and
the waveguide produces a transfer of stored energy in the
ring to the output port of the waveguide. A small size ring
resonator are modeled using Gallium Indium Arsenide
Phosphide laser (GaInAsP) which offers a high index
contrast between core and cladding to ensure a high
confinement of light and hence reducing the bending losses

**Error! Reference source not found.**.

In WDM the free spectral range is required to be as high as possible (30 nm) [9] in the C-window and around 1550 nm [10]. This implies that using a small radius in the range of 5 µm, but reducing the radius of the ring will result in increasing of bending loss to unacceptable levels. However using rings of varying radii coupled with each other to two outer straight waveguides represents an alternative for increasing FSR [11].

II. RING RESONATOR TRANSFER FUNCTIONS

**International Journal of Emerging Technology and Advanced Engineering **

**Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 9, September 2015)**

15

The transfer function is affected by the characteristics of coupling regions represented by the power coupling coefficients and through coefficients with and as their conjugate. For lossless coupling;

(2)

For a single unidirectional mode of the resonator, the coupling is lossless so single polarization is considered implying that the various kinds of losses occurring along the propagation of light in the ring resonator filter are incorporated in the attenuation constant represented by the matrix [12].

(3)

For further simplification the modal is chosen to be 1. Then the round trip in the ring will be

. The resonance condition will be the round trip phase shift a multiple of . The phase shift is a function of wavelength, propagation constant and the length of the resonator

(4) Where is the propagation constant which is a function of wavelength and structure specification represented by effective refractive index. The complex mode amplitudes

*E*are normalized so that their squared magnitude
corresponds to the modal power.

Where α is the circulating loss factor of the ring

The power transfer characteristics are described by

The circulating power *Pi2* is given by

Two features of equation (8) illustrate most of potential applications; (1) there exists a special condition,

when the internal losses are equal to the

coupling losses for which the transmitted

power is zero, (2) at the transmission is much higher. It is apparent that in high-Q resonators , small modulation of α or , causes large modulation of the transmitted power. This can be used to construct electro-optic modulators. In addition to the under-coupled region , as the gain is increased, the power transmission decreases until the critical coupling point

[13].

Ei1 Et1

α Et2

-k*

Ei2

k t

[image:2.612.323.571.127.536.2]t*

**Fig 2: Model of a single ring resonator with one waveguide **

Et2 Ei2

t2

k2

Ei1 Et1

α Et

-k1*

k1

t1

t1

*

-k*

2

t2*

**Fig 3: Model of a single ring resonator with two waveguide **

R1

R2

1µm _{n}

1

n2

n3

**Fig 1: A ring with radii R1 and R2 with refractive indices increasing **

**International Journal of Emerging Technology and Advanced Engineering **

**Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 5, Issue 9, September 2015)**

16

From equation (1) to (8), it is possible to get a good idea of the behavior of a simplified basic ring resonator filter configuration consisting of only one waveguide and a ring.

An addition of a second waveguide leads to the following amplitude mode power output from the drop port as in Fig 2.

The throughput port mode amplitude in the first waveguide is given by [14];

The mode amplitude in the ring has to pass the second coupler to become the new dropped mode amplitude . This mode is given by;

During resonance the output power

from the drop port will be;

*A. free spectral range (FSR) *

This is the difference in position between two consecutive resonant peaks and can be defined either in frequency or wavelength domain

respectively.

Where R is the radius of the ring, *c* the speed of light in
vacuum

*n*

*is the group refractive index and , is the difference in position between two consecutive resonant peaks frequency and wavelength respectively.*

_{g}*B. Finesse F *

This is the ratio of the FSR and the width of a specific wavelength ; for a full width at half maximum (FWHM) [15].

*C. Quality factor Q *

It is the measure of the sharpness of the resonance. It is defined as the ratio of the operation wavelength and the resonance width.

III. RESONANT MODES

[image:3.612.329.562.428.553.2]The tested structures had radii of between RC = 5 μm and RC =200 μm. The following resonant modes were achieved: resonance wavelength of 1.2955μm and real angular frequency of 1.45 x 1015 rad/s.

**Table Ι **

**CALCULATED RESONANT MODES OF A RING RESONATOR **

Radius λres [μm] λres [μm] *Re(ω) *(x 10
15

rad/s) * *

R [μm] [15] calculated [15] calculated

5 1.29192 1.285 1.4590 1.4664 20 1.29763 1.277 1.4526 1.4766

40 1.29859 1.297 1.4509 1.4531
60 1.30006 1.314 1.4499 1.4343
80 1.30074 1.299 1.4491 1.4509
100 1.29965 1.316 1.4504 1.4327
200 1.30008 1.301 1.4499 1.4486
*A. Variation in dielectric constant (ε) *

The nature of the dielectric material affects how light passes through the guide. This dielectric is characterized by a dielectric constant (ε), which is directly proportional to the square of the refractive index of the material.

2

0

*n x y*

### ( , )

###

###

### (15)

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When the dielectric of the material increases the refractive index is increased therefore interfering with the total internal reflection inside the guide. This lead to a reduction in the amplitude of light as indicated in Fig 4.

**20** **40** **60** **80** **100** **120** **140**
**-0.12**

**-0.10**
**-0.08**
**-0.06**
**-0.04**
**-0.02**
**0.00**
**0.02**
**0.04**
**0.06**
**0.08**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

**Epsilon 8**
**Epsilon 10**
**Epsilon 14**

**Fig 4: Variation in dielectric in a waveguide**

*B. Perfect matched layers (PML) *

It is a list of PML objects which can be two or more and
are placed inside the guide. The objects absorb waveguide
modes and also reduce the numerical reflections **Error! **
**Reference source not found.**. The PML was varied from 1
μm to 3 μm and it was evident that as the thickness
increases the transmitted power fades away as shown in Fig
5.

**40** **60** **80** **100** **120** **140**

**-0.010**
**-0.005**
**0.000**
**0.005**
**0.010**
**0.015**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

**Epsilon 12.1**
** Thickness 1.0**
** Thickness 1.5**
** Thickness 2.0**
** Thickness 3.0**

**Fig 5: Variations in perfect matched layers in a waveguide **

C. *Frequency of the source current *

The frequency was varied from 0.1 PHz, 0.15 PHz, 0.20 PHz and 0.25 PHz. As the frequency value was increasing the amplitude also increased though the smooth nature of the curve was being lost as in Fig6.

**20** **40** **60** **80** **100** **120** **140**

**-0.35**
**-0.30**
**-0.25**
**-0.20**
**-0.15**
**-0.10**
**-0.05**
**0.00**
**0.05**
**0.10**
**0.15**
**0.20**
**0.25**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

**f= 0.1**
**f= 0.15**
**f= 0.20**
**f= 0.25**

**Fig 6: Variations in frequency of source current**

When the reciprocal of each frequency is obtained, 10, 6.67, 5.0 and 4.0 respectively are obtained; this value corresponds to the period of the wave. and because in the MEEP calculation ε = 12 to relate the wavelength with the frequency the wavelength is now divided with 12 giving 1.2, 1.8, 2.4 and 3.0 respectively, this values state how much the wavelength equals to the current source. The value 1.2 means that the waveguide is roughly equal to the wavelength of the current source, 1.8 indicate that the waveguide is half the current wavelength, while 2.4 and 3.0 correspond to a third of the wavelength of the current source.

For frequency values of 0.1 PHz and 0.15 PHz makes the current source to be single mode and thus the mode is analytically solvable and corresponds to a frequency of 0.15076 PHz [16].

*D.Two waveguides with a Ring*

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**20** **40** **60** **80** **100** **120** **140** **160**
**0.000**

**0.002**
**0.004**
**0.006**
**0.008**
**0.010**
**0.012**
**0.014**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

**Fig 7: Power in the ring which is maximum at resonance **

*E. Variation of the length between the waveguide and the *
*ring *

A continuous source frequency f=0.15 PHz was placed at point (-7, 3.8) in the lower waveguide. For this broadband source the resonance frequency was found to be in the range 0.068 GHz – 5.026 GHz.

When the length between the ring and the straight guide was increased by 0.5 μm from an initial 0.5 μm interspace, there is an upward shift in the power that goes through the ring at points 54 μm and 104 μm. In this case, as the length is increased the filter characteristics of the optical band pass resonator fades away; as shown in Fig 8.

**20** **40** **60** **80** **100** **120** **140**
**-0.08**

**-0.06**
**-0.04**
**-0.02**
**0.00**
**0.02**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

** d=0.5**
** d=1.0**
** d=1.5**
** d=2.0**

**Fig 8: Variations in length between the ring and the straight guide **

Going to the ring which is between the wavelength of 54 μm and 104 μm, the power that is coupled in the ring increases as the length between the guide and the ring is minimized. This length was varied from 0.5 μm to 0.9 μm.

However for a minimum of 0.5 μm the coupled power was high as indicated in Fig 9.

**20** **40** **60** **80** **100** **120** **140** **160**
**0.000**

**0.002**
**0.004**
**0.006**
**0.008**
**0.010**
**0.012**
**0.014**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

** d=0.5**
** d=0.7**
** d=0.9**

**Fig 9: Comparison of power coupled in the ring on varying length**

*F. Variation of the computation cell *

The physical parameters were n=3.4, r=2.8 μm, pulse frequency 0.15, PML=1.0μm and w=1.0 μm. The run time was set at 200. The results gave a shift in the wavelength with a decrease in the output power. This is the power that is coupled in the ring. In this a case a tunable [5] [10] filter was achieved.

**50** **100** **150** **200** **250** **300**

**0.000**
**0.002**
**0.004**
**0.006**
**0.008**
**0.010**
**0.012**
**0.014**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

** 16x16x0**
** 24x24x0**
** 30x30x0**
** 36x36x0**

**Fig 10: different computation cells showing a shift in a tunable filter**

*G. Power variation in different types of Rings *

**International Journal of Emerging Technology and Advanced Engineering **

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As the rings were increased the power coupled during resonance kept on reducing. The outcome is as shown in Fig 11 and Fig 12.

**20** **40** **60** **80** **100** **120**
**0.000**

**0.005**
**0.010**
**0.015**
**0.020**
**0.025**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

** single straight guide with a ring**
** Two straight guides with a ring**

**Fig 11: Single and double rings at resonance**

For three rings and two waveguides

**20** **40** **60** **80** **100 120 140 160 180 200 220**

**0.0000**
**0.0002**
**0.0004**
**0.0006**
**0.0008**
**0.0010**

**In**

**te**

**n**

**s**

**it**

**y**

**Wavelength (****m)**

**Two straight guides with two rings**
**Two straight guides with three rings**

**Fig 12: double and triple rings at resonance **

IV. CONCLUSION

A demonstration of how ring resonators can be used for optical filtering in WDM is presented. The validation of analytical analysis using MEEP was carried out in this paper. It has been shown that ring resonators allow for compact channel filters in WDM and represent a key component in modern optical networks. Different factors affecting filter performance are demonstrated.. It was shown that tunable ring resonator based filters present a good candidate for add/drop multiplexing for high capacity in DWDM optical fibre transmissions in metro access networks.

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