Design of Add-Drop Multiplexers

Betül Küçüköz

Department of Physics, Bilkent University, Bilkent, Ankara, 06800

Abstract—

We discuss the design optical add-drop multiplexers. This includes the design of single mode waveguides as well couplers and

ring resonators. We study the add-drop characteristics of this design as a function of various parameters. An add or drop multiplexer is a

wavelength selective device that separates a given optical wavelength from a bus waveguide as well as adding a desired wavelength to the

bus waveguide. As an element of add-drop multiplexer, waveguides are used in form of rib or ridge waveguide structure in our work. Also

we used disc or ring resonator as a coupler. Further, this paper includes overview of design of the waveguide and importance of parameters

to prevent losses on.

INTRODUCTION

Wavelength selection is an important

function in the design of optical circuits. Selecting

and separating a given wavelength to detect a signal or to reroute is critical to operation of many

photonic circuits. Optical add-drop multiplexers are

class of wavelength selection devices. Their main

effective usage is as an optical filter. Add-drop

multiplexer is composed of two bus waveguide that

can carry many wavelength separates by a ring or

disc resonator that performs the wavelength

selection. As the name implies, a ring resonator is a

resonant device to which only resonant wavelength

can couple. Once coupled from a bus waveguide it

is possible to drop this signal to another bus

waveguide. In this work, we concentrate on the design

of single mode waveguides since they are the basic

elements of an add-drop multiplexer,

Fig.1

Optical ring resonators are used as a building block in filter

delay line, channel dropping, combining or compensation of

dispersion on a system. [1]

In addition we use standard design rules to design ring resonators. Ring resonators have a significant

place on recent technological improvement. Its key

factors are functionality and compactness. One can

couple multiple ring resonators to build wide range

of different filters. Compactness refers the fact that

ring resonators with radii less than m25 can lead

to large scale integration of devices with densities

on the order of 541010 devices per square

centimeter. [2]

A ring resonator is a circular device built from the single mode waveguide designed earlier.

An attenuation constant, is used to include losses

to the ring resonator. Also we used power coupling

coefficient K to describe the coupling of the optical

of the optical signal from the bus waveguide to the

resonator and back. The working principle of add-

drop multiplexer is as follow; light at many

different wavelengths propagate in the single mode

bus waveguide and at a specific design wavelength, evanescent field of the propagating filed couples to

the ring. A ring resonator can be thought of as a

Fabry-Perot resonator in which only those

wavelengths that satisfy the constructive

interference of back folding waves survive.

To calculate circumference of resonator we will use

LNwg

relation where N is mode number, is

free space wavelength and L is circumference of

resonators.

Fig.2 Single ring resonator with two ports [1]

Fig.3 Single ring resonator with four ports [1]

Normally, most of the signals, which are around

specific frequency, are injected into resonators from

port 1 and they exit from port 3. However, some of

input signals enter from port 1 and they leave from

port 2. Hence, we mark

port 1 as input port, port 2 as through port,

port 3 as drop port, and

port 4 as add port. This effect of ring resonator is

called an add-drop filter or an add-drop multiplexer.

In our design of add-drop multiplexer (ADM) we ignore some effects. For example, we omit facet

discontinuity, waveguide scattering or waveguide

wall surface roughness. Since they are very small in

magnitude, one can ignore them without any

problem. [1]

Fig.4 Schematic of ring resonator [8]

In ring resonators one should observe sharp dip

when signals drop. During an ADM experiment,

one can detect these deeps for specific wavelength

and interpret relation between these wavelengths

and radius of ring resonator or any other

parameters.

Fig.5 Transmission measurement of a typical ring resonator. A

dip occurs around 1524 nm wavelength. [3]

METHOD

There are many methods, which we used for design

of single mode waveguide. These are effective

index method, beam propagation method and finite

difference method, which can be listed among

many ways of analyzing an optical circuit. So we

have employed these methods in our design.

I. Effective Index Method:

Effective index method is basically a method to

solve 2 dimensional waveguide‟s effective index as a whole structure. Let us visualize a rib waveguide;

Fig.6 Domain by domain separation of a typical rib waveguide

for effective index calculation. [4]

In this method we divide in three slab waveguides

vertically. Firstly we solved these three, and then

we solved other three slab waveguide which are

dividing horizontally. We use these equations to

calculate propagation constant,

21

)tan(

f

sc

f

sc

fh

for TE mode, and

)

)tan(

22

4

2

2

2

2

2

sc

sc

f

f

c

c

f

s

s

f

f

f

nn

n

n

n

n

n

h

for TM modes. From these equations we obtain

propagation constant, β for each slab waveguide,

now by using these β‟s we calculate effective index

thanks to neff = β/k0 relation. We calculate effective

index for first structure then we repeat same

calculation for original structure rib waveguide. We

wrote a MathCAD file to calculate effective index

by numeric solver. [4] (See Appendix)

II. Finite Difference Method

In space, we assume there is no current flowing and

we obtain six equations from Maxwell relation.

However, four of them are symmetric and others

are like this;

y

E

z

E

t

Hzyx

1

x

H

z

H

t

Ezxy

1

The FDTD (Finite Difference Time Domain)

Method solves Maxwell equations by discretize

according to time parameter. Common method to

discretize time parameter and solve these equations

according to this is Yee‟s mesh method.

Fig.11 Yee‟s mesh points [7]

On this mesh method, E and H field components at

all grid points spaced x , y and z . However this

broken into t so in E field time tnt . and in H

field tnt .21 where n is integer. As a result

of these six equations, method computes all mesh

points, such as

=

These two complicated equations are two of the six

of Maxwell equations.

The commercial software “FullWAVE” solves

these equations by using FDTD method. We can

differ between TE or TM modes in this software.

We could not see on BeamPROP.

III. Beam Propagation Method

For complicated optical systems Beam Propagation

Method (BPM) is a better way for numerical

solution. This method is generally used for its wide

range of simulation capabilities. We employed a

commercially available software package, RSoft

BeamPROP. This software allows us fast analysis

of complicated designs. Moreover, we can

determine easily geometry and index distribution of

our optical system.

Fundamentally, BeamPROP solves three

dimensional Helmholtz wave equations. There are

limitations to solve this wave equation; one of

which is parabolic or paraxial approximation and

the other limitation is that we neglect polarization

and solve for scalar wave function, and final limitation is that BPM approach neglects backward

reflection. It allows only one way wave equation

solution, and does not allow positive and negative

waves traveling on the same path. Basically, we

solve exact wave equation for monochromatic

waves with these approximations. This is well-

known Helmholtz equation for monochromatic

waves;

0),,(2

2

2

2

2

2

2

zyxk

zyx

Where scalar electric field is tt

ezyxtzyxE

),,(),,,(

and notation introduce as a

),,(),,(0

zyxnkzyxk also we know

20k .

Electric field can be separated in two parts. As we

define electric field is ikz

ezyxuzyx ),,(),,( .

The envelope term is ),,( zyxu , which changes

slowly and fast term is ikze where z is propagation

direction. If we assume u and z sufficiently slow

it is also referred paraxial or parabolic

approximation. After these assumptions, equation reduces to

ukk

y

u

x

u

k

i

z

u)(

2

22

2

2

2

2

. (1)

In early implementations of BPM method, split-step

Fourier method was employed. However, as more

complicated circuit designs need for

telecommunication industry finite difference

method which is based on Crank- Nicholson

scheme was employed more often. Combination of

this method and BPM method start called as

FDBPM.

For u denoted the field where i transverse grid

and longitudinal plane is n . If we substitute known

plane n unknown plane 1n in equation (1) we

obtain

2,

2

12

2

212

21 n

i

n

i

ni

n

i

n

iuu

kzxkxk

i

z

uu

This equation is valid at some boundary conditions

1i and N refers to unknown quantities outside

domain since we know that it is represented in

finite computational domain. The most important

and critical issue is artificial reflection of light

incident on the boundary back into the

computational domain. “Simply requiring the field

to vanish on the boundary is in sufficient since it is

equivalent to placing perfectly reflecting walls at

the edge of the domain.”[6] Several work based on absorbing material on the edges of domain. (to

minimize reflection) Because of these results

boundary conditions are called Transparent

Boundary Condition (TBC). At the beginning we

assumed scalar Helmholtz equation now we

analyze if electric field E is a vector, we should

solve vector wave equation instead of scalar

Helmholtz equation. We assumed before our wave

equation should be paraxial. If paraxial approach

reduce to derivation of basic BPM by ignoring 22

zu term. For a wide-angle BPM we introduced

first order equation.

uPkiz

u11

This equation refers to one

way wave equation because first derivative allow

only forward traveling. This information is related with background of BeamPROP software.

In this software we can easily determine

the index of layers or interface also we can arrange

how signal launched. As shown in Fig.7 we can

arrange all parameters in edit windows.

Fig. 7. Parameter window in BeamPROP software

Fig.8 Layer editor in BeamPROP software

Calculate the mode spectrum of the waveguide and

see whether it is single mode. Also we can sketch

its profile and spectrum. An example is shown

below

Fig.9 Single mode computed spectrum

and single mode profile is shown in figure as an

example,

Fig.10 Transverse mode profile

During calculation of single mode or design of slab

waveguide, there are some parameters which

determine the main property of waveguide. One can

use as a substrate material Si whose index is 3.45.

Over of it we placed SiO2 as quartz and its index is

1.46. Finally one can filled channel of waveguide

with bcb polymer (benzo-cyclo-butene). Index of bcb is 1.54. One can focus on design part of this

paper.

Fig.11 Ring resonator schema with our parameter

DESIGN

In design part, first; waveguide design was

analyzed and secondly we focused on how ring

resonator designed. Since ring resonators also

include two slab waveguides, investigation of

design of slab waveguides is reasonable.

I. Single Mode Waveguide Design

Design of waveguide is restricted with slab

waveguide and BPM software will be used for this

design in this paper. Figure 7 and Figure 8

represent parameters of slab waveguide which is

single mode. Its counter map of transverse index

profile is

Fig.12. Single mode waveguide counter map profile

Its mode profile is also given before in Fig.10. If

this waveguide is schematized basically with its

material, Fig.13 is good enough to represents

materials and index difference between these

materials.

Fig.13. Schematic of single mode waveguide

Choice of index is very important part of obtaining

single mode waveguide progress also another significant points are determination of slab height

and waveguide width. Light will propagate on bcb

polymer layer as we expect due to its index. Light

confined in high index layer if all parameters

arrange good enough with respect to high or width.

For instance, if height of SiO2 is small, light

penetrates from SiO2 and confined on Si which is

index is 3.45. This situation is possible if all

parameters are not set with sensitively.

Furthermore, during choosing material progress,

properties of growth of material should be taken

into account for fabrication of waveguide. For

instance, in this waveguide substrate is silicon and on top of it SiO2 is growth as quartz. To filled

channel, bcb polymer is used with index 1.54. Bcb

polymer is useful for filling or covering the channel

because of its flowing material and soft to shape

easily. These kinds of properties are also important

while designing a waveguide or ring resonator to

make fabrication easy.

II. Add-Drop Multiplexer Design

In ring resonator design there are two slab waveguide and at the middle of them there is one

ring. Ring is made up of two lenses. These two

lenses are placed on same origin. These lens shapes

are cylindrical and lens which is at the middle has

index difference 0 but index difference of bigger

lens is 2. Thanks to arrangement of index difference

these two lenses behave as a ring. Before the

designing ring resonator, determination of distance

between ring and slab waveguide is important. Due

to this, let‟s start with demonstration of interaction

between distance and coupling. Gap distance between them can arrange from parameter window

in symbol part as seen in figure 14.

Fig.14. parameter window to change gap

We simulate for 0.1μm-0.2μm-0.3μm gap distance

and we observe effect on coupling. Result is

convenient with expected result. While gap distance

become larger coupling reduce obviously. On

figure 15-16 and 17 this result can be observed.

Fig.15. coupling simulation for 0.1 μm gap distance

Fig.16. coupling simulation for 0.2μm gap distance

Fig.17. coupling parameter for 0.3μm gap distance

However, we should take into account time

parameter, when we compare these simulations. We

observe coupling changed in time and if we want to

compare we should took screen shut at same time

as seen on left part at the bottom of each three

figure. After demonstration of effect of gap

distance on coupling, we can design ring resonator.

Figure 18 represents ring resonator structure as seen

on FullWAVE file

Fig.18 Structure of ring resonator in FullWAVE file

Radius of yellow lens is 1.6μm and waveguide

width is 3.2 μm. Bigger lens has radius which is 1.8

μm and index difference between these two lenses

is zero. Moreover, distance between ring and slab

waveguide is 0.1 μm. Both slab waveguides have

same distance to the ring. Ring radius and distance from waveguides are critical parameter for

observation of light coupling. Light coupling is

specified for wavelength of light, already it is

known that coupling can be used for switch button.

For light, obtaining of switch button is not easy

because we should not keep light waiting with any

block. It can be absorb or reflect. However to wait

light with its same direction and its same power,

ring resonator could be used.

During design of ring resonator structure is taken as

2D structure. Parameter window of figure18 is in

figure 19 (below).

Fig.19 parameter window of 2D ring resonator in FullWAVE

Again choice of parameter is very sensitive also

indexes should be real because of fabrication; due

to the fact that, arrangement should made on heights, widths or distance parameters. The main

idea of design of ring resonator is obtaining of

wavelength range by coupling. In simulation of this

resonator, some specific wavelength points there

are some deeps which represent drops of light at

these wavelengths.

RESULTS

As mentioned until result part of this paper

arrangement of parameters are main part of this design. After this point, analyses of simulations and

analyses of data come. These simulations are very

exciting because there is no assumption for which

wavelengths will drop. For simulation, another

arrangement should be done on simulation

parameter window which is seen on figure 20.

Fig.20 Parameter window for simulation

As seen in this window, software will scan lambda

which represents wavelength range from 0.797μm

to 2.39μm, also this work will do for 11 steps. During simulation, propagation of light and

coupling of light can be observed obviously. In

figure 21-22 simulation of coupling at different

time,

Fig.21. simulation of ring resonator art less coupling wavelength

As can see on these figures, there are two monitors

for detect to light. Add-drop multiplexer can be

easily observed on these simulations. Below the

simulation figure there is a graph which is monitor

value with respect to time. The blue line on this

graph represents throughput monitor value and

green line represents drop monitor value where

light comes from -2 points on x axis.

Fig.22. high coupling simulation during 11 steps

These two figures were taken during computation

in progress. When computation completed, there

were occur a graph which is y axis value with

respect to lambda. Figure 23 is this graph. This

graph represent monitor 1 which is on throughput

port.

Fig.23. Final graph after simulation (power versus l wavelength)

for 11steps

Fig.24.Final graph after simulation it represents second monitor

In figure 24, we observed second monitor output

power. It is opposite of figure 23. Since second

monitor is drop port, we expect this result.

CONCLUSION

In these simulations expected things are detect

some specific wavelengths. In these wavelengths

light drops and on simulation graphs it creates some

deeps on these lambda values. Design of add-drop

multiplexer has important place on technological

view. It is useful not only for detect wavelength but

also it can also use as a switch button of light. This

paper includes many figures for simulation; these

figures represent light propagation in ring resonator

and effects of distance between ring and slab waveguides. However, these simulations restricted

with FullWAVE simulation. It is also possible that,

observing these simulation on other programs.

Methods of calculation and design of ring

resonators explained according to FullWAVE

software. In this design, we study on micron or

nanometer scale because these resonators are using

in complicated optical device system and length

parameters are as small as possible. Further we

used laser whose wavelength is 1.55μm. This is

also telecom wavelength and we scanned

wavelength from 0.797μm to 2.39μm. Moreover, waveguides and ring resonators also measure

polarization, intensity and phase.

ACKNOWLEGMENT

I would also like to thank to Ertuğrul KARADEMİR

for his help during this whole work and his

friendship. I have learned a lot from him.

I would like to thank Gonca ARAS from whom I

have learned the MATHCAD software and properties of material which I used in this work.

REFERENCES

[1] Schwelb, O: „Transmission, Group Delay, and

Dispersion in Single-Ring Optical Resonators and

Add/Drop filters-A Tutorial Overview‟, J.

Lightwave Technol., 2004, 22, (5), pp. 1380-1394

[2] Vörckel, A. , Mönster, M. , Henschel, W. ,

Bolivar, P.H. , Kurz, H: „Asymmetrically Coupled Silicon-On-Insulator Microring Resonators for

Compact Add-Drop Multiplexers‟, IEEE Photonics

Technol. Lett., 2003, 15, (7), pp. 921-923

[3] Morand, A., Zhang,Y., Martin,B., Huy,K.P.,

Amans,D., Benech,P: „Ultra-Compact Microdisk

Resonator Filters on SOI Substrate‟ , Optical

Express, 2006, 14, (16), pp. 12814-12821

[4] Kiyat,I., „Monolithic and Hybrid Silicon-on-

Insulator Integrated Optical Devices‟ Ankara:

Bilkent University,(2005)

[5] Kocabas,C., „Integrated Optical Displacement

Sensors for Scanning Force Microscopies‟ Ankara:

Bilkent University, (2003)

[6] Scarmozzino,R., Osgood, R.M., Jr. “Comparison of finite-difference and Fourier-

transform solutions of the parabolic wave equation

with emphasis on integrated-optics applications”, J.

Opt. Soc. Amer. A 8, (724) (1991).

[7] Microwaves, Antennas and Propagation,

Retrieved from

http://www.brunel.ac.uk/about/acad/sed/sedres/tele

com/wncc/research/antenna.bspx on November

12,2009

[8] Optical Ring Resonator, Retrieved from

http://www.photond.com/products/fimmprop/fimm

prop_applications_01.htm on December 29, 2009

APPENDIX

EFFECTIVE INDEX METHOD

ηf 1.54:= λ 1.55 104−

⋅:=

ηs 1.46:= ko2 π⋅

λ:=

ηc 1:=μo 4 π⋅ 10

2−⋅:=

h1

1.5 104−

⋅:=

εo10

18−

36 π⋅:=

h2

2 104−

⋅:=

ωoko

μo εo⋅:=

h3

5 104−

⋅:=

The guided light is polarized in the x direction. The field will be a TM mode in the

horizontal structures.

II. Region

κmax ko2ηf

2⋅ ko

2ηs

2⋅−:=

γc κ( ) ko2ηf

2⋅ κ

2−( ) ko

2ηc

2⋅−:= κ 0 κmax..:=

γs κ( ) ko2ηf

2⋅ κ

2−( ) ko

2ηs

2⋅−:=

ftm κ( ) tan κ h2

⋅( ):=

gtm κ( )

ηf( )2 γc κ( )

ηc( )2

⋅ ηf2 γs κ( )

ηs( )2

⋅+

κ⋅

κ2 ηf( )

4γc κ( )⋅ γs κ( )⋅

ηc( )2ηs( )

2⋅

−

:=

0 5 103

× 1 104

× 1.5 104

× 2 104

×

10−

5−

0

5

10

ftm κ( )

gtm κ( )

κ

κ 10 103

⋅:= trial value to find first root

κtmx2 root ftm κ( ) gtm κ( )− κ, ( ):=

κtmx2 1.212 104

×=

βtmx2 ko2ηf

2⋅( ) κtmx2

2−:=

βtmx2 6.124 104

×=

γtmx2 ko2ηf

2⋅ κtmx2

2−( ) ko

2ηs

2⋅−:=

γtmx2 1.573 104

×=

nefftm2βtmx2

ko:=

nefftm2 1.511=

I. & III. Region ( Horizontal Slab )

qmax ko2ηf

2⋅ ko

2ηs

2⋅−:=

Γc q( ) ko2ηf

2⋅ q

2−( ) ko

2ηc

2⋅−:= q 0 qmax..:=

Γs q( ) ko2ηf

2⋅ q

2−( ) ko

2ηs

2⋅−:=

Ztm q( ) tan q h1

⋅( ):=

Γtm q( )

ηf( )2 Γc q( )

ηc( )2

⋅ ηf2 Γs q( )

ηs( )2

⋅+

q⋅

q2 ηf( )

4Γc q( )⋅ Γs q( )⋅

ηc( )2ηs( )

2⋅

−

:=

0 5 103

× 1 104

× 1.5 104

× 2 104

×

10−

5−

0

5

10

Ztm q( )

Γtm q( )

q

q 1.5 104

⋅:= trial value to find first root

qtm1 root Ztm q( ) Γtm q( )− q, ( ):=

qtm1 1.478 104

×=

βtm1 ko2ηf

2⋅ qtm1( )

2−:=

βtm1 6.065 104

×=

nefftm1βtm1

ko:=

nefftm1 1.496=

For TE( Vertical Structure ),

Umax ko2

nefftm22

⋅ ko2

nefftm12

⋅−:=

( )

γyte u( ) ko2

nefftm22

⋅ u2

−( ) ko2

nefftm12

⋅−:= u 0 Umax..:=

Umax 8.462 103

×=r u( ) tan

u h3

⋅

2

:=

b u( )γyte u( )

u:=

0 2 103

× 4 103

× 6 103

× 8 103

× 1 104

×

10−

5−

0

5

10

r u( )

b u( )

u

u 2 103

⋅:= trial value to find first root

uy root r u( ) b u( )− u, ( ):=

uy 4.204 103

×=

βyte ko2

nefftm22

⋅ uy2

−:=

βyte 6.109 104

×=

γyte ko2

nefftm22

⋅ uy2

−( ) ko2

nefftm12

⋅−:=

γyte 7.344 103

×=

For x direction

TEX1 x( )cos κtmx2 x⋅( )

cos

κtmx2 h2

⋅

2

:=

TEX2 x( ) e

γtmx2− xh2

2−

⋅

:=

TEX3 x( ) e

γtmx2 xh2

2+

⋅

:=

Amptexβtmx2

2 μo⋅ ωo⋅∞−

h2−

2

xTEX3 x( )( )2

⌠⌡

d

h2−

2

h2

2

xTEX1 x( )( )2

⌠⌡

d+

h2

2

∞

xTEX2 x( )( )2⌠

⌡

d+

⋅

−

:=

Amptex 2.198 106

×=TEX x( ) TEX1 x( )

h2

−

2x<

h2

2<if

TEX2 x( ) x

h2

2≥if

TEX3 x( ) x

h2

−

2≤if

:=

Solving in y direction

TEY1 y( )cos uy y⋅( )

cos

uy h3

⋅

2

:= TEY2 y( ) e

γyte− yh3

2−

⋅

:= TEY3 y( ) e

γyte yh3

2+

⋅

:=

Ig3

h3

2

∞

yTEY2 y( )( )2⌠

⌡

d:=Ig2

h3−

2

h3

2

yTEY1 y( )( )2

⌠⌡

d:=Ig1

∞−

h3−

2

yTEY3 y( )( )2

⌠⌡

d:=

Ampteyβyte

2 μo⋅ ωo⋅Ig1 Ig2+ Ig3+( )⋅

1−

2

:= Amptey 1.871 106

×=

TEY y( ) TEY1 y( ) y

h3

−

2>

y

h3

2<

∧if

TEY2 y( ) y

h3

2≥

if

TEY3 y( ) y

h3

−

2≤

if

:=

1− 103−

× 5− 104−

× 0 5 104−

× 1 103−

×

0

2 106

×

4 106

×

6 106

×

8 106

×

Amptex TEX x( )⋅

Amptey TMX x( )⋅

x

The guided light is polarized in the y direction. The field will be a TE mode in the

horizontal structures.

II. Region

fte κ( ) tan κ h2

⋅( ):=

gte κ( )γc κ( ) γs κ( )+

κ 1γc κ( ) γs κ( )⋅

κ2

−

⋅

:=

0 5 103

× 1 104

× 1.5 104

× 2 104

×

10−

5−

0

5

10

fte κ( )

gte κ( )

κ

κtex2 root fte κ( ) gte κ( )− κ, ( ):=

κtex2 1.143 104

×=

βtex2 ko2ηf

2⋅( ) κtex2

2−:=

βtex2 6.137 104

×=

γtex2 ko2ηf

2⋅ κtex2

2−( ) ko

2ηs

2⋅−:=

γtex2 1.624 104

×=

neffte2βtex2

ko:=

neffte2 1.514=

I. & III. Region ( Horizontal Slab )

Zte q( ) tan q h1

⋅( ):=

Rte q( )γc q( ) γs q( )+

q 1γc q( ) γs q( )⋅

q2

−

⋅

:=

0 5 103

× 1 104

× 1.5 104

× 2 104

×

10−

5−

0

5

10

Zte q( )

Rte q( )

q

qte1 root Zte q( ) Rte q( )− q, ( ):=

qte1 1.383 104

×=

βte1 ko2ηf

2⋅ qte1( )

2−:=

βte1 6.087 104

×=

neffte1βte1

ko:=

neffte1 1.502=

For TM( Vertical Structure ),

Kmax ko2

neffte22

⋅ ko2

neffte12

⋅−:=

k 0 Kmax..:=γytm k( ) ko

2neffte2

2⋅ k

2−( ) ko

2neffte1

2⋅−:=

Kmax 7.799 103

×=

l k( ) tan

k h3

⋅

2

:=

n k( )γytm k( )

k

neffte22

neffte12

⋅:=

0 1 103

× 2 103

× 3 103

× 4 103

×

10−

5−

0

5

10

l k( )

n k( )

k

k 4 103

⋅:= trial value to find first root

ky root l k( ) n k( )− k, ( ):=

ky 4.099 103

×=

βytm ko2

neffte22

⋅ ky2

−:=

βytm 6.123 104

×=

γytm ko2

neffte22

⋅ ky2

−( ) ko2

neffte12

⋅−:=

γytm 6.635 103

×=

TMX1 x( )cos κtex2 x⋅( )

cos

κtex2 h2

⋅

2

:=

TMX2 x( ) e

γtex2− xh2

2−

⋅

:=

TMX3 x( ) e

γtex2 xh2

2+

⋅

:=

Amptmxβtex2

2 εo⋅ ωo⋅

∞−

h2−

2

x1

ηs2

TMX3 x( )( )2

⋅

⌠⌡

d

h2−

2

h2

2

x1

ηf2

TMX1 x( )( )2

⋅

⌠⌡

d+

h2

2

∞1

ηc2

⌠⌡

+

⋅

:=

1−

2

Amptmx 1.038 103−

×=TMX x( ) TMX1 x( )

h2

−

2x<

h2

2<if

TMX2 x( ) x

h2

2≥if

TMX3 x( ) x

h2

−

2≤if

:=

TMY1 y( )cos ky y⋅( )

cos

ky h3

⋅

2

:= TMY2 y( ) e

γytm− yh3

2−

⋅

:= TMY3 y( ) e

γytm yh3

2+

⋅

:=

In1

∞−

h3−

2

y1

neffte12

TMY3 y( )( )2

⋅

⌠⌡

d:= In2

h3−

2

h3

2

y1

neffte22

TMY1 y( )( )2

⋅

⌠⌡

d:= In3

h3

2

∞⌠⌡

:=

Amptmyβytm

2 εo⋅ ωo⋅In1 In2+ In3+( )⋅

1−

2

:= Amptmy 7.785 104−

×=

TMY y( ) TMY1 y( )

h3

−

2y<

h3

2<if

TMY2 y( ) y

h3

2≥if

TMY3 y( ) y

h3

−

2≤if

:=

0.01− 5− 103−

× 0 5 103−

× 0.01

0

5 104−

×

1 103−

×

1.5 103−

×

Amptmx TEY y( )⋅

Amptmy TMY y( )⋅

y

x2

TMX2 x( )( )2

⋅ d

1−

2

3

2

∞

y1

neffte12

TMY2 y( )( )2

⋅ d