Modes and Threshold Gain of a Planar Circularly Symmetric Distributed Feedback Semiconductor Laser Using Open Source Finite Element Software

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)

545

Modes and Threshold Gain of a Planar Circularly Symmetric

Distributed Feedback Semiconductor Laser Using Open Source

Finite Element Software

Dr. Ndagije Charles

Faculty of applied fundamental sciences, Institute of Higher Education (INES- Ruhengeri), P.O. Box 155 Ruhengeri (Rwanda)

Abstract- Computer modeling and simulations of physical processes play an important role in analysis of different physical and engineering phenomena. In this paper we analyzed the threshold conditions of a circular symmetric distributed feedback semiconductor laser formed in planar waveguide. The distribution of magnetic field in circular symmetric distributed feedback (CS-DFB) laser has been analyzed using open source finite element software from the OneLab project: Gmsh and GetDP. The resulting eignefrequencies have been compared to that of unperturbed structure. It has been found that the critical angular frequency for no perturbed model is real while for grating model is complex.

Keywords-- Simulation, Maxwell’s equations, circular symmetric distributed feedback semiconductor laser, grating, finite element method, weak formulation, discrete formulation, Galerkin Method.

I. INTRODUCTION

It was realized that Modeling and simulation can help to better understand different physical processes including the operation principle of the circular symmetric distributed feedback semiconductor laser. The threshold behavior of linear distributed feedback laser has been analyzed by H. Kogelnik and C.V. Shank by using coupled mode theory [16]. Circular symmetric distributed feedback lasers were analyzed first time by Erdogan and Hall in 1990 by using the coupled mode theory [15], many other authors have also analyzed the threshold conditions of this laser by using the coupled mode theory. We have also analyzed the threshold conditions of this laser by combining the coupled mode theory with the low parameter method [17]. This combination facilitated to better analyze the operation principle of that laser. All those methods conducted to the use of different approximations which complicate the result analysis.

Circular symmetric distributed feedback lasers are used in integrated optics and are occupied a small space comparing to the linear structure.

To facilitate the understanding of the operation principle of this laser, we used the finite element method which is an appropriate numerical method for solving partial differential equations in limited structures.

Electromagnetic field distribution in distributed feedback semiconductor laser is described by inhomogeneous partial differential equations (PDEs) that are often difficult or impossible to solve analytically, which complicates the analysis of the process. In this paper we propose to set up parametric finite element models to compute the mode distribution of electromagnetic fields in three layers planar circular symmetric distributed feedback semiconductor lasers. Such structures are widely used in fiber optics communication because they facilitate the mode coupling with the fiber optics. By using second order grating, those lasers can radiate from the surface and this propriety facilitates their coupling with a fiber optics.

Modeling such planar waveguides requires the solution of Maxwell’s equations, which can be obtained numerically by using the finite element method [14], [11]. We propose to evaluate the threshold conditions in those structures. The main advantages of using the finite element method for analyzing the threshold conditions of circular symmetric distributed feedback semiconductor laser is that, we directly find the eignevalue which contains the frequency of lasing mode and its threshold gain; apart from that with this method it is easy to visualize the pattern of mode distribution in the structure while by using analytical methods it is only possible to predict the frequency of lasing mode and its threshold gain.

The results of this work showed that there is no radiation in the center of the structure, and this is explained by the fact that the reflection of Hankel function takes place at the big argument, i.e., far from the center.

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Since the software tools are freely available on the internet, different researchers can use them without restrictions [12].

II. MATHEMATICAL FORMULATION OF THE PROBLEM

[image:2.612.52.293.240.358.2]

The simulation of electromagnetic fields in CS-DFB laser is analyzed by solving Maxwell’s equations with consideration of the boundary conditions.

Figure 1: Cross section of analyzed circular symmetric DFB semiconductor laser

In this structure

h

₀ is the thickness of the planar active medium,

n

_f is the refractive index of the film which is greater than the refractive index of the cladding (

n

_c ) and that of the substrate (

n

_s

)

. The permittivity of active medium depends on coordinates as follow [17], [15].

)

,

(

)

,

(

r

z



0

n

2

r

z





,

where



₀is the permittivity of the free space,









,

0 (

)

,

(3)

;

(r)

,

);

(

,

)

,

(

(2)

;

0 ,

;

0 ,

;

,

)

(

(1)

)

,

(

)

(

)

,

(

0 2 2 0 2 2 0 2 2 2 0 0 0 2 2 0 2





















































i f f f i f f c c i f s f c

in

n

h

r

h

z

n

in

n

h

z

h

n

r

h

z

h

n

in

n

z

r

n

z

n

h

z

n

h

z

n

z

n

z

r

n

z

n

z

r

n

and

n

_i



n

_f

.

The refractive index of active medium (guiding medium) is complex.

The main objective of our work is to determine the longitudinal propagation constant of lasing modes and their threshold gains. The use of finite element method gives the complex angular frequency

'







i

.

The values to determine are found by using the following mathematical expressions [18],





















2 '

'

th eff eff eff

g

i

N

c

N

c

i

N

c









(4)

Where



is the longitudinal propagation coefficient,

eff

N

is the effective refractive index ,

g

_th is the threshold gain of the active medium;



is the confinement factor,

c

is the speed of light in free space,



'

and



'

are respectively the real and imaginary parts of the lasing mode angular frequency



and

i

is the imaginary unit

.

The resolution of Maxwell’s equation by using finite element method, Gmsh and GetDP open source software is one of the effective methods used to find the frequency of generation



[2][12]. From the equation (4) we find that

















f eff s eff th eff

n

N

n

c

N

g

N

c

'

2 '





(5)

Maxwell’s equations in linear media read:













                t D j H t H E B E            , , 0 . , .   

(6) R n_f(Active layer)

nc(Cladding)

n_s(Substrate)

h(r) h0

1

(3)

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Where

E



(

V

/

m

)

is the electrical field intensity,

)

/

(

Coul

m

3



is the electrical charge density,



is the permittivity of the medium,

B



(

Web

/

m

2

)

is the magnetic flux density,



is the permeability of the medium,

)

/

(

A

m

H



is the magnetic field intensity,



j

(

A

/

m

2

)

is the conduction current density. If the considered medium is the air, the permittivity becomes



₀, and the permeability becomes



₀, where₀8.854.1012Farad/mand

m henry/ 7 10 . 4

0

  

 . For a semiconductor material,

0 



j



.

By using the properties of vector analysis and considering the variation of refractive index as a function of coordinate, the system (6) of first order partial differential equations (PDEs) can be rewritten as the following second order equation:

(7) 0 2 2 )] , ( 2 ) , ( 2 0 [ 0

0 _ 

 

 

   

t E z r n z r n E

 





This equation can be written as follow

(8)

)

,

(

)

,

(

₂

2 2 0 0 2 2 2 0 0 0

t

E

z

r

n

t

E

z

r

n

E































Which is an non- homogeneous Helmholtz wave equation.

The magnetic field

E



must satisfy the boundary conditions, i.e., its value must be equal to zero on the surface of the structure for avoiding the lateral radiation, meaning that the following condition has to be satisfied [17]:

0 )

(

r



R



E



.Where

R

is the exterior radius of the structure.

For effective analysis of threshold condition for operation of a laser it is very important to know the radiation frequency and the threshold gain. In order to know which waves (modes) are able to be generated, we need to compute the eigenfrequencies f[Hz] of the structure. This problem can be solved by using finite element method.

III. FINITE ELEMENT METHOD

The finite element method is a powerful computational technique which can help to approximate solutions to PDEs encountered in scientific and engineering applications [10],[12].

With the finite element method the solution of PDEs is found by following different steps. The first step is the preprocessing which includes the definition of the geometric domain of the problem, the definition of elements type to use for the domain discretization, the definition of material properties of the elements, and the definition of the physical constraints. The second step is the solution phase. During this step the finite element software assembles the governing algebraic equations in matrix form and computes the unknown values. This step requires the discrete formulation of the problem through the weak formulation of the problem (see section 4). For thediscrete formulation of the problem we used the Galerkin method [10], [7] as implemented in the open source software GetDP [2].

[image:3.612.334.556.422.620.2]

The last step of finite element method is the post processing which is focused on analysis and evaluation of the solution results. In this study this step was carried out by using the Gmsh open source software [1].

Figure 2: Geometric representation of the section of CS-DFB laser

On this figure,



is the period of the grating

.

The next figure represents the discretization of the domain of resolution which is the section of a circularly symmetric medium with a periodic variation of the refractive index at the point

z



h

₀

.

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548

Figure 3: Discretized section of CS-DFB laser

IV. WEAK FORMULATION OF THE PROBLEM

For the weak formulation of the problem, we used the method of weighted residual [10]. According to this method, instead of solving (8) directly, we look for

E



such that





















         

 . 0

2 2 ) , ( 2 0 [ 0

0

(

,

)]

2 d U t E z r n

E

n

r

z



   





(9)

holds for appropriately chosen weight (test) function

U

. By using the properties of vector analysis the equation (9) can be rewritten as follows:

(10) 0 . 2 2 ) , ( 2 0 0 ) .( ) .(                 

 Ud

t E z r n d U A d A U          

Where AE. By using the divergence theorem, the expression (10) becomes

(11) 0 . 2 2 ) , ( 2 0 0 ) ).( ( ) ( ). (                  

 Ud

t E z r n d U E d n A U           

Where  is a curve delimiting the surface_{. By} using the property of vector analysis, we get

(U A n d). ( ) A n U d.( ) ( ) 0

         

  (12)

Because it is necessary to chose the weight function which also satisfies the boundary conditions such that

0 n U

 

.

Finally the integral in (10) can be written as follows:

(13) 0 . 2 2 ) , ( 2 0 0 ) ).( (          

 t Ud

E z r n d U E        

This expression represents the weak formulation of the problem.

V. DISCRETE FORMULATION OF THE PROBLEM

The discrete formulation consists in approximating the electric field

E



using a finite number of shape functions (basis functions)_{W j}:

   N

j CjWj E 1  

(14)

The GetDP code uses so-called Whitney edge elements to approximate the electric field. In this case, the functions

W j have continuous tangential components across element borders, but discontinuous normal components and the coefficients _{C j} are the circulations of E along each edge of the mesh.

N

is the total number of edges in the mesh. By using the Galerkin method, the weight functions

U

are chosen identical to the form functionsW , i.e.,

N k

k W

U   , 1,2,3,... (15) Introducing (15) and (14) into (13) leads to a system of N linear equations:

(16) 0

1 1 2

2 ) , ( 2 0 0 .                    N j N j d k W t j W z r n j C d k W Curl j W Curl j C       or (17) 0 . 1 2 2 0

0 (, )

2 _                  _    



   N j k j k j

j W d

t W d W Curl W Curl

C n rz 

 







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The numerical resolution aims to find those values. Once they are found, the electrical field

E



can be obtained by using the expression (14).

VI. NUMERICAL SOLUTION

We tested the method by solving the equation (17) with the variation of refractive indexn(r)

.

The numerical used parameters are indicated in table (II).

(18) 0 .

1

2 2

0

0 (, )

2 _

  

 

  

 

   

 

 _

  







 

N j

k j k

j

j W d

t W d

W Curl W Curl

C n rz 

 



 

A. Case of unperturbed passive medium

We first tested the case of mode distribution in unperturbed passive medium. In this case we considered that the refractive index n(r,z) is a constant (

6 . 3 )

,

(r z n_f 

n

)

and we found the frequency by solving the following equation

.

(19) 0 .

1

2 2

0 0

2 _

  

 

  

 

   

 

 _

  







 

N j

k j k

j

j W d

t W d

W Curl W Curl C

f n

  



[image:5.612.326.564.127.329.2]

 

[image:5.612.49.290.316.608.2]

Figure 4: Distribution of TM01-mode

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550

Figure 7: Distribution of TM13- mode

Figure 8: Distribution of TM23-mode Table I

Eignenfrequency And Normed Prpopagarion Constant For A Passive Unperturbed Medium

This table shows that for the unperturbed passive medium, the imaginary party of the eignefrequency is very small comparing to its real party, so that we can neglect it

.

The constant



is the radial propagation constant and

R

is the exterior radius of the structure.

B.Case of perturbed active medium

In this case, we tested the following equation

(18)

0 [

.

1

2 2

0

0 ()]

2 ) ( 2

0 

  

 

  

 

   

 

 _

  







 

 

N j

k j k

j

j W d

t W d

W Curl W Curl

C n r n r

  



[image:6.612.47.290.131.502.2]

 

Table II

The Used Parameters For Numerical Solutions

Parameter Numerical value

The period of grating (



) 5m The maximum radius of the structure ( R)

m

50

The effective refraction index active medium

3.58

The confinement parameter 0.5 The index of the cladding and substrate

3.55

The refractive index of the active medium without grating

3.6

The imaginary part of refractive index

0.1

[image:6.612.48.289.133.312.2]

The confinement factor(

)

0.5

[image:6.612.325.563.179.702.2]

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Figure 10 : Distribution of TM12- mode

[image:7.612.324.563.122.336.2]

Figure 12: Distribution of TM13- mode

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552

Figure 14: Variation of first type Bessel’s functions with respect to the

coordinate

In center of the laser, there is a dark spot because as it is indicated on figure 14, at the origine (r0

)

, the value of high order Bessel’s function is always equal to zero.

Table III

Eignenfrequency For Lasing Modes, Their Normed Prpopagarion Constants And Threshold Gains For Active Perturbed Medium

Type of

mode



(

rad

/

s

)



R

g

th

R

01

TM

₅_.₁₅_.₁₀12__i₃_.₆₉_₁₀10 2.704 0.08 12

TM

7.09.1012i5.011010 3.723 0.105 02

TM

₁_.₀₈_.₁₀13__i₇_.₂₅_₁₀10 5.671 0.152 13

TM

₁_.₃₅_.₁₀13__i₉_.₈₄_₁₀10 7.088 0.207 23

TM

1.63.1013i1.011011 8.543 0.212

This table shows that in case of perturbed active medium, the found eignevalues are complex and the threshold gain increases with the increasing of the order of the lasing mode.

VII. C

ONCLUSION

In this study, the electromagnetic field distributions in unperturbed circular symmetric structure and in circular symmetric distributed feedback semiconductor laser have been described and simulated, by using the finite element method.

The distribution of magnetic field intensity along the radius reminds the behavior of the variation of the Bessel’s functions with respect to the coordinate. The found results show that the lowest threshold gain acquires for the lowest mode.

It was demonstrated that the finite element method has not only the capacity of visualization of results which are sometimes an abstract concept but also it is an effective method of solving non-homogeneous partial differential equation. For our next work, we will use the found results in this study, to solve the rate equations of the DFB semiconductor laser.

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[5] Robert E. Collin, Foundations for Microwave Engineering, second edition, John Willey &Sons, Inc., New York, 2001

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[7] J.N. Reddy, An introduction to the finite element method, second edition, McGraw-Hill, In., Texas, 1993.

[8] Jean-Philippe Grivet, Méthodes numériques appliquées pour le scientifique et l’ingénieur, EDP Sciences, Paris 2009.

[9] Benoît Meys, Modélisation des champs électromagnétiques aux hyperfréquences par la méthode des éléments finis. Application au problème du chauffage diélectrique, Thèse présentée à la Faculté des sciences Appliquées, Université de Liège, 1999.

[10] David V. Hutton, Fundamental of finite element analysis, McGraw- Hill, New York, 2004.

[11] C. J. Reddy, Manohar D. Deshpande, C.R. Cockrell, and Fred B. Beck; Finite Element Method for Eigenvalue Problems in Electromagnetics, NASA Technical paper 3485, Virginia 23681- 0001, 1994.

[12] C. Ndagije and C. Geuzaine, Simulation of Electromagnetic Field Distribution in Metallic Waveguides Using Open Source Finite Element Software, International Journal of Emerging Technology & Advanced Engineering, Volume3, Issue2, Page 91-98, February, 2013.

[13] Oussama Zahwe, Conception et Réalisation d’un Circulateur coplanaire à couche Magnétique de YIG en Bande X pour des Applications en Télécommunications ; Thèse pour obtenir le grade de Docteur de l’Université Jean Monnet de Saint-Etienne, 2009. [14] Matthew N. O. Sakidu, Numerical Techniques in Electromagnetics,

Second edition, CRC Press LLC, Florida, 2001.

[15] T.Erdogan and D.G. Hall, Circularly symmetric distributed feedback laser: An analysis, Journal of Applied Physics, Vol 68,N0_{4, 15} August 1990.

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 5 10 15 20 25

) Jn(x

x r

x

) J0(x

) J1(x

) J2(x

[image:8.612.47.286.377.493.2]

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[16] H.Kogelnik and C.V.Shank . Coupled-wave theory of distributed feedback lasers, Journal of Applied physics , Vol. 43, N0_{5, May} 1972.

[17] Ndagije Charles, Cylindrical- symmetric distributed feedback laser, A thesis presented to Friendship University in the fulfillment of the requirements for the degree of Doctor of Philosophy in Physical mathematical science-specialty of Radiophysics, Moscow, 2000.