Outline of the thesis

This thesis has been organized in the following arrangement. The first chapter of the thesis will introduce the different types of edge emission and surface emission lasers and the theory of Bragg diffraction in 2D photonic crystal. Chapter 2 begins by laying out the theoretical dimensions of the research, the couple wave theory and the finite difference method for square lattice and triangular lattice. Chapter 3 describes the simulation results, characteristics and calculated result of PCSELs. The last chapter assesses the conclusion and future work.

11

References

[1] A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958) [2] T. H. Maiman, Nature 187, 493 (1960)

[3] R. N. Hall, G. E. Fenner, J. D. Kingsley, T. J. Soltys, Phts. Rev. Lett. 9, 366 (1962)

[4] N. Holonyak Jr. and S. F. Bevacqua, Appl. Phys. Lett. 1, 82 (1962)

[5] T. M. Quist, R. H. Rediker, R. J. Keyes, W. E. Krag Lax, A. L. McWhorter, H. J.

Zeigler, Appl. Phys. Lett 1, 91 (1962)

[6] 盧廷昌、王興宗，「半導體雷射導論」，五南出版社，2008

[7] M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D.

Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999)

[8] M. Imada, A. Chutinan, S. Noda and M. Mochizuki, Phys. Rev. B. 65, 195306 (2002)

[9] I. Vurgaftman and J. Meyer, IEEE J. Quantum. Electronics. 39, 689 (2003) [10] Ohnishi, D., Okano, T., Imada, M. and Noda, S, Opt. Express. 12, 1562 (2004) [11] Kim, M, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman and J. R. Meyer, Appl. Phys. Lett. 88, 191105 (2006)

[12] H. Matsubara, S Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka and S. Noda, Science. 319, 445 (2008)

[13] T. C. Lu, S. W. Chen, L. F. Lin, T. T. Kao, C. C. Kao, P. Yu, H. C. Kuo, S. C.

Wang and S. H. Fan, Appl. Phys. Lett. 92, 011129 (2008)

[14] K. Sakoda, K. Ohtaka and T. Ueta, Opt. Express. 4, 481 (1999)

[15] Y. H. Lee, H. Y. Ryu and M. Notomi, Phys. Rev. B. 68, 045209 (2003) [16] K. Sakai, J. Yue and S. Noda, Opt. Express. 16, 6033 (2008)

[17] K. Sakai, E. Miyai and S. Noda, IEEE J. Quantum Electron. 46, 788 (2010)

12

[18] S. Nojima, J. Appl. Phys. 98, 043102 (2005)

[19] M. Notomi, H. Suzuki, and T. Tamamura, Appl. Phys. Lett., 78, 1325 (2001)

13

Chapter 2

Fundamentals of Photonic Crystal Surface Emitting Lasers

Introduction

Electromagnetic can be divided into two classes by the direction of polarization : transverse electric (TE) polarized light and transverse magnetic (TM) polarized light, as shown in Fig. 2.1. The light's electromagnetic properties are defined by the orientation of its electric and magnetic fields. TE polarized light is characterized by its magnetic field being parallel to the orientation of photonic crystal columns and hence the electric field is perpendicular to the orientation of photonic crystal columns. And TM polarized light is characterized by its electric field being parallel to the orientation of photonic crystal columns. Here we mainly discuss the TE wave in the photonic crystal.

Square lattice photonic crystal and triangular lattice photonic crystal are common two-dimensional (2D) photonic crystal. So we discuss the Square lattice photonic crystal and triangular lattice photonic crystal, respectively.

Fig. 2.1 Schematic diagram of the polarization direction of wave

14

2.1.1 Couple-wave theory for square lattice

[1-3]

We first consider the photonic crystal (PC) structure of a square lattice. The PC of circular holes with period a is in the x−y plane, as shown in Fig. 2.2(a). The structure is assumed to be uniform in the z direction. The dielectric constants of the circular holes is ε_a and the background material is ε_b. The circular holes from a 2D Bravais lattice with sites given by the vectors :

2 2 1

) 1

(t na n a

r =  +  . (2.1)

Here, a₁

and a₂

are the two primitive basis translation vectors of the square lattice.

n1 and n₂ are any integer numbers. The enclosed area of the primitive unit cell of the lattice is A_c = a₁×a₂ =a².

The vectors G(m) of reciprocal lattice which is shown as Fig. 2.2(b) are given by :

2 2 1

) 1

(m mb mb

G  

+

= . (2.2)

(a) (b)

Fig. 2.2 (a) Square lattice photonic crystal structure (b) Schematic

diagram of eight propagation waves in square reciprocal lattice

photonic crystal structure

15

Here, b₁

and b₂

are the two primitive basis translation vectors of this reciprocal lattice. m₁ and m₂ are any integer numbers. The definition of the primitive lattice vectors changes the reciprocal lattice vectors satisfies that

ij

So the two primitive basis translation vectors of this reciprocal lattice are given by

) wave vectors that have contributions at those points of Γ2.

The scalar wave equation for the magnetic field H_z in the TE polarization mode can be written as the form [4] :

We note that the sign of the second term was negative [5]. In the above formula, λ is the wavelength of the light in free space, ε_G is the Fourier coefficient of the

16 which can be expressed as :

0 2

In Eq. (2.9), the periodic variation in the refractive index is included the small perturbation in third term through the Fourier expansion. In the Fourier expansion, the periodic perturbation terms generates an infinite set of diffraction orders. However, as the cavity mode frequency is sufficient close to the Bragg frequency, only the second order diffraction and below can do significant contribution, others can consider to be neglected. We consider the resonance at Γ -point, in which when it is satisfy the second order Bragg diffraction, it will induce 2D optical coupling and result in surface emission. The corresponding coupling coefficient constant

)

β₀ = ²π . Fig. 2.3 shows a schematic illustration of the pairs of wave vectors

17

that are coupled in each of these three cases. Coupling constant κ describes the ₁ intensity of the coupling of two plane waves propagating at 45 to each other. ° Coupling constant κ describes the intensity of the coupling of two plane waves ₂ propagating in directions perpendicular to each other. Coupling constant κ₃ describes the intensity of the coupling of counterpropagating waves to each other, which corresponds to the backward scattering in second-order distributed feedback (DFB) lasers. The coupling constant κ does not exist in the case of a square lattice ₂ with TE polarization. This is because the electric fields of two waves propagating in perpendicular directions are orthogonal to each other and the overlap integral vanishes.

While considering a periodic structure, the magnetic field can be described by the Bloch mode [4] :

( )

∑

^{− + ⋅}

=

G

r G k i G

z r H e

H ( ) . (2.12)

H is the amplitude of each plane wave, k is the wave vector in the first Brillouin G

zone and when it is the Γ point, it comes to zero. However, at the specific Γ point

Fig. 2.3 Diffraction diagram for each coupling constant for square

lattice. White arrows indicate pairs of wave vectors and black arrows

indicate the corresponding reciprocal lattice vectors.

18

discussed for 2D photonic crystal, there are eight propagating waves in PC structure denoted as R_x,S_x,R_y,S_y,F₁,F₂,F₃,F₄ showed in Fig. 2.2(b), the first four items are the complex amplitudes of the four propagating waves along the x,−x,y,−y directions and the other four items are the complex amplitudes of the four propagating waves along Γ−Μ directions, respectively. Those correspond to H in Eq. (2.12). _G Here, we do consider these basic wave vectors along the Γ−Χ directions with

β0

κ+ G = and Γ−Μ directions with κ+ G = 2β₀ [1]. The contribution of the

higher order waves with κ + G ≥2β₀, are considered to be negligible. We should note that the basic waves and higher order waves are partial waves of the Bloch mode, so they have the same eigenvalue β for specific resonant cavity mode.

Using these eight waves, the magnetic field in this case can be rewritten the expression as the following sum :

y

By substituting Eq. (2.9) and Eq. (2.13) into Eq. (2.6), then using Eq. (2.11) and comparing the exponential terms, we obtain eight equations of the form :

4

19

The parameter δ is a normalized frequency defined by ) oscillation frequency ω from the Bragg frequency ω₀. In the above equations, we

assume that β/β₀ ≈1, since we take optical coupling at Γ point and this frequency deviation is small.

One thing we should noted that the coupling coefficient κ , which describes the ₂ intensity of the direct coupling of waves propagating perpendicular to each other along the x and y axes, does not exist in Eq. (2.14) [1]. We can neglect both the first two derivatives and the third terms in each case on the left hand side of Eq.

(2.14e)~Eq. (2.14h), because the amplitudes vary only slowly and α ,δ <<β₀ for the lower-order resonant modes. Then by substituting Eq. (2.14e)~Eq. (2.14h) into Eq.

(2.14a)~Eq. (2.14d) and including diffraction in the direction vertical to the PC plane represented by the coupling constant κ₀ [6][7], we obtain four equations of the form :

20

For the resonant mode in a square lattice PC cavity with TE polarization, the eigenvalues α provide the threshold gain and the eigenvalues δ provide the frequency deviation from the Bragg condition by numerically solving the Eq. (2.16) under some boundary conditions. The wave on the equal sign of the left in the Eq.

(2.16) is meaning that electromagnetic waves in a square lattice PC by moving receive the gain and loss. The wave on the equal sign of the right in the Eq. (2.16) is meaning that electromagnetic waves in a square lattice PC are coupling with R_x ,S_x ,R_y ,S_y, respectively.

The coupling constants for the circular holes are calculated with the formulas [6] : constants of the circular holes and the background material, respectively. The quantity

2 2/ a R

f =π is a hole filling factor and R is the radius of the circular hole. The averaged dielectric constant ε₀ is given by ε₀ = ε_af +ε_b(1− f) . J₁(x) is a Bessel function of the first kind for integer order one. d is the thickness of the _g

grating layer and Γ is its confinement factor. n_g ∆ is the modulation of the real part of the modal index between the waveguide and the bottom of the etched features with perfectly vertical sidewalls. λ is the grating’s resonance wavelength and a_L is the

21

area of the reciprocal lattice primitive cell. We exclusively define the vertical coupling constant κ₀ using the relation κ₀L=2κ₁²L²/500, where L is the length of the PC cavity.

We assume the boundary conditions of zero reflectivity and zero gain perturbation

(

αa −αb

)

in this work. We used the finite difference method as described in the CH2.1.2 for solving the Eq. (2.16). The electric field distribution

( )

^,

(

^, y ^{i t ,0}

)

t i

xe E e

E t r

E = ^{ω ω} is calculated using the time-dependent magnetic field

( )

,

(

0 ,0 ,

)

H r t = H_ze^iωt and Maxwell’s equation

( )

t

t r r E t

r ∂

= ∂

×

∇ ( , )

) ( ,

H ε . (2.19)

The intensity envelope of the resonant mode throughout the PC structure is determined using the sum R_xR^∗_x+S_xS^∗_x+R_yR^∗_y+S_yS^∗_y

22

2.1.2 Finite difference method for square lattice

We discuss the main relationship of the threshold gain α and the frequency deviation δ from the Bragg condition. And Eq. (2.16) are the eigenvalue problems.

So, we change the Eq. (2.16) to the following form :

( )

_x i R_x

(

i

)

S_x i S_y i R_y

Now we can make the matrix of Eq. (2.20), as the following form :

( )

C= . There is a differential item in the matrix.

The differential item is complex and difficult in the matrix operations. The finite difference method that difference is used in place of differential can simplify the problems. Numerical solution of the coupled wave Eq. (2.21) can be found by using the finite difference method. There are several hundred or thousand of the photonic crystal in the x and y directions of the practical devices. However, we cut apart this photonic crystal cavity into a 18×18 matrix for the calculations, as shown in Fig.

2.4. We solve the Eq. (2.21) and get the numerical solution that is the complex amplitudes at each positions by black dots. The value of each white dot can obtain by using the complex amplitudes of the neighboring black dots. For example, the

23

difference equation corresponding to Eq. (2.20a) is written in the form :

( )

the x and y directions, respectively. At all the surrounding boundaries, we set the facet reflection to zero :

0

where L is the length of a square photonic crystal cavity. By solving the eigenvalue problem for the sets of difference equations, we obtain the eigenvalue

(

α −κ₀−iδ

)

and the eigenvectors (R_x(j,k),S_x(j,k),R_y(j,k),S_y(j,k),etc).

24

Fig. 2.4 Schematic diagram for square lattice for the finite difference

method. The target of calculations is carried out at the positions of

the white dots by using the complex amplitudes of the neighboring

black dots.

25

2.2.1 Couple-wave theory for triangular lattice

In this section, we consider the PC structure for triangular lattice. The 2D PC structure investigated here consists of a triangular lattice with circular holes with period a in the x−y plane, as shown in Fig. 2.5(a). The structure is assumed to be uniform in the z direction. The dielectric constants of the circular holes is ε_a and the background material is ε_b. The enclosed area of the primitive unit cell of the

lattice for triangular lattice is

2

translation vectors as 



primitive reciprocal lattice vectors are 



Fig 2.5 (a) Triangular lattice photonic crystal structure (b) Schematic

diagram of six propagation waves in triangular reciprocal lattice

photonic crystal structure

26

The scalar wave equation for the magnetic field H_z in the TE polarization mode can be written as the form [4] : which can be expressed as :

0 2

We consider the resonance at Γ -point, in which when it is satisfy the second order Bragg diffraction, it will induce 2D optical coupling and result in surface emission.

The corresponding coupling coefficient constant κ_i(i=1 ,2 ,3) are denoted as :

27

where

a 3 4

0

β = π . Fig. 2.6 shows a schematic illustration of the pairs of wave vectors

that are coupled in each of these three cases. Coupling constant κ describes the ₁ intensity of the coupling of two plane waves propagating at 60 to each other. ° Coupling constant κ describes the intensity of the coupling of two plane waves ₂ propagating at 120 to each other. Coupling constant ° κ₃ describes the intensity of the coupling of counterpropagating waves to each other, which corresponds to the backward scattering in second-order distributed feedback (DFB) lasers.

While considering periodic structure, the magnetic field can be described by the Bloch mode [4]

( )

∑

^{− + ⋅}

=

G

r G k i G

z r H e

H ( ) (2.30)

H is the amplitude of each plane wave, k is the wave vector in the first Brillouin G

zone and when it is the Γ point, it comes to zero. However, at the specific Γ point discussed in this case, the amplitude H_G with G =β₀ are significant and the other amplitudes are small and can be neglected. For 2D photonic crystal, there are six propagating waves with G =β₀ in PC structure denoted as H₁,H₂,H₃,H₄,H₅,

Fig. 2.6 Diffraction diagram for each coupling constant for triangular

lattice. White arrows indicate pairs of wave vectors and black arrows

indicate the corresponding reciprocal lattice vectors.

28

H . All of these parameters which are shown in Fig. 2.5(b) are considered in our 6

model.

Using these six waves, the magnetic field in this case can be rewritten the expression as the following sum :



By substituting Eq. (2.27), Eq. (2.28) and Eq. (2.31) into wave Eq. (2.24), then using Eq. (2.29), including diffraction in the direction vertical to the PC plane represented by the coupling constant κ₀ [6][7] and comparing the exponential terms, we obtain six equations of the form :

(

_{2 6}

)

²

(

_{3 5}

)

_{3 0 4} distributions of individual light waves propagating in the six equivalent Γ−Μ directions : 0°, +60°, +120°, +180°, +240°, and +300° with respect to the x axis.

3 2

1 ,κ , κ

κ and are the coupling coefficients between light waves propagating at 60°

to each other ( H₁and H₂, H₂ andH₃, and so on ), at 120° ( H₁and H₃,

29

4 2 ,andH

H , and so on ), and at 180° ( H₁and H₄, H₂ and H₅, and so on ), respectively. δ is the deviation of the wave number β (expressed as 2πω /c, where ω is the frequency and c is the velocity of light ) from the fundamental propagation constant β₀ ( equal to 4π/ 3a, where a is the lattice constant ) for each cavity

= , α is the corresponding threshold gain. For the resonant mode in a triangular lattice PC cavity with TE polarization, the eigenvalues α provide the threshold gain and the eigenvalues δ provide the frequency deviation from the Bragg condition by numerically solving the Eq. (2.32) under some boundary conditions. The wave on the equal sign of the left in the Eq.

(2.32) is meaning that electromagnetic waves in a triangular lattice PC by moving receive the gain and loss. The wave on the equal sign of the right in the Eq. (2.32) is meaning that electromagnetic waves in a triangular lattice PC are coupling with

6

H , respectively.

The coupling constants for the circular holes are calculated with the formulas [6]

( ) ( )

constants of the circular holes and the background material, respectively. The quantity

2

function of the first kind for integer order one. The definition of coupling constant κ₀ is the same with that of the square lattice in section 2.1.1.

We assume the boundary conditions of zero reflectivity and zero gain

30

perturbation

(

α_a−α_b

)

in this work. We used the finite difference method as described in the Ch2.2.2 for solving the Eq. (2.32). The electric field distribution

( )

^,

(

^, y ^{i t ,0}

)

t i

xe E e

E t r

E = ^{ω ω} is calculated using the time-dependent magnetic field

( )

^,

(

^{0 ,0 ,}

)

H r t = H_ze^iωt and Maxwell’s equation

( )

t

t r r E t

r ∂

= ∂

×

∇ ( , )

) ( ,

H ε . (2.34)

The intensity envelope of the resonant mode throughout the PC structure is determined using the sum H₁H₁^∗+H₂H₂^∗+H₃H₃^∗+H₄H₄^∗+H₅H₅^∗+H₆H₆^∗

31

2.2.2 Finite difference method for triangular lattice

We discuss the main relationship of the threshold gain α and the frequency deviation δ from the Bragg condition. And Eq. (2.32) are the eigenvalue problems.

So, we change the Eq. (2.32) to the following form :

(

_{2 6}

)

²

(

_{3 5}

)

_{3 0 4}

Now we can make the matrix of Eq. (2.35), as the following form :

( )

respectively. There is a differential item in the matrix. The differential item is complex and difficult in the matrix operations. The finite difference method that difference is

32

used in place of differential can simplify the problems. Numerical solution of the coupled wave Eq. (2.36) can be found by using the finite difference method. There are several hundred or thousand of the photonic crystal in the x−y plane of the practical devices. However, we cut apart this photonic crystal cavity into a matrix for the calculations, as shown in Fig. 2.7. We solve the Eq. (2.36) and get the numerical solution that is the complex amplitudes at each positions by black dots. The value of each white dot can obtain by using the complex amplitudes of the neighboring black dots. For example, the difference equation corresponding to Eq. (2.35a) is written in the form :

At all the surrounding boundaries, we set the facet reflection to zero :

0

where L is the length of a triangular photonic crystal cavity and n is the positive integer. By solving the eigenvalue problem for the sets of difference equations, we obtain the eigenvalue

(

α−κ₀ −iδ

)

and the eigenvectors (H₁(j,k),H₂(j,k),

33

Fig. 2.7 Schematic diagram for triangular lattice for the finite

difference method. The target of calculations is carried out at the

positions of the white dots by using the complex amplitudes of the

neighboring black dots.

34

References

[1] K. Sakai, E. Miyai, and S. Noda, Appl. Phys. Lett. 89, 021101 (2006) [2] K. Sakai, E. Miyai, and S. Noda, Opt. Express 15, 3981 (2007) [3] K. Sakai, J. Yue, and S. Noda, Opt. Express 16, 6033 (2008) [4] M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991) [5] H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969)

[6] I. Vurgaftman and J. R. Meyer, IEEE J. Quantum Electron. 39, 6, 689 (2003) [7] R. F. Kazarinov and C. H. Henry, IEEE J. Quantum Electron. 21, 2, 144 (1985)

35

Chapter 3

Simulation Results of Photonic Crystal Surface Emitting Lasers

Numerical results

In this chapter, we would discuss the numerical results by solving the complex simultaneous equations based on the coupled wave theory for square lattice and triangular lattice, respectively. The normalized frequency deviation from the Bragg condition, threshold gain and near field patterns of the resonant modes in the 2D PC structure for both square and triangular lattice have been calculated. In addition, the relation between the mode pattern and different coupling strengths are also calculated.

Besides, we have evaluated the threshold gain, the normalized frequency deviation and the coupling constants as a function of the hole-filling factor for the fundamental modes. Finally, we would discuss the influence of the coupling constant κ₀ to different band-edge modes.

36

3.1.1 Mode spectra and mode patterns for square lattice

The typical band structure of photonic crystal for square lattice with transverse electric (TE) polarization shows as Fig. 3.1. And Fig. 3.2 shows the detailed band structure around the Γ point where surface emission is obtained. At the band edges of the band structure is easier to form the resonant mode oscillation. The lasing oscillation at the mode will occur with the lowest threshold and the smallest optical loss [1]. By the reason, calculation of the threshold gain for different resonant modes at the band edges point is very important for understanding of the PCSEL characteristics.

We know that the lasing action is easier to achieve the threshold gain at the Γ point of the band structure. There are three fundamental modes that is A mode, B mode and E mode of doubly degenerate as shown in Fig 3.2. We employing the following parameters in the PC model : the holes dielectric constants ε_a =9.8, the

background dielectric constants ε_b =12.0, the gain perturbation (α_a−α_b)=0, the hole filling factor f =0.18, the lattice period a=290nm, and the PC cavity length

m

L=50µ . The hole filling factor f define that a circular hole area in the unit cell occupied area ratio. According to the numerical solution of the calculation, we plot the threshold gain as a function of frequency deviation from the Bragg condition for the resonant modes, as shown in Fig. 3.3(a). We classify the groups of resonances as

,...

3 , 2 , 1± ±

±

N= according to their frequency deviation from the Bragg condition, where N is the mode number [2]. The more detailed plots for modes N =−1 and

=1

N are shown in Fig. 3.3(b) and Fig. 3.3(c), respectively.

The eigenvectors of Eq. (2.20) provide the complex amplitudes such as R , _x S , _x

R , and y S , which are functions of the positions _y x and y. The intensity envelope

37

(mode pattern) of the resonant mode throughout the PC structure can be determined from these amplitudes using the sum R_xR_x^∗+S_xS_x^∗+R_yR^∗_y+S_yS^∗_y.

It could be classified that the mode A (αL=0.46597,δL=−6.1259), mode B )

4.8881 ,

0.57709

(αL= δL=− , and mode E (αL=1.7107,δL=4.4137) are fundamental modes that have a single-lobed intensity pattern throughout the photonic crystal, as shown in Fig. 3.4(a-d). We could understand that the lowest frequency of these three modes is mode A and mode E is doubly degenerate. Modes A and B have twin modes A ₀ (αL=1.6768,δL=4.3506) and B ₀ (αL=1.751,δL=4.4847) ,

respectively. Modes A and ₀ B exhibit vase-like patterns with zero intensity at the ₀ center of the structure, as shown in Fig. 3.4(e-f). The other points in Fig. 3.3(b) and Fig. 3.3(c) correspond to higher order modes which consist of a higher order transverse mode. A lot of higher order modes in the proximity of mode E are almost impossible to distinguish in Fig. 3.3(c). Then, the numerical results of the threshold gain for modes A, B, and E by calculation are α_AL=0.46597, α_BL=0.57709, and

1.7107

=

EL

α , respectively. As a result, mode A has the lowest threshold gain and could easily achieve the lasing oscillation.

Fig. 3.4(g) (ex .αL=0.5522,δL=-5.4595or αL=1.7094,δL=4.4125...) and Fig. 3.4(h) (ex .αL=0.53445,δL=-5.4781or αL=1.7103,δL=4.4133...) illustrates the intensity envelope for the higher order modes around mode E, which have several nodes and antinodes. Fig. 3.4(i) (ex .αL=0.5175,δL=-5.5065or αL=0.59833,

) ...

-5.3942

,δL= and Fig. 3.4(j) (ex .αL=1.3382,δL=-7.7656or αL=1.2426, )

...

-8.3053 L=

δ show like the two-lobe pattern and four-lobe pattern, respectively, which are the intensity envelope for the higher order modes. The envelopes of other higher order modes also exhibit a series of nodes and antinodes.

38

Fig. 3.1 Band structure for square lattice photonic crystal with TE polarization

Fig. 3.2 The detailed band structure in the proximity of the Γ - point

for square lattice

39

) (a

Fig 3.3 (a) Threshold gain as a function of frequency deviation from the Bragg condition for square lattice (b) Magnified plot for modes N

=-1, and (c) for N=1

)

(b (c)

−1

= N

−2

= N

−3

= N

=1 N

=2 N

−1

=

N N=1

A

B

A0

B0

E

40

41

Fig 3.4 (a-d)Mode pattern for square lattice for the fundamental modes (A, B, and E), respectively, (e-f) spatial intensity distributions

A0

and

B₀

, and (g-j) mode pattern for the higher order modes

)

(i ( j)

-8.3053 1.2426

=

= L

L δ α -7.7656 1.3382

=

= L

L δ α -5.3942

0.59833

=

= L

L δ α -5.5065

0.5175

=

= L

L δ

α or or

42

3.1.2 Threshold gain as a function of hole filling factor for square lattice

The coupling constants are a function of filling factor f and hence the threshold gain should also be strongly dependent on filling factor f . Fig. 3.5 shows

The coupling constants are a function of filling factor f and hence the threshold gain should also be strongly dependent on filling factor f . Fig. 3.5 shows

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