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 as a function of the hole filling factor. We note that κ₃ becomes zero at f =0.3, which implies that the backward diffraction vanishes. Fig.

3.7 shows the threshold gain of the fundamental modes A, B, and E as a function of hole filling factor. The threshold gain for modes A and B drastically increases in the region of f =0.3. This is because the degree of backward diffraction becomes very weak and is insufficient for optical oscillation. This result indicates that the coupling constant κ₃ is the dominant factor determining the degree of optical confinement in the current system, a square lattice with TE polarization.

Fig 3.5 Coupling constants as a function of hole filling factor for square lattice

κ0

κ3

κ2

κ1

43

To elucidate the origin of the threshold difference among the fundamental modes, we calculated the threshold gain for zero surface emission (κ₀ =0) as shown in Fig.

3.8 and Fig. 3.9. It is quite obvious that the average of the threshold gain to consider the surface emission κ₀ is higher than the average of the threshold gain not to

consider the surface emission κ₀ for the fundamental modes A, B, and E, as compared with Fig. 3.7 and Fig. 3.9. In this case, the threshold gain of mode E greatly decreased and the threshold gain between mode E and mode A(or B) is similar , as shown in Fig. 3.9. Thus, the major of loss for mode E is surface emission. The difference in threshold gain between modes A and B in Fig. 3.7 indicates that the emission loss from the edges of the cavity differs. The frequency deviation from the Bragg condition don’t have much influence of the surface emission κ₀, as compared with Fig. 3.6 and Fig. 3.8.

In Fig. 3.7, the lowest threshold gains of square PCSELs for A, B and E mode are observed at filling factor =0.6,0.6 and 0.05, respectively. The 2D coupling induces the curves splitting for modes A and B. Therefore, we could observe a highly mode selection with stronger 2D coupling. Besides, in Fig. 3.6, the curves splitting in frequency between mode A (or B) and mode E is induced by backward vector of coupling constant κ₃. This splitting corresponds to the stopband in 1D DFB lasers, which is induced by coupling between the counter propagating waves.

44

Fig 3.6 The frequency deviation as a function of hole filling factor of the fundamental modes A, B and E for square lattice for considering surface emission κ

₀

≠ 0

Fig 3.7 The threshold gain as a function of hole filling factor of the fundamental modes A, B and E for square lattice for considering surface emission κ

₀

≠ 0

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10

0 0.2 0.4 0.6 0.8 1

fr eq ue nc y

filling factor

A B E

0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.2 0.4 0.6 0.8 1

th re sh ol d g ai n

filling factor

A B E

45

Fig 3.8 The frequency deviation as a function of hole filling factor of the fundamental modes A, B and E for square lattice for zero surface emission κ

₀

= 0

Fig 3.9 The threshold gain as a function of hole filling factor of the fundamental modes A, B and E for square lattice for zero surface emission κ

₀

= 0

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10

0 0.2 0.4 0.6 0.8 1

fr eq ue nc y

filling factor

A B E

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.2 0.4 0.6 0.8 1

th re sh ol d g ai n

filling factor

A B E

46

3.2.1 Mode spectra and mode patterns for triangular lattice

The typical band structure of photonic crystal for triangular lattice with transverse electric (TE) polarization shows as Fig. 3.10. Fig. 3.11 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.

We know that the lasing action is easier to achieve the threshold gain at the Γ point of the band structure. There are four fundamental modes including A mode, B mode, C mode and D mode, as shown in Fig 3.11. B mode and D mode are the mode of doubly degenerate, respectively. We import the following parameters in the PC model : the holes dielectric constants ε_a =9.8, the background dielectric constants

0 .

=12

εb , the gain perturbation (α_a −α_b)=0, the hole filling factor f =0.37, the lattice period a=290nm, and the PC cavity length L=50µm. 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 are shown in Fig. 3.12.

The eigenvectors of Eq. (2.36) provide the complex amplitudes such as H , ₁ H , 2 H , ₃ H , ₄ H , and ₅ H , which are functions of the positions ₆ x and y. The intensity envelope (mode pattern) of the resonant mode throughout the PC structure can be determined from these amplitudes using the sum

∗

∗

∗

∗+ _{2 2} + _{3 3} + _{4 4}

1

1H H H H H H H

H +H₅H₅^∗+H₆H₆^∗.

It could be classified that mode A (αL=2.0388,δL=−11.118) , mode B )

2.7631 ,

0.87575

(αL= δL=− , mode C (αL=0.77078,δL=12.3982) and mode D )

2.1771 ,

1.9519

(αL= δL= are fundamental modes, each mode have a single-lobed

47

intensity pattern throughout the photonic crystal, as shown in Fig. 3.13(a-f). We can identify that the lowest frequency of these four modes is mode A, as shown in Fig.

3.11. At the same time, we can find the six mode patterns for fundamental modes which show similarly single-lobed intensity patterns. This observation of the six mode patterns can be found by doubly degenerate into two categories. The frequency of the band structure, as shown in Fig. 3.11, can be helped to distinguish the different modes.

The numerical results of the threshold gain for modes A, B, C, and D are 0388

.

=2

AL

α , α_BL=0.87575, α_CL=0.77078 _and α_DL=1.9519, respectively. As a result, mode C has the lowest threshold gain and could easily achieve the lasing oscillation.

Fig. 3.13(g-j) illustrates the intensity envelope for the higher order modes, which have several nodes and antinodes. Fig. 3.13(g) illustrates that the phase of intensity envelope of the resonant mode is flipped as crossing a given axis. This produces a mode pattern that only exist two lobes. On the other hand, Fig. 3.13(h) illustrates that the phase of intensity envelope of the resonant mode is flipped as crossing two given axes. This produces a mode pattern that exist four lobes. In the Fig. 3.13(i), it shows that the intensity envelope characteristic with six peaks spaced almost evenly around the perimeter of the annulus. For this mode, the phases of the six field components alternate as a function of azimuthal angle that there are not two of adjacent components separated by 60° constructive interference [3]. Finally, Fig. 3.13(j) illustrates that the intensity envelope characteristic forms an annular pattern. The modes shown in Fig. 3.13(g-i) may be classified as out-of-phase, since pronounced destructive interference in the surface-emitted component substantially reduces the output power and also degrades the beam quality. The envelopes of other higher order modes also exhibit a series of nodes and antinodes.

48

Fig. 3.10 Band structure for triangular lattice photonic crystal with TE polarization

Fig. 3.11 The detailed band structure in the proximity of the Γ -

point for triangular lattice

49

Fig 3.12 Threshold gain as a function of frequency deviation from the Bragg condition for triangular lattice

(a) (b)

(c) (d)

B

_δ^α_L^L₌⁼₋^0.87575_2.7631

B '

_δ^α_L^L₌⁼₋^0.87575_2.7631

C

_δ^α_L^L₌⁼_12.3982^0.77078

A

_δ^α_L^L₌⁼₋^2.0388_11.118

50

(e) (f)

(g) (h)

(i) (j)

Fig 3.13 (a-f)Mode pattern for triangular lattice for the fundamental modes (A, B, C, and D), respectively and (g-j) mode pattern for the higher order modes

D

^α_δL^L₌⁼_2.1771^1.9519

D '

_δ^α_L^L₌⁼_2.1771^1.9519

12.8808 L

0.91702 L

=

= δ α

2.6265 L

1.0577 L

−

=

= δ α

13.4993 L

1.0896 L

=

= δ α

13.2719 L

2.27 L

−

=

= δ α

51

3.2.2 Threshold gain as a function of hole filling factor for triangular lattice

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

3.14 shows the coupling constants as a function of the hole filling factor. It should be noted that κ₃ becomes zero at f =0.25 , which implies that the backward diffraction vector is vanished. The threshold gain could be affected by this factor. On the other hand, the κ₀ has the maximum value at f =0.45 which implies that this condition has the maximum radiation loss. We could also obtain that κ₂ and κ₃

have the maximum value at f =0.15, κ₁ and κ₀ have the maximum value at 45

.

=0

f , and κ₂ becomes zero at f =0.35.

Fig. 3.18 shows the threshold gain of the fundamental modes A, B, C and D as a function of hole filling factor. It is clearly the threshold gain for modes C and D

Fig 3.14 Coupling constants as a function of hole filling factor for

triangular lattice

52

drastically increases in the region of f =0.25. It is because the degree of backward diffraction vector becomes very weak and is insufficient for optical oscillation. This result indicates that the coupling constant κ₃ is the mainly factor of determining the degree for optical resonant in the current system.

Fig. 3.15 and Fig. 3.16 show that the frequency deviation and threshold gain as a function of hole filling factor of the fundamental modes A, B, C and D for triangular lattice with considering surface emission κ₀, respectively. The threshold gain of mode A and B have the lowest value at f =0.12 in Fig 3.16. It is because the coupling constant κ₂ and κ₃ are the maximum value and could provide sufficient optical resonant. At the same time, the threshold gain for mode A and D has the local maximum value at f =0.45, because the coupling constant κ₀ and κ₁ are the maximum value which would increase the optical loss. In particular, the threshold gain of fundamental modes becomes larger when the hole filling factor f approach zero or one. In this case, the photonic crystal is without any function.

To elucidate the originally difference among the fundamental modes, we calculated the threshold gain and frequency deviation as a function of filling factor for zero surface emission (κ₀ =0), as shown in Fig. 3.17 and Fig. 3.18. It is obviously that all of the threshold gain with considering surface emission κ₀ is higher than that without surface emission κ₀ for the fundamental modes A, B, C and D. In Fig. 3.18,

by compared with these two conditions (with or without κ₀), the curves of the threshold gain for mode A and D at f =0.45 show smooth tendency instead of violent variation in Fig. 3.16 indicating that the threshold gain for mode A and D are mainly affected by the coupling constant κ₀.The influence of radiation loss κ₀ is

53

larger than the coupling constant κ₁ for these two modes. At the same time, the curve of threshold gain of mode A is similar with mode B for zero surface emission, as shown in Fig. 3.18. The mainly difference between the threshold gain of mode A and B for surface emission is shown in Fig. 3.16. It can be seen the curve of mode A is gradually increased with considering κ₀. Thus, the influence of surface emission factor for mode A is larger than mode B. As for the deviation frequency for mode A, B, C and D without considering κ₀, the tendency of each mode shows similar curves between Fig. 3.15 and Fig. 3.17. It indicates that the frequency deviation from the Bragg condition does not affect by the surface emission factor κ₀.

At last, we could finalize these results of threshold gain for each mode. In Fig.

3.16, the lowest threshold gains of triangular PCSELs for A, B, C and D mode are observed at filling factor =0.1, 0.1, 0.05 and 0.05, respectively. The surface emission coupling induces the curves splitting for modes A and B. Therefore, we could observe a highly mode selection with stronger 2D coupling. In Fig. 3.15, the curves splitting in frequency between mode A (or B) and mode C(or D) is induced by backward vector of coupling constant κ₃. This splitting corresponds to the stopband in 1D DFB lasers, which is induced by coupling between the counter propagating waves.

54

Fig 3.15 The frequency deviation as a function of hole filling factor of the fundamental modes A, B, C and D for triangular lattice for considering surface emission κ

₀

≠ 0

Fig 3.16 The threshold gain as a function of hole filling factor of the fundamental modes A, B, C and D for triangular lattice for considering surface emission κ

₀

≠ 0

-15 -10 -5 0 5 10 15 20

0 15 30 45 60 75

fr eque nc y

filling factor(%)

A B C D

0 0.5 1 1.5 2 2.5 3

0 15 30 45 60 75

th re sh ol d gai n

filling factor(%)

A B C D

55

Fig 3.17 The frequency deviation as a function of hole filling factor of the fundamental modes A, B, C and D for triangular lattice for zero surface emission κ

₀

= 0

Fig 3.18 The threshold gain as a function of hole filling factor of the fundamental modes A, B, C and D for triangular lattice for zero surface emission κ

₀

= 0

-15 -10 -5 0 5 10 15 20

0 15 30 45 60 75

fr eque nc y

filling factor(%)

A B C D

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 15 30 45 60 75

th es h ol d gai n

filling factor(%)

A B C D

56

References

[1] K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, IEEE J.

Sel. Areas Commun., 23, 7, 1335(2005)

[2] H. Kogelnik and C. V. Shank, J. Appl. Phys., 43, 2327 (1972)

[3] I. Vurgaftman and J. R. Meyer, IEEE J. Quantum Electron. 39, 6, 689 (2003)

57

Chapter 4 Conclusion

We have developed the coupled wave theory for square lattice and triangular lattice of photonic crystal lasers with transverse electric polarization. Numerical calculations by solving the eigenvalue problem have shown the threshold gain, the frequency deviation and the mode pattern of the 2D resonant modes. The intensity pattern of the fundamental modes was found to depend on the coupling strength, with peaks in intensity at the ends of the structure for weak coupling and maximum intensity at the center for strong coupling.

For square lattice, the lowest threshold gains of PCSELs for fundamental mode A, B and E are observed at filling factor f =0.6, 0.6 and 0.05, respectively. The surface emission κ₀ and backward coupling κ₃ are the dominant factor of determining the degree of optical confinement in the current system for the fundamental mode. The surface emission coupling constant κ₀ induces the curve splitting between modes A (or B) and E.

As for triangular lattice, the lowest threshold gains of PCSELs for fundamental mode A, B, C and D are observed at filling factor f =0.1, 0.1, 0.05 and 0.05, respectively. The surface emission κ₀ and backward coupling κ₃ are the dominant factor of determining the degree of optical confinement in the current system for the fundamental mode. The surface emission coupling constant κ₀ induces the curve splitting between modes A and B. Therefore, we could observe a highly mode selection with stronger 2D coupling.

The out of plane radiation of photonic crystal has been considered by coupling

58

coefficient κ₀, and it has massive influence to threshold gain of PCSELs. The results obtained in this thesis provide fundamental insight into the 2D DFB effect of the PC lasers. To fabricate low threshold gain PCSELs, coupled wave theory provides us a more convenient and faster method to modify our designs. A further development in designing and optimizing the 2D PC lasers by the current method is envisaged.

59

Appendix Code (in Matlab System)\

Mode patterns of photonic crystal with square lattice

clc clear

L=50*10^(-4); % length of PC cavity Epsa=9.8; % dielectric constants of the circular holes Epsb=12.0; % dielectric constants of the background a=290*10^(-7); % lattice constant b=2*pi/a; % wave number

%---coupling constant--- G=[b sqrt(2)*b 2*b];

z2=1;

for z1=1:3

f(z2)=0.18; % filling factor r(z2)=a*sqrt(f(z2)/(pi));

Eps0(z2)=sqrt(Epsa*f(z2)+Epsb*(1-f(z2)));

kG(z2)=0.2841*(-(pi)*(Epsa-Epsb)/(a*Eps0(z2)))*2*f(z2)*

BESSEL(1,G(z1)*r(z2))/(G(z1)*r(z2));

kG(isnan(kG))=1;

if z1==1

kG1(z2)=kG(z2);

k0(z2)=2*kG1(z2)*kG1(z2)*L/500;

end

k(z1)=kG(z2);

end

%---kappa variable--- n=18; % even

%---boundary & phase shift--- ro=0.0;

P=diag([exp(-i*phs)*ones(1,n/2) exp(i*phs)*ones(1,n/2)]);

M12=kron(eye(n),P*A12);

Q=diag([exp(i*phs)*ones(1,n/2) exp(-i*phs)*ones(1,n/2)]);

M21=kron(eye(n),Q*A21);

A22=(0.5*A+1/d)*eye(n)+diag((0.5*A-1/d)*ones(1,n-1),1);

A22(1,1)=A22(1,1)+(0.5*B)*blro*exp(i*blph);

Mn22=kron(eye(n),A22);

%---

60

TT=[Mn11 M12 M13 M14;M21 Mn22 M13 M14;M31 M32 Mn33 M34;M31 M32 M43 Mn44];

%---

%---Mode pattern Hz--- for z=1:S(1)

61

Sx1=0;

Ry=V(2*n*n+(y-2)*n+x);

Sy1=V(3*n*n+(y)*n+x);

elseif y==1

Rx=V(n*(y-1)+x-1);

Sx1=V(n*n+(y-1)*n+x+1);

Ry=0;

Sy1=V(3*n*n+(y)*n+x);

elseif y==n

Rx=V(n*(y-1)+x-1);

Sx1=V(n*n+(y-1)*n+x+1);

Ry=V(2*n*n+(y-2)*n+x);

Sy1=0;

else

Rx=V(n*(y-1)+x-1);

Sx1=V(n*n+(y-1)*n+x+1);

Ry=V(2*n*n+(y-2)*n+x);

Sy1=V(3*n*n+(y)*n+x);

end

Rx1=V(n*(y-1)+x);

Sx=V(n*n+(y-1)*n+x);

Ry1=V(2*n*n+(y-1)*n+x);

Sy=V(3*n*n+(y-1)*n+x);

SE(x,y)=abs(Rx+Rx1)*abs(Rx+Rx1)+abs(Sx+Sx1)*abs(Sx+

Sx1)+abs(Ry+Ry1)*abs(Ry+Ry1)+abs(Sy+Sy1)*abs(Sy+Sy 1);

end end figure(3)

[X ,Y]=meshgrid(1:n,1:n);

Plot_Result=surf(X,Y,SE);

f0=f(z2)*100;

cd('C:\Users\jky\Desktop\jky\1');

saveas(Plot_Result,strcat(num2str(f(z2)),'gain',num2str(realp art(N(z))),'w', num2str(imagpart(N(z))),'s',num2str(N(z)), '.jpg'));

cd('C:\Users\jky\Desktop\couplingconstant');

end

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