Chapter 3 Simulation Results of Photonic Crystal Surface Emitting Lasers
3.2.2 Threshold gain as a function of hole filling factor for triangular
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
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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.
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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
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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)
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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
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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.
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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);
%---
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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)
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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