Penelusuran literatur. Membentuk perumusan matriks transfer dan distribusi medan untuk modusl kristal fotonik satu dimensi dengan tiga defek

(1)

Lampiran 1. Tahapan penelitian secara lengkap

Berhasil/Tidak

Simulasi dengan program matlab

Analisis Output

Optimasi hasil

Penelusuran literatur

penulisan

Membentuk perumusan matriks transfer dan

distribusi medan untuk modusl kristal fotonik

satu dimensi dengan tiga defek

ya

(2)

Lampiran 2. Penurunan Persamaan Nilai-Eigen

Persamaan Maxwell pertama:

) , ( ) , ( B r t t t r EG G G G G ∂ ∂ − = × ∇ ( , ) ( ) ( , ) E r t r H r t t μ ∂ ∇ × = − ∂ JJG G _{G G} _G _G

bagi kedua ruas dengan

μ( )rG

dan operasikan curl

1 ( , ) ( , ) ( )r E r t t H r t μ ∂ ∇ × = − ∂ JJG G _{G G} _G G 1 ( , ) ( , ) 0 ( )r E r t tH r t μ ⎛ ⎞ ∂ ∇×_⎜ ∇× _⎟+ ∇× = ∂ ⎝ ⎠ JJG G G G _G G _G G

gunakan identitas vektor:

(

)

(

)

1 1 1 ( , ) ( , ) ( , ) ( )r E r t ( )r E r t ( )r E r t μ μ μ ⎛ ⎞ ⎛ ⎞ ∇ ×_⎜ ∇ × _⎟= ∇ × ∇ × + ∇_⎜ _⎟× ∇ × ⎝ ⎠ ⎝ ⎠ G G G _G G G G _G G G G _G G G G

karena

1 0 ( )r μ ⎛ ⎞ ∇ = ⎜ ⎟ ⎝ ⎠ G G

, maka:

(

)

1 ( , ) ( , ) 0 ( )r E r t tH r t

μ

∂ ∇ × ∇ × + ∇ × = ∂ JJG G G G _G G _G G

... 1

Persamaan Maxwell kedua:

) , ( ) , ( ) , ( D r t J r t t t r HG G G G G + ∂ ∂ = × ∇

Asumsi pada bahan tidak terdapat rapat muatan dinamis ,

J r t( , )G

= 0, sehingga:

( , ) ( , ) ( ) H r t E r t r t ε ∂ ∇ × = ∂ G _{G G} _G _G

differensialkan kedua ruas;

2 2 ( , ) ( , ) H r t E r t t

ε

t ∂ ∂ ∇ × = ∂ ∂ G _{G G} _G

... 2

subtitusikan persamaan (2) ke persamaan (1)

(

)

2 2 1 ( , ) ( , ) 0 ( )r E r t

ε

t E r t

μ

∂ ∇ × ∇ × + = ∂ G G _{G G} _G G

(3)

(

)

2 2 ( , ) ( ) ( , ) 0 E r t r E r t t

εμ

∂ ∇ × ∇ × + = ∂ G G _{G G} _G _G

karena bahan bersifat non magnetik maka

μ( )rG =μ₀

(

)

2 0 0 2 ( , ) ( ) ( , ) 0 E r t r E r t t

ε

ε μ

∂ ∇ × ∇ × + = ∂ G G _{G G} _G

mengingat bahwa

2 0 0

1 c

ε μ

=

(

)

2 2 2

1 ( , )

( )

( , ) 0

E r t

r

E r t

c

t

ε

∂

∇ × ∇×

+

=

∂

G

_{G G}

_G

solusi umum persamaan gelombang standar yaitu:

( . ) 0 i k r t

E

=

E e

G G−ω

, maka persamaan

menjadi:

(

)

2 2

( , )

( )

( , ) 0

E r t

r

E r t

c

ω

ε

∇× ∇×

G

G G

+

G

=

bagi kedua ruas dengan ( )

ε

r

(

)

2 2 1 ( , ) ( , ) 0 ( )r E r t c E r t

ω

ε

∇ × ∇ × + = G G _{G G} _G

(4)

Lampiran 3. Penurunan Matriks Transfer

Matriks Transfer untuk satu lapisan:

Pada z = 0

0 0 1 1

A B

+ =

A B

+

(

)

(

0z 0 0 1z 1 1

k

A

−

B

=

k

A

−

B

)

1

Jika dua persamaan diatas dibentuk matriks:

0 1 0 0 0 1 1

1

z z z z

A

k

B

k

B

⎛

⎞⎛

⎞ ⎛

⎞⎛

=

⎜

₋

⎟⎜

⎟ ⎜

₋

⎟⎜

⎝

⎠⎝

⎝

⎠⎝

⎠

⎞

⎟

⎠

1 1 0 1 0 0 0 1 1

1

z z z z

A

B

k

B

−

⎛

⎞ ⎛

⎞⎛

=

⎜

⎟ ⎜

₋

⎟ ⎜

₋

⎟⎜

⎝

⎠⎝

⎝

⎠ ⎝

⎠

⎞

⎟

⎠

0 1 1 0 1 0 1

A

P P

B

−

⎛

⎞

⎛ ⎞

=

⎜

⎟

⎜ ⎟

⎝ ⎠

⎝

⎠

(1)

Pada z = d

1 1 1 1 1 2 1 2 1 1 z 1 z 2 z 2 ik d ik d ik d ik d

A e

+

B e

−

=

A e

+

B e

− z

)

z

(

1 1 1 1

)

(

2 1 2 1 1 1 z 1 z 2 2 z 2 ik d ik d ik d ik d z z

k

A e

−

B e

−

=

k

A e

−

B e

−

Jika dua persamaan diatas dibentuk matriks:

A

0

A

1

A

2

(5)

1 1 1 1 2 1 2 1 1 1 1 1 2 1 2 1 1 2 1 2 1 1 2 2 z z z z z z z z ik d ik d ik d ik d ik d ik d ik d ik d z z z z

A

e

B

k e

− − − −

⎛

⎞

⎛ ⎞

⎛

⎞

=

⎜

₋

⎟

⎜ ⎟

⎜

₋

⎜

⎟

⎝ ⎠

⎝

⎠

⎝

⎠

⎝

⎞

⎟

⎠

1 1 1 1 2 1 2 1 1 1 1 1 2 1 2 1 1 1 2 1 1 1 2 2 2 z z z z z z z z ik d ik d ik d ik d ik d ik d ik d ik d z z z z

A

e

A

B

k e

B

− − − − −

⎛

⎞ ⎛

⎛ ⎞

⎛

⎞

= ⎜

⎟ ⎜

⎜ ⎟

₋

⎜

⎟

⎝ ⎠

⎝

⎠ ⎝

⎝

⎠

1 1 1 1 2 1 1 1 1 1 2 1 1 0

⎞

⎟

⎠

2 2 0 1 1 2 2

1

0

z z z z z z ik d ik d ik d ik d ik d ik d z z z z

A

e

A

B

k e

k

e

B

− − − −

⎛

⎞

⎛

⎞

⎛ ⎞

⎛

⎞

⎛

= ⎜

⎟

⎜

⎟

⎜ ⎟

₋

⎜

₋

⎟

⎜

⎝

⎠

⎝

⎝ ⎠

⎝

⎠

⎝

⎠

⎞

⎟

⎠

0 1 2 1 2 21 0 2

A

Q P R

B

−

⎛ ⎞

⎛

⎞

=

⎜ ⎟

⎜

⎟

⎝

⎠

⎝ ⎠

(2)

Subtitusi persamaan kedua kedalam persamaan pertama:

0 1 1 2 0 1 1 2 21 0 2

A

P PQ P R

B

− −

⎛

⎞

⎛

⎞

=

⎜

⎟

⎜

⎝

⎠

⎝

⎠

⎟

z +

Pada z = d

1

+ d

2 2 ( 1 2) 2 ( 1 2) 1 ( 1 2) 1 ( 1 2) 2 z 2 z 1 z 1 ik d d ik d d ik d d ik d d

A e

+

₊

B e

− +

₌

A e

+

₊

B e

−

(

2 ( 1 2) 2 ( 1 2)

)

(

1 ( 1 2) 1 ( 1 2)

)

2 2 z 2 z 1 1 z 1 z ik d d ik d d ik d d ik d d z z

k

A e

+

−

B e

− +

=

k

A e

+

−

B e

− +

Jika dua persamaan diatas dibentuk matriks:

2 1 2 2 1 2 1 1 2 1 1 2 2 1 2 2 1 2 1 1 2 1 1 2 ( ) ( ) ( ) ( ) 2 1 ( ) ( ) ( ) ( ) 2 1 2 2 1 1 z z z z z z z z ik d d ik d d ik d d ik d d ik d d ik d d ik d d ik d d z z z z

A

e

B

k e

+ − + + − + + + + +

⎛

⎞

⎛

⎞

⎛

⎛ ⎞

=

⎜

₋

⎟

⎜

⎟

⎜

₋

⎜ ⎟

⎝

⎠

⎝ ⎠

⎝

⎠

⎝

⎞

⎟

⎠

2 1 2 2 1 2 1 1 2 1 1 2 2 1 2 2 1 2 1 1 2 1 1 2 1 ( ) ( ) ( ) ( ) 2 1 ( ) ( ) ( ) ( ) 2 2 2 1 1 1 z z z z z z z z ik d d ik d d ik d d ik d d ik d d ik d d ik d d ik d d z z z z

A

e

A

B

k e

B

− + − + + − + + + + +

⎛

⎞ ⎛

⎛

⎞

⎛

= ⎜

⎟ ⎜

⎜

⎟

₋

⎜

⎝

⎠

⎝

⎠ ⎝

⎝

⎞

⎟

⎠

1 2 2 2 1 2 1 2 2 2 2 2 1 1 1 1 2 1 1 1 2 1 2 2 2 2 1 1 1

0

1

0

z z z z z z z z z z ik d ik d ik d ik d ik d ik d z z ik d ik d ik d ik d z z

A

e

B

k e

e

A

e

k

e

B

− − − − − −

⎛

⎞⎛

⎞

⎛

⎞

= ⎜

⎜

⎟⎜

⎟

⎜

_{⎟ ⎜}

₋

_⎟

⎝

⎠

⎝

⎠

⎝

⎠

⎛

⎞

⎛

⎞⎛

⎞

⎛

⎜

⎟

⎜

⎟⎜

⎟

⎜

_⎝

₋

_⎠

_⎝

_⎠⎝

_⎠

⎟

_⎝

⎝

⎠

⎞

⎟

⎠

1 2 2 2 2 2 1 2 2 2 2 2 1 1 1 1 2 1 1 1 2 1 1 2 2 2 2 1 1 1

0

1

0

z z z z z z z z z z ik d ik d ik d ik d ik d ik d z z ik d ik d ik d ik d z z

A

e

B

e

k e

A

e

k

e

B

− − ₋ − − − −

⎛

⎞

⎛

⎞

⎛

⎞

=

_⎜

_⎟

_⎜

_⎟

⎜

⎟

₋

⎝

⎠ ⎝

⎠

⎛

⎞⎛

⎛

⎞

⎛

⎜

⎟⎜

⎜

₋

⎟

⎜

⎝

⎠

⎝

⎠⎝

⎝

⎞

⎟

⎠

1 2 1 1 21 2 1 11 12 2 1

A

R

Q PR R

B

− −

⎛

⎞

⎛

=

⎜

⎟

⎜

⎝

⎠

⎝

⎞

⎟

⎠

(3)

Subtitusi persamaan ketiga kedalam persamaan kedua:

0 1 1 1 1 1 0 1 1 2 21 21 2 1 11 12 0 1

A

P PQ P R R Q PR R

B

− − − −

⎛ ⎞

=

⎜ ⎟

⎝ ⎠

(6)

0 1 1 1 1 0 1 1 2 2 1 11 12 0 1

A

P PQ P Q PR R

B

− − −

⎛

⎞

⎛ ⎞

=

⎜

⎟

⎜ ⎟

⎝ ⎠

⎝

⎠

Lampiran 3. Program Matlab untuk Kurva Transmitansi

clear; %parameter input% c=3*10^8; lambda0=550*10^-6; w0=2*pi*c/lambda0; M=6; N=8; L=2; R=1; K=1000; dwr=0.0008; wr0=0.5; omega0=w0*wr0; counter=0; %indeks bias% n0=1; n1=2.1; n2=1.38; nd1=2.1; nd2=2.1; nd3=2.1; %lebar defek% d1=lambda0/4/n1; d2=lambda0/4/n2; D1=2*lambda0/4; D2=2*lambda0/4; D3=2*lambda0/4; %sudut% p0=0; p1=asin((n0/n1)*sin(p0)); p2=asin((n1/n2)*sin(p1)); pd1=asin((n2/nd1)*sin(p2));

(7)

pd2=asin((n2/nd2)*sin(p2)); pd3=asin((n2/nd3)*sin(p2)); for k=1:K counter=counter+1 wr(k)=dwr*k+wr0; omega(k)=w0*wr(k); lambda(k)=2*pi*c/omega(k); %vektor propagasi% k0(k)=n0*omega(k)*cos(p0)/c; k1(k)=n1*omega(k)*cos(p1)/c; k2(k)=n2*omega(k)*cos(p2)/c; kd1(k)=nd1*omega(k)*cos(pd1)/c; kd2(k)=nd2*omega(k)*cos(pd2)/c; kd3(k)=nd3*omega(k)*cos(pd3)/c; %komponen matriks% P0_11(k)=1; P0_12(k)=1; P0_21(k)=k0(k); P0_22(k)=-k0(k); P1_11(k)=1; P1_12(k)=1; P1_21(k)=k1(k); P1_22(k)=-k1(k); P2_11(k)=1; P2_12(k)=1; P2_21(k)=k2(k); P2_22(k)=-k2(k); Pd1_11(k)=1; Pd1_12(k)=1; Pd1_21(k)=kd1(k); Pd1_22(k)=-kd1(k); Pd2_11(k)=1; Pd2_12(k)=1; Pd2_21(k)=kd2(k); Pd2_22(k)=-kd2(k); Pd3_11(k)=1; Pd3_12(k)=1; Pd3_21(k)=kd3(k); Pd3_22(k)=-kd3(k); M1_11(k)=exp(i*k1(k)*d1); M1_12(k)=exp(-i*k1(k)*d1); M1_21(k)=k1(k)*exp(i*k1(k)*d1); M1_22(k)=-k1(k)*exp(-i*k1(k)*d1); M2_11(k)=exp(i*k2(k)*d2); M2_12(k)=exp(-i*k2(k)*d2); M2_21(k)=k2(k)*exp(i*k2(k)*d2);

(8)

M2_22(k)=-k2(k)*exp(-i*k2(k)*d2); Md1_11(k)=exp(i*kd1(k)*D1); Md1_12(k)=exp(-i*kd1(k)*D1); Md1_21(k)=kd1(k)*exp(i*kd1(k)*D1); Md1_22(k)=-kd1(k)*exp(-i*kd1(k)*D1); Md2_11(k)=exp(i*kd2(k)*D2); Md2_12(k)=exp(-i*kd2(k)*D2); Md2_21(k)=kd2(k)*exp(i*kd2(k)*D2); Md2_22(k)=-kd2(k)*exp(-i*kd2(k)*D2); Md3_11(k)=exp(i*kd3(k)*D3); Md3_12(k)=exp(-i*kd3(k)*D3); Md3_21(k)=kd3(k)*exp(i*kd3(k)*D3); Md3_22(k)=-kd3(k)*exp(-i*kd3(k)*D3); P0(:,:,k)=[P0_11(k) P0_12(k);P0_21(k) P0_22(k)]; P1(:,:,k)=[P1_11(k) P1_12(k);P1_21(k) P1_22(k)]; P2(:,:,k)=[P2_11(k) P2_12(k);P2_21(k) P2_22(k)]; Pd1(:,:,k)=[Pd1_11(k) Pd1_12(k);Pd1_21(k) Pd1_22(k)]; Pd2(:,:,k)=[Pd2_11(k) Pd2_12(k);Pd2_21(k) Pd2_22(k)]; Pd3(:,:,k)=[Pd3_11(k) Pd3_12(k);Pd3_21(k) Pd3_22(k)]; M1(:,:,k)=[M1_11(k) M1_12(k);M1_21(k) M1_22(k)]; M2(:,:,k)=[M2_11(k) M2_12(k);M2_21(k) M2_22(k)]; Md1(:,:,k)=[Md1_11(k) Md1_12(k);Md1_21(k) Md1_22(k)]; Md2(:,:,k)=[Md2_11(k) Md2_12(k);Md2_21(k) Md2_22(k)]; Md3(:,:,k)=[Md3_11(k) Md3_12(k);Md3_21(k) Md3_22(k)]; % matriks n1/n2% TBragg(:,:,k)=P1(:,:,k)*inv(M1(:,:,k))*P2(:,:,k)*inv(M2(:,:,k)); % matriks n2'/n1% TDefect1(:,:,k)=Pd1(:,:,k)*inv(Md1(:,:,k))*P2(:,:,k)*inv(M2(:,:,k)); % matriks n2"/n1% TDefect2(:,:,k)=Pd2(:,:,k)*inv(Md2(:,:,k))*P2(:,:,k)*inv(M2(:,:,k)); % matriks n2"/n1% TDefect3(:,:,k)=Pd3(:,:,k)*inv(Md3(:,:,k))*P2(:,:,k)*inv(M2(:,:,k)); % Transmitansi Init=[1; 0]; h(:,:,k)=inv(P0(:,:,k))*TBragg(:,:,k)^M*TDefect1(:,:,k)*TBragg(:,:,k )^N*TDefect2(:,:,k)*TBragg(:,:,k)^L*TDefect3(:,:,k)*TBragg(:,:,k)^R* P0(:,:,k)*Init; s(k)=abs(1/h(1,1,k)); T(k)=s(k).^2

(9)

end

figure(1);

plot(lambda,T,'-k','LineWidth',2); grid

hold on;

Lampiran 4. Program Matlab untuk Distribusi Medan

clear; %parameter input% wr=0.000003; c=3e14; lambda0=0.55; w0=2*pi*c/lambda0; M=6; N=8; L=2; R=1; K=250; wr0=0.9799; omega0=w0*wr0; counter=0; omega=omega0; %indeks bias% n1=2.1; n2=1.38; nd1=2.1; nd2=2.1; nd3=2.1; n0=1; %lebar lapisan% d1=lambda0/4/n1;d2=lambda0/4/n2; D1=2*lambda0/4; D2=2*lambda0/4; D3=2*lambda0/4; %vektor gelombang% k0=n0*omega/c; k1=n1*omega/c; k2=n2*omega/c;

(10)

kd1=nd1*omega/c; kd2=nd2*omega/c; kd3=nd3*omega/c; %komponen matriks% P0_11=1; P0_12=1; P0_21=k0; P0_22=-k0; P1_11=1; P1_12=1; P1_21=k1; P1_22=-k1; P2_11=1; P2_12=1; P2_21=k2; P2_22=-k2; Pd1_11=1; Pd1_12=1; Pd1_21=kd1; Pd1_22=-kd1; Pd2_11=1; Pd2_12=1; Pd2_21=kd2; Pd2_22=-kd2; Pd3_11=1; Pd3_12=1; Pd3_21=kd3; Pd3_22=-kd3; M0_11=exp(i*k0*d1); M0_12=0; M0_21=0; M0_22=exp(-i*k0*d1); M1_11=exp(i*k1*d1); M1_12=0; M1_21=0; M1_22=exp(-i*k1*d1); M2_11=exp(i*k2*d2); M2_12=0; M2_21=0; M2_22=exp(-i*k2*d2); Md1_11=exp(i*kd1*D1); Md1_12=0; Md1_21=0; Md1_22=exp(-i*kd1*D1);

(11)

Md2_11=exp(i*kd2*D2); Md2_12=0; Md2_21=0; Md2_22=exp(-i*kd2*D2); Md3_11=exp(i*kd3*D3); Md3_12=0; Md3_21=0; Md3_22=exp(-i*kd3*D3); P0(:,:)=[P0_11 P0_12;P0_21 P0_22]; P1(:,:)=[P1_11 P1_12;P1_21 P1_22]; P2(:,:)=[P2_11 P2_12;P2_21 P2_22]; Pd1(:,:)=[Pd1_11 Pd1_12;Pd1_21 Pd1_22]; Pd2(:,:)=[Pd2_11 Pd2_12;Pd2_21 Pd2_22]; Pd3(:,:)=[Pd3_11 Pd3_12;Pd3_21 Pd3_22]; M0(:,:)=[M0_11 M0_12;M0_21 M0_22]; M1(:,:)=[M1_11 M1_12;M1_21 M1_22]; M2(:,:)=[M2_11 M2_12;M2_21 M2_22]; Md1(:,:)=[Md1_11 Md1_12;Md1_21 Md1_22]; Md2(:,:)=[Md2_11 Md2_12;Md2_21 Md2_22]; Md3(:,:)=[Md3_11 Md3_12;Md3_21 Md3_22]; % lapisan n1/n2% TBragg(:,:)=inv(P0(:,:))*P2(:,:)*M2(:,:)*inv(P2(:,:))*P1(:,:)*M1(:,: )*inv(P1(:,:))*P0(:,:); % lapisan n2'/n1% TDefect1(:,:)=inv(P0(:,:))*Pd1(:,:)*Md1(:,:)*inv(Pd1(:,:))*P1(:,:)*M 1(:,:)*inv(P1(:,:))*P0(:,:); % lapisan n2"/n1% TDefect2(:,:)=inv(P0(:,:))*Pd2(:,:)*Md2(:,:)*inv(Pd2(:,:))*P1(:,:)*M 1(:,:)*inv(P1(:,:))*P0(:,:); % lapisan n3"/n1% TDefect3(:,:)=inv(P0(:,:))*Pd3(:,:)*Md3(:,:)*inv(Pd3(:,:))*P1(:,:)*M 1(:,:)*inv(P1(:,:))*P0(:,:); % Transmitansi% Init=[1; 0]; dz1=d1/K; dz2=d2/K; Dz1=D1/K; Dz2=D2/K; Dz3=D3/K; % R Segments for k=1:K+1 counter=counter+1 z(k)=(k-1)*dz1; L110(:,:)=inv(P1(:,:))*P0(:,:)*Init; Field(k)=(L110(1,1)*exp(i*k1*z(k))+L110(2,1)*exp(-i*k1*z(k))); end

(12)

for k=K:2*K counter=counter+1 z(k)=d1+(k-K)*dz2; L121(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*L110(:,:); Field(k)=(L121(1,1)*exp(i*k2*(z(k)-d1))+L121(2,1)*exp(-i*k2*(z(k)-d1))); end % D3 Layer for k=2*K:3*K counter=counter+1 z(k)=(k-2*K)*dz1+(d1+d2); D31(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*L121(:,:); Field(k)=(D31(1,1)*exp(i*k1*(z(k)-(d1+d2)))+D31(2,1)*exp(-i*k1*(z(k)-(d1+d2)))); end for k=3*K:4*K counter=counter+1 z(k)=(k-3*K)*Dz3+(2*d1+d2); D32(:,:)=inv(Pd3(:,:))*P1(:,:)*M1(:,:)*D31(:,:); Field(k)=(D32(1,1)*exp(i*kd3*(z(k)-(2*d1+d2)))+D32(2,1)*exp(-i*kd3*(z(k)-(2*d1+d2)))); end % L Segments for k=4*K:5*K counter=counter+1 z(k)=(k-4*K)*dz1+(2*d1+d2+D3); L11d(:,:)=inv(P1(:,:))*Pd3(:,:)*Md3(:,:)*D32(:,:); Field(k)=(L11d(1,1)*exp(i*k1*(z(k)-(2*d1+d2+D3)))+L11d(2,1)*exp(-i*k1*(z(k)-(2*d1+d2+D3)))); end for k=5*K:6*K counter=counter+1 z(k)=(k-5*K)*dz2+(3*d1+d2+D3); L121(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*L11d(:,:); Field(k)=(L121(1,1)*exp(i*k2*(z(k)-(3*d1+d2+D3)))+L121(2,1)*exp(-i*k2*(z(k)-(3*d1+d2+D3)))); end for k=6*K:7*K counter=counter+1 z(k)=(k-6*K)*dz1+(3*d1+2*d2+D3); L212(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*L121(:,:); Field(k)=(L212(1,1)*exp(i*k1*(z(k)-(3*d1+2*d2+D3)))+L212(2,1)*exp(-i*k1*(z(k)-(3*d1+2*d2+D3)))); end for k=7*K:8*K counter=counter+1 z(k)=(k-7*K)*dz2+(4*d1+2*d2+D3); L221(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*L212(:,:); Field(k)=(L221(1,1)*exp(i*k2*(z(k)-(4*d1+2*d2+D3)))+L221(2,1)*exp(-i*k2*(z(k)-(4*d1+2*d2+D3))));

(13)

end % D2 Layer for k=8*K:9*K counter=counter+1 z(k)=(k-8*K)*dz1+(4*d1+3*d2+D3); D21(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*L221(:,:); Field(k)=(D21(1,1)*exp(i*k1*(z(k)-(4*d1+3*d2+D3)))+D21(2,1)*exp(-i*k1*(z(k)-(4*d1+3*d2+D3)))); end for k=9*K:10*K counter=counter+1 z(k)=(k-9*K)*Dz2+(5*d1+3*d2+D3); D22(:,:)=inv(Pd2(:,:))*P1(:,:)*M1(:,:)*D21(:,:); Field(k)=(D22(1,1)*exp(i*kd2*(z(k)-(5*d1+3*d2+D3)))+D22(2,1)*exp(-i*kd2*(z(k)-(5*d1+3*d2+D3)))); end % N Segments for k=10*K:11*K counter=counter+1 z(k)=(k-10*K)*dz1+(5*d1+3*d2+D3+D2); N11d(:,:)=inv(P1(:,:))*Pd2(:,:)*Md2(:,:)*D22(:,:); Field(k)=(N11d(1,1)*exp(i*k1*(z(k)-(5*d1+3*d2+D3+D2)))+N11d(2,1)*exp(-i*k1*(z(k)-(5*d1++3*d2+D3+D2)))); end for k=11*K:12*K counter=counter+1 z(k)=(k-11*K)*dz2+(6*d1+3*d2+D3+D2); N121(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*N11d(:,:); Field(k)=(N121(1,1)*exp(i*k2*(z(k)-(6*d1+3*d2+D3+D2)))+N121(2,1)*exp(-i*k2*(z(k)-(6*d1++3*d2+D3+D2)))); end for k=12*K:13*K counter=counter+1 z(k)=(k-12*K)*dz1+(6*d1+4*d2+D3+D2); N212(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*N121(:,:); Field(k)=(N212(1,1)*exp(i*k1*(z(k)-(6*d1+4*d2+D3+D2)))+N212(2,1)*exp(-i*k1*(z(k)-(6*d1+4*d2+D3+D2)))); end for k=13*K:14*K counter=counter+1 z(k)=(k-13*K)*dz2+(7*d1+4*d2+D3+D2); N221(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*N212(:,:); Field(k)=(N221(1,1)*exp(i*k2*(z(k)-(7*d1+4*d2+D3+D2)))+N221(2,1)*exp(-i*k2*(z(k)-(7*d1+4*d2+D3+D2)))); end for k=14*K:15*K counter=counter+1 z(k)=(k-14*K)*dz1+(7*d1+5*d2+D3+D2); N312(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*N221(:,:);

(14)

Field(k)=(N312(1,1)*exp(i*k1*(z(k)-(7*d1+5*d2+D3+D2)))+N312(2,1)*exp(-i*k1*(z(k)-(7*d1+5*d2+D3+D2)))); end for k=15*K:16*K counter=counter+1 z(k)=(k-15*K)*dz2+(8*d1+5*d2+D3+D2); N321(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*N312(:,:); Field(k)=(N321(1,1)*exp(i*k2*(z(k)-(8*d1+5*d2+D3+D2)))+N321(2,1)*exp(-i*k2*(z(k)-(8*d1+5*d2+D3+D2)))); end for k=16*K:17*K counter=counter+1 z(k)=(k-16*K)*dz1+(8*d1+6*d2+D3+D2); N412(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*N321(:,:); Field(k)=(N412(1,1)*exp(i*k1*(z(k)-(8*d1+6*d2+D3+D2)))+N412(2,1)*exp(-i*k1*(z(k)-(8*d1+6*d2+D3+D2)))); end for k=17*K:18*K counter=counter+1 z(k)=(k-17*K)*dz2+(9*d1+6*d2+D3+D2); N421(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*N412(:,:); Field(k)=(N421(1,1)*exp(i*k2*(z(k)-(9*d1+6*d2+D3+D2)))+N421(2,1)*exp(-i*k2*(z(k)-(9*d1+6*d2+D3+D2)))); end for k=18*K:19*K counter=counter+1 z(k)=(k-18*K)*dz1+(9*d1+7*d2+D3+D2); N512(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*N421(:,:); Field(k)=(N512(1,1)*exp(i*k1*(z(k)-(9*d1+7*d2+D3+D2)))+N512(2,1)*exp(-i*k1*(z(k)-(9*d1+7*d2+D3+D2)))); end for k=19*K:20*K counter=counter+1 z(k)=(k-19*K)*dz2+(10*d1+7*d2+D3+D2); N521(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*N512(:,:); Field(k)=(N521(1,1)*exp(i*k2*(z(k)- (10*d1+7*d2+D3+D2)))+N521(2,1)*exp(-i*k2*(z(k)-(10*d1+7*d2+D3+D2)))); end for k=20*K:21*K counter=counter+1 z(k)=(k-20*K)*dz1+(10*d1+8*d2+D3+D2); N612(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*N521(:,:); Field(k)=(N612(1,1)*exp(i*k1*(z(k)- (10*d1+8*d2+D3+D2)))+N612(2,1)*exp(-i*k1*(z(k)-(10*d1+8*d2+D3+D2)))); end for k=21*K:22*K counter=counter+1 z(k)=(k-21*K)*dz2+(11*d1+8*d2+D3+D2);

(15)

N621(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*N612(:,:); Field(k)=(N621(1,1)*exp(i*k2*(z(k)- (11*d1+8*d2+D3+D2)))+N621(2,1)*exp(-i*k2*(z(k)-(11*d1+8*d2+D3+D2)))); end for k=22*K:23*K counter=counter+1 z(k)=(k-22*K)*dz1+(11*d1+9*d2+D3+D2); N712(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*N621(:,:); Field(k)=(N712(1,1)*exp(i*k1*(z(k)- (11*d1+9*d2+D3+D2)))+N712(2,1)*exp(-i*k1*(z(k)-(11*d1+9*d2+D3+D2)))); end for k=23*K:24*K counter=counter+1 z(k)=(k-23*K)*dz2+(12*d1+9*d2+D3+D2); N721(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*N712(:,:); Field(k)=(N721(1,1)*exp(i*k2*(z(k)- (12*d1+9*d2+D3+D2)))+N721(2,1)*exp(-i*k2*(z(k)-(12*d1+9*d2+D3+D2)))); end for k=24*K:25*K counter=counter+1 z(k)=(k-24*K)*dz1+(12*d1+10*d2+D3+D2); N812(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*N721(:,:); Field(k)=(N812(1,1)*exp(i*k1*(z(k)- (12*d1+10*d2+D3+D2)))+N812(2,1)*exp(-i*k1*(z(k)-(12*d1+10*d2+D3+D2)))); end for k=25*K:26*K counter=counter+1 z(k)=(k-25*K)*dz2+(13*d1+10*d2+D3+D2); N821(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*N812(:,:); Field(k)=(N821(1,1)*exp(i*k2*(z(k)- (13*d1+10*d2+D3+D2)))+N821(2,1)*exp(-i*k2*(z(k)-(13*d1+10*d2+D3+D2)))); end % D1 Layer for k=26*K:27*K counter=counter+1 z(k)=(k-26*K)*dz1+(13*d1+11*d2+D3+D2); D11(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*N821(:,:); Field(k)=(D11(1,1)*exp(i*k1*(z(k)- (13*d1+11*d2+D3+D2)))+D11(2,1)*exp(-i*k1*(z(k)-(13*d1+11*d2+D3+D2)))); end for k=27*K:28*K counter=counter+1 z(k)=(k-27*K)*Dz2+(14*d1+11*d2+D3+D2); D12(:,:)=inv(Pd1(:,:))*P1(:,:)*M1(:,:)*D11(:,:);

(16)

Field(k)=(D12(1,1)*exp(i*kd1*(z(k)- (14*d1+11*d2+D3+D2)))+D12(2,1)*exp(-i*kd1*(z(k)-(14*d1+11*d2+D3+D2)))); end %M Segments for k=28*K:29*K counter=counter+1 z(k)=(k-28*K)*dz1+(14*d1+11*d2+D3+D2+D1); M11d(:,:)=inv(P1(:,:))*Pd1(:,:)*Md1(:,:)*D12(:,:); Field(k)=(M11d(1,1)*exp(i*k1*(z(k)- (14*d1+11*d2+D3+D2+D1)))+M11d(2,1)*exp(-i*k1*(z(k)-(14*d1++11*d2+D3+D2+D1)))); end for k=29*K:30*K counter=counter+1 z(k)=(k-29*K)*dz2+(15*d1+11*d2+D3+D2+D1); M121(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*M11d(:,:); Field(k)=(M121(1,1)*exp(i*k2*(z(k)- (15*d1+11*d2+D3+D2+D1)))+M121(2,1)*exp(-i*k2*(z(k)-(15*d1+11*d2+D3+D2+D1)))); end for k=30*K:31*K counter=counter+1 z(k)=(k-30*K)*dz1+(15*d1+12*d2+D3+D2+D1); M212(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*M121(:,:); Field(k)=(M212(1,1)*exp(i*k1*(z(k)- (15*d1+12*d2+D3+D2+D1)))+M212(2,1)*exp(-i*k1*(z(k)-(15*d1+12*d2+D3+D2+D1)))); end for k=31*K:32*K counter=counter+1 z(k)=(k-31*K)*dz2+(16*d1+12*d2+D3+D2+D1); M221(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*M212(:,:); Field(k)=(M221(1,1)*exp(i*k2*(z(k)- (16*d1+12*d2+D3+D2+D1)))+M221(2,1)*exp(-i*k2*(z(k)-(16*d1+12*d2+D3+D2+D1)))); end for k=32*K:33*K counter=counter+1 z(k)=(k-32*K)*dz1+(16*d1+13*d2+D3+D2+D1); M312(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*M221(:,:); Field(k)=(M312(1,1)*exp(i*k1*(z(k)- (16*d1+13*d2+D3+D2+D1)))+M312(2,1)*exp(-i*k1*(z(k)-(16*d1+13*d2+D3+D2+D1)))); end for k=33*K:34*K counter=counter+1 z(k)=(k-33*K)*dz2+(17*d1+13*d2+D3+D2+D1); M321(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*M312(:,:); Field(k)=(M321(1,1)*exp(i*k2*(z(k)- (17*d1+13*d2+D3+D2+D1)))+M321(2,1)*exp(-i*k2*(z(k)-(17*d1+13*d2+D3+D2+D1))));

(17)

end for k=34*K:35*K counter=counter+1 z(k)=(k-34*K)*dz1+(17*d1+14*d2+D3+D2+D1); M412(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*M321(:,:); Field(k)=(M412(1,1)*exp(i*k1*(z(k)- (17*d1+14*d2+D3+D2+D1)))+M412(2,1)*exp(-i*k1*(z(k)-(17*d1+14*d2+D3+D2+D1)))); end for k=35*K:36*K counter=counter+1 z(k)=(k-35*K)*dz2+(18*d1+14*d2+D3+D2+D1); M421(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*M412(:,:); Field(k)=(M421(1,1)*exp(i*k2*(z(k)- (18*d1+14*d2+D3+D2+D1)))+M421(2,1)*exp(-i*k2*(z(k)-(18*d1+14*d2+D3+D2+D1)))); end for k=36*K:37*K counter=counter+1 z(k)=(k-36*K)*dz1+(18*d1+15*d2+D3+D2+D1); M512(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*M421(:,:); Field(k)=(M512(1,1)*exp(i*k1*(z(k)- (18*d1+15*d2+D3+D2+D1)))+M512(2,1)*exp(-i*k1*(z(k)-(18*d1+15*d2+D3+D2+D1)))); end for k=37*K:38*K counter=counter+1 z(k)=(k-37*K)*dz2+(19*d1+15*d2+D3+D2+D1); M521(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*M512(:,:); Field(k)=(M521(1,1)*exp(i*k2*(z(k)- (19*d1+15*d2+D3+D2+D1)))+M521(2,1)*exp(-i*k2*(z(k)-(19*d1+15*d2+D3+D2+D1)))); end for k=38*K:39*K counter=counter+1 z(k)=(k-38*K)*dz1+(19*d1+16*d2+D3+D2+D1); M612(:,:)=inv(P1(:,:))*P2(:,:)*M2(:,:)*M521(:,:); Field(k)=(M612(1,1)*exp(i*k1*(z(k)- (19*d1+16*d2+D3+D2+D1)))+M612(2,1)*exp(-i*k1*(z(k)-(19*d1+16*d2+D3+D2+D1)))); end for k=39*K:40*K counter=counter+1 z(k)=(k-39*K)*dz2+(20*d1+16*d2+D3+D2+D1); M621(:,:)=inv(P2(:,:))*P1(:,:)*M1(:,:)*M612(:,:); Field(k)=(M621(1,1)*exp(i*k2*(z(k)- (20*d1+16*d2+D3+D2+D1)))+M621(2,1)*exp(-i*k2*(z(k)-(20*d1+16*d2+D3+D2+D1)))); end % Left Background

(18)

for k=40*K:41*K counter=counter+1 z(k)=(k-40*K)*dz1+(20*d1+17*d2+D3+D2+D1); Left(:,:)=inv(P0(:,:))*P2(:,:)*M2(:,:)*M621(:,:); Field(k)=(Left(1,1)*exp(i*k0*(z(k)- (20*d1+17*d2+D3+D2+D1)))+Left(2,1)*exp(-i*k0*(z(k)-(20*d1+17*d2+D3+D2+D1)))); end g=1/(Left(1,1)); z0=R*(d1+d2)+(d1+D3)+L*(d1+d2)+(d1+D2)+N*(d1+d2)+(d1+D1)+M*(d1+d2) figure(1); plot(z0-z,abs(g*Field),'-k','LineWidth',2); grid hold on;

Lampiran 5. Program Matlab untuk Kurva Indeks Bias

clear; %parameter input% c=3*10^8; A=564.05*10^-6; lambda0=550*10^-6; w0=2*pi*c/lambda0; M=6; N=8; L=2; R=1; omega=2*pi*c/A; counter=0; nd2i=1.33; nd2f=1.5; K=100; ddn2=(nd2f-nd2i)/K; %indeks bias% n0=1; n1=2.1; n2=1.38; nd1=2.1; nd3=2.1; %lebar defek% d1=lambda0/4/n1; d2=lambda0/4/n2; D1=2*lambda0/4; D2=2*lambda0/4; D3=2*lambda0/4;

(19)

for k=1:K nd2=nd2i+k*ddn2; nD2(k)=nd2; counter=counter+1 %vektor propagasi% k0=n0*omega/c; k1=n1*omega/c; k2=n2*omega/c; kd1=nd1*omega/c; kd2(k)=nD2(k)*omega/c; kd3=nd3*omega/c; %komponen matriks% P0_11=1; P0_12=1; P0_21=k0; P0_22=-k0; P1_11=1; P1_12=1; P1_21=k1; P1_22=-k1; P2_11=1; P2_12=1; P2_21=k2; P2_22=-k2; Pd1_11=1; Pd1_12=1; Pd1_21=kd1; Pd1_22=-kd1; Pd2_11(k)=1; Pd2_12(k)=1; Pd2_21(k)=kd2(k); Pd2_22(k)=-kd2(k); Pd3_11=1; Pd3_12=1; Pd3_21=kd3; Pd3_22=-kd3; M1_11=exp(i*k1*d1); M1_12=exp(-i*k1*d1); M1_21=k1*exp(i*k1*d1); M1_22=-k1*exp(-i*k1*d1); M2_11=exp(i*k2*d2); M2_12=exp(-i*k2*d2); M2_21=k2*exp(i*k2*d2); M2_22=-k2*exp(-i*k2*d2);

(20)

Md1_11=exp(i*kd1*D1); Md1_12=exp(-i*kd1*D1); Md1_21=kd1*exp(i*kd1*D1); Md1_22=-kd1*exp(-i*kd1*D1); Md2_11(k)=exp(i*kd2(k)*D2); Md2_12(k)=exp(-i*kd2(k)*D2); Md2_21(k)=kd2(k)*exp(i*kd2(k)*D2); Md2_22(k)=-kd2(k)*exp(-i*kd2(k)*D2); Md3_11=exp(i*kd3*D3); Md3_12=exp(-i*kd3*D3); Md3_21=kd3*exp(i*kd3*D3); Md3_22=-kd3*exp(-i*kd3*D3); P0=[P0_11 P0_12;P0_21 P0_22]; P1=[P1_11 P1_12;P1_21 P1_22]; P2=[P2_11 P2_12;P2_21 P2_22]; Pd1=[Pd1_11 Pd1_12;Pd1_21 Pd1_22]; Pd2(:,:,k)=[Pd2_11(k) Pd2_12(k);Pd2_21(k) Pd2_22(k)]; Pd3=[Pd3_11 Pd3_12;Pd3_21 Pd3_22]; M1=[M1_11 M1_12;M1_21 M1_22]; M2=[M2_11 M2_12;M2_21 M2_22]; Md1=[Md1_11 Md1_12;Md1_21 Md1_22]; Md2(:,:,k)=[Md2_11(k) Md2_12(k);Md2_21(k) Md2_22(k)]; Md3=[Md3_11 Md3_12;Md3_21 Md3_22]; % matriks n1/n2% TBragg=P1*inv(M1)*P2*inv(M2); % matriks n2'/n1% TDefect1=Pd1*inv(Md1)*P2*inv(M2); % matriks n2"/n1% TDefect2(:,:,k)=Pd2(:,:,k)*inv(Md2(:,:,k))*P2*inv(M2); % matriks n2"/n1% TDefect3=Pd3*inv(Md3)*P2*inv(M2); % Transmitansi Init=[1; 0]; h(:,:,k)=inv(P0)*TBragg^M*TDefect1*TBragg^N*TDefect2(:,:,k)*TBragg^L *TDefect3*TBragg^R*P0*Init; s(k)=abs(1/h(1,1,k)); T(k)=s(k).^2 end

(21)

figure(1);

plot(nD2,T,'-k','LineWidth',2); grid