結論 - 應用反向對準型液晶波板研製可調式相位延遲元件之研究

在本論文中，我們研究可調製的相位延遲元件，目的在於此可調製的相位延遲元件，可以利用外加電壓改變元件的相位延遲，達到我們可應用的相位遲延，選定液晶做成的相位板是因為液晶材料的折射率對外加電壓會改變，所以我們使用液晶研究可調製的相位延遲元件。

在本文中，我們深入了解液晶波板的基本參數，與及液晶板做成可調製相位延遲元件的設計參數，最後達到設計理想的可調製相位板，

在這之中，我們利用廣義的瓊斯矩陣，分析整個實驗架構，利用此方法當作我們模擬的基礎，找出影響液晶波板的參數，則可以利用這些參數，達到我們設計理想液晶波板最終目標。

從理論模擬中我們設計了厚度較小的液晶波板，可以有較好的斜向光相位的調變，而在光學實驗中也證實了我們的想法是正確的，所以我們可以應用已經製作完成，厚度為 3.35um 的液晶波板來當可調式的相位延遲元件，對於視角在 11.5 度之內，此元件有很好的相位調變，而且不僅僅只能作全波板與半波板之間的調製，也可以調製其他相位，若是空液晶板的製程很精準，且液晶為 E7，若入射光為氦氖雷射，則根據我們的理論，理想的液晶板的厚度為 3um，不加電壓

時有全波板的功能，加了電壓有其他特殊的相位，所以設計此厚度的液晶波板當作我們理想的相位調製元件。

參考資料

[1] 張博宇,“PQ:PMMA 高分子全像片之製作與全像儲存特性的研究“,國立交通大 學碩士論文,民國 93 年

[2] H.J. Coufal, D. Psaltis, G. T. Sincerbox,“Holographic Data Storage“,Springer, New York (2000)

[3]P.G de Gennes ,J. prost, “The Physics of Liquid crystals“,Oxford university Press,New York(1993)

[4] A. Yariv, P. Yeh,“Optical waves in crystals“, John Wiley

[5]P.Yeh ,C.Gu,“Optics of Liquid Crystal Displays“, Wiley New York(1999), 48-52,390-399

and Sons, New York(1984) ,69-77

[6] P.Yeh,“Eetended Jones matrix method“,J. Opt. Soc. Am. 72, 507-513(1982)

[7]C.Gu ,P. Yeh,“Extended Jones matrix method II“,J. Opt. Soc. Am.A 10, 966-973

[8]P.Yeh,“ Electromagnetic propagation in birefringent layered media“,J.

Opt.Soc.Am.69,724-756(1979)

61 k2 = 9.000e-12;

k3 = 19.500e-12;

el_pal = 19.600*(1/36/pi*1e-9);

el_pen = 5.100*(1/36/pi*1e-9);

lambda =0.63;%光源的波長 K=2*pi/lambda;

ne = 1.73038;e-wave的折射率 no = 1.51891;o-wave的折射率 gamma = (el_pal-el_pen)/el_pen;

kappa = (k3-k1)/k1;

V_c1 = pi.*sqrt(k2)./(sqrt(el_pal - el_pen));截止電壓 d=18.49;液晶的厚度

m =100; 取樣的次數

theda_max=[0.01:0.3:90].*(pi/180);中心厚度的偏折角 z = zeros(m,length(theda_max));

theda3 = zeros(size(z));

V = zeros(length(theda_max),1);

mm=rand(1,m);

phi=pi/2*mm;

for ja=1:length(theda_max);

eta=sin(theda_max);

62 V(ja)=2*f/pi*V_c1;

aa=sqrt((1+kappa*(eta(ja)).^2.*(sin(phi)).^2).*(1+gamma*(eta(ja)).^2) .*...

(1+gamma*(eta(ja)).^2.*(sin(phi)).^2));

bb=sqrt(1-(eta(ja)).^2.*(sin(phi)).^2);

f=sum([aa./bb])/m*pi/2;

Dz=f./(sqrt(el_pal-el_pen)/sqrt(k1)/(el_pen)*d/2);

theda=asin(eta(ja)*sin(phi));

for i = 1:m;

x = i;

theda2 = 0 + (theda(x)-0).*mm;

a1=sqrt(k1*(cos(theda2)).^2+k3*sin(theda2).^2);

a2=1./(el_pal*(sin(theda2)).^2+el_pen*(cos(theda2)).^2);

a3=1./(el_pal*(eta(ja)).^2+el_pen*(cos(theda_max(ja))).^2);

f=(sum([a1./sqrt(a2-a3)]))/m.*(theda(x)-0);

z(i,ja)=(f./Dz);

theda3(i,ja)=theda(x);

end

zz=sort([z;z.*-1+d]./d);

zz1=zz*d;

figure(5)

tl_angle = [sort(theda3);inverse(sort(theda3),m)];

plot(zz, tl_angle.*180./pi, '-','linewidth',2);

grid on

n=((sin(tl_angle)).^2./(no).^2+(cos(tl_angle)).^2./(ne).^2).^(-1/2);%

xlabel('z/d','fontsize',16);

ylabel('Tilted angle(DEG)','fontsize',16);

for i1=1:2*m-1;

dz(i1+1,ja)=zz1(i1+1,ja)-zz1(i1,ja);

end

dz(1,ja)=zz1(1,ja); dz(2*m+1,ja)=d-zz1(2*m,ja); end for ja4=1:length(theda_max)

n(2*m+1,ja4)=1.75;

for i2=1:2*m+1;

63 for ja1=1:length(theda_max);

f3(i2,ja1)=([n(i2,ja1).*dz(i2,ja1)]);

end

h1=0.001:0.5:5; 改變高度與焦距可以決定入射的角度

%%%newport KBX034 r1=-51.234;半球面曲面半徑 r2=51.234; 半球面曲面半徑 ds=3.79;透鏡的厚度

64 Fresnel 的穿透率公式

tp1=(2.*cos(i1))./(na.*cos(i1)+cos(i2));

ts1=(2.*cos(i1))./(na.*cos(i2)+cos(i1));

tp2=(2.*na.*cos(i3))./(na.*cos(i4)+cos(i3));

ts2=(2.*na.*cos(i3))./(na.*cos(i3)+cos(i4));

入射光的偏振方向 [1./sqrt(2) 1./sqrt(2) ] for j1=1:length(i8);

for j2=1:length(phia);

for j1=1:length(i8);

for j2=1:length(phia);

Eox1(j1,j2)=cos(phia(j2)).*Eox(j1,j2)-sin(phia(j2)).*Eoy(j1,j2);

Eoy1(j1,j2)=sin(phia(j2)).*Eox(j1,j2)+cos(phia(j2)).*Eoy(j1,j2);

end end

k1 = 12.000e-12;液晶的參數 k2 = 9.000e-12;

k3 = 19.500e-12;

el_pal = 19.600*(1/36/pi*1e-9);

el_pen = 5.100*(1/36/pi*1e-9);

lambda =0.63;入射光的波長 K=2*pi/lambda;

ne = 1.73038;e-wave的折射率 no = 1.51891;o-wave的折射率 gamma = (el_pal-el_pen)/el_pen;

kappa = (k3-k1)/k1;

V_c1 = pi.*sqrt(k1)/sqrt(el_pal - el_pen);%截止電壓 d=18.49;液晶的厚度

m =50; 取樣次數

theda_max=[0.01:1:90].*(pi/180); 中心厚度的偏折角 z = zeros(m,length(theda_max));

theda3 = zeros(size(z));

65 V = zeros(length(theda_max),1);

mm=rand(1,m);

phi=pi/2*mm;

for ja=1:length(theda_max);

eta=sin(theda_max);

a=sqrt((1+kappa*(eta(ja)).^2.*(sin(phi)).^2).*(1+gamma*(eta(ja)).^2));

b=sqrt((1-(eta(ja)).^2.*(sin(phi)).^2).*(1+gamma.*(eta(ja)).^2.*(sin(

bb=sqrt(1-(eta(ja)).^2.*(sin(phi)).^2);

f=sum([aa./bb])/m*pi/2;

Dz=f./(sqrt(el_pal-el_pen)/sqrt(k1)/(el_pen)*d/2) theda=asin(eta(ja)*sin(phi));

for i = 1:m;

x = i;

theda2 = 0 + (theda(x)-0).*mm;

a1=sqrt(k1*(cos(theda2)).^2+k3*sin(theda2).^2);

a2=1./(el_pal*(sin(theda2)).^2+el_pen*(cos(theda2)).^2);

a3=1./(el_pal*(eta(ja)).^2+el_pen*(cos(theda_max(ja))).^2);

f=(sum([a1./sqrt(a2-a3)]))/m.*(theda(x)-0);% z(i,ja)=(f./Dz);

theda3(i,ja)=theda(x);

end

zz=sort([z;z.*-1+d]./d);

zz1=zz*d;

figure(5)

tl_angle = [sort(theda3);inverse(sort(theda3),m)];

plot(zz, tl_angle.*180./pi, '-','linewidth',2);

grid on

n=((sin(tl_angle)).^2./(no).^2+(cos(tl_angle)).^2./(ne).^2).^(-1/2);

xlabel('z/d','fontsize',16);

ylabel('Tilted angle(DEG)','fontsize',16);

for i1=1:2*m-1;% dz(i1+1,ja)=zz1(i1+1,ja)-zz1(i1,ja);

end

dz(1,ja)=zz1(1,ja); dz(2*m+1,ja)=d-zz1(2*m,ja);

end

for ja4=1:length(theda_max)

n(2*m+1,ja4)=1.75; end for i2=1:2*m+1;

for ja1=1:length(theda_max);

f3(i2,ja1)=([n(i2,ja1).*dz(i2,ja1)]);

end end

t=phia;%入射光 theda 的角度 r=i8;入射光 alpha 的角度

[tt,rr]=meshgrid(t,r);

[xx,yy]=pol2cart(tt,rr);變為直角座標系

k_x=K.*sin(rr).*cos(tt);%入射光 x 方向的 wavevector k_y=K.*sin(rr).*sin(tt);%入射光 y 方向的 wavevector k_z=K.*cos(rr);% 入射光 z 方向 wavevector

計算 e-wave 在 z 軸 wavevector for ji1=1:2*m;

for ji2=1:length(V);

for ji1=1:2*m;

for ji2=1:length(V);

k_oz(:,:,ji1,ji2)=sqrt(no.^2.*K.^2-k_x.^2-k_y.^2)+tl_angle(ji1,ji2).*

end

67 end

for ji1=1:2*m;

for ji2=1:length(V);

n_eff(:,:,ji1,ji2)=sqrt(k_x.^2+k_y.^2+kez(:,:,ji1,ji2).^2)./K;

end

for j5=1:2*m+1;

for j6=1:length(V);

R1(:,:,j5,j6)=k_oz(:,:,j5,j6).*dz(j5,j6);

R2(:,:,j5,j6)=k_ez(:,:,j5,j6).*dz(j5,j6);

end

xp=exp(-j.*ppp1);

xp1=exp(-j.*sss1);

for ji2=1:length(V);

Eox2(:,:,ji2)=Eox1(:,:).*xp1(:,:,ji2);

Eoy2(:,:,ji2)=Eoy1(:,:).*xp(:,:,ji2);

end

for j1=1:length(i8);

for j2=1:length(phia);

for ji2=1:length(V);

Eox3(j1,j2,ji2)=cos(phia(j2)).*Eox2(j1,j2,ji2)+sin(phia(j2)).*Eoy2(j1 ,j2,ji2);

Eoy3(j1,j2,ji2)=-sin(phia(j2)).*Eox2(j1,j2,ji2)+cos(phia(j2)).*Eoy2(j 1,j2,ji2);

end end end

for j1=1:length(i8);

for j2=1:length(phia);

for ji2=1:length(V);

Eox4(j1,j2,ji2)=sin(phia(j2)).*ts1(j1).*ts2(j1).*Eox3(j1,j2,ji2)+cos(

phia(j2)).*tp1(j1).*tp2(j1).*Eoy3(j1,j2,ji2);

Eoy4(j1,j2,ji2)=-cos(phia(j2)).*ts1(j1).*ts2(j1).*Eox3(j1,j2,ji2)+si(

phia(j2)).*tp1(j1).*tp2(j1).*Eoy3(j1,j2,ji2);

end end end

Eox5=1./2.*Eox4-1./2.*Eoy4;經過檢偏板 Eoy5=-1./2.*Eox4+1./2.*Eoy4;

uuu1=(angle(Eox5)-angle(Eoy5)).*180./pi;

ccc1=abs(Eox5)./abs(Eoy5);

uuu=(angle(Eoy4)-angle(Eox4));

ccc=abs(Eox4)./abs(Eoy4);

E1=sqrt(Eox4.*conj(Eox4)+Eoy4.*conj(Eoy4));

I1=E1.^2;

E2=sqrt(Eox5.*conj(Eox5)+Eoy5.*conj(Eoy5))電場的總合 I2=E2.^2;intensity

mesh(fa.*tan(xx),fa.*tan(yy),T2(:,:,50));選定不同的電壓值

在文檔中應用反向對準型液晶波板研製可調式相位延遲元件之研究 (頁 67-77)