Guided Modes Of Sted Index Fiber...... Solution and MATLAB code with Graphic representation...

1. Assignment-2
PHL791
Ajay Singh
Que. Consider a parabolic index fiber with a = 14 m and = 0.0045 operating at 1310 nm. Assuming the
formulation for an infinitely extended parabolic index fiber to be valid,
a) Obtain the various LPlmmodes supported by this fiber.
b) Obtain the corresponding effective indices and group velocities.
c) Plot the corresponding transverse field distributions along the radial direction.
d) From the propagating modes calculate the intermodal dispersion of the fiber.
You may assume that the cladding is made of pure silica.

2. MATLAB CODE
i=(-1)^0.5;
a=14e-6; /Core Radius /
n2=1.446804317113822; /cladding refractive index/
delta =0.0045; /Δ = delta/
lembda=1310e-9; /lambda= wavelength λo/
c=3e8; /Light velocity in free space/
w=2*pi*c/lembda ; /frequency ωo/
K=2*pi/lembda; /wave number in free space/
r=0:0.00000007:15e-6;
n1=(n2*n2/(1-2*delta))^0.5 /Core refractive index/
V=K*a*n1*((2*delta)^0.5); /V number for the fiber/
x=V.*(r.^2)/(a.^2); /x= ɤ2
.r2
/
for m=1:1:10;
for l=0:1:9;
if(0<(2*m+l-1) & (2*m+l-1)<V/2)
LPlm= [l m] /Values of ‘l’ and ‘m’ corresponding to the mode/
n=m-1;
syms p ;
L =symsum((((-1).^p)* gamma(n+l+1).*(x).^p)./ (((factorial(n-p)).*
(gamma(p+l+1)).*(factorial(p)))),0,n);
/ Assiciated Lagguerre Ploynomials
/
N=(((V.^0.5)./a).^(l+1)).*((2*(factorial(m-1))./gamma(m+l))).^0.5;
/ Normalization Factor /
R=N.*(r.^l).*(exp(-x)/2).*L; /Field as function or ‘r’/
beta=(K*n1)*sqrt(1-((2*((2*m+l-1))*sqrt(2*delta))/(K*n1*a)));
digits(10)

3. vpa(beta) / β corresponding to that mode/
Neff = beta/K ; /Effective Refractive index for the mode/
digits(10)
vpa(Neff)
Vg = (n1./c +(2*c*2*delta*((2*m+l-1)^2)./(w*w*n1*a*a)))^-1;
digits(10)
vpa(Vg) /Group velocity for that mode /
plot(r,R);
hold on
end
end
end
Dispersion = n1*delta*delta*(10^3)/(2*c)
/Internalmodal Dispersion /
RESULTS:
Mode Β(m-1
) neff Vg(m/s)
LP01 6964003.04160 1.45194571519 206417952.166
LP02 6950423.99045 1.44911457841 206414831.240
LP11 6957216.82897 1.45053083750 206416781.808
LP12 6943624.50662 1.44769693380 206412100.508
LP21 6950423.99045 1.44911457841 206414831.240
LP31 6943624.50662 1.44769693380 206412100.508
Internalmodal Dispersion = 49.05 pS/KM

4. fig. Field distribution with Radial coordinate r.