Physics 4510 Optics - University of Colorado Boulder · Physics 4510 Optics Homework Assignment...
Transcript of Physics 4510 Optics - University of Colorado Boulder · Physics 4510 Optics Homework Assignment...
Physics 4510 Optics Homework Assignment Problem Set #2 (Turn in before class by Oct 6th)
1. (20%) A 10 cm optical crystal with a refractive index of n=(2+10z). Calculate the optical path length of the crystal. How long does it take for an optical pulse to propagate through the crystal, assuming that the optical pulse is infinitely short? For the same amount of time, what is the distance light propagates in free space? How thick is the crystal if the propagating time in the crystal is doubled?
2. (20%) (a) For a symmetric planar waveguide to support only one mode, proof that
the core thickness d=2a is
21
22
0
4 nna
−<
λ
(b) Proof that for a multimode fiber, the number of modes n is
)2(1πaVIntn +=
3. (15%) What is the largest thickness d of a planar symmetric dielectric waveguide
with refractive indices 50.12 =n and 46.11 =n for which there is only one TE mode at mµλ 3.10 = ? What is the number of modes if a waveguide with this thickness is used at mµλ 6.00 = instead?
4. (25%) A multi-mode planar waveguide has a core thickness of mµ4.2 , with the
refractive indexes of the core is 1.5 and of the cladding is 1.45. Assume an optical pulse with a wavelength of mµλ 5.10 = was coupled into the fiber and simultaneously excited the fundamental and the 1st eigenmode of the fiber. Due to modal dispersion, the pulse will be splitted. Calculate the time difference between the two splitted pulses exiting the fiber if the fiber length is 10km. (programming is needed to solve the propagation constant of the two modes)
5. (20%) Plot the electric field distribution )(xE of the fundamental and the 1st
eigenmode of the fiber in Problem 4.
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3
This program is written and calucalated with Mathematica
Data Initization
In[1]:= lamda0 = 1.5
Out[1]= 1.5
In[2]:= n1 = 1.45
Out[2]= 1.45
In[3]:= n2 = 1.5
Out[3]= 1.5
In[4]:= a = 1.2
Out[4]= 1.2
Define the symmetric mode function and the corresponding V-number function
In[5]:= Gma = h ∗Tan@h ∗aD
Out[5]= h [email protected] hD
In[6]:= fun = Gma^2 + h^2 − Hn2^2 − n1^2L∗H2∗Piêlamda0L^2
Out[6]= −2.58803 + h2 + h2 [email protected] hD2
Evaluate h for the fundamental mode
In[7]:= FindRoot@fun 0, 8h, 1<D
Out[7]= 8h → 0.846957<
In[8]:= h0 = h ê. %
Out[8]= 0.846957
Calculate the value of the first beta
In[9]:= beta0 = Sqrt@n2^2 ∗H2 ∗Piêlamda0L^2 − h0^2D
Out[9]= 6.22584
Calculate the speed of the fundmanetal mode using the first beta
In[10]:= v0 = H2 ∗PiêHlamda0∗ beta0LL ∗3∗10^8
Out[10]= 2.01842× 108
Define the symmetric mode function and the corresponding V-number function
prob4.nb 1
In[11]:= Gma = −h∗ Cot@h ∗aD
Out[11]= −h [email protected] hD
In[12]:= fun = Gma^2 + h^2 − Hn2^2 − n1^2L∗H2∗Piêlamda0L^2
Out[12]= −2.58803 + h2 + h2 [email protected] hD2
Evaluate h for the 1st order mode
In[13]:= FindRoot@fun 0, 8h, 1<D
Out[13]= 8h → 1.54477<
In[14]:= h1 = h ê. %
Out[14]= 1.54477
Calculate the value of the second beta
In[15]:= beta1 = Sqrt@n2^2 ∗H2 ∗Piêlamda0L^2 − h1^2D
Out[15]= 6.09033
Calculate the speed of the 1st order mode using the second beta
In[16]:= v1 = H2 ∗PiêHlamda0∗ beta1LL ∗3∗10^8
Out[16]= 2.06333× 108
Calculte the speed difference for the two modes after propagating a 10Km fiber
In[17]:= dt = 10000∗H1êv0 − 1ê v1L
Out[17]= 1.07837× 10−6
Therefore, the speed difference of the two modes is 1us
prob4.nb 2
C:\Docs\MOptics\HW\Chap2\prob5.mJuly 7, 2005
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%problem 5 written in matlab
clear all;
beta0 = 6.22584;beta1 = 6.09033;
n1 = 1.45;n2 = 1.5;lamda = 1.5;a = 1.2;
k0 = 2*pi/lamda;
h0 = sqrt(n2^2*k0^2-beta0^2);gma0 = sqrt(beta0^2-n1^2*k0^2);
h1 = sqrt(n2^2*k0^2-beta1^2);gma1 = sqrt(beta1^2-n1^2*k0^2);
x = -5:.01:5;y0 = x;y1 = x;
Ar = cos(h0*a);j=1;
for i=x if (i<=-a) y0(j) = Ar*exp(gma0*(i+a)); elseif (i>-a)&(i<=a) y0(j) = cos(h0*i); else y0(j) = Ar*exp(-gma0*(i-a)); end j=j+1;end Ar = sin(h1*a);j=1;
for i=x if (i<=-a) y1(j) = -Ar*exp(gma1*(i+a)); elseif (i>-a)&(i<=a) y1(j) = sin(h1*i); else y1(j) = Ar*exp(-gma1*(i-a)); end
C:\Docs\MOptics\HW\Chap2\prob5.mJuly 7, 2005
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j=j+1;end
plot(x,[y0' y1']);
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