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.

\

Ct- rytz) *d>

prtzp,^q ol,rn 4t'n*- - -.5I I U

v*dz

e--h-S-z=t*

o , z 9

4x l o8--=--8-3-ne--

-- e,l;, k n ut -

-q_ _ _ = W X l o

=__o__L$__(_ioj-*_

/ s L l

---+ - - k{}:-p- p E----e.g-- :-",*

tr-czy-dz

-->-T--:-o,|7a^1 I--_

Xon

h az

*hu-*t (*^+-+q-_

-4---ha --*, -- ao cl-- -r.*- *= .o-t --- --*- cot- f{-)_-'------ )-

--T n -4i----c-@sL --

m""{" & -fu--,*,,"U*tz-

--Ya. -=-o-

- --{+-ff'=t}1:--Tr

).Y a.

- r-AA-6v442-- *-'ih --*o_ Aau& A.g a

#*- -+^-vA-{-!Y4-+L -TT

t

\eu---ft-g4-JFn,'

1 , 3I A

_ _-_3J rr , c l '- ct.q 6)'_ _o3**-C,,-;)--

--g-f t=- t+ J^t ( QJt

T+-rtrrr.Ts)

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 Tan@1.2 hD

In[6]:= fun = Gma^2 + h^2 − Hn2^2 − n1^2L∗H2∗Piêlamda0L^2

Out[6]= −2.58803 + h2 + h2 Tan@1.2 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 Cot@1.2 hD

In[12]:= fun = Gma^2 + h^2 − Hn2^2 − n1^2L∗H2∗Piêlamda0L^2

Out[12]= −2.58803 + h2 + h2 Cot@1.2 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

Page 13:06:15 PM

%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

Page 23:06:15 PM

j=j+1;end

plot(x,[y0' y1']);

5 4 3 2 1 0 1 2 3 4 51

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