physics 4510 optics homework assignment … 4510 optics homework assignment problem set #5 (turn in...
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Physics 4510 Optics Homework Assignment Problem Set #5 (Turn in before class by Dec 8th, no late homework will be accepted)
1. (15%) Prove that the secondary maxima of a single-slit diffraction pattern occur
when ββ tan= .
2. (15%) A single-slit Fraunhofer diffraction pattern is formed using white light. For what wavelength does the second minimum coincide with the third minimum produced by a wavelength of 400=λ nm.
3. (20%) For an apodized slit which has a transmission function of
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−=
2
2exp21)(ay
ayf
π,
proof that the Fraunhofer diffraction pattern is
)8
exp()(22akdyeyfE iky −=∝ ∫
∞
∞−
by using the relation π=∫∞
∞−
− due u2
.
Note that the diffraction pattern has no sidebands.
4. (15%) Consider a double slits aperture, each slit has a width b, and are separated by a distance h. Prove that the intensity of the Fraunhofer diffraction pattern is
γββ 22
0 cos)sin(II =
where θβ sin21 kb= and θγ sin
21 kh= .
5. (15%) Continue on problem #4, the first term of the diffraction pattern of a
double slits aperture 2)sin(ββ typically accounts for diffraction pattern of a single
slit, and the terms corresponds to the interference between the two slits. Missing orders occur at those values of
γ2cosθsin which satisfy, at the same time, the
condition for interference maxima and the condition for diffraction minima. Show
that this leads to the condition integer.)( =bh
6. (20%) Plot the diffraction pattern )(θI of the double slits aperture for θ from -
180o to 180o in Problem #5, where 10 =I , mb µ1= , mh µ3= and the optical wavelength nm632=λ .
Fo. A
ytLwn,
+' 1;J
l-lw # ss;"61n slit D;{(^ct,"^
S i"r
rL&d F
P*'f) 0
Sf o.'t,;,Fro^"k*(rn
t sin (? 1zt (1 r
t na f , i n r * r .a , :e t # =o
= r. a(f f)#r i f l ir l Dl -
- F s 'n / t( - |
f B c"E[7L t {
I- o
+o
>s*n f(
ze;^ f
*'f )II
T
f le*
= S ; n f
= * . o Ft 4
?=
iirritI t l
i i tI t t
t l lJ t l
iiiI t l
tititit : l
i l !i i ti t it : i
I t tiltr J i, i ; i
l a t* :i f l
liit*li t r
t ' ii i il l i
iirr : i: i ii i i
! t ii : i:. ,,: i 1
f, Fo" sht d,:{Nno*ta4, tL^ ;^-f*nsr1V
tt(
lin c-p
r , si i{ l ) 'r o t f
)
occLr o-Sl*y f= vv ,T I I r n -_ t l , t Z . - -
* t I s i ' r o
+ b s;no
(t -- * Lsr,, s =
.)\
s i n t r = r n A
+1*r'$r*
s i^6lu
T
p r i n i m a
At min irrra ,
anol,
3 \
+ \ '
b
bb
^ \
s r ' n& = eAs r ' n F = J \ /
e A r
* \
6 un y wt I
l r
ut'urz' A =Voonyrt
To, Fr^^ntrr{^r /,ff^drLn,I
I**4,*> I '*!^.^rf ^ f lE -a ( *=) _1 .1 .L
) ^ a J T e e v o l \
(^ -(+)"- tk? ,I *g
" o l 7I l a
t _Iq J T T
sincr (ry)'*ik.* = (*f+iK%+ (f)' f#t=tt * ' f t1 '+*"
-, ) tr (* ̂ -(ry *i$t-(+e)d a J n . } _ : ,
\ - - ' - _ , , t : , _ ' d r r |
= *H - t r
[ e- (#nsf ^ ,= \ F Q. u
I_e-*T - "f ' , , l f ?-.f ]
r:olot,, ̂f e - X t
J - e d , x = f n- k=o'/ g
E-\-)
l)',U tl*
eE
Tl* F nn^n hrfi* /tfi*.t;t^, p^t*c,t,. is
E e( ( ;F t5{no . ! , ,r=
J J"^L[ u]it e \r ag
: f t
n,kt,'^oort + Jl-t e ,k efls;nu uW
J o
= I f ib^ i [e ls in$i K " ^ o L i " e
o o ( ( t Q s i ^ o )
+ f h+b ,
i K6s,,^ $ tL;kg ,ing) ]
) ^ (
I - ihLq,ur$ 'K(h+bls i*$ i f ihsra$
L K r , " o L e - l + e a J| / ^ i k L s i , i $ i i < h s , . , $ \
r k - ^ e ( e l ) ( \ + e )
4t I KW: )s;"$ iK[hl'k;-'S
e*+t
:b I {nLs,'r$
* ktn^e e e
* rus,ns = 1ll F i v
E o{. fb e' e-
tl.+* iuo*a*#&
i k(*)s'' pq +- ikt*)s;,'s
e
i f khrr"s ht+ kbsns) cor{{rh*^$
* n n l
sin (i
F
i t.l', 5in I = |"
CrS X
t *i"tr
1l*afuru ,
'T -TJ- ".Lo
Fo. Tra^^hr1* d;.ll^r1,"^I'aLl< Ett , +tl-
' irj,,^s;tq
T ( o ) = T o + * o r ' $
r ' o l * r o P = * r b s i n 8 a n * t
M ,ss; 1 o.lrnr oao&fs r^rlrnt(
i nfnn {o't
n11u- yv\^Ki6ry\^
dntg
is
*o
sin F
l , Y t : O , t l , t 7 - -
n = t l , t Z
f , = i r l
r
57
{t\
*h
,t ifi.".*un h a l n t n t , + tvl
,^ \rr r \
YN= - =
n
ru\^x Ic*= f I
V -- rnnm i r \ f + )\ F "
= nTT
in*tq,q.r-a h
or
+ ?rP
hot^ TD
% Problem #6% plotting the double slit diffraction pattern
clear all;
theta_angle = -180:.01:180;theta = theta_angle/180*pi;
lambda = 632e-9;b=1e-6;h=3e-6;
k=2*pi/lambda
beta = .5*k*b*sin(theta);gamma = .5*k*h*sin(theta);
I=(sin(beta)./beta.*cos(gamma)).^2;
plot(theta_angle, I);xlabel('Angle');ylabel('Diffraction Pattern');