Transcript Fraunhofer Diffraction

Fraunhofer Diffraction
Last Lecture
• Dichroic Materials
• Polarization by Scattering
• Polarization by Reflection from Dielectric Surfaces
• Birefringent Materials
• Double Refraction
• The Pockel’s Cell
This Lecture
• Fraunhofer versus Fresnel Diffraction
• Diffraction from a Single Slit
• Beam Spreading
• Rectangular and Circular Apertures
Optical Diffraction
•
Diffraction is any deviation from geometric optics that results from the
obstruction of a light wave, such as sending a laser beam through an
aperture to reduce the beam size. Diffraction results from the interaction
of light waves with the edges of objects.
• The edges of optical images are blurred by diffraction, and this represents
a fundamental limitation on the resolution of an optical imaging system.
• There is no physical difference between the phenomena of interference
and diffraction, both result from the superposition of light waves.
Diffraction results from the superposition of many light waves,
interference results from the interference of a few light waves.
Optical Diffraction
Hecht,
Optics,
Chapter 10
Fraunhofer versus Fresnel Diffraction
•
The passage of light through an aperture or slit and the resulting
diffraction patterns can be analyzed using either Fraunhofer or Fresnel
diffraction theory. In Fraunhofer (far-field) diffraction theory the source is
far enough from the aperture that the wavefronts are planar at the
aperture, and the image plane is far enough from the aperture that the
wavefronts are planar at the image plane.
• If the curvature of the optical waves must be taken into account at the
aperture or image plane, then we must use Fresnel (near-field) diffraction
theory.
• The Huygens-Fresnel principle is used in diffraction theory, in that every
point of a given wavefront of light can be considered as a source of
secondary wavelets. To analyze two-slit interference, we assumed that
the slits were point sources. To analyze diffraction, we need to consider
the generation of wavelets at different spatial positions within the slit.
Fraunhofer versus Fresnel Diffraction
•
We can move to the Fraunhofer regime by placing lenses on each side of
the aperture. Lens 1 is place a focal length away from the point source
so that the wavefronts are planar at the aperture. The observation
screen is in the focal plane of lens 2 so that the diffraction pattern is
imaged at infinity.
Fraunhofer Diffraction from a Single Slit
•
Consider the geometry shown below. Assume that the slit is very long in
the direction perpendicular to the page so that we can neglect diffraction
effects in the perpendicular direction.
Fraunhofer vs. Fresnel diffraction
• In Fraunhofer
diffraction, both
incident and diffracted
waves may be
considered to be plane
(i.e. both S and P are a
large distance away)
• If either S or P are close
enough that wavefront
curvature is not
negligible, then we have
Fresnel diffraction
S
P
7
Hecht 10.2
Hecht 10.3
Fraunhofer Vs. Fresnel Diffraction
Now calculate variation in (r+r’) in going from one side of aperture
to the other. Call it 
  d '2 h     d 2  h     d '2  h'2  d 2  h 2
2
2
 1  h'  2   1  h    2 
 1  h'  2   1  h  2 
  d 1  
  d ' 1      d 1    
  d ' 1  


 2  d'    2  d  
 2  d'   2  d  

 


 

11 1  2
 h' h 
          
2  d d'
 d' d 
8
Fraunhofer diffraction limit
• If aperture is a square -  X 
• The same relation holds in azimuthal plane
and 2 ~ measure of the area of the aperture
• Then we have the Fraunhofer diffraction if,
2
d

or ,
d
area of
aperture

Fraunhofer or far field limit
9
Fraunhofer, Fresnel limits
• The near field, or Fresnel, limit is
2
d

10
Fraunhofer Diffraction from a Single Slit
The contribution to the electric field amplitude
at point P due to the wavelet emanating from
the element ds in the slit is given by
 dE 
dEP   0  exp i  kr  t  
 r 
Let r  r0 for the source element ds at s  0.
Then for any element
 dE0 
dEP  
 exp i  k  r0     t 
  r0    


We can neglect the path difference  in the amplitude term, but not in the phase term.
We let dE0  EL ds, where EL is the electric field amplitude, assumed uniform over the width of the slit .
The path difference   s sin  . Substituting we obtain
 E ds 
dEP   L  exp i  k  r0  s sin    t 
 r0 

Integrating we obtain

E
EP   L
 r0
E 
EP   L  exp i  kr0  t  
 r0 
 exp  i k s sin   

 exp i  kr0  t   

i k sin 


 b / 2
b/2

b/2
b / 2
exp  i k s sin   ds
Fraunhofer Diffraction from a Single Slit
Evaluating with the integral limits we obtain
 exp  i    exp  i   
E 
EP   L  exp i  kr0  t   

i k sin 
 r0 


where
 
1
k b sin 
2
Rearranging we obtain
E 
b
EP   L  exp i  kr0  t  
exp  i    exp  i   
2i  
 r0 
E 
E 
b
b sin 
  L  exp i  kr0  t  
 2 i sin     L  exp i  kr0  t 
2i 

 r0 
 r0 
The irradiance at point P is given by
2
 E b  sin 2 
1
1
sin 2 
I =  0 c EP EP*   0 c  L 

I
 I 0 sinc 2 
0
2
2
2
2
r


 0 
Fraunhofer Diffraction from a Single Slit
The irradiance at point P is given by
1
1
I =  0 c EP EP*   0 c
2
2
2
 EL b  sin 2 
sin 2 
 I0
 I 0 sinc 2 


2
2


 r0 
The sinc function is 1 for   0,
lim sinc   lim
 0
 0
The zeroes of irradiance occur when sin   0, or when
sin 

 
1
1
k b sin   m 
2
m   1,  2,
Fraunhofer Diffraction from a Single Slit
In terms of the length y on the observation screen,
y  f sin  , and in terms of wavelength   2 / k ,
we can write

1 2
y by
b 
2 
f
f
Zeroes in the irradiance pattern will occur when
by
 m
f
y
m f
b
The maximum in the irradiance pattern is at β = 0.
Secondary maxima are found from
d  sin   cos  sin   cos   sin 


0


d   

2
2

 
sin 
 tan 
cos 
Fraunhofer Diffraction from a Single Slit
Note: x- and y-axes
switched in book, Figs. 165a (here) and Fig. 16-1 do
not match.
Beam Spreading
The angular width of the central
maximum is defined as the angular
separation Δθ between the first minima on
either side of the central maximum,
y
 sin   
f
The first min ima in the irradiance pattern
will occur when
y
 1  f
m f

b
b

Δθ 
2
b
The width W of the diffraction pattern thus
increases linearly with distance from the slit,
in the regions far from the slit where Fraunhofer
diffraction applies
W = L Δθ 
2L
b
Rectangular Apertures
When the length a and width b of the
rectangular aperture are comparable,
a diffraction pattern is observed in
both the x - and y - dimensions, governed
in each dimension by the formula we
have already developed. The irradiance
pattern is
x
I  I 0  sinc 2  sinc 2  
where
y
1
  k a sin 
2
Zeroes in the irradiance pattern are observed
when
y
m f
b
or x 
m f
a
Square Apertures
Fraunhofer Diffraction from General
Apertures
For the general aperture
E
dEP  A exp i t  kr  dA
r
where
dA  dy dz
2
2
r   X 2  Y  y    Z  z  


R   X 2  Y 2  Z 2 
1/ 2
1/ 2
Fraunhofer Diffraction from General
Apertures
We can combine the relations for r and R to obtain

r  R 1 

 y2  z2 
R2
1/ 2
2 Yy  Zz  


R2

In the far field R is very large compared to the aperture dimensions, and
y
we can neglect the term
2 Yy  Zz  

r  R 1 

R2


1/ 2
2
+ z2 
R2
, and we can write

Yy  Zz    R  Yy  Zz 
 R 1 

R2 
R

Therefore, the total electric field at point P is given by
E
EP  A exp i t  kR    exp ik Yy  Zz   dA
R
aperture

EA
exp i t  kR    exp ik Yy  Zz   dy dz
R
aperture
Fraunhofer Diffraction from Circular Apertures
Now we specialize to a circular aperture of radius a. At this point we switch to cylindrical coordinates
z   cos 
y   sin 
Z  q cos 
Y  q sin 
dA   d  d
Substituting into our expression for EP we obtain
EP 

  a   2
 ik  q sin  sin   q cos  cos   
EA
exp i t  kR    
exp 
  d  d
 0  0
R
R


  a   2
EA
i k q 

exp i t  kR    
exp 
cos       d  d


0


0
R
 R

Fraunhofer Diffraction from Circular Apertures
Because of symmetry our solution will be the same for any angle . Choosing Φ = 0, we obtain
EP 
But
  a   2
EA
i k q 

exp i t  kR    
exp 
cos    d  d
 0  0
R
 R

  2

J 0 (u ) 
0
 kq

exp i
cos   d
 R

1
2

v  2
v 0
is in the form of a Bessel function.
exp i u cos v  dv is a Bessel function of the order zero.
We can write
EP 
 a
EA
 k q 
exp i t  kR   2  J 0 
  d
 0
R
 R 
In general , a Bessel function of the order m is given by
im
J m (u ) 
2

v  2
v 0
exp i  mv  u cos v   dv
Fraunhofer Diffraction from Circular Apertures
A useful recurrence relation for Bessel functions is
d
u m J m  u    u m J m1  u 
du
When m = 1, we can integrate the expression to find

u
0
J 0  u  du  u J1  u 
Define w 
R
k q
, then d     dw gives us
R
 kq 
 a
R
E
E
 k q 
EP  A exp i t  kR   2  J 0 
 d   A exp i t  kR   2  

 0
R
R
 R 
 kq 
From the recurrence relation we obtain
EP 
 R   kaq 
EA
exp i t  kR   2 a 2 
 J1 

R
 kaq   R 
The irradiance is given by
1
I  c  0 EP EP*
2
2
 E A  2  R    kaq  
  J1 


 A 
 R 
 kaq    R  
2
2
2
w  kaq / R

0
J 0  w  w dw
Fraunhofer Diffraction from Circular Apertures
Assuming that R is essentially constant over the observation screen and recognizing that
sin   q / R, we can write the irradiance as
 2 J  kaq R  
 2 J1  k a sin   
I  I  0  1
  I 0 

  kaq R  
  k a sin   
2
where we have used the
relation that lim
u 0
J1  u  1

u
2
recognizing that u  0 when
q  0.
2
Fraunhofer Diffraction from Circular Apertures:
Bessel Functions
Fraunhofer
Diffraction from
Circular Apertures:
The Airy Pattern
First minimum in the Airy pattern is at
 2   D 
k a sin   k a   3.83  
    min
   2 
 min 
1.22 
D
where D  2a.
Circular Apertures