Download F - Technion moodle

Course outline
1. Maxwell Eqs., EM waves, wave-packets
2. Gaussian beams
3. Fourier optics, the lens, resolution
4. Geometrical optics, Snell’s law
5. Light-tissue interaction: scattering, absorption
Fluorescence, photo dynamic therapy
‫ חבילות גלים‬,‫ גלים אלקטרומגנטים‬,‫ משואות מקסוול‬.1
‫ קרניים גאוסיניות‬.2
‫ הפרדה‬,‫ העדשה‬,‫ אופטיקת פורייה‬.3
‫ חוק סנל‬,‫ אופטיקה גיאומטרית‬.4
,‫ פלואורסנציה‬,‫ בליעה‬,‫ פיזור‬:‫רקמה‬-‫ אינטראקציה אור‬.5
‫דינמי‬-‫טיפול פוטו‬
6. Fundamentals of lasers
‫ עקרונות לייזרים‬.6
7. Lasers in medicine
‫ לייזרים ברפואה‬.7
8. Basics of light detection, cameras
9. Microscopy, contrast mechanism
10.Confocal microscopy
‫ מצלמות‬,‫ עקרונות גילוי אור‬.8
‫ ניגודיות‬,‫ מיקרוסקופיה‬.9
‫ מיקרוסקופיה קונפוקלית‬.10
Fourier optics and imaging
• Linear optical systems
• Fresnel diffraction
• Fraunhofer diffraction
• The lens
• Optical resolution
Light propagation - Intuition
Light from a point source propagates in spherical waves:
CCD array
Point source
The action of a lens
CCD array
spot size  1.22  
clipping

resolution
drop
Point source
f
D
a Van-leeuwenhoek microscope
Conventional microscopy
Can light interact with itself ?
Fourier optics
An arbitrary wave in free space may be analyzed as a superposition of plane
waves.
If it is known how a linear optical system modifies plane waves, the principle of
superposition can be used to determine the effect of the system on any
arbitrary wave.
The linear-systems approach
The complex amplitudes in two planes normal to the optical (z) axis are regarded as the input
and output of the system. A linear system may be characterized by:
- its impulse response function - the response of the system to an impulse (a point) at the input.
- its transfer function - the response to spatial harmonic functions.
f  x, y   U  x, y , 0 
g  x, y   U  x, y , d 
Spatial harmonics
An arbitrary function f(t) may be analyzed as a sum of harmonic functions of
different frequencies and complex amplitudes.

f  t    F   ei 2 t d

In two (spatial) dimensions:*
f  x, y    F  x , y  e

 i 2  x x  y y

* Our definitions of temporal and spatial Fourier transforms differ in the sign of the exponent. The choice of this
signs is arbitrary, as long as opposite signs are used in the Fourier and inverse Fourier transforms. With
different signs in the spatial (2D) and temporal (1D) cases, the traveling wave exp[i(2t-kzz)] represents wave
that moves in +z direction as time propagates.
A little on Fourier transform
Example 1
f(x,y)
|F(x,y)|2
Example 2
f(x,y)
|F(x,y)|2
Log (|F(x,y)|2)
Example 3
f(x,y)
|F(x,y)|2
Log (|F(x,y)|2)
P(x,y)
Example 4
f(x,y)
|F(x,y)|2
Log (|F(x,y)|2)
Spatial harmonics & Plane waves
Consider a plane wave:
(t=0, monochromatic)
with a wavevector
U  x, y, z   Ae

i kx x  k y y k z z

A: Complex amplitude
k   kx , k y , kz 
k  k  k x2  k y2  k z2 
2

.
This wave propagates at angles:
 x  sin 1 k x k
 y  sin 1 k y k
 x = 0 means that there is no component of
k in the x-axis (kx=0).
At z=0:
We also know (previous slides) that
U is comprised of spatial harmonics:
U  x, y, 0   Ae

i kx x  k y y

U  x, y, 0    F  x , y  e

 i 2  x x  y y

Spatial harmonics & Plane waves

A single plane wave:
U  x, y, 0   Ae
i kx x  k y y
A single spatial harmonic:
U  x, y, 0   Fe


 i 2  x x  y y

 x  k x 2
 
 y  k y 2
(!)  x , y  k x , y 2 → Spatial frequency [cycles/mm]
  kc 2
→ Optical frequency [cycles/sec]
 x  sin 1 k x k
 y  sin 1 k y k
 x  sin 1   x 
 y  sin 1   y 
The angles of inclination of the wavevector are then directly proportional to the spatial
frequencies of the corresponding harmonic function. Apparently, there is a one-to-one
correspondence between the plane wave U(x,y,z) and the harmonic function f(x,y).
Spatial harmonics & Plane waves
plane wave U(x,y,z)  harmonic function f(x,y)
Given one, the other can be readily determined, provided the wavelength  is known:
- The harmonic function f(x,y) is obtained by sampling at the z0 plane, f(x,y) = U(x,y,z0).
- Given the harmonic function f(x,y), on the other hand, the wave U(x,y,z) is constructed
by using the relation:
U  x, y, z   f  x, y  eikz z
With:
k z   k 2  k x2  k y2  k 
 x  sin 1 k x k  sin 1   x 
2

 y  sin 1 k y k  sin 1   y 
A condition for the validity of this correspondence is that kz is real ( kx  k y  k ).
This condition implies that x < 1 and y < 1, so that the angles x and y exist.
2
2
2
The  signs represent waves traveling in the forward and backward directions. We shall be
concerned with forward waves only.
Spatial-spectral analysis
With a single spatial frequency:
U  x, y , z   f  x , y  e
 ik z z
U  x , y , 0   f  x, y   e

e
 i 2  x x  y y


 i 2  x x  y y

e ikz z
If the transmittance of the optical element f(x,y) is the sum of several harmonic functions of
different spatial frequencies, the transmitted optical wave is also the sum of an equal number
of plane waves dispersed into different directions. The amplitude of each wave is proportional
to the amplitude of the corresponding harmonic component of f(x,y).
Spatial-spectral analysis
Mathematically, if f(x,y) is a superposition integral of harmonic functions,
f  x, y    F  x , y  e

 i 2  x x  y y

d x d y

with frequencies x and y, and amplitudes F(x,y), the transmitted wave U(x,y,z) is the
superposition of plane waves,
U  x, y, z    F  x , y  e

 i 2  x x  y y

eikz z d x d y

with complex envelopes F(x ,y) and k z
For any z:
 k 2  k x2  k y2 =2  2  x2  y2 .
 f  x, y   F  ,  e i 2  x x  y y  d d
x
y
 x y


 F  x , y   f  x, y  ei 2  x x  y y  dxdy


Spatial-spectral analysis
A thin optical element of complex amplitude transmittance f(x,y) decomposes an incident plane
wave into many plane waves. Each wave travels at angles x = sin-1(x) and y = sin-1(y) and
has a complex envelope F(x,y).
This process of "spatial spectral analysis" is analogous to the angular dispersion of different
temporal-frequency components (wavelengths) by a prism. Free-space propagation serves
as "spatial prism“, sensitive to the spatial rather than temporal frequencies of the waves.
The transfer function
input plane
f  x, y   U  x, y,0
output plane
g  x, y   U  x, y, d 
We regard f(x,y) and g(x,y) as the input and output of a linear system. The system
is linear since the Helmholtz equation, which U(x,y,z) must satisfy, is linear.
Linear systems
Shift-invariant system: Invariance of free space to displacement of the coordinate system.
A linear shift-invariant system is characterized by its impulse response function h(x,y) or
by its transfer function H(x,y).
Impulse response
Transfer function
Transfer function of free space
consider a single harmonic input function,
 i 2  x x  y y 
f  x, y    F  x , y  e

which corresponds to a plane wave:
d x d y  A  e
U  x, y, z   A  e


 i 2  x x  y y
i kx x  k y y  kz z

where: k z  k 2  k x2  k y2 =2  2  x2  y2
 x  k x 2
 y  k y 2
After propagating a distance d:
g  x, y   A  e

i kx x  k y y  kz d

Thus transfer function of free space:
g  x, y 
 i 2 d
 ik z d
 H  x , y  
e
e
f  x, y 
 2  x2  y2

Transfer function of free space
The (complex) transfer function of free space:
H  x , y   e
i 2 d  2  x2  y2
sphere
Fourier optics and imaging
• Linear optical systems
• Fresnel diffraction
• Fraunhofer diffraction
• The lens
• Optical resolution
Fresnel approximation
H  x , y   e
i 2 d  2  x2  y2
If the input function f(x,y) contains only spatial frequencies that are much smaller than the
cutoff frequency -1, so that
2
2
2
 x  y   ,
the plane wave components of the propagating light then make small angles xx and
yy , corresponding to paraxial rays.
  x2  y2 
2
2
d
 x  y
d

 i 2 1

i
2

1

2
2
2
2 


 i 2 d   x  y
2 

 2

*
x
y
H  ,
H  x , y
*
e
e
e

2
2
 ikd i d  x  y
e
e

1   1  2
parabola
sphere
Input-output relation (Fresnel)
Given the input function f(x,y), the output function g(x,y) may be determined as follows:
1. we determine the Fourier transform
F  x , y    f  x, y  e

i 2  x x  y y

dxdy

2. the product H(x,y) F(x,y) gives the complex envelopes of the plane wave components
in the output plane.
3. the complex amplitude in the output plane is the sum of the contributions of these plane
waves:
 i 2  x x  y y 
g  x, y    H  x , y  F  x , y  e
d x d y

H  x , y   e
Using the Fresnel approximation for H(x,y), we have:
g  x, y   e
 ikd
 F 

x
, y  e

i d  x2  y2

e

 i 2  x x  y y


2
2
 ikd i d  x  y
e
d x d y

Impulse response of free space
The impulse response function h(x,y) of the system of free-space propagation is the
response g(x,y) when the input f(x,y) is a point at the origin (0,0).
F  x , y    f  x, y  e

i 2  x x  y y


dxdy     x, y  e

i 2  x x  y y

g  x, y    1 H  x , y  e

which is the inverse Fourier transform of
H  x , y   e
i
h  x, y   g  x, y  
e
d

 i 2  x x  y y

2
2
 ikd i d  x  y
e

x2  y 2
 ik
 ikd
2d
e
Thus, each point in the input plane generates a paraboloidal wave;
all such waves are superimposed at the output plane.

d x d y
(exercise):

dxdy  1
Free space propagation as a
convolution (Fresnel)
Knowing h, we can regard f(x,y) as a superposition of different points (delta functions), each
producing a paraboloidal wave.
The wave originating at the point (x’,y’) has an amplitude f(x’,y’) and is centered about
(x’,y’) so that it generates a wave with amplitude f(x’,y’)h(x-x’,y-y’) at the point (x,y) in the
output plane. The sum of these contributions is the two-dimensional convolution:
g  x, y    f  x, y  h  x  x, y  y  dxdy,

which in the Fresnel approximation
h  x, y  
i
e
d
i ikd
g  x, y  
e  f  x, y  e
d

x2  y 2
 ik
 ikd
2d
e
becomes:
2
2
x  x    y  y  

 i
d
dxdy.
Unlike previous derivation of g from f, here no Fourier transform is involved.
Convolution

f h 
 f   g  t    d

Fresnel approximation: summary
Within the Fresnel approximation, there are two approaches for determining the complex
amplitude g(x,y) in the output plane, given the complex amplitude f(x,y) in the input plane:
1. space-domain approach in which the input wave is expanded in terms of paraboloidal
elementary waves.
i ikd
g  x, y  
e  f  x, y  e
d

2
2
x  x    y  y  

 i
d
dxdy
2. frequency-domain approach in which the input wave is expanded as a sum of plane
waves:
g  x, y   e
 ikd
 F 

x
, y  e

i d  x2  y2

e

 i 2  x x  y y

d x d y
Fourier optics and imaging
• Linear optical systems
• Fresnel diffraction
• Fraunhofer diffraction
• The lens
• Optical resolution
Fraunhofer approximation
Start with the space-domain approach of Fresnel approximation:
i  ikd
g  x, y  
e  f  x, y  e
d


i  ikd
e  f  x, y  e
d

2
2
x  x   y  y 

 i
d
x
 i
2
dxdy
 

 y 2  x2  y 2  2 xx  yy  
d
dxdy
If f(x,y) is confined to a small area of radius b, and if the distance d is sufficiently large so
2
2
that b2/d is small, then the phase factor  x  y    d  is negligible:
i
g  x, y  
e
d

i
e
d
x2  y 2
 i
 ikd
d
e


 f  x, y e
i 2
xx yy 
d
dxdy
x



 x  d

  y
 y  d

x2  y 2
 i
 ikd
d
e

f  x, y  e

i ikd i x dy  x y 

e e
F
,

d

d

d


2
2

i 2  x x  y y 

dxdy
 x  sin 1   x 
sin  x   x
sin  x  tan  x 
x
x
  x   x 
d
d
Fraunhofer approximation
 x y 
g  x, y   h0 F 
,

 d d 
i
h0 
e
d
x2  y 2
 i
 ikd
d
e
In the Fraunhofer approximation, the complex amplitude g(x,y) of a wave of wavelength 
in the z=d plane is proportional to the Fourier transform F(x,y) of the complex amplitude
f(x,y) in the z=0 plane, evaluated at the spatial frequencies x =x/d and y =y/d.
The approximation is valid if f(x,y) at the input plane is confined to a circle
 2
of radius b satisfying:
2
x

d
y

max
 d
f  x, y 
2
b2
“Fraunhofer region”
2
g  x, y 
Fresnel region
2b
Fraunhofer region
Fraunhofer – FT in the far-field
One slit. The width of the silts is varied.
Two slits. The width of the silts is constant
and the distance between them is varied.
Periodic objects - gratings
f  x   1  cos  2 Gx 
F  x    f  x, y  e

i 2  x x  y y 

dxdy
1/G
G: lines per meter

i 2  x 
  1  cos  2 Gx   e  x  dx
g  x

1
1
   x     x  G     x  G 
2
2
 x 
 x  1  x
 1  x

F





G



G







 d 
 d  2  d
 2  d

g  x
2
1
 x y 
 2 2 F
,
 d

d
 d 


1
 2d 2
b2
2
d
2b
f  x
2
  x  1  x
 1  x





G



G



  d  4  d


 4  d

 
g  x
2
1/(2d2)
1/(42d2)
-dG
0
dG
x
Sarcomere contractions
Sarcomeres are multi-protein complexes composed of different filament systems.
http://highered.mcgraw-hill.com/sites/0072495855/student_view0/chapter10/animation__sarcomere_contraction.html
Sarcomere contractions
-dG
dG
x
d
Helium-Neon laser
 = 632 nm
Fourier optics and imaging
• Linear optical systems
• Fresnel diffraction
• Fraunhofer diffraction
• The lens
• Optical resolution
Angular spectrum - definition
Definition: The angular spectrum A(x,y,z) of a wave U(x,y,z) emerging from an object:
AU FT pair:
 A  , , z   U  x, y, z  ei 2  x x  y y  dxdy
x
y



U  x, y, z   A  x , y , z  ei 2  x x  y y  d x d y


Thus A(x,y,z) is simply the equivalent for the Fourier transform F of the object f(x,y):
Angular spectrum
Fourier transform of the object
A  x , y , z   F  x , y  eikz z
U  x, y, z   f  x, y  e ikz z
Complex wave
Object complex transmission
Propagation of angular spectrum
Substitute U   A  x , y , z  ei 2  x y  d x d y into Helmholtz equation 2U  x, y, z   k 2U  x, y, z   0
x
y

And executing the derivatives of the x and y coordinates gives:
 d2
 i 2  x x  y y 
2
2
2
2
A

,

,
z

k

4




A

,

,
z
d x d y  0
 x y   x y  e
  dz 2  x y 



=0
Which yields a differential equation:


d2A
2
2
2
2

k

4





x
y A0
2
dz
with a solution:
A  x , y , z   A  x , y , 0  e

 iz k 2  4 2  x2  y2
2
2
2
Fresnel approximation:  x  y  
1   1  2

 A  x , y , 0  e
 A  x , y , 0  e

 izk 1  2  x2  y2

 1

 izk 1  2  x2  y2 
 2

Propagation of the angular spectrum (Fresnel approximation)
A  x , y , z   A  x , y ,0  e

2
2
ikz i z  x  y
e



Phase transformation with a thin lens
From straight-forward geometrical considerations:
R1


x2  y 2 
x2  y 2 
  x, y   t0  R1 1  1 
  R2 1  1 

2
2
R
R
Lens thickness




1
2
R2
t0
Paraxial approximation
R >> lens diameter:
n
Lensmaker’s equation for a thin
lens in air:

 1 1 
f   n  1    
 R1 R2  

1
The pupil function:
P(x,y) =
1, inside the aperture
0, otherwise
x2  y 2
x2  y 2
1
 1
2
R
2R2
x2  y 2  1 1 
   x, y   t0 
  
2  R1 R2 
The phase transformation  (x,y) by the lens is given by (k=2/):
  x, y   kn  x, y   k t0    x, y    kt0  k  n 1   x, y 
glass
  x, y   knt0 
air
k
x2  y 2 

2f
Therefore, the phase transformation of a perfect thin lens is:
t  x, y   e
iknt0
i
e

k 2 2
x y
2f

Fraunhofer diffraction by a lens
The field after a (thin) lens (neglecting the exp(iknt0) term):
U  x, y , 0

  U  x, y , 0  P  x, y  e

Using the Fresnel integral:
U  x, y , z  
i  ikz
e  U  x, y, 0  e
z

2
2
x  x    y  y  

 i
z
i

k
x2  y 2
2f
U  x, y , 0  

U  x, y , 0  
f
U  x, y, f 
dxdy
2
k
i  x2  y 2  i  xx  yy  
i  ikz  i 2kz  x2  y 2 

z
2z



e e
U
x
,
y
,
0
e
e
dxdy



z

i
 x
i  ikz  i 2kz  x2  y 2 

2f

e e
U  x, y, 0  P  x, y  e

z

k
U  x, y , f  

i
f
i
f
e ikf e
i

k 2 2
x y
2f

2
 y 2

2
 xx yy 
f
e



U
x
,
y
,
0

 P  x, y e

i



U
x
,
y
,
0

 P  x, y e

i 2  x x  y y 
i

k
x 2  y 2
2z

e
e
 ikf
e

k 2 2
x y
2f


2
 xx yy
z
dxdy
dxdy

i
i

dxdy

Lens  Fraunhofer diffraction of U(x,y,0-) multiplied by the pupil function
x



 x f


  y
 y  f
Fourier transform with a lens
U  x, y, f  
ie
 ikf
f
e

k 2 2
i
x y
2f




U
x
,
y
,0

 P  x, y e


i 2  x x y y

U  x, y , 0  
U  x, y , 0  
dxdy

d1
f
Assume U(x,y,0-) has an extent less than P(x,y):
i
x
ie  ikf

2f
U  x, y , f  
A  x , y , 0  e
f
k
2
 y2

Propagation of the angular spectrum from -d1 to 0-:
A  x , y ,0

  A  ,
 U  x, y, f  
x
ie
y , d1  e
 ik  f  d1 
f

2
2
ikd1 i d1  x  y
e
U  x, y, d1  U  x, y, f 
Reminders:
A  x , y , z    U  x, y, z  e

i 2  x x  y y
A  x , y , z   A  x , y ,0  e

A  x , y , d1  e
i
k  d1  2 2
 1 x  y
2f  f 



dxdy

2
2
ikz i z  x  y
e
x
f
y
y 
f
x 
d1  f
iei 2 kf
U  x, y, f  
A  x , y ,  f 
f
The field at the focal plane of the
lens is the 2D Fourier transform
of the field at z = -f

Fourier transform with a lens
iei 2 kf
U  x, y, f  
A  x , y ,  f 
f
Fourier transform
property of a lens:
A  x , y , z   F  x , y  eikz z
U  x, y, z   f  x, y  e ikz z
 x y 
iei 2 kf
g  x, y  
F
,

f

f

f


 The complex amplitude of light at a point (x,y) in the back focal plane of a lens of focal
length f is proportional to the Fourier transform of the complex amplitude in the front focal
plane evaluated at the frequencies x/λf, y/λf. This relation is valid in the Fresnel
approximation. Without the lens, the Fourier transformation is obtained only in the
Fraunhofer approximation, which is more restrictive.
Image formation with a lens
U  x, y , 0  
Assume a positive, aberration-free thin lens and monochromatic light.
Free space propagation as a convolution (Fresnel):
g  x, y    f  x, y  h  x  x, y  y  dxdy

ieikd

d

f  x, y  e
i


 x  x 2   y  y  2
d
U  x, y , 0  
x1 , y1
d1
  
dx dy
di
U  x, y, d1  U  xi , yi , di 

To find h, we replace f(x’,y’)  U(x,y,-d1)  (x-x1,y-y1,-d1):
U  x, y , 0   
U  x, y , 0

ie
 ikz
z
   x  x1 , y  y1 , d1  e
2
2
x  x   y  y 

 i
z
dxdy   e

  U  x, y , 0  P  x, y  e

i

k 2 2
x y
2f

U  xi , yi , di   h  xi , yi , x1 , y1    U  x, y , 0  e
i
k
2 di
 x  x   y  y  
2
i
2
i
dxdy

  P  x, y  e

k 1 1 1 
i     x2  y 2
2  d1 di f 


e
 x x   y y  
ik  i  1  x   i  1  y 
 di d1   di d1  
dxdy
2
2
x  x1    y  y1 

 ik
2 d1
Image formation with a lens
h   P  x, y  e
k 1 1 1 
i     x2  y 2
2  d1 di f 


=0

e
 x x   y y  
ik  i  1  x  i  1  y 
 di d1   di d1  
1 1 1
 
di f d1
dxdy
h  xi , yi , x1 , y1    P  x, y  e
d1
The lens law
2
 x  Mx1  x   yi  My1  y 
 di  i
  e
di
U  x, y, d1  U  xi , yi , di 
M
dxdy

Neglect P(x,y):
U  x, y , 0  
x1 , y1
In this case, the impulse response becomes
i
U  x, y , 0  
di
d1
Magnification
x

i 2  x1  i
M

yi
 
 x   y1  M
 
 
 y
 
dxdy

xi
yi 

   x1  , y1  
M
M

Imaging by a lens:
y
 x

 U  xi , yi , di   U   1 ,  1 , d1 
 M M




ei 2 xx dx    x 
• Inversion
• Magnification
Imaging examples
The lens law
U  x, y 
f
f
f
f
 x y
U  , 
 4 4
U  x, y 
d1  5 f 4
F
f
f
f
U  x, y 
F
f
 x y 
A
,

f
 f 

U  x, y 
F
f
f
f1
f1
f
M
di
d1
M 
f2
f1
U   x,  y 
f
f
 x y 
A
,

f f 
f1
f
Magnification
F
di  5 f
 x y
U  , 
 2 2
f2=2f
f2
f2
F
F
F
d1  2 f
1 1 1
 
di f d1
F
di  2 f
U   x,  y 
The 4-f system
Imaging examples
A  x , y 
U  x, y 
f
Object
f
f
Mask
U   x,  y 
f
Image (inverted)
Fourier optics and imaging
• Linear optical systems
• Fresnel diffraction
• Fraunhofer diffraction
• The lens
• Optical resolution
Point-spread function (PSF)
PSF ↔ Impulse response
Diffraction limited (also “Fourier limited”) system:
Perfect spherical wave
Point object
Image
Point-spread function (PSF)
U  x, y , 0  
PSF ↔ Impulse response
h  xi , yi , x1 , y1    P  x, y  e
k 1 1 1 
i     x2  y 2
2  d1 di f 


e
 x x   y y  
ik  i  1  x  i  1  y 
 di d1   di d1  
dxdy

d1
di
U  x, y, d1  U  xi , yi , di 
Consider:
• An impulse at x1=y1=0
• An imaging condition:
x1 , y1

0, 0
U  x, y , 0  
1 1 1
 
d1 d i f
 h  xi , yi    P  x, y  e
 xx y y
i 2  i  i 
  di  di 

  P   di x,  di y  e

Circular pupil function:
(D - lens diameter)

1, x 2  y 2  D 2
P  x, y   

0, otherwise
  x2  y 2
1,   D 2
P  
0, otherwise
dxdy
x’=x/λdi
(neglect constants)
i 2  xi x  yi y 
dxdy
2D Fourier transform
of the scaled pupil
PSF
FT P    
J1  2  

  D 
2 J1 


d
i 

h  x, y   h  0, 0 
 D
 di
 s  1.22
di
h
di
D
2
yi

d
 s  1.22 i
D
“Airy disk”
xi
Airy disk of microscope objectives

D 
D
NA  n  sin   n sin  arctan
n
2f 
2f

f
1
In photography: “angular aperture” or “f-number”
F#  
D 2  NA
D
f
n
θ
 n 1
  32
  48
  58
NA  0.6
NA  0.8
NA  1.3
Reminder
Gaussian beams - properties
Beam divergence
z
 z 
W  z   W0 1   
 z0 
W
z0  W  z   0 z
z0
NA for Gaussian beams
2
W0
U  r   A0
e
W  z
W0 W0

0 



2
NA  n sin  0  n
z0  W0  W0
 W0
4 
Thus the total angle is given by 2 0 
 2W0

2
W 2  z
e
 z 
W  z   W0 1   
 z0 
 ikz ik
2
2 R z 
2
  z0  2 
R  z   z 1    
  z  
  z   tan 1
W0 
z
z0
 z0

2W0  2 0 
4

 i  z 
Resolution
The final image is a convolution of the perfect image with
the system’s impulse response:
y
 x

U  xi , yi , di   h  xi , yi  U   i ,  i , d1 
M
 M

Convolution

f h 
 f   g  t    d

2D convolution examples
Quantifying resolution
The Rayleigh criterion
Rayleigh criterion
“Two point sources are just resolved if they have an angular
separation equal to the angular radius of the Airy disk.”
1
sin  r  1.22  
D
f
For an ideal lens: l  1.22  
D
1
For a microscope objective: l  0.61  
NA
NA  n  sin   n
D
2f
Effect of noise on resolution
Summary
An arbitrary wave may be analyzed as a superposition of plane waves.
U(x,y,0)=f(x,y)e-ikz can be represented
as a combination of spatial harmonics:
Fresnel approximation
ieikz
U  xi , yi , z  
z
Propagation of the angular spectrum
 U  x, y, 0  e
i
 
 x  x 2  yi  y 2 
 z  i

A  x , y , z   A  x , y ,0  e
At its focal plane a lens performs a Fourier transform
of the incoming field
 x y 
iei 2 kf
g  x, y  
F
,
f

f
 f 

The 4-f system allows Fourier domain image manipulations
 s  1.22
di
D

2
2
ikz i z  x  y
ieikz i 2kz  xi2  yi2 
e
FT U  x, y, 0 
Fraunhofer approximation U  xi , yi , z  
z
The PSF of a lens is
limited by its pupil function
dxdy
e
