z - Technion moodle

Report
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
Reminder
Plane waves
Solutions for the Helmholtz equation:
(proof in next slide)
The real electric field:
E  r   E0e

ik r
H  r   H 0eik r

E  r , t   Re E  r  eit


 Re E0eik r eit
E0 is independent of r

infinite field!
r k:
r k:
“wavelength”


E  r , t   E0 cos k  r  t


E  r , t0   E0 cos k  r  t0
E  r , t0   constant

k
2
k
Gaussian beams – the paraxial wave
A0 ikr
e
Spherical wave: U  r  
r
Plane wave: E  r   E0 eik r
A paraxial wave is a plane wave traveling mainly along the z direction (e-ikz, with k=2π/λ),
modulated by a complex envelope that is a slowly varying function of position, so that its
complex amplitude is given by:
ikz
U  r   A r  e
‘Carrier’ plane wave
Slowly varying complex amplitude (in space)
The paraxial Helmholtz equation
Substitute the paraxial wave
into the Helmholtz equation:
2U  k 2U  0
 2U  2U  2U
2



k
U r   0
2
2
2
x
y
z
2
 ik  z
 2 A  r   ik  z  2 A  r  ik z   A  r  e 
2
 ik  z
e

e


k
A
r
e
0


2
2
2
x
y
z
Paraxial wave
U  r   A r  eikz
 2 A  r  ik z  2 A  r  ik z  2 A  r  ik z
A  r  ik z
e

e

e
 2ik
e
 k 2 A  r  eik z  k 2 A  r  eik z  0
2
2
2
x
y
z
z
2 A 2 A 2 A
A



2
ik
0
2
2
2
x
y
z
z
 A  A  A
A



2
ik
0
2
2
2
x
y
z
z
2
2
2
Paraxial wave
U  r   A r  eikz
We now assume that the variation of A(r) with z is
slow enough, so that:
2
 A
2

k
A
 z 2
 2
  A  k A
 z 2
z

These assumptions are
equivalent to assuming
that sin   
and tan   
2 A 2 A
A
 2  2ik
0
2
x
y
z
Paraxial Helmholtz equation:
Transverse Laplacian:
A
2
T A  2ik
0
z
2
2

A

A
T2  2  2
x
y
Gaussian beams
A
 A  2ik
0
z
Paraxial Helmholtz equation
2
T
One solution to the paraxial Helmholtz equation of the slowly varying complex
2
amplitude A, has the form:
 ik
A1
Ar  
e
q z
Where q  z   z  iz0
and
2 q z 
z0: “Rayleigh range”
 2  x2  y2
q(z) can be separated into its real and imaginary parts:
1
1


i
q  z  R  z  W 2  z 
Where
W(z): beam width
R(z): wavefront radius of curvature
Gaussian beams
The full Gaussian beam:
2
2
W0 W 2  z  ikz ik 2 R z  i  z 
U  r   A0
e
e
W  z
With beam parameters:
 z 
W  z   W0 1   
 z0 
2
A0  A1 iz0
  z0  2 
R  z   z 1    
  z  
z
  z   tan 1
z0
W0 
 z0

A0 and z0 are two independent parameters which are determined from the boundary
conditions. All other parameters are related to z0 and  by these equations.
Gaussian beams - properties
Intensity
I r   U r 
I 0  A0
2
2
W0
U  r   A0
e
W  z
2
 W0 
 I   , z   I0 
 e
W  z  

22
W
2
2
W 2  z
e
 z 
W  z   W0 1   
 z0 
z
 ikz ik
2
  z0  2 
R  z   z 1    
  z  
At any z, I is a Gaussian function of . On the beam axis:
  z   tan 1
2
 W0 
I0
I    0, z   I 0 


2
W
z



 1   z z0 
I

(Lorentzian)
W0 
z
z0
 z0

1/2
0
z0
z
- Maximum at z=0
- Half peak value at z = ± z0
z=0
z=z0
1
1
z=2z0
1
2
2 R z 
i  z 
Gaussian beams - properties
Beam width
W0
U  r   A0
e
W  z

2
W 2  z
e
 z 
W  z   W0 1   
 z0 
2
  z0  2 
R  z   z 1    
  z  
 z
W  z   W0 1   
 z0 
1 e line 1 e2 in intensity 
2W0
2
  z   tan 1
W0 
 z0

z
z0
 ikz ik
2
2 R z 
i  z 
Gaussian beams - properties
Beam divergence
z
z0
 z 
W  z   W0 1   
 z0 
W
 W z  0 z
z0
2
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  
 W02
z0 

W0 W0

0 


2
z0  W0  W0
Thus the total angle is given by
W0
U  r   A0
e
W  z

4 
20 
 2W0
  z   tan 1
W0 
z
z0
 z0

2W0  2 0 
4

i  z 
Gaussian beams - properties
Depth of focus
 z0
W0 

 W02
2 z0  2

W0
U  r   A0
e
W  z

2
W 2  z
e
 z 
W  z   W0 1   
 z0 
2
  z0  2 
R  z   z 1    
  z  
  z   tan 1
W0 
z
z0
 z0

z0: Rayleigh range
The total depth of focus is often defined as twice the Rayleigh range.
 ikz ik
2
2 R z 
i  z 
Gaussian beams - properties
Phase
W0
U  r   A0
e
W  z
W0
U  r   A0
e
W  z

2
W 2  z
e
 ikz ik
2
2 R z 
i  z 
z
  0, z   kz  tan
z0
1

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

A. Ruffin et al., PRL (1999)
The total accumulated excess retardation as the wave travels from - to  is .
This phenomenon is known as the Gouy effect.
i  z 
Gaussian beams - properties
Wavefront
2
z
k

   , z   kz  tan 1 
z0 2R  z 
W0
U  r   A0
e
W  z

2
W 2  z
e
 z 
W  z   W0 1   
 z0 
 ikz ik
2
2 R z 
i  z 
2
  z0  2 
R  z   z 1    
  z  
  z   tan 1
W0 
z
z0
 z0

~ spherical wave
~ plane wave
  z0  2 
R  z   z 1    
  z  
z 
R  z  
z
Gaussian beams - properties
Propagation
W0
U  r   A0
e
W  z

2
W 2  z
e
 z 
W  z   W0 1   
 z0 
W
W0
R
 ikz ik
2
2 R z 
i  z 
2
  z0  2 
R  z   z 1    
  z  
  z   tan 1
z
W0 
z
z0
 z0

Consider a Gaussian beam whose width W and radius of curvature R are known at a particular
point on the beam axis.
The beam waist radius is given by
W0 
W
1   W  R 
2
located to the right at a distance
z
R
1    R W

2 2
2
Gaussian beams - properties
Propagating through lens
The complex amplitude induced by a thin lens of focal length f is proportional to exp(-ik2/2f).
When a Gaussian beam passes through such a component, its complex amplitude is multiplied
by this phase factor. As a result, the beam width does not change (W'=W), but the wavefront
does.
W0
U  r   A0
e
W  z

2
W 2  z
e
 ikz ik
2
2 R z 
i  z 
Consider a Gaussian beam centered at z=0, with waist radius W0, transmitted through a thin
lens located at position z. The phase of the emerging wave therefore becomes (ignore sign):
kz  k
2
2R
  k
2
2f
 kz  k
2
2R '

Where
1 1 1
 
R' R f
 The transmitted wave is a Gaussian beam with width W'=W and radius of curvature R'. The
sign of R is positive since the wavefront of the incident beam is diverging whereas the opposite
is true of R'.
Gaussian beams - properties
Propagating through lens
The magnification factor M plays an
important role: The waist radius is
magnified by M, the depth of focus is
magnified by M2, and the divergence
angle is minified by M.
Gaussian beams - properties
Beam focusing
For a lens placed at the waist of a Gaussian beam (z=0),
the transmitted beam is then focused to a waist radius
W0’ at a distance z' given by:
W0 ' 
z'
W0
z
1   z0 f 
2
f
z
1   f z0 
M  z  0 
f
0


f
f
f
W0 
W

z0
 W02  0  W0
f
0

f
2
1
1   z0 f 
z
2
W0 
f
0


f
z0
f
 W0
Gaussian beams - properties
The ABCD law
Reminder:
A
Ar   1 e
q z
 ik
2
2 q z 
, where q  z   z  iz0
1
1



i
or:
q  z  R  z  W 2  z 
The ABCD Law
The q-parameters, q1 and q2, of the incident
and transmitted Gaussian beams at the input
and output planes of a paraxial optical system
described by the (A,B,C,D) matrix are related
by:
Aq1  B
q2 
Cq1  D
Example: transmission Through Free Space
When the optical system is a distance d of free space (or of any homogeneous medium),
the ray-transfer matrix components are (A,B,C,D)=(1,d,0,1) so q2 = q1 + d.
*Generality of the ABCD law
The ABCD law applies to thin optical components as well as to propagation in a
homogeneous medium. Since an inhomogeneous continuously varying medium may be
regarded as a cascade of incremental thin elements, the ABCD law applies to these systems
as well, provided that all rays (wavefront normals) remain paraxial.

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