336533 סרוק
14:00-15:00 ' ג םוי : הלבק תועש , ןילי ריבד ר " ד : הצרמ
.
ךשמהב ועבקי הלבק תועש , ןלוג רואיל : לגרתמ
Saleh & Teich, “Fundamentals of Photonics”, 2 nd edition : רפס
.) לוגרת םייתעש , רועיש םייתעש ( תוכז תודוקנ 3 : ףקיה
.
םיליגרת 20% , הניחב 80% : ןויצ הנבמ
.
םאתהב םינויצ ונתניו , תורעה םע ורזחוי , וקדבי םיליגרתה לכ .
לכה ךסב םיליגרת השימחכ ונתני *
.
תומדוק תוניחבלו םיליגרתל תונורתפ ומסרופי אל .
ספא ןויצ לבקי דעומב שגוי אלש ליגרת
תולאש עבראכ לולכת , םידיינ םיבשחמו םירפס דבלמ חותפ רמוח םע ) תועש שולשכ ( הניחבה *
.
םיליגרתה דחא ךותמ הלאש לע תססובמה תחא ןכותמ , תוחותפ

“… the market for biomedical optics doubled from 1985 to 1995, and then tripled from
1995 to 2005 to reach just over $6 billion
… 5-fold increase is expected over the next five years.
… One example is the PillCam developed by Given Technology in Israel, which optics.org reported on a few months ago. The 11 ×31 mm capsule contains automatic lighting control, as well as tiny cameras at each end that capture four images per second for up to 10 hours. Although the technology is nothing new, the device enables clinicians to see parts of the body that cannot be reached by an endoscope.
“… to succeed in this market companies must be sure that their optics-based technology addresses a specific medical need. Indeed, analysis by medical device maker Johnson & Johnson reveals that some 87% of all innovations originate from the clinicians working in hospitals
… companies must therefore attempt to focus on highvalue solutions that address real-life problems …”
David Benaron, CEO of Spectros Corp. 2007

Photonics West conference in San Jose, CA (BIOS, LASE,
MOEMS-MEMS, OPTO)
BIOS (biomedical optics) conference
2006: 65/260 pages
2007: 81/284 pages
2008: 81/308 pages
2009: 95/324 pages
2010: moving to a larger convention center in San Francisco.

Laser invention
Optical technologies
CCD technologies
Better light handling
Improved detection, imaging
Biomedical applications

, תויטפוא הדידמ תוטישב , הקיטפואב םיבחר תודוסי קינעהל
טנדוטסל םילכ תתל תנמ לע , הימדהבו
האופר ויבב
הקיטפוא
תויטפוא הימדהו הדידמ תוטישב
חותיפלו רקחמל תושדח תויגולונכט
ןיבהל
שמתשהל
םימייק םייטפוא םיעצמא תונשל
חתפל •
•
•
•

1. Maxwell equations, wave equation
2. Electromagnetic waves, Gaussian beams
3. Fourier optics, the lens, resolution
4. Light-matter interaction: scattering, absorption
5. Fluorescence, photo dynamic therapy
6. Fundamentals of lasers
7. Lasers in medicine
8. Basics of light detection, cameras
9. Microscopy, contrast mechanism
10. Confocal microscopy, laser scanning microscopy
11. Nanoparticles in biomedical optics
12. Optical fibers and waveguides
13. Endoscopy
14. Advanced microscopy techniques, super resolution
םילג תואוושמ , לווסקמ תואושמ
תויניסואג םיינרק , םיטנגמורטקלא םילג
הדרפה , השדעה , היירופ תקיטפוא
העילב , םוטאה , רוזיפ : המקר רוא היצקארטניא
ימניד וטופ לופיט , היצנסרואולפ
םירזייל תונורקע
.1
.2
.3
.4
.5
.6
האופרב םירזייל
תומלצמ , רוא יוליג תונורקע
.7
.8
תוידוגינ , היפוקסורקימ .9
קרוס רזייל תיפוקסורקימ , תילקופנוק היפוקסורקימ .10
תיאופר ויב הקיטפואב םיקיקלח וננ
.11
םיטפוא םיביס .12
היפוקסודנא .13
היצולוזר רפוס , תמדקתמ היפוקסורקימ .14

1. Maxwell equations
2. The wave equation
3. Maxwell equations in medium
4. Helmholtz equation
5. Electromagnetic waves: plane, spherical, Gaussian beams
6. Properties of Gaussian beams

An electromagnetic field is described by two related vector fields that are functions of position and time:
Electric field:
Magnetic field:
E
H
 
 
In free space:
1.
2.
 H  
0

0
 E
 t
 H
 t
3.
0
4.
  H 
0 where
 E
 x x


 E
 y z

 E y
 y

 E y
 z

 E
 z z


,


 E
 z x

 E
 x z


,


 E y
 x

 E
 y x



0
 
1 36
  
10

9
F m
Electrical permittivity of free space in MKS units:

0

4
 
10

7
H m
Magnetic permeability of free space

1.
2.
 H  
0

0
 E
 t
 H
 t

E
 
     
0
 H
 t


3.
0
4.
  H 
0
Speed of light c in free space:
0

1
 
0 0
  8
 
E 

      2   
 H
 t
3
1
 2
E   
0
 2
E   
0



H


 t
 
0
E t

 t m s
 2 E   
0 0
 2
E
 t 2

0
 2 E 
1
 2
E c
0
2
 t
2

0

E
  
 2
E
For each component:
 2
E i

1
 2 E i c
0
2  t
2

0 i

, ,
Similar procedure is followed for H

Assuming no free electric charges or currents.
Electric flux density:
Magnetic flux density:
D
B
 
 
D  
0
E P
B  
0
H  
0
M
In source-free media:
1.
2.
 H 
 B
 t
 D
 t
3.
 D 
0
4.
0
In free space:
P 
0
M 
0
D  
0
E
B  
0
H
E
D
P
Nucleus
Electron cloud

1. A dielectric medium is said to be linear if the vector field P ( r,t ) is linearly related to the vector field E ( r,t ) . The principle of superposition then applies.
2. The medium is said to be nondispersive if its response is instantaneous, i.e., if P at time t is determined by E at the same time t and not by prior values of E . Nondispersiveness is clearly an idealization since all physical systems, no matter how rapidly they may respond, do have a response time that is finite.
3. The medium is said to be homogeneous if the relation between P and E is independent of the position r .
4. The medium is said to be isotropic if the relation between the vectors P and E is independent of the direction of the vector E , so that the medium exhibits the same behavior from all directions. The vectors P and E must then be parallel.

P   
0
E
D  
0
E P   
0
E 
0
E
Or: where:
D   E

1
  
0
-
E
D
P
+
E p
E int
Electric permittivity Electric susceptibility
In isotropic media:
E int
 E - E
P
!
 We will assume that P is linear with E , which is valid for low field intensities.

n
The wave equation (in a medium):
 2 E 
1
 2 E c
2  t
2

0
Speed of light in free space: c
0

1
 
0 0
  8 m s where the speed of light in the medium is denoted c : c

1

The ratio of the speed of light in free space to that in the medium, c
0
/ c , is defined as the refractive index n : n
 c c
0

 
 
0 0
For a nonmagnetic material,

=

0 and:
 
0

1
   n



0

1
 

In a homogeneous medium, all components of the fields
E, D, H, B are continuous functions of position. At the boundary between two dielectric media, in the absence of free electric charges and currents, the tangential components of the electric
E and magnetic
H fields, and the normal components of the electric
D and magnetic
B flux densities must be continuous.
E =
0

The flow of electromagnetic power is governed by the Poynting vector :
Which is orthogonal to both
E and
H
.
The optical intensity I (power flow across a unit area normal to the vector
S
) is the magnitude of the time-averaged Poynting vector
S
.
H
E
  
S
S
The average is taken over times that are long in comparison with an optical cycle.

For the case of monochromatic electromagnetic waves in an optical medium, all components of the fields are harmonic functions of time with the same frequency

.
 
2

Angular frequency frequency
E
H
  
Re
 
  
Re
 
Similarly:
P
D
M
B
  
Re
  
Re
  
Re
  
Re
 

 

 
“complex-amplitude” vectors


Re

 H 


 D


 t
Re t
  The Maxwell equations become
(source-free medium, monochromatic):
1.

H
 
If the operations on the complex fields are linear, one may drop the symbol Re and operate directly with the complex functions. The real part of the final expression will represent the physical quantity in question.
 e
 


 
 




 t
D r

 

 

  t e
2.
3.
4.
E

0
0
Also,
D
 
0
E

P
B
 
0
H
 
0
M



1
4
Re
    
2
 *
Ee E e



2
 * 
E
*   e i 2
 t   e
 i 2
 t
E
* 
H
*

 *
He H e

The last two terms on the right oscillate at optical frequencies and are therefore will be washed out by the averaging process, which is slow in comparison with an optical cycle:
S 
1
4

 * 
E
* 
H


1
2

S

S
*


Re
 
Where the new vector S

1
2
 may be regarded as a complex Poynting vector.
*

The optical intensity is the magnitude of the vector S : I

S


H
 
If we use the “material equations” for monochromatic waves:
1.
D
 
E
2.
E

3.
0
B
 
H
4.
0
We obtain the Maxwell's equations which depend solely on the complex-amplitude vectors E and H :
1.

H
 i

E
2.
E i

H
3.
0
4.
 
H

0
* linear, non-dispersive, homogenous, isotropic, source-free medium, monochromatic.

Substitute the complex amplitude notation
E   
Re
  into the wave equation yields:
 2 E 
 2
E

1 c
2  t
 
2 c
2
 2 E
2
E


0
0
 2
E
 
2
E

0

2

2
E k E

0 c

1
 k
 nk
0
   or:  2
U
 2 k U

0
Helmholtz equation
U
   represents the complex amplitude of any of the components of the electric and magnetic fields:
U
 x
, y
, z
, x
, y
, H z

Assumptions:
Medium: linear, homogenous, non-dispersive, isotropic.
Light: monochromatic
- Plane waves
- Spherical waves
- Gaussian waves

Solutions for the wave equation:
The real electric field:
  
E e
0
  
H e
0
E   
Re


Re
E
0

 
E e
0
 e

 cos

   t
 k r k
 r
:
E 
,
0
  cos

   t
0

:
E 
,
0
  constant
“wavelength”
 
2
 k

Proof: Plane waves satisfies the Helmholtz equation:
  2
E k E

0
  
E e
0 ik r
 2
E e
0
  k
2

E e
0
 
2
 2 ik E e k E e
0 0

0

0 k
2 k
2 
0 k
  nk
0
Implying that the length of the wave vector parameter k in Helmholtz equation: k
 nk
0
 k
  

 n
 c c
0
Also: k
 n
2


0
 n c
0
2


 n
2
 c
0
 n
 c
0

  
H e
0
  
E e
1.
 
H
 i

E
2.
E i

H
3.
4.
0
 
H

0 Substitute in Maxwell equations 1 & 2 yields (exercise): k

H
0
  
E
0 k

E
0
 
H
0
E is perpendicular to both k and H
H is perpendicular to both k and E
Transverse electromagnetic ( TEM ) wave:

The ratio between the amplitudes of the electric and magnetic fields is known as the impedance of the medium:
 
E
0
H
0
   the impedance of free space:

0
  
0 0

377
 
120
     

0 n
The complex Poynting vector:
S

1
2
 *

I

S

1
2
E H
*
0 0

E
0
2

2
Example: an intensity of 10 W/cm 2 in free space corresponds to an electric field of 87 V/cm:
E
0

2

I

2 377 10

87
V cm

Another simple solution (proof in exercise) of the Helmholtz equation is the scalar spherical wave:
U is spherically symmetric:
  
A r
0 e ikr
    
Reminder: Laplacian in radial coordinates:
1

2
 r
2  r  r
 r 
An oscillating dipole radiates a wave with features that resemble the scalar solution. For points at distances from the origin much greater than a wavelength ( r
»λ or kr
»2π ), the complexamplitude vectors may be approximated by:
  
E
0
 sin
    ˆ
  
H
0
 sin
    ˆ

Spherical wave:
Plane wave:

A
0 r
  
E e
0 e ikr
A paraxial wave is a plane wave traveling along the z direction ( e -ikz , with k= 2
π/λ ), modulated by a complex envelope that is a slowly varying function of position, so that its complex amplitude is:
Paraxial waves:
      ikz
‘Carrier’ plane wave
Slowly varying complex amplitude (in space)

 2
U
 2 k U

0
Substitute the paraxial wave into the Helmholtz equation:
 2
 x 2 e

 2
 y 2 e

 2


2 x
U
2



2 y
U
2



2 z
U
2

 z 2
 2
2
 
  
0

0
Paraxial wave
     ik z

2
 x 2 e


2
 y 2 e


2
 z 2 e

2 ik
  
 z e
      
0


2 x
A
2



2 y
A
2



2 z
A
2

2 ik

A
 z

0



2 x
A
2



2 y
A
2



2 z
A
2

2 ik

A
 z

0
Paraxial wave
     ik z
We now assume that the variation of A ( r ) with z is slow enough, so that:  
 
2 z
A




 
2 z
A
2
2

 k
2 k A


A z



2 x
A
2



2 y
A
2

2 ik

A
 z

0
These assumptions are equivalent to assuming that and sin tan
 
 
Paraxial Helmholtz equation:
 2 
T
A 2 ik

A
 z

0
Transverse Laplacian:
 
T


2 x
2
A



2 y
A
2


2

T
A 2 ik

A
 z

0
Paraxial Helmholtz equation
Substitute the paraxial wave yields a solution of the form:
Where and
     ik z

A
1
  into the paraxial Helmholtz equation e
 ik
2
 2
 
   
0

2  x
2  y
2 z
0
: Rayleigh range q ( z ) can be separated into its real and imaginary parts:
1
 

1
 
 i


W
2
 
Where W ( z ) : beam width
R ( z ) : wavefront radius of curvature

The full Gaussian beam:
  
A
0
With beam parameters:
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 i
  
A
0

A iz
1 0
  
W
0
1

 z z
0


2
   z



1


 z z
0


2



W
0
 tan

1 z z
0


 z
0
A
0 and z
0 are two independent parameters which are determined from the boundary conditions. All other parameters are related to z
0 and
 by these equations.

  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  
     2
I
0

A
0
2

I
 
, z

I


W
0



2 e

W
2
2
 2
 
At any z , I is a Gaussian function of

. On the beam axis:
I
  
I
0



W
0
 



2

1
 
I
0 z z
0

2
  
W
0
1

 z z
0


2
   z



1

 z
0
 2
 z




W
0
 tan

1 z z
0


 z
0
- Maximum at z= 0
- Half peak value at z=±z
0 z= 0
1 z=z
0
1 z= 2 z
0
1

  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  
  
W
0
1 z
0
2
  
W
0
1

 z z
0


2
   z



1

 z
0
 2
 z 



W
0
 tan

1 z z
0


 z
0
2 W
0

  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  
  
W
0
1
 z
 2 z z

0
 z
0

  
W
0
1

 z z
0


2
   z



1

 z
0
 2
 z 



W
0
 tan

1 z z
0


 z
0

0

W
0 z
0 z

W
0
 z
0


W
0
W
0
2



W
0
Thus the total angle is given by 2

0

4


2 W
0
2 W
0

2

0

4



  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  
W
0


 z
0
2 z
0

2

W
0
2

  
W
0
1

 z z
0


2
   z



1

 z
0
 2
 z 



W
0
 tan

1 z z
0


 z
0 z
0
: Rayleigh range

  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  
  
A
0
W
0
  e

 2
W
2
 
 e
 ikz ik
2
 2
 
 i
  
    kz
 tan

1 z z
0
  
W
0
1

 z z
0


2
   z



1

 z
0
 2
 z 



W
0
 tan

1 z z
0


 z
0
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 .

  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  

, z
  kz
 tan

1 z z
 k
0
2

2
    
W
0
1

 z z
0


2
   z



1

 z
0
 2
 z 



W
0
 tan

1 z z
0


 z
0
~ spherical wave

  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  
Plane wave
  
W
0
1

 z z
0


2
   z



1

 z
0
 2
 z 



W
0
 tan

1 z z
0


 z
0
Spherical wave
Gaussian beam

  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  
  
W
0
1

 z z
0


2
   z



1

 z
0
 2
 z 



W
0
 tan

1 z z
0


 z
0
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
W
0

W
1



W
2

R

2 located to the left at a distance z

1

R

 
W
2

2

  
A
0
W
0
  e

 2
W
2
   e
 ikz ik
2
 2
 
 i
  
The complex amplitude induced by a thin lens of focal length f is proportional to exp ( ik

2 /2 f ) .
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 altered ( W'=W ), but the wavefront does.
Consider a Gaussian beam centered at z= 0 , with waist radius W
0
, transmitted through a thin lens located at position z . The phase of the emerging wave therefore becomes: kz
 k

2
2 R
 k
2 f
2
 kz
 k

2
 
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' .

The magnification factor M evidently plays an important role. The waist radius is magnified by M , the depth of focus is magnified by M 2 , and the divergence angle is minified by M .

For a lens placed at the waist of a Gaussian beam ( z= 0 ), the transmitted beam is then focused to a waist radius
W
0
’ at a distance z' given by:
W
0
'

1

W
0
 z
0 f

2 z '

1
 f
 f z
0

2
M

1
1
  z f
0

2 f z
0 f z
0
W
0 f


A
1
  e
 ik
2
 2 q z
,
   
The ABCD Law
The q -parameters, q
1 and q
2
, 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: q
2

Aq B
1

Cq D
1

1
 

1
 
 i


W
2
 
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 q
2
= q
2
+ d .
*Generality of the ABCD law
The ABCD law applies to thin optical components as well as to propagation in a homogeneous medium.
All of the paraxial optical systems of interest are combinations of propagation in homogeneous media and thin optical components such as thin lenses and mirrors. It is therefore apparent that the ABCD law is applicable to all of these systems. Furthermore, 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.

Maxwell equation: linear, non-dispersive, homogenous, isotropic, source-free medium, monochromatic light
 
H
 i

E
E i

H
0
 
H

0
Helmholtz equation
 2
U
 k U

0
H
Plane waves:
  
E e
0
Spherical waves:
  
A r
0 e ik r
Paraxial Helmholtz equation:
 2 
T
A 2 ik
The Gaussian beam:
  
A
0
W
0

A
 z
 

0 e

 2
W
2
 
 e
 ikz ik
2
 2
 
 i
  
E k