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‫קורס ‪336533‬‬
‫‪Fundamentals of biomedical‬‬
‫‪optics and photonics‬‬
‫יסודות אופטיקה ופוטוניקה ביו‪-‬רפואית‬
‫מרצה‪:‬‬
‫ד"ר דביר ילין‪,‬‬
‫שעות קבלה‪ :‬יום ג' ‪14:00-15:00‬‬
‫מתרגל‪:‬‬
‫ליאור גולן‪,‬‬
‫שעות קבלה יקבעו בהמשך‪.‬‬
‫ספר‪:‬‬
‫‪Saleh & Teich, “Fundamentals of Photonics”, 2nd edition‬‬
‫היקף‪:‬‬
‫‪ 3‬נקודות זכות (שעתיים שיעור‪ ,‬שעתיים תרגול)‪.‬‬
‫מבנה ציון‪:‬‬
‫‪ 80%‬בחינה‪ 20% ,‬תרגילים‪.‬‬
‫* ינתנו כחמישה תרגילים בסך הכל‪ .‬כל התרגילים יבדקו‪ ,‬יוחזרו עם הערות‪ ,‬וינתנו ציונים בהתאם‪.‬‬
‫תרגיל שלא יוגש במועד יקבל ציון אפס‪ .‬לא יפורסמו פתרונות לתרגילים ולבחינות קודמות‪.‬‬
‫* הבחינה (כשלוש שעות) עם חומר פתוח מלבד ספרים ומחשבים ניידים‪ ,‬תכלול כארבע שאלות‬
‫פתוחות‪ ,‬מתוכן אחת המבוססת על שאלה מתוך אחד התרגילים‪.‬‬
“… 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
‫מטרת הקורס‬
‫להעניק יסודות רחבים באופטיקה‪ ,‬בשיטות מדידה אופטיות‪,‬‬
‫ובהדמיה‪ ,‬על מנת לתת כלים לסטודנט‬
‫• להבין אופטיקה‬
‫• להשתמש בשיטות מדידה והדמיה אופטיות בביו‪-‬רפואה‬
‫• לשנות אמצעים אופטיים קיימים‬
‫• לפתח טכנולוגיות חדשות למחקר ולפיתוח‬
Course outline
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
‫ משוואות גלים‬,‫ משואות מקסוול‬.1
‫ קרניים גאוסיניות‬,‫ גלים אלקטרומגנטים‬.2
‫ הפרדה‬,‫ העדשה‬,‫ אופטיקת פורייה‬.3
‫ בליעה‬,‫ האטום‬,‫ פיזור‬:‫רקמה‬-‫ אינטראקציה אור‬.4
‫דינמי‬-‫ טיפול פוטו‬,‫ פלואורסנציה‬.5
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, laser scanning microscopy
11. Nanoparticles in biomedical optics
12. Optical fibers and waveguides
13. Endoscopy
14. Advanced microscopy techniques, super resolution
‫ מצלמות‬,‫ עקרונות גילוי אור‬.8
‫ ניגודיות‬,‫ מיקרוסקופיה‬.9
‫ מיקרוסקופית לייזר סורק‬,‫ מיקרוסקופיה קונפוקלית‬.10
‫רפואית‬-‫חלקיקים באופטיקה ביו‬-‫ ננו‬.11
‫ סיבים אופטים‬.12
‫ אנדוסקופיה‬.13
‫רזולוציה‬-‫ סופר‬,‫ מיקרוסקופיה מתקדמת‬.14
Lectures 1-2
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
Maxwell equations
An electromagnetic field is described by two related vector fields that are functions
of position and time:
Electric field: E  r , t 
Magnetic field: H  r , t 
In free space:
E
  H  0
t
H
2.  E   
0
t
1.
3.
E  0
4.
 H  0
where
 E 
E x E y E z


x
y
z
 E E y   E x E z   E y E x 
 E   z 


,

,

y

z

z

x

x
y 
 

 
 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
E
t
H
2. E   0
t
1.   H   0
The wave equation
3. E  0

H 
    E      0


t 

 H 
2
    E   E    0  

 t 




4.   H  0


3
1
 E   0
2
 E   0
Speed of light
in free space:
c0 
1
0 0
 3 108

2

 H




  E      E   2E

t
  0 E t

t
 2E
 E   0 0 2  0
t
2
m
s
Similar procedure is followed for H
For each component:
2
1

E
 2E  2 2  0
c0 t
1  2Ei
 Ei  2
0
2
c0 t
2
i  x, y , z
Maxwell equations in medium
D   0E  P
Assuming no free electric charges or currents.
Electric flux density:
Magnetic flux density:
D r ,t 
B r ,t 
B  0H  0M
E
In source-free media:
B
t
D
2.   H 
t
1.
D
 E  
3.
D  0
4.
B  0
In free space:
P 0
M0
P
+
D   0E
B  0H
Nucleus
Electron cloud
Electromagnetic waves in dielectric media definitions
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.
Induced polarization
E
P   0 E
D
- P +
D   0E  P   0E   0 E
Or:
where:
D  E
   0 1   
Electric permittivity
Electric susceptibility
 We will assume that P is linear with E,
which is valid for low field intensities.
+
Ep
Eint
!
-
In isotropic media: Eint  E - E P
The refractive index (n)
The wave equation (in a medium):
1  2E
 E  2 2 0
c t
2
Speed of light in free space:
1
m
c0 
 3 108
s
 0 0
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, c0/c,
is defined as the refractive index n:
c0
 
n 

c
 0 0
For a nonmagnetic material, =0 and:

n
 1 
0
   0 1   
Boundary conditions
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
Poynting vector
The flow of electromagnetic power is governed
by the Poynting vector:
Which is orthogonal to both E and H.
S  E H
H
E
S
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.
I r ,t   S
The average is taken over times that are
long in comparison with an optical cycle.
Monochromatic EM waves
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 .
Angular
frequency


H  r , t   Re H  r  e 
E  r , t   Re E  r  eit
  2
frequency
Similarly:
it


D  r , t   Re  D  r  e 
M  r , t   Re M  r  e 
B  r , t   Re  B  r  e 
P  r , t   Re P  r  eit
it
it
it
“complex-amplitude” vectors
Maxwell equations in medium
(Complex amplitude)
D
t

 Re D  r  eit
t
H 

  Re H  r  e
it



The Maxwell equations become
(source-free medium, monochromatic):
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.

it


   H  r  e    D  r  eit 
t

eit   H  r   D  r  eit
t
  H  r   i D  r 
Also,
D  0E  P
B  0 H  0 M
1.
  H  i D
2.
  E  i B
3.
D  0
4.
B  0
Intensity and power
(Complex amplitude)
S  E H




 Re Eeit  Re Heit 


 
1
1
Eeit  E *e it  Heit  H *e it
2
2

1
E  H *  E *  H  e i 2 t E  H  e  i 2 t E *  H *
4


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:





1
1
*
*
S  E  H  E  H  S  S *  Re S
4
2
1
Where the new vector
S  E H*
2
may be regarded as a complex Poynting vector.
The optical intensity is the magnitude of the vector S:
I S
Linear, nondispersive,
homogenous, and isotropic media
1.
  H  i D
D E
2.
  E  i B
B  H
3.
D  0
4.
B  0
If we use the “material equations” for monochromatic waves:
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.
E  0
4.
H  0
* linear, non-dispersive,
homogenous, isotropic,
source-free medium,
monochromatic.
Helmholtz equation

E  r , t   Re E  r  eit
Substitute the complex amplitude notation
into the wave equation
yields:
2
1

E
 2E  2 2  0
c t
i 

2
 E
2
c2
E0
c
 2 E   2 E  0
 U k U 0
2
1

k  nk0   
2 E  k 2 E  0
or:

2
Helmholtz equation
where U  U  r  represents the complex amplitude of any of the components of
the electric and magnetic fields:
U  Ex , E y , Ez , H x , H y , H z
Elementary electromagnetic waves
Assumptions:
Medium: linear, homogenous, non-dispersive, isotropic.
Light: monochromatic
- Plane waves
- Spherical waves
- Gaussian waves
Plane waves
“wavelength”
E  r   E0eik r
|k |

2
Solutions for the wave equation:
H  r   H 0eik r
The real electric field:

E  r , t   Re E  r  eit


 Re E0 eik r eit


 E0 cos k  r  t

k r:
E  r , t0   cos k  r  t0
k r:
E  r , t0   constant


k
Plane waves
Proof: Plane waves satisfies the Helmholtz equation:
2 E  k 2 E  0
E  r   E0eik r
 2  E0 eik r   k 2  E0eik r   0
 
ik
2
E0 eik r  k 2 E0eik r  0
k 2  k 2  0
k  k  nk0
Implying that the length of the wave vector
parameter k in Helmholtz equation:
Also:
k of the plane wave must be equal to the
k  nk0    
k n
2
0
n

c
n

c0
2
2

n
n
c0 
c0
c0
Plane waves
H  r   H 0e
E  r   E0e
1.   H  i E
ik r
2.   E  i H
ik r
3.   E  0
Substitute in Maxwell equations 1 & 2 yields (exercise):
4.   H  0
k  H 0   E0
E is perpendicular to both k and H
k  E0   H 0
H is perpendicular to both k and E
Transverse electromagnetic (TEM) wave:
Intensity of TEM waves
The ratio between the amplitudes of the electric and magnetic fields is known as the
impedance of the medium:
  E0 H 0   
the impedance of free space:
The complex Poynting vector:
0  0  0  377     120   
 
0
n
1
S  E H*
2
E0
1
*
 I  S  E0 H 0 
2
2
2
Example: an intensity of 10 W/cm2 in free space corresponds to an electric field of 87 V/cm:
V
E0  2 I  2  377 10  87
cm
Spherical waves
Another simple solution (proof in exercise) of the Helmholtz equation
is the scalar spherical wave:
0 ikr
A
U r   e
r
U is spherically symmetric: U  r   U  r 
Reminder: Laplacian
in radial coordinates:
2 
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  r   E0  sin  U  r   ˆ
H  r   H 0  sin  U  r   ˆ
Gaussian beams – the paraxial wave
A0 ikr
e
Spherical wave: U  r  
r
ik r
Plane wave: E  r   E0 e
A paraxial wave is a plane wave traveling 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:
ikz
Paraxial waves:
U  r   A r  e
‘Carrier’ plane wave
Slowly varying complex amplitude (in space)
The paraxial Helmholtz equation
 2U  k 2U  0
Substitute the paraxial wave
into the Helmholtz equation:
 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
2
 ik  z
2
 ik  z
e

e

e

2
ik
e

k
A
r
e

k
A
r
e
0




x 2
y 2
z 2
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
2
T
Paraxial Helmholtz equation
Substitute the paraxial wave U  r   A  r  eik z into the paraxial Helmholtz equation
2
yields a solution of the form:
 ik
A1
Ar  
e
q z
Where q  z   z  iz0
and
2 q z 
z0: Rayleigh range
 2  x2  y 2
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
z
 0


W0 
z
z0
 z0

- 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 
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
W0
U  r   A0
e
W  z
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 W0

0 


2
z0  W0  W0
Thus the total angle is given by

4 
2 0 
 2W0
W0 
z
z0
 z0

2W0  2 0 
4

 i  z 
Gaussian beams - properties
Depth of focus
W0
U  r   A0
e
W  z
 z0
W0 

 W02
2 z0  2

z0: Rayleigh range

2
W 2  z
e
 z 
W  z   W0 1   
 z0 
2
  z0  2 
R  z   z 1    
  z  
  z   tan 1
W0 
 z0

z
z0
 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 
  0, z   kz  tan 1

2
W 2  z
e
 z 
W  z   W0 1   
 z0 
z
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
   , z   kz  tan
1
z
k2

z0 2 R  z 
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 
 z0

~ spherical wave
z
z0
 ikz ik
2
2 R z 
 i  z 
Gaussian beams - properties
Wavefront
W0
U  r   A0
e
W  z

2
W 2  z
e
 z 
W  z   W0 1   
 z0 
Plane wave
  z0  2 
R  z   z 1    
  z  
  z   tan 1
W0 
Spherical wave
Gaussian beam
2
 z0

z
z0
 ikz ik
2
2 R z 
 i  z 
Gaussian beams - properties
Propagation
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

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 left at a distance
z
R
1    R W

2 2
2
Gaussian beams - properties
Propagating through lens
W0
U  r   A0
e
W  z

2
W 2  z
e
 ikz ik
2
2 R z 
 i  z 
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 altered (W'=W), but the wavefront
does.
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:
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 evidently
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'
M
W0
z
1   z0 f 
2
f
z
1   f z0 
1
1   z0 f 
f
f
W0
z0
0

f
2
z
2
f
0


f
0


f
z0
Gaussian beams - properties
The ABCD low
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 = q2 + 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.
Summary
Maxwell equation:
linear, non-dispersive,
homogenous, isotropic,
source-free medium,
monochromatic light
  H  i E
  E  i H
k
E  0
H
H  0
Helmholtz equation
 2U  k 2U  0
Plane waves:
E  r   E0eik r
Spherical waves:
U r  
A0 ik r
e
r
Paraxial Helmholtz equation: T2 A  2ik
The Gaussian beam:
E
A
0
z
W0
U  r   A0
e
W  z

2
W 2  z
e
 ikz ik
2
2 R z 
 i  z 