Download Maxwell equation - Technion moodle

‫קורס ‪336533‬‬
‫‪Fundamentals of biomedical‬‬
‫‪optics and photonics‬‬
‫יסודות אופטיקה ופוטוניקה ביו‪-‬רפואית‬
‫מרצה‪:‬‬
‫ד"ר דביר ילין‬
‫חדר ‪265‬‬
‫שעות קבלה‪ :‬יום ג' ‪15:00-16:00‬‬
‫מתרגל‪:‬‬
‫ליאור גולן‬
‫חדר ‪259‬‬
‫שעות קבלה‪ :‬יום ד' ‪16:30-17:30‬‬
‫חדר ‪259‬‬
‫שעות קבלה‪ :‬ימי א' ‪14:30-15:30‬‬
‫בודק תר'‪ :‬עובדיה אילגייב‬
‫ספר‪:‬‬
‫)‪B.E.A. Saleh & M.C. Teich, Fundamentals of Photonics, 2nd edition, Wiley (2007‬‬
‫היקף‪:‬‬
‫‪ 3‬נקודות זכות (שעתיים הרצאה‪ ,‬שעתיים תרגול)‪.‬‬
‫מבנה ציון‪:‬‬
‫‪ 80%‬בחינה‪ 20% ,‬תרגילים‪.‬‬
‫* ינתנו כחמישה תרגילים בסך הכל‪ .‬כל התרגילים שיוגשו במועד יבדקו ויוחזרו עם הערות וציונים‪ .‬לא יפורסמו פתרונות‬
‫לתרגילים ולבחינות קודמות‪.‬‬
‫* הבחינה תארך כשלוש שעות ותכלול כארבע שאלות פתוחות‪ ,‬מתוכן אחת המבוססת על שאלה מתוך אחד התרגילים‪.‬‬
‫הבחינה תהיה עם חומר פתוח‪ ,‬מלבד ספרים ומחשבים ניידים‪.‬‬
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
Maxwell equations & waves
• Maxwell equations
• The wave equation
• Maxwell equations in a medium
• Helmholtz equation
• Electromagnetic waves: plane, spherical
• Wave-packets: pulse beams
• Propagation in medium
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
1.   H   0
t
H
2.  E   
0
t
3.
E  0
4.
 H  0
where
 E 
E x E y E z


x
y
z
”Divergence”
 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
”Curl”
Magnetic permeability
of free space
Maxwell equations & waves
• Maxwell equations
• The wave equation
• Maxwell equations in a medium
• Helmholtz equation
• Electromagnetic waves: plane, spherical
• Wave-packets: pulse beams
• Propagation in medium
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
Laplace
operator

t
  0 E t

t
 2E
 E   0 0 2  0
t
2
m
s
For each component:
1  2E
 E  2 2 0
c0 t
2
Similar procedure is followed for H
1  2Ei
 Ei  2
0
2
c0 t
2
 2Ei  2Ei  2Ei
 2  2
x 2
y
z
i  x, y , z
Maxwell equations in a medium
Assuming no free electric charges or currents:
Electric flux density:
D  r , t  : D   0E  P
Magnetic flux density:
B  r , t  : B  0 H  0 M
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.
 The assumption of linearity is valid for fields
which are weak compared to the atomic fields.
2. The medium is said to be non-dispersive 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
So:
D   0E  P   0E   0 E
Or:
D  E
where:
   0 1   
Electric permittivity
D
- P +
+
Ep
Eint
!
-
Electric susceptibility
0: Electrical permittivity of free space
In isotropic media: Eint  E - E P
! By definition, electric fields are directed
from ‘+’ to ‘-’ .
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 non-magnetic material, =0 and:

n
 1 
0
   0 1   
Thus, a material with no susceptibility (such as the vacuum) would have n=1.
BONUS
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.
or “irradiance”
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.
S  E H
H
E
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  (“new”).
real!
complex


H  r , t   Re H  r  e 
E  r , t   Re E  r  eit
  2
Angular frequency
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
 This notation is done to
ease the calculation of the
fields, avoiding complicated
trigonometry.
it
“complex-amplitude” vectors
Maxwell equations & waves
• Maxwell equations
• The wave equation
• Maxwell equations in a medium
• Helmholtz equation
• Electromagnetic waves: plane, spherical
• Wave-packets: pulse beams
• Propagation in medium
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 Poynting vector)
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 therefore will be washed
out by the time-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 (or “irradiance”) 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 can 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.
Maxwell equations & waves
• Maxwell equations
• The wave equation
• Maxwell equations in a medium
• Helmholtz equation
• Electromagnetic waves: plane, spherical
• Wave-packets: pulse beams
• Propagation in medium
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   
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
Maxwell equations & waves
• Maxwell equations
• The wave equation
• Maxwell equations in a medium
• Helmholtz equation
• Electromagnetic waves: plane, spherical
• Wave-packets: pulse beams
• Propagation in medium
Elementary electromagnetic waves
Assumptions:
Medium: linear, homogenous, non-dispersive, isotropic.
Light: monochromatic
An oscillation is a time-varying disturbance:
A wave is a time-varying disturbance that also
propagates in space. A wave transports energy
without any permanent transfer of the medium.
Plane waves
Solutions for the Helmholtz equation:
(proof in next slide)
The real electric field:
E  r   E0eik r
H  r   H 0e


ik r
E  r , t   Re E  r  eit


 Re E0 eik r eit


 E0 cos k  r  t
r k:
r k:


E  r , t0   E0 cos k  r  t0
E  r , t0   constant
“wavelength”

k
2
k
Plane waves
Proof: Plane waves satisfy the Helmholtz equation:
2 E  k 2 E  0
E  r   E0eik r
 2  E0 eik r   k 2  E0eik r   0
 
ik
2
E0eik r  k 2 E0eik r  0
k 2  k 2  0
k k
Implying that the length of the wave-vector
parameter k in Helmholtz equation:
   c
  2
Also:
k of the plane wave must be equal to the
k    
k 
2



c
n

c0
 nk0
2 2


 n  nk0
c  c0 n
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:
Irradiance 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 irradiance 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
Polarization of plane waves
A monochromatic plane wave propagating in the +z direction*: E
 r , t   Re E0eik r eit 
* Sign: The choice of signs in the exponent is arbitrary (no mathematical proof). With different
signs in the spatial and temporal exponents, the traveling wave exp[i(t-kz)] represents
a wave that moves in +z direction as time propagates.
Such wave can have its vector k in the z axis only, and field component in the x-y plane:
k   0, 0, k  , E0   Ax , Ay , 0 
( complex envelope:
E0  Ax xˆ  Ay yˆ

 E  r , t   Re E0e 
)
i  kz t 

 z

 i  t  c  

 Re  E0e





To describe the polarization of this wave, we trace the endpoint of the vector E(z,t) at each
position z as a function of time.
k

c
Polarization ellipse
Expressing Ax and Ay in terms of their magnitudes and phases:
i
Ax  ax eix , Ay  a y e y ,


and substituting into E0  Ax xˆ  Ay yˆ and E  r , t   Re E0ei t  z c  , we finally obtain:
E  z, t   E x xˆ  E y yˆ ,
where:
  z

E x  ax cos   t     x 
  c

  z

E y  a y cos   t     y 
  c

are the x and y components of the (real) electric-field vector E(z,t). The components Ex and Ey
are periodic functions of t-z/c that oscillate at the same angular frequency .
Polarization ellipse
  z

E x  ax cos   t     x 
  c

  z

E y  a y cos   t     y 
  c

At a fixed value of z, the tip of the electric-field vector rotates periodically in the x-y plane,
tracing out the shape of an ellipse. At a fixed time t, the tip of the electric-field vector follows
a helical trajectory in space that lies on the surface of an elliptical cylinder. The electric field
rotates as the wave advances, repeating its motion periodically for each distance
corresponding to a wavelength = 2 c / .
Linearly polarized light
If one of the components vanishes (ax=0, for example), the light is linearly polarized (LP)
in the direction of the other component (the y direction).
The wave is also linearly polarized if the phase difference y-x = 0 or  :
  z

E x  ax cos   t     x 
  c

  z

E y  a y cos   t     y 
  c

 y  x  0
Ey  
ay
ax
Ex
 y  x  
which is the equation of a straight line of slope ay /ax . In these cases the elliptical cylinder
in previous slide collapses into a plane.
If ay =ax the plane of polarization makes an angle 45° with the x axis.
If ax=0 the plane of polarization is the y-z plane.
Circularly polarized light
If y-x = /2 and ay=ax =0 :
  z

E x  ax cos   t     x   a0 cos   t  z c    x 
  c

  z

E y  a y cos   t     y    a0 sin   t  z c    x 
  c

Ex 2  E y 2  a02
which is the equation of a circle. The wave is said to be circularly polarized.
In the case y-x = +/2, the electric field at a fixed position z rotates in a clockwise direction
when viewed in the direction from which the wave is approaching (‘behind’ the wave). The
light is then said to be right circularly polarized (RCP).
The case y-x = -/2 corresponds to counterclockwise rotation and left circularly
polarized (LCP) light.
Matrix representation
The Jones Vector
We saw that a monochromatic plane wave of angular frequency  traveling in the z direction is
fully characterized by the complex envelopes
Ax  ax eix , Ay  a y e
i y
of the x and y components of the electric-field vector. These complex quantities may be written
in the form of a column matrix known as the Jones vector:
 Ax 
J 
 Ay 
Given J, we can determine the total irradiance of the wave:

I  Ax  Ay
2
2
 2
 
and use the ratio r  Ay Ax and the phase difference   arg Ay  arg  Ax  to determine
the orientation and shape of the polarization ellipse.
the intensity in each case
has been normalized:
2
Ax  Ay  1
2
and the phase of the x
component is taken to
be x =0.
Matrix representation
Consider the transmission of a plane wave of arbitrary polarization through an optical
system that maintains the ‘plane-wave’ nature of the wave, but alters its polarization:
 A1x 
J1   
 A1 y 
 A2 x 
J2   
 A2 y 
The system is assumed to be linear  the principle of superposition is obeyed.
The complex envelopes of the two electric field components of the input and output
(transmitted or reflected) waves are related by the weighted superpositions:
A2 x  T11 A1x  T12 A1 y
A2 y  T21 A1x  T22 A1 y
where
 A2 x  T11 T12   A1x 
A   
 A 
T
T
 2 y   21 22   1 y 
Jones matrix
J2  TJ1
Matrix representation
Simple example: the linear
polarizer
The system represented by the Jones matrix
1 0 
T

0
0


transforms a wave of components (A1x, A1y) into a wave of components (A1x, 0) by eliminating
the y component, thereby yielding a wave polarized along the x direction:
This system is a linear polarizer with its transmission axis pointing in the x direction.
* More examples include quarter-wave and half-wave retarders (Exercise).
Cascaded polarization devices
The action of cascaded optical systems on polarized light may be conveniently determined by
using matrix multiplication. A system characterized by the Jones matrix T1 followed by another
characterized by T2 are equivalent to a single system characterized by T=T2T1.
! The matrix of the system through which light is first transmitted must stand to the right in the matrix
product since it is the first to affect the input Jones vector.
Polarization at interfaces
Reflection and refraction at the boundary between two dielectric media:
‘P’
‘S’
t x 0 
t
,
0
t
y

 rx
r
0
• The x-polarized mode is called the S-polarization (orthogonal to the plane of incidence).
• The y-polarized mode is called the P-polarization (parallel to the plane of incidence).
Fresnel equations (Exercise):
0
ry 
Polarization at interfaces
The Brewster
angle:
The absence of reflection of the P wave at the Brewster angle
is useful for making polarizers from simple glasses.
B  tan 1  n2 n1 
BONUS
Spherical waves
Another simple solution of the Helmholtz equation
is the scalar spherical wave:
A0 ikr
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
Maxwell equations & waves
• Maxwell equations
• The wave equation
• Maxwell equations in a medium
• Helmholtz equation
• Electromagnetic waves: plane, spherical
• Wave-packets: pulse beams
• Propagation in medium
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 E0 eik r eit


E  r , t   E0 cos k  r  t
r k:
r k:
“wavelength”


E  r , t0   E0 cos k  r  t0
E  r , t0   constant

k
2
k
The monochromatic (plane) wave

vp
Monochromatic wave*:
n

E  r , t   E0 cos k  r  t

* Sign: The choice of signs in the cosine argument is arbitrary, as long as opposite signs are used in the
Fourier and inverse-Fourier transforms. With different signs in the spatial and temporal terms, the
traveling wave exp[i(t-kz)] represents a wave that moves in +z direction as time propagates.
Assuming propagation in the z axis and linear polarization in the x axis:
k   0, 0, k  , E0   A, 0, 0 
 E  r , t   A cos  kz  t 
k
2 n

  2 f
The phase velocity of a monochromatic wave is given by:
d
d
 dz 
dz
k
d
d

 dt  
dt

k
Optical frequency
dz
d k

2 f
f  c0
 vp 

 


dt d  k 2 n 
n
n
“v”
f   (new)
Polychromatic and pulsed light
Since the wavefunction of monochromatic light is a harmonic function of time, extending over
all time (from - to ), it is an idealization that cannot be met in reality.
Most waves have arbitrary time dependence, including optical pulses of finite time duration.
Such waves are polychromatic rather than monochromatic.
Temporal and Spectral Description
Although a polychromatic wave is described by a wavefunction E(r,t) with non-harmonic time
dependence, it may be expanded as a superposition of harmonic functions, each represents a
monochromatic wave. Since we already know how monochromatic waves propagate in free
space, we can determine the effect of optical systems on polychromatic light by using the
principle of superposition.
Fourier methods permit the expansion of an arbitrary function of time E(t), representing the
wavefunction E(r,t) at a fixed position r, as a superposition integral of harmonic functions of
different frequencies, amplitudes, and phases:

E  t    v   ei 2 t d

Where v() is determined by carrying out the Fourier transform

v     E  t  ei 2 t dt.

since E(t) is real, v(-)=v*(). Thus, the negative-frequency components are not independent;
they are simply conjugated versions of the corresponding positive-frequency components.
Polychromatic and pulsed light
Complex – amplitude & phase
v  1 
v  2 
v  n 

E  t    v   ei 2 t d

Complex representation
U  r, t 

E  t    v   ei 2 t d


v     E  t  e

 i 2 t
dt
Reminder:

E  r , t   Re E  r  eit
(for monochromatic waves)

It is convenient to represent the real function E(t) as the real part of a complex function U(t):

U  t   2 v   ei 2 t d
0
that includes only the positive-frequency components (multiplied by a factor of 2), and
suppresses all the negative frequencies. The Fourier transform of U(t) is therefore a function
V()=2v() for  > 0, and 0 for  < 0.

U  t    V   ei 2 t d


V     U  t  e i 2 t dt

The real function E(t) can be determined from its complex representation U(t) by simply taking
its real part:
E  t   ReU  t .
As a simple example, the complex representation of the real harmonic function E(t)=cos(t) is
the complex harmonic function U(t)=exp(it). The complex representation of a polychromatic
wave is simply a superposition of the complex representations of each of its monochromatic
Fourier components.
Pulsed plane wave
The simplest example of pulsed light is a pulsed plane wave.
The complex wavefunction U(r,t) has the form:
U  r , t   A t  z c  e
 z
i 2 0  t  
 c
where the complex envelope A(t) is a time-varying function
and 0 is the central optical frequency.
U  r , t   Ae
Note: the monochromatic plane wave is a special case for which A(t)=A:
 z
i 2 0  t  
 c
 A eik0 z ei0t
If A(t) is of finite duration , then at any fixed position z the wave lasts for a time period , and at
any fixed time t it extends over a distance c. It is therefore a wavepacket of fixed extent traveling
in the z direction.
(example: a pulse of duration  = 1 ps extends over a distance c = 0.3 mm in free space)
The Fourier transform of the above U(r,t) is
2 z
V  r ,   A   0  e
i
c
where A() is the Fourier transform of A(t).
A   0 
Gaussian pulse
The transform-limited Gaussian pulse
A transform-limited Gaussian pulse has a complex envelope with constant phase and Gaussian
t2
magnitude:

A  t   A0 e
where  is a real time constant. The intensity
I  t   I 0e

2
2t 2
2
is also a Gaussian function with peak value I0= |A0|2, 1/e full width of 2, and FWHM:
 FWHM  2ln 2  .
The Fourier transform of the complex envelope A(t) is also a Gaussian function
A    e
and so is the spectral intensity
The FWHM of the spectral intensity is
S    e
 
 2 2 2
2 2 2   0 
0.375


2
0.44
 FWHM
so that the product of the temporal and spectral widths is
  FWHM  0.44
.
A broad spectrum is needed
to create a short pulse!
,
Gaussian pulse
Chirped Gaussian Pulse
A more general Gaussian pulse has the form
A  t   A0 e

t2

2
i
e
at 2
2
.
The magnitude of the complex envelope is a Gaussian function |A0|2exp(-t2/2) and the intensity
is also Gaussian. The phase is a quadratic function =at2/2 so that the instantaneous
frequency is a linear function of time; i.e., the pulse is linearly chirped with chirp parameter a.
The pulse is up-chirped for positive a, down-chirped for negative a, and unchirped )“transformlimited”) for a=0.
The Fourier transform of the complex envelope is also a Gaussian function of frequency. The
spectral intensity S() is Gaussian with FWHM
 
0.375

1  a2 
0.44
 FWHM
1  a2 .
The product of the FWHM temporal and spectral widths is
  FWHM  0.44 1  a 2
so that the unchirped Gaussian pulse (a=0) has the least temporal and spectral-width product.
Gaussian pulse
Gaussian pulse
Further reading: S&T page 943
Maxwell equations & waves
• Maxwell equations
• The wave equation
• Maxwell equations in a medium
• Helmholtz equation
• Electromagnetic waves: plane, spherical
• Wave-packets: pulse beams
• Propagation in medium
Group velocity
Reminder: since monochromatic wave has no beginning and no end, it cannot send a signal.
Information can only travel if the wave is modulated → wavepackets:
A wavepacket consists of several k components
(wavelengths) that are all “in phase”:
d
 z
 t  vg t
dk
where
d
vg 
dk
(k): “dispersion relation”. For example:
In vacuum n  n(k), therefore:
d d
d
0

 kz  t   z  t
dk dk
dk
(“group velocity”)
c
n
 k
d c
vg 
  vp
dk n
However, in most materials, n=n()constant:
n

Group velocity
1
vg 
1
1 

c0
  k 




n


   


n
k   
c


 0

n    

n *
n
  n 
ng  n     
 n      
  n    


   
vg 
c0 c0
  vp
ng n
Refractive index of Silica
ng
n
* n  n   n  2 c   n 2 c    n
     
  2
 
0°C (blue)
100°C (black)
200°C (red)
Pulse in a dispersive medium
c c
vg 
  vp
ng n
The phase fronts of different frequency components propagate at different velocities and
the pulse propagates with the group velocity, which is lower than all the phase velocities.
Spectral phase
Reminder: Ignoring space coordinates, an optical pulse is fully described by its magnitude
and phase:
i  2 0   t 
U  t   I  t e
or equivalently, by the magnitude and phase of its Fourier transform
V    S  ei  
Spectral phase: the phase
of each spectral (wavelength)
component.
|V|
E  t   cos t   U  t   eit
t




E t 
V
t
|V|


Time delay  linear spectral phase

U  t    V   ei 2 t d


V     U  t  e i 2 t dt

or (in ):

U  t    V   e d 
it

V    A  
IFT

A t 
IFT
V    A   e i 
A t  
proof:


U  t    V   e d   A   e  i eit d

it


  A   e i t   d  A  t   

! Linear spectral phase
corresponds to time delay
V  
A

0

Dispersion formalism

U  t    V   ei 2 t d


V     U  t  e i 2 t dt

The general form of V():
V    A   ei    A   eik   z
=k()z
V  
A
0

U(t) will be determined by both the amplitude (A) and the phase () of the spectrum. Apparently,
the spectral phase has a critical role on the pulse. Expand k into a Taylor series around 0:
1
1
2
3
k    k 0   k ' 0   0   k ''    0   k '''    0   ...
2
6
Propagation
constant at 0
Inverse group velocity
vg 
d
1
 k ' 0  
dk
vg
group velocity dispersion
(GVD): variation in group
velocity with frequency.
“Chirp”
Third-order dispersion.
Produces asymmetric
distortion of the pulse.
GVD
1
1
2
3
k    k 0   k ' 0   0   k ''    0   k '''    0   ...
2
6
Neglecting the third order dispersion k’’’, pulse broadening is usually described using the 2nd
order dispersion which accounts for the GVD. In the literature there are several terms for this
coefficient:
1. The “GVD coefficient”:
03 d 2 n  s2 
D  2 k ''  2
c0 d 02  m 
! The propagation constant k is
often referred as  (k’’=’’).
2. Sometimes, the GVD coefficient is given in units of time/distance2 and is called D *. In that
case, D can be computed using:
02
* Sometimes D is given
D   D s m2 
in units of [ps/(kmnm)]
c0
3. The “chirp parameter” b [s2] includes the propagation distance z:
d 2k
b  2 2 z  2k '' z
d
b, k’’ and D are related by: b  2k '' z 
D

z
s 2 
Chirp filtering of a Gaussian pulse
The propagation of a pulse in a dispersive medium may be approximated by two effects: a
time delay associated with the group velocity vg = 1/k' and a chirp filter with chirp parameter
b = 2k"z proportional to the propagation distance z. The parameters k' and k" are the
derivatives of the propagation constant k with respect to the angular frequency .
The chirp filter is a phase-only filter whose phase is a quadratic function of frequency:
A  t   A10e

t2
A  t   A20e
12
He  He  f  e
 ie  f 
 1 e

t2
 22
i
e
a2t 2
 22
ib 2 f 2
chirp coefficient [s2]
Chirp filtering of a Gaussian pulse

A10e
t2
12
 A20e

t2
 22
i
e
a2t 2
 22
Upon transmission through a chirp filter, an unchirped Gaussian pulse remains Gaussian and its
properties are modified as follows:
 The pulse width is increased by a factor (1+a22)=(1+b2/14).
o For |b|=12 this factor is 2.
o For |b|»12, i.e., for large chirp coefficient (or narrow original pulse), 2=|b|/1, so that narrower pulses
undergo greater broadening.
 The initially transform-limited pulse becomes chirped with a chirp parameter a2 that is directly
proportional to b and inversely proportional to 12. The filtered pulse will be up-chirped if b is
positive, and down-chirped if b is negative.
 The spectral intensity (and width) of the pulse remains unchanged (the chirp filter is a
spectral-phase-only filter).
Propagation in a dispersive medium
z0: Dispersion length - the distance
in which the pulse broadens by
a factor 2.
The pulse remains Gaussian, but its
width  (z) expands, and it becomes
chirped with an increasing chirp
parameter a(z).
Normal and anomalous dispersion
 ''  0, D  0, D  0
 ''  0, D  0, D  0
 In a medium with normal dispersion the shorter-wavelength components of the pulse arrive
later that those with longer wavelength. A medium with anomalous dispersion exhibits the
opposite behavior.
 Normal dispersion – “positive chirp”.
 Most glasses exhibit normal dispersion in the visible region of the spectrum.
 At longer wavelengths, the dispersion often becomes anomalous.
Example: Pulse broadening in fused silica
From the graph:
D  800nm   100
D  

2
0
c0
D 
ps
s
 100 106 2
km×nm
m
800 109 
3 108
2
6
100 10  2.13 10
25
s2
m
2
D
26 s
 '' 
 3.4 10
2
m
For a slab of thickness 1 cm, this corresponds
to a chirp coefficient:
b  z  1cm   2  '' z  6.4 1028 s 2   25 fs 
2
This means that when a Gaussian pulse of width
25 fs crosses the slab, its width expands by a
factor of 2 (  2  1 1  b2 14 ).
2
4
15
1   6.4 1028 
For a 10 fs pulse:  2   1 1  b  1  10 10
2
For a 50 fs pulse:  2   1 1  b 2  14  50 1015 1   6.4 1028 
2
10 
14 4
 65 fs
5 1014   52 fs
4
Summary
Maxwell equation:
linear, non-dispersive,
homogenous, isotropic,
source-free medium,
monochromatic light
  H  i E
  E  i H
E  0
H  0
E
Helmholtz equation
Plane waves:
 2U  k 2U  0
E  r   E0eik r
k
H
Complex wavefunction of a polychromatic wave: U  r , t   A  t  z c  e
V  r ,   A   0  e
Pulse broadening in medium:
 z
i 2 0  t  
 c
i
2 z
c