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יסודות אופטיקה ופוטוניקה ביו-רפואית
Fundamentals of biomedical optics and photonics
15:00-16:00שעות קבלה: יום ג' 265חדר ד"ר דביר ילין מרצה:
16:30-17:30שעות קבלה: יום ד' 259חדר ליאור גולן מתרגל:
14:30-15:30שעות קבלה: ימי א' 259חדר עובדיה אילגייב בודק תר':
B.E.A. Saleh & M.C. Teich, Fundamentals of Photonics, 2nd edition, Wiley (2007)ספר:
נקודות זכות (שעתיים הרצאה, שעתיים תרגול).3היקף:
תרגילים.20% בחינה, 80%מבנה ציון:
* 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.
336533 קורס
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
6.Fundamentals of lasers
7.Lasers in medicine
8.Basics of light detection, cameras
9.Microscopy, contrast mechanism
10.Confocal microscopy
Course outlineמשואות מקסוול, גלים אלקטרומגנטים, חבילות 1.
גלים
קרניים גאוסיניות2.
אופטיקת פורייה, העדשה, הפרדה3.
אופטיקה גיאומטרית, חוק סנל4.
אינטראקציה אור-רקמה: פיזור, בליעה, 5.
טיפול פוטו-דינמיפלואורסנציה,
עקרונות לייזרים6.
לייזרים ברפואה7.
עקרונות גילוי אור, מצלמות8.
מיקרוסקופיה, ניגודיות9.
מיקרוסקופיה קונפוקלית10.
•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 & waves
Maxwell equations
0 t
EH
0 t
H
E
0
E
0
H
An electromagnetic field is described by two related vector fields that are functions
of position and time:
Electric field:
Magnetic field:
,
,
r t
r t
E
H
In free space:
90 1 36 10
F
m
70 4 10
H
m
Electrical permittivity of free
space in MKS units:
, ,
yx z
y yx xz z
x y z
y z z x x y
EE EE
E EE EE EE
1.
2.
3.
4.
where
Magnetic permeability
of free space
”Divergence”
”Curl”
•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 & waves
2
E E E
0 t
EH
0 t
H
E
0
E
0
H
1.
2.
3.
4.
22
2 20
10
c t
E
E
80
0 0
13 10
mc
s
Speed of light in free space:
22
2 20
10i
i c t
E
E
, ,i x y zSimilar procedure is followed for
The wave equation
For each component:
H
0
20
20
020
t
t
t
t
t
HE
HE E
HE
EE
22
0 0 20
t
E
E
3
1
Laplace operator
2 2 2
2 2 2
i i i
x y z
E E E
Maxwell equations in a medium
Electric flux density:
0
0 0
, :
, :
r t
r t
D D E P
B B H M
In source-free media:
t
DH
t
B
E
0
D
0
B
1.
2.
3.
4.
Nucleus
Electron cloud
+ -
E
P
D
In free space: 0
0
P
M
0
0
D E
B H
Magnetic flux density:
Assuming no free electric charges or currents:
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 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.
Electromagnetic waves in dielectric media - definitions
The assumption of linearity is valid for fields which are weak compared to the atomic fields.
+ -
E
P
D
int
E
p
E
Induced polarization
Or:
where:
0
0 0 0
0 1
P E
D E P E E
D E
Electric permittivity
int P
E E - EIn isotropic media:
+-
Electric susceptibility
!So:
0: Electrical permittivity of free space
! By definition, electric fields are directed from ‘+’ to ‘-’ .
The refractive index (n)
22
2 2
10
c t
E
E8
0
0 0
13 10
mc
s
Speed of light in free space:The wave equation (in a medium):
1c
where the speed of light in the medium is denoted c:
The ratio of the speed of light in free space to that in the medium, c0/c, is defined as the refractive index n:
0
0 0
cn
c
For a non-magnetic material, =0 and:
0
1n
0 1
Thus, a material with no susceptibility (such as the vacuum) would have n=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.
0
E =
BONUS
Poynting vector
S E H
H E
S
,I r t S
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.
The average is taken over times that are long in comparison with an optical cycle.
or “irradiance”
Monochromatic EM waves
, Re
, Re
i t
i t
r t E r e
r t H r e
E
H
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”).
“complex-amplitude” vectors
, Re
, Re
, Re
, Re
i t
i t
i t
i t
r t P r e
r t D r e
r t M r e
r t B r e
P
D
M
B
2
Similarly:
Angular frequency
frequency
real! complex
This notation is done to ease the calculation of the fields, avoiding complicated trigonometry.
•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 & waves
Maxwell equations in medium(Complex amplitude)
0
0
H i D
E i B
D
B
1.
2.
3.
4.
Re Re
i t i t
i t i t
i t i t
t
H r e D r et
H r e D r et
e H r D r et
H r i D r
DH
0
0 0
D E P
B H M
The Maxwell equations become: (source-free medium, monochromatic)
Also,
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.
*1
2S E H
I S
* *
* * 2 2 * *
1 1Re Re
2 21
4
i t i t i t i t i t i t
i t i t
Ee He Ee E e He H e
E H E H e E H e E H
S E H
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 1Re
4 2E H E H S S S
S
Where the new vector
may be regarded as a complex Poynting vector.
Intensity and power (Complex Poynting vector)
The optical intensity (or “irradiance”) is the magnitude of the vector S:
Linear, nondispersive, homogenous, and isotropic media
D E
B H
0
0
H i E
E i H
E
H
1.
2.
3.
4.
linear, non-dispersive, homogenous, isotropic, source-free medium, monochromatic.
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:
0
0
H i D
E i B
D
B
1.
2.
3.
4.
•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 & waves
where represents the complex amplitude of any of the components of the electric and magnetic fields:
Helmholtz equation , Re i tr t E r e
E
2
22
2 2
2 2
0
0
0
iE E
c
E E
E k E
k
or: 2 2 0U k U
U U r
, , , , ,x y z x y zU E E E H H H
Substitute the complex amplitude notation
Helmholtz equation
22
2 2
10
c t
E
E
yields:
into the wave equation
1c
•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 & waves
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
0
0
ik r
ik r
E r E e
H r H e
Solutions for the Helmholtz equation: (proof in next slide)
0
0
, Re
Re
cos
i t
ik r i t
r t E r e
E e e
E k r t
E
0 0 0
0
: , cos
: , constant
r k r t E k r t
r k r t
E
E
The real electric field:
k
2 k
“wavelength”
Implying that the length of the wave-vector of the plane wave must be equal to the
parameter k in Helmholtz equation:
Proof: Plane waves satisfy the Helmholtz equation:2 2 0E k E
2 20 0
22
0 0
2 2
0
0
0
ik r ik r
ik r ik r
E e k E e
ik E e k E e
k k
k k
0
ik rE r E e
k
Also:2
c
Plane waves
00
00 0
2 2 2
k n nkc c
k n nkc c n c
Substitute in Maxwell equations 1 & 2 yields (exercise):
0
0
H i E
E i H
E
H
1.
2.
3.
4.
0 0
0 0
k H E
k E H
E is perpendicular to both k and H
H is perpendicular to both k and E
Transverse electromagnetic (TEM) wave:
0
0
ik r
ik r
H r H e
E r E e
Plane waves
Irradiance of TEM waves
The ratio between the amplitudes of the electric and magnetic fields is known as the impedance of the medium:
the impedance of free space: 0 0
00 0 0 377 120
E H
n
*1
2S E H
The complex Poynting vector:
2
0*0 0
1
2 2
EI S E H
Example: an irradiance of 10 W/cm2 in free space corresponds to an electric field of 87 V/cm:
0
V2 2 377 10 87
cm E I
Polarization of plane waves
A monochromatic plane wave propagating in the +z direction*:
k
c
00,0, , , ,0
x yk k E A A
0 0 , Re Re
zi t
i kz t cr t E e E eE
* 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.
0, Re ik r i tr t E e eE
0 ˆ ˆ
x yE A x A y
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.
( complex envelope: )
Such wave can have its vector k in the z axis only, and field component in the x-y plane:
Polarization ellipseExpressing Ax and Ay in terms of their magnitudes and phases:
and substituting into and , we finally obtain:
, , yxii
x x y yA a e A a e
0, Re i t z cr t E eE0 ˆ ˆ
x yE A x A y
ˆ ˆ, ,
cos
cos
x y
x x x
y y y
z t x y
za t
c
za t
c
E E E
E
E
where:
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 .
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 / .
Polarization ellipsecos
cos
x x x
y y y
za t
c
za t
c
E
E
Linearly polarized lightIf 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 :
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.
cos
cos
x x x
y y y
za t
c
za t
c
E
E y
y xx
a
aE E
0 y x
y x
If y-x = /2 and ay=ax =0 :
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.
Circularly polarized light
0
0
cos cos
cos sin
x x x x
y y y x
za t a t z c
c
za t a t z c
c
E
E
2 2 20 x y aE E
Matrix representationThe Jones VectorWe saw that a monochromatic plane wave of angular frequency traveling in the z direction is fully characterized by the complex envelopes
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:
, yxii
x x y yA a e A a e
J
x
y
A
AGiven J, we can determine the total irradiance of the wave:
and use the ratio and the phase difference to determine the orientation and shape of the polarization ellipse.
222 x yI A A
y xr A A arg arg y xA A
the intensity in each case has been normalized:
and the phase of the x component is taken to be x =0.
221 x yA A
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:
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:
where
Matrix representation
2 11 1 12 1
2 21 1 22 1
x x y
y x y
A T A T A
A T A T A
1
11
J
x
y
A
A2
22
J
x
y
A
A
2 111 12
2 121 22
x x
y y
A AT T
A AT T 2 1J TJ
Jones matrix
Matrix representationSimple example: the linear polarizerThe system represented by the Jones matrix
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.
1 0T
0 0
Reflection and refraction at the boundary between two dielectric media:
• 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).
Polarization at interfaces
‘S’‘P’
0 0,
0 0
x x
y y
t rt r
t r
Fresnel equations (Exercise):
Polarization at interfacesThe Brewster angle:
The absence of reflection of the P wave at the Brewster angleis useful for making polarizers from simple glasses.
1B 2 1tan n n
Spherical waves
0
0
ˆsin
ˆsin
E r E U r
H r H U r
0 ikrA
U r er
Another simple solution of the Helmholtz equation is the scalar spherical wave:
U is spherically symmetric: U r U r 2 2
2
1r
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 complex-amplitude vectors may be approximated by:
Reminder: Laplacian in radial coordinates:
BONUS
•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 & waves
Plane waves
0
0
ik r
ik r
E r E e
H r H e
Solutions for the Helmholtz equation: (proof in next slide)
0
0
, Re
Re
, cos
i t
ik r i t
r t E r e
E e e
r t E k r t
E
E
0 0 0
0
: , cos
: , constant
r k r t E k r t
r k r t
E
E
The real electric field:
k
2 k
“wavelength”
Reminder
0
2
2
p
d dk dz
cdz d k f fdz k vd d dt d k n n n
dtdt
The monochromatic (plane) wave
The phase velocity of a monochromatic wave is given by:
Monochromatic wave*:
2
2
nk
f
n
pv
0, cosr t E k r t
E
00,0, , ,0,0k k E A
, cosr t A kz t
E
* 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:
Optical frequencyf (new)
“v”
Polychromatic and pulsed lightSince 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 DescriptionAlthough 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:
Where v() is determined by carrying out the Fourier transform
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.
2v i tt e d
E
2v .i tt e dt
E
Polychromatic and pulsed light
2v i tt e d
E
1v
2v
v n
Complex – amplitude & phase
It is convenient to represent the real function E(t) as the real part of a complex function U(t):
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.
The real function E(t) can be determined from its complex representation U(t) by simply taking its real part:
As a simple example, the complex representation of the real harmonic function E(t)=cos(t) is the complex harmonic function U(t)=exp(it). The complex representation of a polychromatic wave is simply a superposition of the complex representations of each of its monochromatic Fourier components.
Complex representation
, Re i tr t E r e
E
,U r t
2
2
v
v
i t
i t
t e d
t e dt
E
E
2
02 v i tU t e d
Re .t U tE
Reminder:(for monochromatic waves)
2
2
V
V
i t
i t
U t e d
U t e dt
The simplest example of pulsed light is a pulsed plane wave. The complex wavefunction U(r,t) has the form:
where the complex envelope A(t) is a time-varying function and 0 is the central optical frequency.
Note: the monochromatic plane wave is a special case for which A(t)=A:
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
where A() is the Fourier transform of A(t).
Pulsed plane wave
02
,z
i tcU r t t z c e
A
00 0
2
,z
i tik z i tcU r t e e e
A A
2
0V ,
z
icr A e
0 A
Gaussian pulseThe transform-limited Gaussian pulse
A transform-limited Gaussian pulse has a complex envelope with constant phase and Gaussian magnitude:
where is a real time constant. The intensity
is also a Gaussian function with peak value I0= |A0|2, 1/e full width of 2, and FWHM:
The Fourier transform of the complex envelope A(t) is also a Gaussian function
and so is the spectral intensity
The FWHM of the spectral intensity is
so that the product of the temporal and spectral widths is
2
2
0
t
t A e
A
2
2
2
0
t
I t I e
2ln 2 . FWHM
2 2 2
A e
22 202 . S e
0.375 0.44 ,
FWHM
0.44FWHM
A broad spectrum is needed to create a short pulse!
Gaussian pulseChirped Gaussian Pulse
A more general Gaussian pulse has the form
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 (“transform-limited”) 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
The product of the FWHM temporal and spectral widths is
so that the unchirped Gaussian pulse (a=0) has the least temporal and spectral-width product.
2 2
2 2
0 .
t at
i
t A e eA
2 20.375 0.441 1 .
FWHM
a a
20.44 1FWHM a
Gaussian pulse
Further reading: S&T page 943
Gaussian pulse
•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 & waves
Group velocityReminder: 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”: 0
d d dkz t z t
dk dk dk
g
dz t v t
dk
(k): “dispersion relation”. For example:c
kn
In vacuum n n(k), therefore: g p
d cv v
dk n
However, in most materials, n=n()constant:
n
g
dv
dkwhere (“group velocity”)
11
0
0
1
g
ckv n
nk c n
2
2 2n n n c n c n
Group velocity
g
n n nn n n n
*
*
0 0 g pg
c cv v
n n 0°C (blue)100°C (black)200°C (red)
ng
n
Refractive index of Silica
Pulse in a dispersive medium
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.
g pg
c cv v
n n
Spectral phase
t
|V|
t
Spectral phase: the phase of each spectral (wavelength) component.
Reminder: Ignoring space coordinates, an optical pulse is fully described by its magnitude and phase:
or equivalently, by the magnitude and phase of its Fourier transform
02i tU t I t e
V iS e
V
|V|
cos i tt t U t eE
tE
A
0
Slope:
Time delay linear spectral phase
! Linear spectral phase corresponds to time delay
or (in ):
2
2
V
V
i t
i t
U t e d
U t e dt
V
i tU t e d
IFT
IFTi
V A t
V A e t
A
A
V
i t i i t
i t
U t e d A e e d
A e d tA
proof:
V
2 3
0 0 0 0 0
1 1' '' ''' ...
2 6k k k k k
Dispersion formalism
Propagation constant at 0
Inverse group velocity
group velocity dispersion(GVD): variation in group velocity with frequency.
Third-order dispersion. Produces asymmetric distortion of the pulse.
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:
“Chirp”
A
0
=k()z
0
1'
gg
dv k
dk v
2
2
V
V
i t
i t
U t e d
U t e dt
i ik zV A e A e
The general form of V():
V
GVD
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:
2 3
0 0 0 0 0
1 1' '' ''' ...
2 6k k k k k
3. The “chirp parameter” b [s2] includes the propagation distance z:
3 2 202 20 0
s2 ''
m
d nD k
c d
22
22 2 '' s
d kb z k z
d
1. The “GVD coefficient”:
2. Sometimes, the GVD coefficient is given in units of time/distance2 and is called D *. In that case, D can be computed using: 2
20
0
s m D Dc
2 ''
Db k z zb, k’’ and D are related by:
! The propagation constant k is often referred as (k’’=’’).
* Sometimes D is given in units of [ps/(kmnm)]
Chirp filtering of a Gaussian pulseThe 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 .
2 2
1 ei f ib fe eH H f e e
The chirp filter is a phase-only filter whose phase is a quadratic function of frequency:
chirp coefficient [s2]
2
21
10
t
t A e
A 22
22 22 2
20
a tti
t A e e
A
22 22
2 2 21 2 2
10 20
a tt t
i
A e A e e
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+b2/1
4).
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 1
2. 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).
Chirp filtering of a Gaussian pulse
Propagation in a dispersive medium
The pulse remains Gaussian, but its width (z) expands, and it becomes chirped with an increasing chirp parameter a(z).
z0: Dispersion length - the distance in which the pulse broadens by a factor 2.
Normal and anomalous dispersion
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.
'' 0, 0, 0 D D
'' 0, 0, 0 D D
Example: Pulse broadening in fused silica
From the graph:
For a slab of thickness 1 cm, this corresponds to a chirp coefficient:
This means that when a Gaussian pulse of width 25 fs crosses the slab, its width expands by a factor of 2 ( ).
For a 10 fs pulse:
For a 50 fs pulse:
62
292 26 250
80
226
ps s800nm 100 100 10
km×nm m
800 10 s100 10 2.13 10
3 10 m
s'' 3.4 10
2 m
D
D Dc
D
228 21cm 2 '' 6.4 10 s 25 fs b z z
2 42 1 11 b
2 42 4 15 28 142 1 11 10 10 1 6.4 10 10 65 fs b
2 42 4 15 28 142 1 11 50 10 1 6.4 10 5 10 52 fs b
Summary
0
0
H i E
E i H
E
H
Maxwell equation: linear, non-dispersive, homogenous, isotropic, source-free medium, monochromatic light
2 2 0U k U Helmholtz equation
Complex wavefunction of a polychromatic wave:
0ik rE r E e
Plane waves:
Pulse broadening in medium:
k
E
H
02
2
0
,
V ,
zi t
c
zi
c
U r t t z c e
r A e
A
