Pulse description --- a propagating pulse
description
Transcript of Pulse description --- a propagating pulse
Pulse description --- a propagating pulse
A Bandwidth limited pulse No Fourier Transform involved
Fourier transforms review
Slowly Varying Envelope Approximation
OPTICS OF SHORT PULSES
with minimum of equations, maximum of analogies and hand waving.
time0
Electric fieldamplitude
Many frequencies in phase construct a pulse
A Bandwidth limited pulse
FREQUENCY
Time and frequency considerations: stating the obvious
TIME
E
A Bandwidth limited pulse
FREQUENCY
The spectral resolution of the cw wave is lost
TIME
E
A Bandwidth limited pulse
z
t
z = ctz = vgt
A propagating pulse
t
A Bandwidth limited pulse
We may need the Fourier transforms (review)
0
Shift
Derivative
Linear superposition
Specific functions: Square pulse Gaussian Single sided exponential
Real E(E*(-
Linear phase
Product Convolution
Derivative
Properties of Fourier transforms
Construct the Fourier transform of
0
Description of an optical pulse
Real electric field:
Fourier transform:
Positive and negative frequencies: redundant information Eliminate
Relation with the real physical measurable field:
Instantaneous frequency
We have to return to Maxwell's propagation equation:
In frequency
How to correctly propagate an ultrashort pulse without phase and group velocity
It is only if That the pulse propagates unchanged at velocity n/c
Group velocity is a concept that is clearly related to the SVEA
Maxwell’s equations, linear propagation
Propagation of the complex field
Maxwell’s equations, nonlinear propagation
Pulse broadening, dispersion
Maxwell’s equations, linear propagation
Dielectrics, no charge, no current:
Medium equation:
can be a tensor birefringence
In a linear medium:
Maxwell’s equations, nonlinear propagation
Maxwell’s equation:
Since the E field is no longer transverse
Gadi Fibich and Boaz Ilan PHYSICAL REVIEW E 67, 036622 (2003)
Is it important?
Only if
20 0
02
n nE P
z c t z c t t
2 2 2 20
02 2 2 2
nE P
z c t t
22
2F FP P
t
Study of propagation from second to first order
From Second order to first order (the tedious way)
( ) ( )kz kz
2 2 2 20 i t i t
02 2 2 2
ne P e
z c t t
2 2 22
2 2 2 2 2
22
0 0 02
1 2ik 2ik
c z c t c t z
P i P Pt t
01 i cP
z c t 2
(Polarization envelope)
Pulse broadening, dispersion
Solution of 2nd order equation
22
02
( ) ( , ) 0E zz
0( ) (1 ( ))
( )( , ) ( , ) ik zE z E 0 e
( ) ( )2 20k
0( )P E Propagation through medium
No change in frequency spectrum
To make F.T easier shift in frequencyExpand k value around central freq l
l
( )( , ) ( , ) lik zz 0 e ε εz
Z=0
1( , ) ( , ) ( )
2i tE t z E z e d
1
0gz v t
ε ε
Study of linear propagation
Expand k to first order, leads to a group delay:
Expansion orders in k(Material property
l
l
2| 22
1( , ) ( ,0) (1 | ( ) ) ( )
2l
dkiik z i td d k
t z e e e i z dd
ε ε
( )( , ) ( , ) lik zz 0 e ε εll
| ( )| ( )( , )
22
2 l
1 d kdk i zi z ik z2d d0 e e e
ε
l
l
| ( )( , ) ( | ( ) ) l
dk 2i z 2 ik zd2
1 d k0 e 1 i z e
2 d
ε
22
2
( ) 1( ) ( )
2ixtt
x x e d xt
ε ε
2 2
2 2
10
2g
i d k
z v t d t
ε ε ε
Study of linear propagation
Propagation in dispersive media: the pulse is chirped and broadening
Propagation in nonlinear media: the pulse is chirped
Combination of both: can be pulse broadening, compression,Soliton generation
Propagation in the time domain
PHASE MODULATION
n(t)or
k(t)
E(t) = (t)eit-kz
(t,0) eik(t)d (t,0)
DISPERSION
n()or
k()() ()e-ikz
Propagation in the frequency domain
Retarded frame and taking the inverse FT:
PHASE MODULATION
DISPERSION
Townes’soliton
Eigenvalue equation (normalized variables. Solution of type:
2D nonlinear Schroedinger equation
Normalization: and
Soliton equation in space
In space:
In time
Back to linear propagation: Gaussian pulse
-6 -4 -2 0 2 4 6
-1
0
1
-20 -10 0 10 20
Delay (fs)
Pulse propagation through 2 mm of BK7 glass
Pulse duration, Spectral width
Two-D representation of the field: Wigner function
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
Time TimeF
requ
ency
Fre
quen
cy
-2 -1 0 1 2
-2
-1
0
1
2
-2 -1 0 1 2
-2
-1
0
1
2
Time TimeF
requ
ency
Fre
quen
cyGaussian Chirped Gaussian
Wigner Distribution
Wigner function: What is the point?
Uncertainty relation:
Equality only holds for a Gaussian pulse (beam) shape free of anyphase modulation, which implies that the Wigner distribution for aGaussian shape occupies the smallest area in the time/frequencyplane.
Only holds for the pulse widths defined as the mean square deviation
A Bandwidth limited pulse
Some (experimental) displays of electric field versus time
-6 -4 -2 0 2 4 6
-1
0
1
-20 -10 0 10 20
Delay (fs)
How was this measured?
A Bandwidth limited pulse
Some (experimental) displays of electric field versus time
-20 -10 0 10 20
Delay (fs)
Chirped pulse
Construct the Fourier transform of
Pulse Energy, Parceval theorem
Poynting theorem
Pulse energy
Parceval theorem
Intensity?
Spectral intensity