Spatiotemporal distortions

1
Spatio-temporal characteristics of light and how
to model them

• Spatio-temporal distortions
• Ray matrices
• The Gaussian beam
• Complex q and its propagation
• Ray-pulse Kosten-bauder matrices
• The prism pulse compressor
• Gaussian beam in space and time and the complex Q
  matrix

Optical system ? 4x4 Ray-pulse matrix
2
Spatio-temporal distortions
Ordinarily, we assume that the pulse-field
spatial and temporal factors (or their
Fourier-domain equivalents) separate
where the tilde and hat mean FTs with respect to
t and x, y, z
3
Angular dispersion is an example of a
spatio-temporal distortion.
In the presence of angular dispersion, the
off-axis k-vector component kx depends on w
where kx0(w) is the mean kx vs. frequency w.
4
Spatial chirp is a spatio-temporal distortion in
which the color varies spatially across the beam.

• Propagation through a prism pair produces a beam
  with no angular dispersion, but with spatial
  dispersion, often called spatial chirp.

Prism pairs are inside nearly every ultrafast
laser. A third and fourth prism undo this
distortion, but must be aligned carefully.
5
Spatial chirp is difficult to avoid.

• Simply propagating through a tilted window causes
  spatial chirp!

Because ultrashort pulses are so broadband, this
distortion is very noticeableand often
problematic!
6
How to think about spatial chirp
Suppose we send the pulse through a set of
monochromatic filters and find the beam center
position, x0, for each frequency, w.
where x0 is the center of the beam component of
frequency w.
7
Pulse-front tilt is another common
spatio-temporal distortion.
Phase fronts are perpendicular to the direction
of propagation. Because the group velocity is
usually less than phase velocity, pulse fronts
tilt when light traverses a prism.
Angularly dispersed pulse with
pulse-front tilt
Undistorted input pulse
Prism
Angular dispersion causes pulse-front tilt.
8
Angular dispersion causes pulse-front tilt even
when group velocity is not involved.
Diffraction gratings also yield pulse-front tilt.
Angularly dispersed pulse with
pulse- front tilt
The path is simply shorter for rays that impinge
on the near side of the grating. Of course,
angular dispersion and spatial chirp occur, too.
Undistorted input pulse
Diffraction grating
Gratings have about ten times the dispersion of
prisms, and they yield about ten times the tilt.
9
Modeling pulse-front tilt
Pulse-front tilt involves coupling between the
space and time domains
For a given transverse position in the beam, x,
the pulse mean time, t0, varies in the presence
of pulse-front tilt. Pulse-front tilt occurs
after pulse compressors that arent aligned
properly.
10
Angular dispersion always causes pulse-front tilt!
Angular dispersion means that the off-axis
k-vector depends on w
where g dkx0 /dw
Inverse Fourier-transforming with respect to kx,
ky, and kz yields
using the shift theorem
Inverse Fourier-transforming with respect to w-w0
yields
using the shift theorem again
which is just pulse-front tilt!
11
The combination of spatial and temporal chirp
also causes pulse-front tilt.
The theorem we just proved assumed no spatial
chirp, however. So it neglects another
contribution to the pulse-front tilt.
The total pulse-front tilt is the sum of that due
to dispersion and that due to this effect.
Xun Gu, Selcuk Akturk, and Erik Zeek
12
A pulse with temporal chirp, spatial chirp, and
pulse-front tilt.
Suppressing the y-dependence, we can plot such a
pulse
where the pulse-front tilt angle is
Well need a nice formalism for calculating these
distortions!
13
Spatio-temporal distortions can be useful or
inconvenient.
Good They allow pulse compression. They help
to measure pulses (tilted pulse fronts). They
allow pulse shaping. They can increase bandwidth
in nonlinear-optical processes. Bad They
usually increase the pulse length. They reduce
intensity. They can be hard to measure.
14
Ray Optics
axis
We'll define light rays as directions in space,
corresponding, roughly, to k-vectors of light
waves. Each optical system will have an axis,
and all light rays will be assumed to propagate
at small angles to it. This is called the
Paraxial Approximation.
15
The Optic Axis
A mirror deflects the optic axis into a new
direction. This ring laser has an optic axis
that scans out a rectangle.
Optic axis
We define all rays relative to the relevant optic
axis.
16
The Ray Vector
xin, qin
xout, qout

• A light ray can be defined by two co-ordinates

its position, x its slope, q
optical ray
q
x
Optical axis
These parameters define a ray vector,
which will change with distance and as the ray
propagates through optics.
17
Ray Matrices

• For many optical components, we can define 2 x 2
  ray matrices.
• An elements effect on a ray is found by
  multiplying its ray vector.

Ray matrices can describe simple and com- plex
systems.
Optical system ? 2 x 2 Ray matrix
These matrices are often called "ABCD Matrices."
18
Ray matrices as derivatives
Since the displacements and angles are assumed to
be small, we can think in terms of partial
derivatives.
We can write these equations in matrix form.
19
For cascaded elements, we simply multiply ray
matrices.
O2
O1
O3
Notice that the order looks opposite to what it
should be, but it makes sense when you think
about it.
20
Ray matrix for free space or a medium

• If xin and qin are the position and slope upon
  entering, let xout and qout be the position and
  slope after propagating from z 0 to z.

Rewriting these expressions in matrix notation
21
Ray Matrix for an Interface

• At the interface, clearly
• xout xin.
• Now calculate qout.
• Snell's Law says n1 sin(qin) n2
  sin(qout)
• which becomes for small angles n1 qin n2
  qout

Þ qout n1 / n2 qin
22
Ray matrix for a curved interface
At the interface, again xout xin. To
calculate qout, we must calculate q1 and q2. If
qs is the surface-normal slope at the height xin,
then q1 qin qs and q2 qout qs
If R is the surface radius of curvature, the
surface z coordinate will be
23
Ray matrix for a curved interface (contd)
q1 qin xin / R and q2 qout xin / R
Snell's Law n1 q1 n2 q2
Now the output angle depends on the input
position, too.
24
A thin lens is just two curved interfaces.
Well neglect the glass in between (its a really
thin lens!), and well take n1 1.
This can be written
where
The Lens-Makers Formula
25
Ray matrix for a lens
The quantity, f, is the focal length of the lens.
Its the single most important parameter of a
lens. It can be positive or negative. Its
possible to extend the Lens Makers Formula to
lenses of greater thickness.
R1 gt 0 R2 lt 0
R1 lt 0 R2 gt 0
If f gt 0, the lens deflects rays toward the axis.
If f lt 0, the lens deflects rays away from the
axis.
26
A lens focuses parallel rays to a point one focal
length away.
A lens followed by propagation by one focal
length
Assume all input rays have qin 0
At the focal plane, all rays converge to the z
axis (xout 0) independent of input
position. Parallel rays at a different angle
focus at a different xout.
27
Types of lenses
Lens nomenclature
Which type of lens to use (and how to orient it)
depends on the aberrations and application.
28
Ray Matrix for a Curved Mirror

• Consider a mirror with radius of curvature, R,
  with its optic axis perpendicular to the mirror

Like a lens, a curved mirror will focus a beam.
Its focal length is R/2. Note that a flat mirror
has R 8 and hence an identity ray matrix.
29
Laser Cavities
Mirror curvatures matter in lasers.
Two flat mirrors, the flat-flat laser cavity,
is difficult to align and maintain aligned.
Two concave curved mirrors, the stable laser
cavity, is easy to align and maintain aligned.
Two convex mirrors, the unstable laser cavity,
is impossible to align!
30
A system images an object when B 0.

• When B 0, all rays from a point xin arrive at a
  point xout, independent of angle.

xout A xin
A is the magnification.
31
The Lens Law

• From the object to
• the image, we have
• 1) A distance d0
• 2) A lens of focal length f
• 3) A distance di

32
Lenses can also map angle to position.

• From the object to
• the image, we have
• 1) A distance f
• 2) A lens of focal length f
• 3) A distance f

So And this arrangement maps position to
angle
33
If an optical system lacks cylindrical symmetry,
we must analyze its x- and y-directions
separately Cylindrical lenses

• A "spherical lens" focuses in both transverse
  directions.
• A "cylindrical lens" focuses in only one
  transverse direction.

When using cylindrical lenses, we must perform
two separate ray-matrix analyses, one for each
transverse direction.
34
Large-angle reflection off a curved mirror also
destroys cylindrical symmetry.
The optic axis makes a large angle with the
mirror normal, and rays make an angle with
respect to it.
tangential ray
Rays that deviate from the optic axis in the
plane of incidence are called tangential. Rays
that deviate from the optic axis to the plane
of incidence are called sagittal. (We need a 3D
display to show one of these.)
35
Ray Matrix for Off-Axis Reflection from a Curved
Mirror

• If the beam is incident at a large angle, q, on a
  mirror with radius of curvature, R

tangential ray
q
where Re R cosq for tangential rays and Re
R / cosq for sagittal rays
36
But lasers are Gaussian beams, not rays.

• Real laser beams are localized in space at the
  laser and hence must diffract as they propagate
  away from the laser.

• The beam has a waist at z 0, where the spot
  size is w0. It then expands to w w(z) with
  distance z away from the laser.
• The beam radius of curvature, R(z), also
  increases with distance far away.

37
Gaussian beam math
The expression for a real laser beam's electric
field is given by

• where
• w(z) is the spot size vs. distance from the
  waist,
• R(z) is the beam radius of curvature, and
• y(z) is a phase shift.
• This equation is the solution to the wave
  equation when we require that the beam be well
  localized at some point (i.e., its waist).

38
Gaussian beam spot size, radius, and phase

• The expressions for the spot size,radius of
  curvature, and phase shift

where zR is the Rayleigh Range (the distance over
which the beam remains about the same diameter),
and it's given by
39
Gaussian beam collimation

• Twice the Rayleigh range is the
• distance over which the beam
• remains about the same size,
• that is, remains collimated.
• _____________________________________________
• .225 cm 0.003 km 0.045 km
• 2.25 cm 0.3 km 5 km
• 22.5 cm 30 km 500 km
• _____________________________________________
• Tightly focused laser beams expand quickly.
  Weakly focused beams expand less quickly, but
  still expand.As a result, it's very difficult to
  shoot down a missile with a laser.

Collimation
Collimation Waist spot Distance
Distance size w0 l 10.6 µm
l 0.633 µm
Longer wavelengths expand faster than shorter
ones.
40
Gaussian beam divergence
Far away from the waist, the spot size of a
Gaussian beam will be

• The beam 1/e divergence half angle is then w(z) /
  z as z

The smaller the waist and the larger the
wavelength, the larger the divergence angle.
41
Focusing a Gaussian beam

• A lens will focus a collimated Gaussian beam to a
  new spot size
• wfocus l f / pwinput
• So the smaller the desired focus, the BIGGER the
  input beam should be!

42
The Guoy phase shift
The phase factor yields a phase shift relative to
the phase of a plane wave when a Gaussian beam
goes through a focus.
Phase relative to a plane wave
Recall the i in front of the Fresnel integral,
which is a result of the Guoy phase shift.
43
The Gaussian-beam complex-q parameter
We can combine these two factors (theyre both
Gaussians)
q completely determines the Gaussian beam.
where
44
Ray matrices and the propagation of q
Wed like to be able to follow Gaussian beams
through optical systems. Remarkably, ray matrices
can be used to propagate the q-parameter.
Optical system
This relation holds for all systems for which ray
matrices hold
Just multiply all the matrices first and use this
result to obtain qout for the relevant qin!
45
Important point about propagating q
Use
to compute qout.
But use matrix multiplication for the various
components to compute the total system ray
matrix. Dont compute
for each component. Youd get
the right answer, but youd work much harder than
you need to!
46
Propagating q an example
Free-space propagation through a distance z
The ray matrix for free-space propagation is
Then
47
Propagating q an example (contd)
Does q(z) q0 z? This is equivalent to
1/q(z) 1/(q0 z).
LHS
Now
so
RHS
which is just this.
So
48
Propagating q another example
Focusing a collimated beam (i.e., a lens, f,
followed by a distance, f )
A collimated beam has a big spot size (w) and
Rayleigh range (zR), and an infinite radius of
curvature (R), so qin i zR
So
But
The well-known result for the focusing of a
Gaussian beam
49
Now consider the time and frequencyof a light
pulse in addition
Wed like a matrix formalism to predict such
effects as the group-delay dispersion ?t/?w
angular dispersion ?kx /?w or ?q /?w spatial
chirp ?x/?w pulse-front tilt ?t/?x time vs. angle
?t/?q. where weve dropped0 subscripts
forsimplicity.
This pulse has all of these distortions!
Well need to consider, not only the position (x)
and slope (q ) of the ray, but also the time (t)
and frequency (w ) of the pulse.
50
Propagation in space and time Ray-pulse
Kostenbauder matrices

• Kostenbauder matrices are 4x4 matrices that
  multiply 4-vectors comprising the position,
  slope, time (group delay), and frequency.

where each vector component corresponds to the
deviation from a mean value for the ray or pulse.
Optical system ? 4x4 Ray-pulse matrix
A Kostenbauder matrix requires five additional
parameters, E, F, G, H, I.
51
Kostenbauder matrix elements
As with 2x2 ray matrices, consider each element
to correspond to a small deviation from its mean
value (xin x x0 ). So we can think in terms
of partial derivatives.
52
Some Kostenbauder matrix elements are always zero
or one.
53
Kostenbauder matrix for propagation through free
space or material
The ABCD elements are always the same as the ray
matrix. Here, the only other interesting element
is the GDD I ?tout/?nin
So
The 2p is due to the definition of K-matrices in
terms of n, not w.
where L is the thickness of the medium, n is its
refractive index, and k is the GVD
54
Example Using the Kostenbauder matrix for
propagation through free space
Apply the free-space propagation matrix to an
input vector
The position varies in the usual way, and the
beam angle remains the same.
The group delay increases by kLwin
The frequency remains the same.
Because the group delay depends on frequency, the
pulse broadens. This approach works in much more
complex situations, too.
55
Kostenbauder matrix for a lens
The ABCD elements are always the same as the ray
matrix. Everything else is a zero or one.
So
where f is the lens focal length. The same holds
for a curved mirror, as with ray matrices. While
chromatic aberrations can be modeled using a
wavelength-dependent focal length, other lens
imperfections cannot be modeled using
Kostenbauder matrices.
56
Kostenbauder matrix for a diffraction grating
Gratings introduce magnification, angular
dispersion and pulse-front tilt
So
angular magnification
spatial magnification
no spatial chirp (yet)
angular dispersion
pulse-front tilt
time is independent of angle
no GDD (yet)

• where b is the incidence angle, and b is the
  diffraction angle.
• The zero elements (E, H, I) will become nonzero
  when propagation follows.

57
Kostenbauder matrix for a general prism
All new elements are nonzero.
L
angular dispersion
spatial chirp
angular magnification
spatial magnification
time vs. angle
GDD
pulse-front tilt
where
is the GVD,
and
58
Kostenbauder matrix for a Brewster prism

• If the beam passes through the apex of the prism
• (this simplifies the calculation a lot!)

Brewster angle incidence and exit
Use if the prism is oriented as above use if
its inverted.
Just angular dispersion and pulse-front tilt. No
GDD etc.
where
59
Using the Kostenbauder matrix for a Brewster prism
This matrix takes into account all that we need
to know for pulse compression.
Dispersion changes the beam angle.
Pulse-front tilt yields GDD.
When the pulse reaches the two inverted prisms,
this effect becomes very important, yielding
longer group delay for longer wavelengths (D lt
0 and use the minus sign for inverted prisms).
60
Modeling a prism pulse compressor using
Kostenbauder matrices
Use only Brewster prisms
1
7
2
6
Kprism
Kprism
Kprism
Kprism
Kair
Kair
3
5
Kair
4
K K7 K6 K5 K4 K3 K2 K1
61
Free space propagation in a pulse compressor
There are three distances in this problem.
L1
L3
L2
n 1 in free space
62
K-matrix for a prism pulse compressor
K K7 K6 K5 K4 K3 K2 K1
Spatial chirp unless L1 L3.
Negative GDD!
Time vs. angle unless L1 L3.
The GDD is negative and can be tuned by changing
the amount of extra glass in the beam (which we
havent included yet, but which is easy).
63
Propagating spot size, radius of curvature,
pulse length, and chirp
To follow beams that are Gaussian in both space
and time
We could propagate Gaussian beams in space
because theyre quadratic in space (x and y)
A Gaussian pulse is quadratic in time. And the
real and imaginary parts also have important
meanings (pulse length and chirp)
64
The complex-Q matrix
We define the complex Q-matrix so that the space
and time dependence of the pulse can be written
These complex matrix elements contain all the
parameters of beams/pulses that are Gaussian in
space and time.
65
The complex-Q matrix (contd)
When the off-diagonal elements, , are zero
spatial complex-q parameter for Gaussian beams
temporal complex-q parameter the pulse length
and chirp parameter for Gaussian pulses
When the off-diagonal components are not zero,
there is pulse-front tilt
66
K-matrices and the propagation of Q
Kostenbauder matrices can be used to propagate
the Q-matrix.
This relation holds for all systems (multiply the
component matrices together first and then use
this complex result)
This is actually more elegant than it
looks...
Division means multiplication by the inverse.
67
Propagating the Q-matrix
A
B
D
C
Notice the symmetry in the 2x2 matrices in the
Q-propagation equation.
In terms of these 2x2 matrices
68
Important point about propagating Q
Use
to compute Qout.
But use matrix multiplication for the various
components to compute the total system ray
matrix. Dont compute
for each component. As with q,
youd get the right answer, but youd work much
harder than you need to!
69
How to compute pulse distortions
The K-matrix elements are derivatives of the
output parameters with respect to the input
parameters, and the spatio-temporal distortions
are derivatives of means with respect to the
output pulse parameters. But because nout nin,
the E dxout/dvin, F dqout/dvin, and I
dtout/dvin elements clearly indicate the (added)
spatial chirp, angular dispersion, and
group-delay dispersion. This doesnt work for
pulse-front tilt, G dtout/dxin, however.
assuming theres zero distortion to begin with
Spatial chirp E element of the
K-matrix Angular dispersion F element of the
K-matrix Group-delay dispersion I element of
the K-matrix Pulse-front tilt You must look
for an off-diagonal Q-1 element
because xout may not be xin
70
How to really compute pulse distortions
In general, you really should look at the
complete Q-matrix. Lets write
where
and
So
Now the pulse-front tilt is the real part of the
off-diagonal -element.
71
How to really compute pulse distortions
What about the other spatio-temporal
distortions? Fourier-transform
with
respect to t
Complex Gaussians transform to complex Gaussians!
The spatial chirp is the real part of the
off-diagonal R element.
Now Fourier-transform E(x,t) into the kw, and kt
domains
Angular dispersion
Time vs. angle
72
The imaginary parts of the pulse distortions
Spatio-temporal phase distortions
The imaginary parts of the spatio-temporal
distortions are new and interesting. Consider the
imaginary part of
Wave-propagation direction
A movie of the electric field vs. x and z.
This distortion is called wave-front rotation.
73
The imaginary parts of the pulse distortions
Spatio-temporal phase distortions
Now consider the imaginary part of Rxw
Plots of the electric field vs. x and z for
different colors.
This distortion is called wave-front tilt
dispersion.
74
Kostenbauder matrices can model very general
systems using the pulse Wigner Distribution.
Okay, the complex-Q matrix tells us what happens
to Gaussian pulse/beams. But what about more
complex pulse/beams? Ordinarily, youd have to
numerically solve the Fresnel diffraction
integral, which can yield a very complex computer
computation. But theres a simpler approach. It
uses the Wigner Distribution, a different way to
represent a pulses dependence on time
The Wigner Distribution converts the pulse into a
plot of intensity vs. time (delay) and frequency.
75
We can invert the Wigner Distribution to obtain
the pulse field.
Inverse-Fourier transform the Wigner Distribution
with respect to w
Setting t t/2
So we cant determine the absolute phase, but
simple inverse Fourier transforming yields the
rest of E(t).
76
Examples of Wigner Distributions
Linearly chirped Gaussian
Quadratically chirped Gaussian
The Wigner Distribution is always real, but it
usually goes negative.
Double pulse
77
Properties of the Wigner Distribution
The marginals (integrals) of the Wigner
Distribution yield the pulse intensity vs. time
and the spectrum vs. frequency.
The Wigner Distribution has many other nice
properties.
78
We can define spatial and space-time Wigner
Distributions, too.
dropping the x subscript on k
A spatial Wigner Distribution is in terms of x
and kx
The Wigner Distribution converts the beam into a
plot of intensity vs. space and spatial
frequency. Then we can define a space-time
Wigner Distribution
79
In terms of Wigner Distributions, Kostenbauder
matrices can describe a general optical system!
If the K-matrix of the system and the input-pulse
Wigner Distribution are known
where the output Wigner Distribution and its
parameters are determined from the input
parameters, the K-matrix of the system, and the
input pulse Wigner Distribution. The quantity, m,
is a magnification.
J. Paye and A. Migus, Spacetime Wigner
functions and their application to the analysis
of a pulse shaper, J. Opt. Soc. Am. B, 12, 8, p
1480, August 1995.