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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 “Kostenbauder” matrices
The prism pulse
compressor
Optical system ↔ 4x4 Ray-pulse matrix
⎡ xin ⎤
⎢θ ⎥
⎢ in ⎥
⎢ tin ⎥
⎢ ⎥
⎢⎣ν in ⎥⎦
⎡A
⎢C
⎢
⎢G
⎢
⎣0
B
0
D
0
H
1
0
0
E⎤
F ⎥⎥
I⎥
⎥
1⎦
Gaussian beam in space and time and the complex Q matrix
⎡ xout ⎤
⎢θ ⎥
⎢ out ⎥
⎢ tout ⎥
⎢
⎥
⎣⎢ν out ⎦⎥
Spatio-temporal distortions
Ordinarily, we assume that the pulse-field spatial and temporal
factors (or their Fourier-domain equivalents) separate:
E ( x, y, z , t ) = Exyz ( x, y, z ) Et (t )
Eˆ% (k x , k y , k z , ω ) = Eˆ xyz (k x , k y , k z ) E% t (ω )
where the tilde and
hat mean FTs with
respect to t and x, y, z
In pulses with spatio-temporal distortions, they don’t:
We’ll use
dt0
coordinates
xt : E ( x, y, z, t ) → E[ x, y, z, t +
( x − x0 )]
such as x – x0,
dx
that are (small)
dx0
deviations from
%
%
xω : E ( x, y, z , ω ) → E[ x +
(ω − ω0 ), y, z, ω ]
their means.
dω
dk x 0
ˆ%
ˆ%
kω : E (k x , k y , k z , ω ) → E0 [k x +
(ω − ω0 ), k y , k z , ω ]
dω
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.
Angular dispersion is an example
of a spatio-temporal distortion.
dk x 0
dω
In the presence of angular dispersion, the off-axis k-vector
component kx depends on ω:
dk x 0
ˆ
ˆ
%
%
(ω − ω0 ), k y , k z , ω ]
E (k x , k y , k z , ω ) = E0 [k x +
dω
where kx0(ω) is
the mean kx vs.
frequency ω.
x
z
Input
pulse
Prism
Angularly
dispersed
output pulse
Note that the
mean off-axis
k-vector
component, kx0,
depends on ω.
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 pair
Input
pulse
Spatially chirped output
pulse
Prism pairs are inside nearly every ultrafast laser. A third and fourth
prism undo this distortion, but must be aligned carefully.
Spatial chirp is difficult to avoid.
Simply propagating through a tilted window causes spatial chirp!
Tilted
window
Input
pulse
Spatially chirped
output pulse
Because ultrashort pulses are so broadband, this distortion is
very noticeable—and often problematic!
How to think about ∂x0
spatial chirp ∂ω
x
x0(ω9)
x0(ω8)
x0(ω7)
x0(ω6)
x0(ω5)
x0(ω4)
x0(ω3)
x0(ω2)
x0(ω1)
z
Suppose we send the pulse
through a set of monochromatic
filters and find the beam center
position, x0, for each frequency, ω.
E% ( x, y, z, ω ) →
dx0
%
E[ x +
(ω − ω0 ), y, z , ω ]
dω
where x0 is the center of
the beam component of
frequency ω.
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.
Undistorted
input
pulse
Angularly
dispersed pulse
with pulse-front tilt
Prism
Angular dispersion causes pulse-front tilt.
Angular dispersion causes pulse-front tilt
even when group velocity is not involved.
Diffraction gratings also yield pulse-front tilt.
Angularly dispersed
pulse with pulsefront tilt
Undistorted
input
pulse
Diffraction
grating
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.
Gratings have about ten times the dispersion of prisms, and they
yield about ten times the tilt.
Modeling
pulse-front tilt
dt0
dx
Pulse-front tilt involves coupling
between the space and time domains:
dt0
E ( x, y, z, t ) → E[ x, y, z, t +
( x − x0 )]
dx
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 aren’t aligned
properly.
Angular dispersion always causes pulsefront tilt!
Angular dispersion means that the off-axis k-vector depends on ω:
ˆ
ˆ
E% (k x , k y , k z , ω ) = E% 0 [k x + γ (ω − ω0 ), k y , k z , ω ]
where γ = dkx0 /dω
Inverse Fourier-transforming with respect to kx, ky, and kz yields:
i γ (ω −ω0 ) x
%
%
E ( x, y, z , ω ) = E0 ( x, y, z , ω ) e
using the shift theorem
Inverse Fourier-transforming with respect to ω−ω0 yields:
⇒ E ( x, y, z , t ) = E0 ( x, y, z , t + γ x)
using the shift theorem again
which is just pulse-front tilt!
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.
Dispersive
medium
Spatially chirped
input pulse
Spatially
chirped pulse
with pulse-front tilt,
but no angular
dispersion
vg(red) > vg(blue)
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
A pulse with temporal chirp, spatial
chirp, and pulse-front tilt.
Suppressing the y-dependence, we can plot such a pulse:
4.5 mrad
ψ = 11.3
mrad
797
803 nm
nm
where the pulsefront tilt angle is:
∂t 0
ψ ≡c
∂x
xx [[m
mm
m]
]
]
tt [[ffss]
775 nm
777 nm
We’ll need a nice formalism for calculating these distortions!
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.
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
A ray propagating
through this system
We define all rays relative to the relevant optic axis.
The Ray
Vector
xin, θin
xout, θout
A light ray can be defined by two co-ordinates:
its position, x
ray
l
a
c
i
t
op
its slope, θ
θ
x
Optical axis
These parameters define a ray vector,
which will change with distance and as
the ray propagates through optics.
⎡x⎤
⎢θ ⎥
⎣ ⎦
Ray Matrices
For many optical components, we can define 2 x 2 ray matrices.
An element’s effect on a ray is found by multiplying its ray vector.
Ray matrices
can describe
simple and complex systems.
Optical system ↔ 2 x 2 Ray matrix
⎡ xin ⎤
⎢θ ⎥
⎣ in ⎦
⎡A
⎢C
⎣
B⎤
D ⎥⎦
⎡ xout ⎤
⎢θ ⎥
⎣ out ⎦
These matrices are often called "ABCD Matrices."
Ray matrices as
derivatives
Since the displacements and angles
are assumed to be small, we can
think in terms of partial derivatives.
spatial
magnification
∂xout
∂xin
xout
∂xout
∂xout
=
xin +
θin
∂xin
∂θin
θout
∂θout
∂θout
=
xin +
θin
∂xin
∂θin
∂xout
∂θin
⎡ xout ⎤ ⎡ A B ⎤ ⎡ xin ⎤
⎢θ ⎥ = ⎢C D ⎥ ⎢θ ⎥
⎦ ⎣ in ⎦
⎣ out ⎦ ⎣
∂θout
∂xin
∂θout
∂θin
angular
magnification
We can write
these equations
in matrix form.
For cascaded elements, we simply
multiply ray matrices.
⎡ xin ⎤
⎢θ ⎥
⎣ in ⎦
O2
O1
⎡ xout ⎤
⎪⎧
⎢θ ⎥ = O3 ⎨O2
⎣ out ⎦
⎪⎩
⎛
⎜ O1
⎝
O3
⎡ xout ⎤
⎢θ ⎥
⎣ out ⎦
⎡ xin ⎤ ⎞ ⎪⎫
⎡ xin ⎤
⎢θ ⎥ ⎟ ⎬ = O3 O2 O1 ⎢θ ⎥
⎣ in ⎦ ⎠ ⎪⎭
⎣ in ⎦
Notice that the order looks opposite to what it should be,
but it makes sense when you think about it.
Ray matrix for free space or a medium
If xin and θin are the position and slope upon entering, let xout and
θout be the position and slope after propagating from z = 0 to z.
xout θout
xin, θin
xout = xin + z θin
θ out = θin
Rewriting these expressions
in matrix notation:
z
z=0
Ospace
⎡1 z ⎤
= ⎢
⎥
0
1
⎣
⎦
⎡xout ⎤
⎡1 z⎤ ⎡xin ⎤
⎢θ ⎥ = ⎢0 1⎥ ⎢θ ⎥
⎣
⎦ ⎣ in ⎦
⎣ out ⎦
Ray Matrix for an Interface
At the interface, clearly:
θin
xout = xin.
xin
xout
n1
Now calculate θout.
Snell's Law says:
θout
n2
n1 sin(θin) = n2 sin(θout)
which becomes for small angles: n1 θin = n2 θout
⇒ θout = [n1 / n2] θin
Ointerface
0 ⎤
⎡1
=⎢
⎥
n
n
0
/
⎣
1
2⎦
Ray matrix for a curved interface
At the interface, again:
xout = xin.
To calculate θout, we must
calculate θ1 and θ2.
If θs is the surface-normal
slope at the height xin, then
θs
θin
R
θ2
θ1
xin = xout
n1
n2
θout
θs
z
θ1 = θin+ θs and θ2 = θout+ θs
If R is the surface radius of curvature, the surface z coordinate will be:
z = R − R 2 − xin2 = R − R 1 − ( xin / R) 2 ≈ R − R ⎡⎣1 − 12 ( xin / R) 2 ⎤⎦
= 12 ( xin2 / R)
dz
⇒ θs ≈
≈ xin / R
dxin
Ray matrix for a curved interface (cont’d)
θ1 = θin+ xin / R and θ2 = θout+ xin / R
Snell's Law:
n1 θ1 = n2 θ2
θ1
θ2
⇒ n1 (θin + xin / R) ≈ n2 (θ out + xin / R)
⇒ θ out ≈ (n1 / n2 )(θin + xin / R) − xin / R
n1
⇒ θ out ≈ (n1 / n2 )θin + (n1 / n2 − 1) xin / R
Ocurved
interface
1
0 ⎤
⎡
=⎢
⎥
n
n
R
n
n
−
(
/
1)
/
/
⎣ 1 2
1
2⎦
Now the output angle depends on the input position, too.
n2
A thin lens is just two curved interfaces.
We’ll neglect the glass in between (it’s a
really thin lens!), and we’ll take n1 = 1.
Ocurved
interface
1
0 ⎤
⎡
=⎢
⎥
(
/
1)
/
/
n
n
−
R
n
n
1
2⎦
⎣ 1 2
R1
n=1
R2
n≠1
n=1
1
0⎤ ⎡
1
0 ⎤
⎡
Othin lens = Ocurved Ocurved = ⎢
⎥ ⎢ (1/ n − 1) / R 1/ n ⎥
(
1)
/
−
n
R
n
interface 2
interface 1
2
1
⎣
⎦⎣
⎦
1
0 ⎤ ⎡
1
0⎤
⎡
=⎢
=⎢
⎥
⎥
(
n
1)
/
R
n
(1/
n
1)
/
R
n
(1/
n
)
(
n
1)
/
R
(1
n
)
/
R
1
−
+
−
−
+
−
⎣
2
1
⎦ ⎣
2
1
⎦
1
0⎤
⎡
=⎢
⎥
(
n
1)(1/
R
1/
R
)
1
−
−
⎣
2
1
⎦
This can be written:
where: 1/ f = (n − 1)(1/ R1 − 1/ R2 )
⎡ 1
⎢ −1/ f
⎣
0⎤
1 ⎥⎦
The Lens-Maker’s Formula
Ray matrix for a lens
Olens
1/ f = (n − 1)(1/ R1 − 1/ R2 )
⎡ 1
= ⎢
⎣-1/f
0⎤
⎥
1⎦
The quantity, f, is the focal length of the lens. It’s the single most
important parameter of a lens. It can be positive or negative.
In a homework problem, you’ll extend the Lens Maker’s Formula to
lenses of greater thickness.
R1 > 0
R2 < 0
f>0
If f > 0, the lens deflects
rays toward the axis.
R1 < 0
R2 > 0
f<0
If f < 0, the lens deflects
rays away from the axis.
A lens focuses parallel rays to a point
one focal length away.
For all rays
A lens followed by propagation by one focal length:
xout = 0!
f ⎤⎡xin ⎤ ⎡ 0 ⎤
⎡xout ⎤ ⎡1 f ⎤⎡ 1 0⎤⎡xin ⎤ ⎡ 0
⎢θ ⎥ = ⎢0 1⎥⎢−1/ f 1⎥⎢ 0 ⎥ = ⎢−1/ f 1⎥⎢ 0 ⎥ = ⎢−x / f ⎥
⎦⎣
⎦⎣ ⎦ ⎣
⎦⎣ ⎦ ⎣ in ⎦
⎣ out ⎦ ⎣
f
f
Assume all input
rays have θin = 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.
Types of lenses
Lens nomenclature
Which type of lens to use (and how to orient it) depends on the
aberrations and application.
Ray Matrix for a Curved Mirror
Consider a mirror with radius of curvature, R, with its optic axis
perpendicular to the mirror:
θ1 = θin − θ s
θ s ≈ xin / R
R
θout
θs
θin
θ out = θ1 − θ s = (θin − θ s ) − θ s
≈ θin − 2 xin / R
θ1
θ1
xin = xout
z
0⎤
⎡ 1
⇒ Omirror = ⎢
⎥
−
2
/
1
R
⎣
⎦
Like a lens, a curved mirror will focus a beam. Its focal length is R/2.
Note that a flat mirror has R = ∞ and hence an identity ray matrix.
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!
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.
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
⎡1 di ⎤ ⎡ 1
O=⎢
⎥ ⎢ −1/ f
0
1
⎣
⎦⎣
⎡1 d i ⎤ ⎡ 1
=⎢
⎥ ⎢ −1/ f
0
1
⎣
⎦⎣
B
⎡ A
=⎢
⎣ −1/ f 1 − d 0 /
0 ⎤ ⎡1 d 0 ⎤
1 ⎥⎦ ⎢⎣ 0 1 ⎥⎦
d0 ⎤
1 − d 0 / f ⎥⎦
⎤
f ⎥⎦
B = d 0 + di − d 0 di / f =
d 0 di [1/ d 0 + 1/ di − 1/ f ] =
0 if
1 1 1
+ =
d 0 di f
⎡1 1⎤
A = 1 − di / f = 1 − di ⎢ + ⎥
⎣ d 0 di ⎦
d
⇒
Mag = − i
d0
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
0⎤ ⎡1 f ⎤ ⎡xin ⎤
⎡xout ⎤ ⎡1 f ⎤ ⎡ 1
⎢θ ⎥ = ⎢0 1 ⎥ ⎢−1/ f 1⎥ ⎢0 1 ⎥ ⎢θ ⎥
⎦⎣
⎦⎣
⎦ ⎣ in ⎦
⎣ out ⎦ ⎣
f ⎤ ⎡xin ⎤
⎡1 f ⎤ ⎡ 1
=⎢
⎥ ⎢−1/ f 1 ⎥ ⎢θ ⎥
0
1
⎣
⎦⎣
⎦ ⎣ in ⎦
f ⎤ ⎡xin ⎤ ⎡ f θin ⎤
⎡ 0
=⎢
=⎢
⎢
⎥
⎥
⎥
x
f
θ
/
−
f
1/
0
−
⎣
⎦ ⎣ in ⎦ ⎣ in ⎦
So
xout ∝ θ in
And this arrangement
maps position to angle:
θ out ∝ xin
If an optical system lacks cylindrical
symmetry, we must analyze its x- and ydirections 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.
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.
Optic axis
before reflection
tangential
ray
Optic axis after
reflection
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.)
Ray Matrix for Off-Axis Reflection from
a Curved Mirror
If the beam is incident at a large angle, θ, on a mirror with radius of
curvature, R:
tangential ray
Optic axis
θ
⇔
R
where Re = R cosθ for tangential rays
and Re = R / cosθ for sagittal rays
⎡ 1
⎢ −2 / R
e
⎣
0⎤
1 ⎥⎦
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.
Gaussian
beam
math
The expression for a real laser beam's electric field is given by:
E ( x, y , z ) ∝
%
exp [ −ikz − iψ ( z ) ]
w( z )
⎡ x2 + y 2 π x2 + y 2 ⎤
exp ⎢ − 2
−i
⎥
λ
w
(
z
)
R
(
z
)
⎣
⎦
where:
w(z) is the spot size vs. distance from the waist,
R(z) is the beam radius of curvature, and
ψ(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).
Gaussian
beam spot
size, radius,
and phase
The expressions for
the spot size,
radius of curvature,
and phase shift:
w( z ) = w0 1 + ( z / z R )
2
R ( z ) = z + z R2 / z
ψ ( z ) = arctan( z / z R )
where zR is the Rayleigh Range (the distance over which the
beam remains about the same diameter), and it's given by:
z R ≡ π w0 / λ
2
Gaussian beam collimation
Twice the Rayleigh range is the
distance over which the beam
remains about the same size,
that is, remains “collimated.”
2 z R = 2π w02 / λ
Collimation
Collimation
Waist spot
Distance
Distance
λ = 10.6 µm
λ = 0.633 µm
size w0
_____________________________________________
.225 cm
0.003 km
0.045 km
2.25 cm
0.3 km
5 km
Longer wavelengths
expand faster than
shorter ones.
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.
Gaussian beam
divergence
Far away from the waist, the
spot size of a Gaussian beam
will be:
w( z ) = w0 1 + ( z / z R ) ≈ w0
2
( z / zR )
2
= w0 z / z R
The beam 1/e divergence half angle is then w(z) / z as z → ∞ :
θ1/ e
⇒
w0 z w0
w0
=
=
=
z R z z R π w02 / λ
θ1/ e = λ / (π w0 )
The smaller the waist and the larger the wavelength, the larger
the divergence angle.
Focusing a Gaussian beam
winput
f
wfocus
f
A lens will focus a collimated Gaussian beam to a new spot size:
wfocus ≈ λ f / πwinput
So the smaller the desired focus, the BIGGER the input beam should be!
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.
⎧ π / 2 when z → +∞
ψ ( z ) = arctan( z / z R ) → ⎨
⎩−π / 2 when z → −∞
ψ(z)
Phase relative
to a plane wave:
π/2
-zR
zR
−π/2
Recall the i in front of the Fresnel integral, which is a result of the
Guoy phase shift.
The Gaussian-beam complex-q parameter
⎡ π x2 + y2 ⎤
⎡ x2 + y2 ⎤
E ( x, y, z ) ∝ exp ⎢ −i
exp ⎢ − 2
⎥
⎥
λ
R
z
w
z
(
)
(
)
⎣
⎦
⎣
⎦
We can combine these two factors (they’re both Gaussians):
⎡ π x2 + y 2 ⎤
E ( x, y, z ) ∝ exp ⎢ −i
⎥
(
)
λ
q
z
⎣
⎦
where:
1
1
λ
≡
−i
q( z )
R ( z ) π w2 ( z )
q completely determines
the Gaussian beam.
Ray matrices and the propagation of q
We’d like to be able to follow Gaussian beams through optical systems.
Remarkably, ray matrices can be used to propagate the q-parameter.
⎡A B⎤
O=⎢
⎥
C
D
⎣
⎦
Optical system
This relation
holds for all
systems for
which ray
matrices hold:
qout
Aqin + B
=
Cqin + D
Just multiply all
the matrices first
and use this result
to obtain qout for
the relevant qin!
Important point about propagating q
Use
qout
Aqin + B
=
Cqin + D
to compute qout.
But use matrix multiplication for the various components to
compute the total system ray matrix.
Don’t compute
qout
Aqin + B
=
Cqin + D
for each component.
You’d get the right answer, but you’d work much harder than
you need to!
Propagating q: an example
qout
Aqin + B
=
Cqin + D
Free-space propagation through a distance z:
The ray matrix for free-space propagation is:
Ospace
1q(0) + z
= q(0) + z
Then: q(z) =
0 q(0) + 1
⎡1 z ⎤
=⎢
⎥
0
1
⎣
⎦
Propagating q: an example (cont’d)
Does q(z) = q0 + z?
LHS:
This is equivalent to: 1/q(z) = 1/(q0 + z).
1
1
λ
≡
− i 2
πw (z)
R(z)
q(z)
1
1
1
1
1
≡
− i
=
− i
2
2
2
2
2
q(z)
z + zR / z
zR (1+ z / zR )
z + zR / z
zR + z /zR
Now:
RHS:
π w20
q(0) = i
= i zR
λ
so
q(0) + z = izR + z
1
z − iz
z
1
iz R
=
= 2 R2 = 2
−
q(0) + z
z + zR
z + z2R
z 2 + zR2
z + izR
=
1
1
−
i
z + z2R / z
z2 / zR + zR
1
=
q(z)
So:
which is just this.
q(z) = q(0) + z
Propagating q:
another example
winput
Focusing a collimated beam (i.e., a
lens, f, followed by a distance, f ):
wfocus
f
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
OspaceOlens
⎡1
=⎢
⎣0
Cqin + D
1
=
qout
Aqin + B
So:
[
Im 1/ q focus
But:
[
]
Im 1/ q focus ≡ −
⇒
]
f ⎤⎡ 1
1 ⎥⎦ ⎢⎣ −1/ f
1
qafter lens
=
0⎤ ⎡ 0
=⎢
⎥
1 ⎦ ⎣ −1/ f
f⎤
1 ⎥⎦
(−1/ f )(iz R ) + 1
1 − iz R / f
=
0 (izR ) + f
f
π winput
zR
= − 2 =−
f
λf 2
2
λ
πw
2
focus
⇒ w focus =
λf
π winput
The well-known result
for the focusing
of a Gaussian beam
Now consider the time and frequency
of a light pulse in addition
We’d like a matrix formalism to predict such effects as the:
group-delay dispersion ∂t/∂ω
angular dispersion ∂kx /∂ω or ∂θ /∂ω
spatial chirp ∂x/∂ω
pulse-front tilt ∂t/∂x
time vs. angle ∂t/∂θ.
This pulse has all of
these distortions!
where we’ve dropped
“0” subscripts for
simplicity.
We’ll need to consider, not only the position (x) and slope (θ ) of the ray,
but also the time (t) and frequency (ω ) of the pulse.
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.
Optical system ↔ 4x4 Ray-pulse matrix
⎡ xin ⎤
⎢θ ⎥
⎢ in ⎥
⎢ tin ⎥
⎢ ⎥
⎣⎢ν in ⎦⎥
⎡A B 0 E⎤
⎢C D 0 F ⎥
⎢
⎥
⎢G H 1 I ⎥
⎢
⎥
⎣0 0 0 1⎦
⎡ xout ⎤
⎢θ ⎥
⎢ out ⎥
⎢ tout ⎥
⎢ ⎥
⎣⎢ν out ⎦⎥
where each vector
component
corresponds to the
deviation from a
mean value for the
ray or pulse.
A Kostenbauder matrix requires five additional parameters, E, F, G, H, I.
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.
the usual 2x2
ray matrix
angular dispersion
⎡ xout ⎤
⎡A
⎢θ ⎥
⎢C
⎢ out ⎥ = ⎢
⎢ tout ⎥
⎢G
⎢ ⎥
⎢
⎢⎣ν out ⎥⎦
⎣0
pulsefront tilt
∂tout
∂xin
B
D
H
0
∂θ out
∂ν in
E ⎤ ⎡ xin ⎤ ∂xout
∂ν in
⎢
⎥
⎥
0 F ⎥ ⎢θ in ⎥
1 I ⎥ ⎢ tin ⎥
⎥⎢ ⎥
0 1 ⎦ ⎢⎣ν in ⎥⎦
0
time vs. angle
∂tout
∂θ in
spatial
chirp
GDD
∂tout
∂ν in
Some Kostenbauder matrix elements are
always zero or one.
⎡ xout ⎤
⎡A B
⎢θ ⎥
⎢C D
out
⎢ ⎥ = ⎢
⎢ tout ⎥
⎢G H
⎢ ⎥
⎢
⎢⎣ν out ⎥⎦
⎣0 0
0 E ⎤ ⎡ xin ⎤
0 F ⎥⎥ ⎢⎢θ in ⎥⎥
1 I ⎥ ⎢ tin ⎥
⎥⎢ ⎥
0 1 ⎦ ⎢⎣ν in ⎥⎦
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:
So:
^ material
⎡1 L / n
⎢0
1
⎢
=
⎢0
0
⎢
0
⎣0
⎤
⎥
⎥
1 2π Lk ′′⎥
⎥
0
1 ⎦
0
0
0
0
I = ∂tout/∂νin
The 2π is due to
the definition of
K-matrices in
terms of ν, not ω.
where L is the thickness of the medium, n is its refractive index,
and k” is the GVD:
d 2k
k ′′ ≡
=
2
dω ω
0
λ3 d 2n
2πc 2 dλ 2
Example: Using the Kostenbauder matrix
for propagation through free space
Apply the free-space propagation matrix to an input vector:
⎡1
⎢0
⎢
⎢0
⎢
⎣0
⎤ ⎡ xin ⎤ ⎡ xin + Lθ in ⎤
⎥
⎥ ⎢θ ⎥ ⎢
θ
in
⎥
⎥ ⎢ in ⎥ = ⎢
0 1 2π Lk ′′⎥ ⎢ tin ⎥ ⎢tin + 2π Lk ′′ν in ⎥
⎥
⎥⎢ ⎥ ⎢
ν in
0 0
1 ⎦ ⎢⎣ν in ⎥⎦ ⎢⎣
⎥⎦
L 0
1 0
0
0
The position varies
in the usual way,
and the beam angle
remains the same.
The group delay
increases by k”Lωin
The frequency
remains the same.
Because the group delay depends on frequency, the pulse broadens.
This approach works in much more complex situations, too.
Kostenbauder matrix for a lens
The ABCD elements are always the same as the ray matrix.
Everything else is a zero or one.
So:
^ lens
⎡ 1
⎢ −1/ f
= ⎢
⎢ 0
⎢
⎣ 0
0
1
0
0
0
0
1
0
0⎤
0 ⎥⎥
0⎥
⎥
1⎦
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 wavelengthdependent focal length, other lens imperfections cannot be
modeled using Kostenbauder matrices.
Kostenbauder matrix for a diffraction grating
Gratings introduce magnification, angular dispersion and pulse-front tilt:
So:
angular
magnification
spatial magnification
cos( β ′)
⎡
−
⎢
cos( β )
⎢
⎢
0
⎢
^ grating = ⎢
⎢ sin( β ) − sin( β ′)
⎢
c cos( β )
pulse⎢
front
⎢
0
⎣
tilt
0
−
cos( β )
cos( β ′)
0
0
0
1
0
0
no spatial
chirp (yet)
⎤
⎥
⎥
λ[sin( β ') − sin( β )] ⎥
⎥
c cos( β ')
⎥
⎥
0
⎥
⎥
⎥
1
⎦
0
time is independent of angle
angular
dispersion
no GDD (yet)
where β is the incidence angle, and β’ is the diffraction angle.
The zero elements (E, H, I) will become nonzero when propagation follows.
Kostenbauder matrix for a general prism
ϑin
ϑout
L
ψ in
ψ out
angular
dispersion
angular
magnification
^ =
spatial magnification
⎡
⎢
⎢
⎢ −2π
⎢
⎢⎣
min mout
Lmout / nmin
0
1/ min mout
dn
dω
min (tanψ in + tanψ out ) / λ0
0
−2π
dn
dω
L tanψ out / nmin λ0
0
pulse-front tilt
d 2k
where k ′′ ≡
dω 2
All new elements are
nonzero.
ω0
Lmout tanψ in / n
⎤
⎥
0
−2π ddnω (tanψ in + tanψ out ) / mout
⎥
2 dn 2
1 4π ( dω ) L tanψ in tanψ out / nλ0 + 2π Lk ′′⎥
⎥
0
1
⎥⎦
0
time vs. angle
is the GVD, min ≡
−2π
spatial chirp
dn
dω
GDD
cosψ in
cos ϑin
and mout
cos ϑout
≡
cosψ out
Kostenbauder matrix for a Brewster prism
Brewster angle
incidence and exit
If the beam passes
through the apex
of the prism:
ψ in = ψ out ≡ ψ
min = 1/ mout
L→0
(this simplifies the
calculation a lot!)
^ prism
where
W = − 4π (dn /d ω ) tanψ / mout = −4π min (dn /d ω ) tanψ
0
1
0
0
0
0 ⎤
0 ±W ⎥⎥
1
0 ⎥
⎥
0
1 ⎦
Use + if the prism is
oriented as above;
use – if it’s inverted.
⎡
⎢
= ⎢
⎢ ±W / λ0
⎢
⎣ 0
1
0
Just angular dispersion
and pulse-front tilt. No
GDD etc.
Using the Kostenbauder
matrix for a Brewster prism
Brewster angle
incidence and exit
This matrix takes into account all that we
need to know for pulse compression.
⎡ 1
⎢ 0
⎢
⎢ ±W / λ0
⎢
⎣ 0
xin
0 0
0 ⎤ ⎡ xin ⎤ ⎡
⎤
1 0 ±W ⎥⎥ ⎢⎢θin ⎥⎥ ⎢⎢ θ in ± W ν in ⎥⎥
=
0 1
0 ⎥ ⎢ tin ⎥ ⎢tin ± xinW / λ0 ⎥
⎥
⎥⎢ ⎥ ⎢
ν in
0 0
1 ⎦ ⎣ν in ⎦ ⎣
⎦
When the pulse reaches the two inverted
prisms, this effect becomes very important,
yielding longer group delay for longer
wavelengths (D < 0; and use the minus
sign for inverted prisms).
Dispersion
changes the
beam angle.
Pulse-front tilt
yields GDD.
Modeling a prism pulse compressor
using Kostenbauder matrices
Use only Brewster
prisms
1
^prism
2
^air
7
^prism ^prism 6
^prism
^air
3
^air 4
5
K = ^7 ^6 ^5 ^4 ^3 ^2 ^1
Free space propagation in a pulse compressor
There are three distances in this problem.
L1
L3
L2
^
free − space
⎡1 Li
⎢0 1
=⎢
⎢0 0
⎢
⎣0 0
0
0
1
0
0⎤
0 ⎥⎥
0⎥
⎥
1⎦
n = 1 in free space
K-matrix for a prism pulse compressor
K = ^7 ^6 ^5 ^4 ^3 ^2 ^1
Spatial chirp unless L1 = L3.
^
=
L1 + L2 + L3
⎡1
⎢0
1
⎢
⎢
W
( L1 − L3 )
⎢0 −
λ0
⎢
⎢⎣0
0
Time vs. angle unless L1 = L3.
W ( L1 − L3 ) ⎤
⎥
0
0
⎥
⎥
W 2
1 −
( L1 + L3 ) ⎥
λ0
⎥
⎥⎦
0
1
Negative GDD!
0
The GDD is negative and can be tuned by changing the amount of extra
glass in the beam (which we haven’t included yet, but which is easy).
Propagating spot size, radius of curvature,
pulse length, and chirp
To follow beams that are Gaussian in both space and time:
Qout
Qin
We could propagate Gaussian beams in space because they’re
quadratic in space (x and y):
E ( x, y ) ∝ exp[ −(1/ w2 + iπ / λ R) ( x 2 + y 2 )]
A Gaussian pulse is quadratic in time. And the real and imaginary
parts also have important meanings (pulse length and chirp):
E(t ) ∝ exp[−(1/τ G + i β )t ]
2
2
The complex-Q matrix
We define the complex Q-matrix so that the space and time
dependence of the pulse can be written:
⎧ π
E ( x, t ) ∝ exp ⎨−i [ x, −t ] Q −1
⎩ λ
⎡Q11−1 + Q12−1 ⎤ ⎡ x ⎤ ⎫⎪
⎡ x⎤⎫
⎪⎧ π
⎢ t ⎥ ⎬ = exp ⎨−i λ [ x, −t ] ⎢ −1
−1 ⎥ ⎢ ⎥ ⎬
⎣ ⎦⎭
⎪⎩
⎣Q21 + Q22 ⎦ ⎣ t ⎦ ⎪⎭
⎧⎪ π
⎡Q11−1 x + Q12−1t ⎤ ⎫⎪
= exp ⎨−i [ x, −t ] ⎢ −1
−1 ⎥ ⎬
λ
+
Q
x
Q
⎪
22 t ⎦ ⎭
⎣ 21
⎩⎪
−1
−1
Note : Q21 = − Q12
⎧ π −1 2
⎫
= exp ⎨−i [Q11 x + 2Q12−1 xt − Q22−1t 2 ]⎬
⎩ λ
⎭
These complex matrix elements contain all the parameters of
beams/pulses that are Gaussian in space and time.
The complex-Q matrix (cont’d)
−1
When the off-diagonal elements, Q12 , are zero:
⎧ π −1 2
⎫
E ( x, t ) ∝ exp ⎨−i [Q11 x − Q22−1t 2 ]⎬
⎩ λ
⎭
Q11−1 ≡ 1/ qx =
−1
22
Q
1
λ
−i
π w2 ( z )
R( z )
λ
≡ 1/ qt ≡ ( β + i /τ G2 )
π
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:
Pulse-front tilt =
Im ⎡⎣Q12−1 ⎤⎦
off-diagonal Q-1 element
Im ⎡⎣Q22−1 ⎤⎦
temporal component of Q-1
K-matrices and the propagation of Q
Kostenbauder matrices can be used to propagate the Q-matrix.
Qout
Qin
This relation holds for all systems (multiply the component matrices
together first and then use this complex result):
Qout
⎡A
⎢G
= ⎣
⎡C
⎢0
⎣
0⎤
⎡B E / λ⎤
Qin + ⎢
⎥
⎥
1⎦
⎣H I / λ ⎦
0⎤
⎡D F / λ⎤
Qin + ⎢
⎥
⎥
0⎦
1 ⎦
⎣0
Division means multiplication by the inverse.
This is
actually
more
elegant
than it
looks...
[A]
Propagating
the Q-matrix
⎡A
⎢C
^ =⎢
⎢G
⎢
⎣0
Qout
B
0
D
H
0
1
0
0
E /λ⎤
⎥
F /λ⎥
I /λ ⎥
⎥
1 ⎦
In terms of these 2x2 matrices:
0⎤
⎡ B E / λ⎤
Qin + ⎢
⎥
⎥
/
1⎦
λ
H
I
⎣
⎦
0⎤
⎡D F / λ ⎤
Qin + ⎢
⎥
⎥
0⎦
0
1
⎣
⎦
⎡A
⎢G
= ⎣
⎡C
⎢0
⎣
[C]
[D]
Notice the symmetry in the
2x2 matrices in the Qpropagation equation.
Qout =
[ A] Qin + [ B ]
[C ] Qin + [ D ]
[B]
Important point about propagating Q
Use Qout =
[ A] Qin + [ B ]
[C ] Qin + [ D ]
to compute Qout.
But use matrix multiplication for the various components to
compute the total system ray matrix.
Don’t compute
Qout =
[ A] Qin + [ B ]
[C ] Qin + [ D ]
for each component.
As with q, you’d get the right answer, but you’d work much
harder than you need to!
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 νout = νin, the E = dxout/dvin, F = dθout/dvin, and I = dtout/dvin
elements clearly indicate the (added) spatial chirp, angular dispersion,
and group-delay dispersion.
This doesn’t work for pulse-front tilt, G = dtout/dxin, however.
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
assuming
there’s zero
distortion to
begin with
because xout
may not be xin
Pulse-front tilt: You must look for an off-diagonal Q-1 element
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, you’d have to numerically solve the Fresnel diffraction
integral, which can yield a very complex computer computation.
But there’s a simpler approach. It uses the Wigner Distribution, a
different way to represent a pulse’s dependence on time:
∞
t
t
WEt (ω,τ ) ≡
Et (τ + ) Et *(τ − ) exp(−iω t ) dt
2
2
−∞
∫
The Wigner Distribution converts the pulse into a plot of intensity
vs. time (delay) and frequency.
We can invert the Wigner Distribution to
obtain the pulse field.
Inverse-Fourier transform the Wigner Distribution with respect to ω :
Y
{
-1
}
1
WEt (ω,τ ) =
2π
∞
t
t
Et (τ + ) Et *(τ − )exp(−iω t ) dt exp(iω t′) dω
2
2
−∞
∫∫
−∞
∞
t
t ⎧1
= Et (τ + ) Et *(τ − ) ⎨
2
2 ⎩ 2π
−∞
∫
∞
⎫
′
exp[−iω( t − t )] dω ⎬ dt
−∞
⎭
∫
∞
∞
t
t
= Et (τ + ) Et *(τ − ) δ ( t − t′) dt
2
2
−∞
t′
t′
= Et (τ + ) Et *(τ − )
2
2
∫
Setting τ = t’/2:
= Et (t′) Et *(0)
So we can’t determine the
absolute phase, but simple
inverse Fourier transforming
yields the rest of E(t).
Examples of
Wigner
Distributions
The Wigner
Distribution is
always real, but
it usually goes
negative.
Linearly
chirped
Gaussian
Quadratically
chirped
Gaussian
τ
ω
τ
ω
τ
ω
Double pulse
Properties of the Wigner Distribution
The marginals (integrals) of the Wigner Distribution yield the pulse
intensity vs. time and the spectrum vs. frequency.
I (τ ) =
∫
∞
−∞
WE (ω ,τ ) dω
I(τ)
S (ω ) =
∫
∞
−∞
WE (ω ,τ ) dτ
S(ω)
τ
ω
The Wigner Distribution has many other nice properties.
We can define spatial and space-time
Wigner Distributions, too.
A spatial Wigner Distribution is in terms of x and kx:
dropping the x
subscript on k
x′
x′
WEx (k , x) ≡
Ex ( x + ) Ex *( x − ) exp(−ikx′) dx′
2
2
−∞
∫
∞
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:
WE (k , x, ω,τ ) ≡
∞
x′
t
x′
t
E ( x + ,τ + ) E *( x − ,τ − )
2
2
2
2
−∞
exp(−ikx′ − iωt ) dx′ dt
∫∫
−∞
∞
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:
WEout ( xout , kout , ωout ,τ out ) = m WEin ( xin , kin , ωin ,τ in )
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, “Space–time Wigner functions and their application to the
analysis of a pulse shaper,” J. Opt. Soc. Am. B, 12, #8, p 1480, August 1995.