Ray matrices

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

• 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
Ray Optics
xin, qin
axis
xout, qout
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 the axis.
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Ray Optics
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 will change with distance and as
the ray propagates through optics.
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Ray Matrices Ray Vectors

• For many optical components, we can define 2x2
  "Ray Matrices."
• The effect on a ray is determined by multiplying
  its "Ray Vector."

Ray matrices can describe simple and com- plex
systems.
Optical system ? 2x2 Ray matrix
These matrices are often called "ABCD Matrices."
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Ray matrices as derivatives
Since the displacements and angles are assumed to
be small, we can think in terms of partial
derivatives.
spatial magnification
angular magnification
Its easy to evaluate these derivatives for most
optical components.
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Rays in free space or a medium

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

Rewriting this expression in matrix notation
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Rays at a lens
If the lens is thin, then only the ray slope
changes.
xout xin qout (1/f) xin qin
Rewriting this expression in matrix notation
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Ray Matrices for Space and Lenses

• Free space

z
Lens
f
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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
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Ray Matrices for Curved Mirrors

• Consider a mirror with radius of curvature, R
• On axis
• Off axis
• where Re R cosq if in plane of incidence
  ("tangential")
• and Re R / cosq if to plane of incidence
  ("sagittal")

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For cascaded elements, multiply ray matrices
O2
O1
O3
Notice that the order looks opposite to what it
should be.
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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
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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

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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
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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.

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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).

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Gaussian Beam Spot, 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

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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.
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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.

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Focusing a Gaussian Beam

• A lens will focus a collimated Gaussian beam to a
  new spot size
• w0 l f / pw
• So the smaller the desired focus, the BIGGER the
  input beam should be!

21
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 Irradiance
(for reference)
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The Gaussian-Beam Complex q Parameter
We can combine these two factors (theyre both
Gaussians)
q completely determines the Gaussian beam.
where
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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
Nice, eh?
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Propagating q an example
Free-space propagation through a distance z
The ray matrix for free-space propagation is
Then
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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
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Propagating q another example
Focusing a collimated beam (i.e., a lens, f,
followed by a distance, f )
winput
wfocus
f
A collimated beam has big spot size (w) and
Rayleigh range (zR), and an infinite radius of
curvature (R), so qin i zR
After some algebra, we find
But
A well-known result for the focusing of a
Gaussian beam
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Now consider the time and frequencyof a light
pulse in addition
Wed like a matrix formalism to predict such
effects as angular dispersion ?q
/?w group-delay dispersion ?t/?w spatial chirp
?x/?w pulse-front tilt ?t/?x time vs. angle ?t/?q.
This pulse has all of these distortions!
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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.
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Kostenbauder matrix elements
As with 2x2 ray matrices, consider each element
to correspond to a small deviation from its mean
value. So we can think in terms of partial
derivatives.
angular dispersion
spatial chirp
the usual 2x2 ray matrix
time vs. angle
GDD
pulse-front tilt
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Some Kostenbauder matrix elements are always zero
or one.
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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
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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.
This approach works in much more complex
situations, too.
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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.
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Kostenbauder matrix for a diffraction grating
Gratings introduce magnification, angular
dispersion and pulse-front tilt
angular magnification
spatial magnification
no spatial chirp (yet)
So
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.

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Kostenbauder matrix for a general prism
All new elements are nonzero.
angular dispersion
spatial chirp
angular magnification
spatial magnification
time vs. angle
GDD
pulse-front tilt
where
is the GVD,
and
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Kostenbauder matrix for a Brewster Prism
Brewster angle incidence and exit
Use if the prism is oriented as above use if
its inverted.
Just dispersion and pulse-front tilt. No GDD or
spatial chirpyet.
where
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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).
38
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 K 7 K 6 K 5 K 4 K 3 K 2 K 1
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Free space propagation in a pulse compressor
There are three distances in this problem.
L1
L3
L2
n 1 in free space
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K-Matrix for a prism pulse compressor
K K 7 K 6 K 5 K 4 K 3 K 2 K 1
Spatial chirp unless L1 L3.
Negative GDD!
The GDD is negative and can be tuned by changing
the amount of extra glass in the beam.
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Propagating spot size, radius of curvature,
pulse length, and chirp!
Wed like to be able to follow beams that are
Gaussian in both space and time through optical
systems.
What allowed us to propagate Gaussian beams in
space was the fact that 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)
42
The complex Q matrix
We define the complex Q-matrix so that the space
and time dependence of the pulse can be written
(It gets considerably more complicated when
theyre not. For example, what is the pulse
length of a tilted pulse?)
the complex q parameter for Gaussian beams
pulse length and chirp parameter for Gaussian
pulses
43
K-matrices and the propagation of Q
Kostenbauder matrices can be used to propagate
the Q-matrix.
This relation holds for all systems
This is actually more elegant than it looks.
Division means multiplication by the inverse.
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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
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A case for complex Q and K-matrices

• K-matrices can model even very complex
  situations. Here is a case with a little of
  everything

• The tendency of different colors to propagate
• differently can cause the pulse to have
• severe spatio-temporal distortions.
• Beam divergence angle q depends on l
• q 2l/pw, where w beam spot size.
• So if l ranges from 400 nm to 1600 nm,
• varies by a factor of 4.
• The lens focal length will also depend on l.
• Now send this light into a pulse compressor
• Were actually doing this experiment!