spatio-temporal characteristics of light pulses and … · more ray matrices the gaussian beam ......
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More ray matrices
The Gaussian beam
Complex q and its propagation
Ray-pulse Kosten-bauder matrices
The prism pulse compressor
Spatio-Temporal Characteristics of Light Pulses and How to Model Them II
in
in
in
in
x
t
out
out
out
out
x
t
001
0 0 0 1
A B EC D FG H I
Optical system ↔ 4x4 Ray-pulse matrix
Ray MatricesAn optical element’s effect on a ray is found by multiplying the ray vector by the element’s ray matrix.
Ray vector before lens
after lens before lens
after lens before lens
x xA BC D
Lens ↔ 2 x 2 ray matrix
For many optical components, we can define 2 x 2 ray matrices.
Distance ↔ 2 x 2 ray matrix
Ray matrix for lens
Ray vector after lens
xin , in
xout , out
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 When B = 0, A is the magnification.
0out in in
out in in in
x x AxACx DC D
Lens
ImageObject
do di
f
/
1/ 1/ 1/0
o i o i
o i o i
B d d d d fd d d d f
if
1 1 0 10 1 1/ 1 0 1
11
1/ 1 /0 1
1 / /
1/ 1 /
i o
oi
o
i o i o i
o
d dO
f
ddf d f
d f d d d d ff d f
The Lens Law
From the object to the image, we have:1) A distance do2) A lens of focal length f3) A distance di
1 1 1
o id d f
This is the Lens Law.
Lens
ImageObject
do di
f
Imaging Magnification
1 1 1
o id d f
1 11 / 1i io i
A d f dd d
i
o
dd
If the imaging condition,
is satisfied, then:
1 / 01/ 1 /
i
o
d fO
f d f
1 11 / 1o oo i
D d f dd d
1/o
i
d Md
01/ 1/M
Of M
So:
Lens
ImageObject
do di
f
Angular magnification
M magnification
f
f
f
f
And it maps input position to angle:
Lenses can simultan-eously map angle to position and position to angle.
From input to output, use:1) A distance f2) A lens of focal length f3) Another distance f
1 1 0 10 1 1/ 1 0 1
1 1
0 1 1/ 1
0
/1/ 0
out in
out in
in
in
in in
in in
x xf ff
xf ff
x ffx ff
out inx
out inx
f
So this arrangement maps input angle to position:
independent of input position
independent of input angle
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.
Real laser beams are localized in space at the laser and hence must diffract as they propagate away from the laser.
But lasers are Gaussian beams, not rays.
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.
x
Collimated region
R(z) = wave-front radius of curvature
w(z)
z
Beam radius w(z)
Gaussian Beam Math
The expression for a real laser beam's electric field is given by:
w(z) is the spot size vs. distance from the waist,R(z) is the beam radius of curvature, and(z) is a phase shift.
This is the solution to the wave equation or, equivalently, the Fresnel diffraction integral.
2 2 2 2
2
exp ( )( , , ) exp
( ) ( ) 2 ( )ikz i z x y x yE x y z ikw z w z R z
zw0
w(z)
R(z)
zw0
w(z)
R(z)zR
Gaussian Beam Spot, Radius, and Phase
The expressions for the spot size, radius of curvature, and phase shift:
20
2
( ) 1 /
/( ) arctan( / )
R
R
R
w z w z z
R z z z zz z z
20 /Rz w
where zR is the Rayleigh range, and is given by:
Collimation CollimationWaist spot Distance Distancesize w0 = 10.6 µm = 0.633 µm
_____________________________________________
.225 cm 0.003 km 0.045 km
2.25 cm 0.3 km 5 km
22.5 cm 30 km 500 km____________________________________________
Longer wavelengths and smaller waists expand faster than shorter ones.
w0
20 /Rz w
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.
Twice the Rayleigh range is the distance over which the beam remains about the same size, that is, remains collimated.
Gaussian Beam Collimation
w(z)Gaussian beam divergence
2 20 0 0( ) 1 / / /R R Rw z w z z w z z w z z
The smaller the waist and the larger the wavelength, the larger the divergence angle.
Far away from the waist, the spot size of a Gaussian beam will be:
w0
z
0 0 01/ 2
0 /eR R
w z w wz z z w
1/ 0 /e w
The beam 1/e divergence half angle is then w(z) / z as z :
A lens will focus a collimated Gaussian beam to a new spot size:
wfocus f / winput
So the smaller the desired focus, the BIGGER the input beam should be!
Focusing a Gaussian beam
2winput 2wfocus
f
f
The Gaussian-beam complex-q parameter
2 2 2 2
2( , , ) exp exp( ) ( )
x y x yE x y z iR z w z
2 2
( , , ) exp( )
x yE x y z iq z
We can combine these two factors (they’re both Gaussians):
where:2
1 1( ) ( ) ( )
iq z R z w z
q completely determinesthe Gaussian beam.
x
Collimated region
R(z) = wave-front radius of curvature
w(z)
z
Beam radius w(z)
Ray matrices and the propagation of qWe’d like to be able to follow Gaussian beams through optical systems.Remarkably, ray matrices can be used to propagate the q-parameter.
Optical system
A BO
C D
qout Aqin BCqin D
This relationholds for allsystems for which ray matrices hold:
Just multiply all the matrices first and use this result to obtain qout for the relevant qin!
Important point about propagating q
qout Aqin BCqin D
Use
But use matrix multiplication for the various components to compute the total system ray matrix.
to compute qout.
qout Aqin BCqin D
Don’t compute for each component.
You’d still get the right answer, but you’d work much harder than you need to!
Propagating q: an example
Free-space propagation through a distance z:
10 1space
zO
Then: q(z) 1q(0) z0 q(0) 1
q(0) z
qout Aqin BCqin D
The ray matrix for free-space propagation is:
Propagating q: an example (cont’d)
1q(z)
1
R(z) i
w2(z)
1q(z)
1
z zR2 / z
i1
zR(1 z2/ zR2 )
1
z zR2/ z
i1
zR z2 /zR
So: q(z) q(0) z
1q(0) z
1
z izR
z izR
z2 zR2
zz2 zR
2 izR
z 2 zR2So:
Does q(z) = q0 + z? This is equivalent to: 1/q(z) = 1/(q0 + z).
q(0) iw0
2
i zR q(0) z izR zRHS: so
1
z zR2 / z
i1
z2 / zR zR
1
q(z)
which is just this.
LHS:
Propagating q: another exampleFocusing a collimated beam (i.e., a lens, f, followed by a distance, f ):
A collimated beam has a big spot size (winput) and Rayleigh range (zR), and an infinite radius of curvature (R), so: qin = i zR
wfocus f
winput
2winput 2wfocus
f
f
1 1 0 00 1 1/ 1 1/ 1space lens
f fO O
f f
So: 2
2 2Im 1/ inputRfocus
wzqf f
The well-known result for the focusingof a Gaussian beam2Im 1/ focus
focus
qw
But (from the definition of q):
( 1/ )( ) 1 1 /10 ( )
R R
focus R
f iz iz fq iz f f
1 in
out in
Cq Dq Aq B
Now consider the time and frequencyof a light pulse in addition
We’d like a matrix formalism to predict such effects as the:
group-delay dispersion ∂t/∂angular dispersion ∂kx /∂ or ∂/∂spatial chirp ∂x/∂pulse-front tilt ∂t/∂xtime vs. angle ∂t/∂.
where we’ve dropped“0” subscripts for simplicity and used partial derivatives in case more than one effect is present.
This pulse has all of these distortions!
We’ll need to consider, not only the position (x) and slope () of the ray, but also the time (t) and frequency () of the pulse.
Propagation in space and time: Ray-pulse Kostenbauder matricesKostenbauder matrices are 4x4 matrices that multiply 4-vectors comprising the position, slope, time (group delay), and frequency.
A Kostenbauder matrix requires five additional parameters, E, F, G, H, I.
in
in
in
in
x
t
out
out
out
out
x
t
001
0 0 0 1
A B EC D FG H I
Optical system ↔ 4x4 Ray-pulse matrixwhere each vector component corresponds to the deviation from a mean value for the ray or pulse.
the usual 2x2 ray matrix
Kostenbauder matrix elementsAs with 2x2 ray matrices, consider each element to correspond toa small deviation from its mean value (xin = x – x0 ). So we can think in terms of partial derivatives.
001
0 0 0 1
out in
out in
out in
out in
I
x xA BC DG Ht t
FE
angular dispersion out
in
GDD out
in
t
spatialchirp
out
in
x
time vs. angleout
in
t
pulse-front tilt
out
in
tx
001
0 0 0 1
out in
out in
out in
out in
I
x xA BC DG Ht t
FE
Some Kostenbauder matrix elements are always zero or one.
Kostenbauder matrix for propagation through free space or material
1 / 0 00 1 0 00 0 1 20 0 0 1
material
L n
Lk
^
The ABCD elements are always the same as the ray matrix.
Here, the only other interesting element is the GDD: I = ∂tout /∂in
where L is the thickness of the medium, n is its refractive index,and k” is the GVD:
0
2 3 2
2 2 22d k d nkd c d
So:
The 2 is due to the definition of K-matrices in terms of , not .
Example: Using the Kostenbauder matrix for propagation through free space
1 0 00 1 0 0
20 0 1 20 0 0 1
in in in
in in
in in in
in in
x x LL
t t LkLk
Apply the free-space propagation matrix to an input vector:
Because the group delay depends on frequency, the pulse broadens.
This approach works in much more complex situations, too.
The position varies in the usual way, and the beam angle remains the same.
The group delay increases by k”Lin
The frequency remains the same.
Kostenbauder Matrix for a Lens
1 0 0 01/ 1 0 00 0 1 00 0 0 1
lens
f
^
The ABCD elements are always the same as the ray matrix.Everything else is a zero or one.
To include the GDD of a lens, just multiply by a Kostenbaudermaterial-propagation matrix for the thickness of the lens.
So: where f is the lens focal length.
As with ray matrices, the same holds for a curved mirror.
While chromatic aberrations can be modeled using a wavelength-dependent focal length, other lens imperfections cannot be modeled using Kostenbauder matrices.
no spatial chirp (yet)
Kostenbauder matrix for a diffraction grating
cos( ) 0 0 0cos( )
cos( ) [sin( ') sin( )]0 0cos( ) cos( ')
sin( ) sin( ) 0 1 0cos( )
0 0 0 1
gratingc
c
^
Gratings introduce magnification, angular dispersion and pulse-front tilt:
where is the incidence angle, and ’ is the diffraction angle (note that Kostenbauder uses different angle definitions in his paper).The zero elements (E, H, I) become nonzero when propagation follows.
So:
time is independent of angle no GDD (yet)
pulse-front tilt
angulardispersion
spatial magnification angularmagnification
Kostenbauder matrix for a general prism
All new elements are nonzero.
0
2
2
d kkd
is the GVD,
spatial chirp
220 0 0
/ 0 2 tan /0 1/ 0 2 (tan tan ) /
2 (tan tan ) / 2 tan / 1 4 tan tan / 20 0 0 1
dnin out out in out ind
dnin out in out outd
dn dn dnin in out out in in outd d d
m m Lm nm Lm nm m m
m L nm L n Lk
^
time vs. angle GDDpulse-front tilt
angulardispersion
where
in out
Lin out
coscos
inin
in
m
coscos
outout
out
m
and
spatial magnificationangularmagnification
Just angular dispersionand pulse-front tilt. No GDD etc.
Kostenbauder matrix for a Brewster prism
4 ( / ) tan / 4 ( / ) tanout indn d m m dn d W
If the beam passes through the apex of the prism:
(this simplifies the calculation a lot!)
0L
in out
1/in outm m
where
Use + if the prism is oriented as above; use – if it’s inverted.
Brewster angle incidence and exit
0
1 0 00 1 0
0 10 0 0
01
/
0
prism
^
W
W
0L
Using the Kostenbaudermatrix for a Brewster prism
This matrix takes into account all that we need to know for pulse compression.
When the pulse reaches the two inverted prisms, this effect becomes very important, yielding longer group delay for longer wavelengths (W < 0; and use the minussign for inverted prisms).
Pulse-front tiltyields GDD.
Dispersionchanges the beam angle.
Brewster angle incidence and exit
00 0
1 0 00 1 0
0 10 0 0 1
//
0 in in
in in in
in in in
in in
x x
t t x
W
W
W
W
Modeling a prism pulse compressor using Kostenbauder matrices
^prism ^prism^prism^prism^air ^air
^air
1
2
3 4 5
6
7
^ = ^7 ^6 ^5 ^4 ^3 ^2 ^1
Use only Brewster prisms
Free space propagation in a pulse compressor
1 0 00 1 0 00 0 1 00 0 0 1
i
free space
L
^
There are three distances in this problem.
n = 1 in free space
L1
L2
L3
^ = ^7 ^6 ^5 ^4 ^3 ^2 ^1
K-matrix for a prism pulse compressor
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 (which we haven’t included yet, but which is easy).
Time vs. angle unless L1 = L3.
1 2 3 1 3
1 3
2
01 3
0
1 0 ( )0 1 0 0
0 ( ) 1 (
0 0 0 1
)
L L L L L
L L L L
W
W
^ W
What does the pulse look like inside a pulse compressor?
If we send an unchirped pulse into a pulse compressor, it emerges with negative chirp.
Note all the spatio-temporal distortions.