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IMPRS: Ultrafast Source Technologies
1
Franz X. Kärtner & Thorsten Uphues, Bldg. 99, O3.097 & Room 62/322
Email & phone: [email protected], 040 8998 6350
[email protected], 040 8998 2706
Lectures: Tuesday 09.00-11.00 Geb 99, Sem Raum IV (1.OG)
Thursday 09.00-11.00 Geb 99, Sem Raum IV (1.OG)
Start: 18.02.2014
Course Secretary: Christine Berber
O3.095, phone x-6351, E-mail: [email protected].
Class website: http://desy.cfel.de/ultrafast_optics_and_x_rays_division/
teaching/ imprs_summer_semester_2014/lecture_notes
Recommended Texts:
Fundamentals of Photonics, B.E.A. Saleh and M.C. Teich, Wiley, 1991.
Ultrafast Optics, A. M. Weiner, Hoboken, NJ, Wiley 2009.
Ultrashort Laser Pulse Phenomena, Diels and Rudolph,
Elsevier/Academic Press, 2006
Optics, Hecht and Zajac, Addison and Wesley Publishing Co., 1979.
Principles of Lasers, O. Svelto, Plenum Press, NY, 1998.
Waves and Fields in Optoelectronics, H. A. Haus, Prentice Hall, NJ, 1984.
Prerequisites: Basic course in electrodynamics
Text: Class notes will be distributed in class.
Grade: None
Gratings, Mirrors and Slits: Beamline Design for Soft X-Ray Synchrotron Radiation
Sources,
W. B. Peatman, Gordon and Breach Science Publishers, 1997.
Soft X-ray and Extreme Ultraviolet Radiation, David Attwood, Cambridge University
Press, 1999
General Information
2
Schedule
Lecturer Topic Time Location
Kärtner Classical Optics Tuesday
Feb. 18, 9-11am
Geb. 99
Raum IV, 1.OG
Uphues X-ray Optics Thursday
Feb. 20, 9-11am
Geb. 99
Raum IV, 1.OG
Kärtner Ultrafast Lasers Tuesday
Feb. 25, 9-11am
Geb. 99
Raum IV, 1.OG
Uphues Synchrotron
Radiation
Thursday
Feb. 27, 9-11am
Geb. 99
Raum IV, 1.OG
Kärtner Electron Sources
and Accelerators
Tuesday
Mar. 04, 9-11am
Geb. 99
Raum IV, 1.OG
Uphues High Order
Harmonic Gen.
Thursday
Mar. 06, 9-11am
Geb. 99
Raum IV, 1.OG
3
4
2 Classical Optics
2.1 Maxwell’s Equations and Helmholtz Equation
2.1.1 Helmholtz Equation
2.1.2 Plane-Wave Solutions (TEM-Waves) and Complex Notation
2.1.3 Poynting Vectors, Energy Density and Intensity
2.2 Paraxial Wave Equation
2.3 Gaussian Beams
2.4 Ray Propagation
2.5 Gaussian Beam Propagation
2.6 Optical Resonators
Classical Optics
2.1 Maxwell’s Equations of Isotropic Media
Maxwell’s Equations: Differential Form
Material Equations:
Ampere’s Law
Faraday’s Law
Gauss’s Law
No magnetic charge
Polarization
Magnetization
Current due to free charges
Free charge density
Bring Life into Maxwell’s Equations
2. Classical Optics
5
Vector Identity:
Vacuum speed of light:
6
Classical Optics
No free charges, No currents from free charges, Non magnetic
Every field can be written as the sum of tansverse and longitudinal fields:
Only free charges create a longitudinal electric field:
Pure radiation field
Simplified wave equation:
Wave in vacuum Source term
7
Classical Optics
2.1.1 Helmholtz Equation
Linear, local medium
dielectric susceptibility
Medium speed of light: with
Refractive Index
8
Real field:
Dispersion relation:
2.1.2 Plane-Wave Solutions (TEM-Waves) and Compl. Notation
Artificial, complex field:
Into wave equation (2.16):
Wavelength
9
10
TEM-Waves
What about the magnetic field?
Faraday’s Law:
TEM-Waves
11
Characteristic Impedance
Vacuum Impedance:
Example: EM-Wave polarized along x-axis and propagation along z-direction:
1 ( )r
12
Backwards Traveling Wave
Backwards
13
2.1.3 Poynting Vector, Energy Density and Intensity
1r relative permittivity:
Example: Plane Wave:
14
15
A plane wave is described by:
zjkyjkxjkEezyxE zyxx exp~
),,(~
0
Although a plane wave is a useful model, strictly speaking it cannot be
realized in practice, because it fills whole space and carries infinite energy.
22220 zyx kkkk free space wavenumber
Maxwell’s equations are linear, so a sum of solutions is also a solution.
An arbitrary beam can be can be formed as a superposition of multiple plane
waves:
yxzyx dkdkzjkyjkxjkEzyxE
exp
~),,(
~0
22220 zyx kkkk
there is no integral over kz because once kx and ky
are fixed, kz is constrained by dispersion relation
2.2 Paraxial Wave Equation
16
Figure 2.2: Construction of paraxial beam by superimposing
many plane waves with a dominant k-component in z-direction
Consider a beam which consists of plane waves propagating at small angles
with z axis
z k
y x
zx kk
zy kk
kz can be approximated as
(paraxial approximation)
such a beam
is called a
paraxial beam
Paraxial Beam
17
yxzyxyx dkdkzjkyjkxjkkkEzyxE
exp),(
~),,(
~0
The beam
can then be expressed as
yxyx
yx
yxzjk
dkdkyjkxjkzk
kkjkkEezyxE
0
22
02
exp),(~
),,(~
0
slowly varying envelope
(changing slowly along z)
quickly varying
carrier wave
Allows to find field distribution at any point in space.
The beam profile is changing as it propagates in free space.
This is called diffraction.
Paraxial Diffraction Integral
18
From the previous slide
yxyx
yx
yxzjk
dkdkyjkxjkzk
kkjkkEezyxE
0
22
02
exp),(~
),,(~
0
At z=0
yxyxyx dkdkyjkxjkkkEzyxE
exp),(
~)0,,(
~0
This is Fourier integral. Using Fourier transforms:
dxdyyjkxjkzyxEkkE yxyx
exp)0,,(
~
)2(
1),(
~20
Given the field at some plane (e.g. z=0), how to find the field at any z?
)0,,(~
zyxEKnowing can find ),(~0 yx kkE and then ),,(
~zyxE at any z.
Paraxial Diffraction Integral – Finding Field at Arbitrary z
19
Choose transverse Gaussian distribution of plane wave amplitudes:
called Rayleigh range.
Performing the integration (i.e. taking Fourier transforms of a Gaussian)
with
(Via Fourier transforms, this is equivalent to choosing Gaussian transversal
field distribution at z=0)
Substitution into paraxial diffraction integral gives
2.3 Gaussian Beam
20
The quantity can be expressed as
)()(
11222 zw
jzRzz
jzz
jzz R
R
R
)/(1 Rjzz
22)(
1
Rzz
z
zR
222 )( R
R
zz
z
zw
where R(z) and w(z) are introduced according to
The pre-factor can be expressed as )/( Rjzzj
))(exp()(
1))(exp(
22zj
zwzzz
zj
jzz
j
RRR
with Rzzz /)(tan
)()(2)(
exp)(
1~),(
~2
02
2
0 zjzR
rjk
zw
r
zwzrE We get
Gaussian Beam
21
Finally, the field is normalized, giving
)()(2)(
exp)(
14),(
~2
02
20
0 zjzR
rjk
zw
r
zw
PZzrE F
Field of a
Gaussian beam
Normalization means that the total power carried by the beam is P:
Final Expression for Gaussian Beam Field
22
Gaussian Beams: Spot Size and Rayleigh Range
)()(2)(
exp)(
14),(
~2
02
20
0 zjzR
rjk
zw
r
zw
PZzrE F
Field of a
Gaussian beam
We derived
222 )( R
R
zz
z
zw
2
2 1)(R
Rz
zzzw
Spot size at
z=0 is min 20
2 )0( wzw R
2
0 1)(
Rz
zwzw
We have
Defines transversal
intensity distribution
20w
zR Rayleigh
range
spot size as a
function of z
)(
2exp~),(
2
2
zw
rzrI )(zw
spot size
23
2
1)(z
zzzR R
Rzzz /arctan)(
22)(
1
Rzz
z
zR
Guoy phase
shift
Gaussian Beams: Phase Distribution
)()(2)(
exp)(
14),(
~2
02
20
0 zjzR
rjk
zw
r
zw
PZzrE F
Field of a
Gaussian beam
We derived
defines phase
distribution We have
Phase fronts (surfaces of constant phase) are parabolic, can be approximated
as spherical for small r. R(z) turns out to be the radius of the sphere.
Radius of
wavefront
curvature
Defines extra phase shift as compared to plane wave
24
Gaussian Beams: All Equations in One Slide
)()(2)(
exp)(
14),(
~2
02
20
0 zjzR
rjk
zw
r
zw
PZzrE F
Gaussian Beam
Field
0w
2
0 1)(
Rz
zwzw min spot size (z=0)
20w
zR Rayleigh range
Spot size
2
1)(z
zzzR R
Rz
zz arctan)( Guoy phase shift
Radius of wavefront curvature
(2.56)
(2.54)
(2.55)
(2.65)
25
Intensity Distribution
Figure 2.3: The normalized beam intensity I/I₀ as a function of the radial
distance r at different axial distances: (a) z=0, (b) z=zR (c) z=2zR.
z=0 z=zR z=2zR
26
Spot Size as a Function of z 2
0 1)(
Rz
zwzwSpot size w(zR)=w0 2At z = zR
Rayleigh range : distance from the
waist at which the spot area doubles
Waist: minimum radius
3 2 1 0 1 2 34
2
0
2
4
4
4
w z( )
w z( )
0
33 z
z.R
0
)(
w
zw)(zw
0w
waist
20w
Rz Rz2 Rz3Rz3 Rz2 Rz 0
zR
27
Figure 2.5: Gaussian beam and its characteristics
Beam Divergence:
Confocal parameter and depth of focus:
RR z
zw
z
zwzw 0
2
0 1)(
For large z
Divergence
angle Rz
w
z
zw 0)(
Divergence Angle and Confocal Parameter
Smaller spot size →
larger divergence
Figure 2.4: Normalized beam intensity I(r=0)/I₀ on the beam axis as a function of
propagation distance z.
r=0 Intensity distribution along Z
axis (r=0)
Axial Intensity Distribution
28
29
Power Confinement
99% of the power is within radius of 1.5 w(z) from the axis
Which fraction of the total power is confined within radius r0 from the axis?
Dependence is
exponential
30
Figure 2.8: Wavefronts of Gaussian beam
Wavefronts, i.e. surfaces of constant phase, are parabolic
constant
Wavefront
31
Figure 2.7: Radius of curvature R(z)
Wavefront radius of curvature:
2
1)(z
zzzR R
Wavefront and Radius of Curvature
32
Comparison to Plane and Spherical Waves
Figure 2.9: Wave fronts of (a) a uniform plane wave, (b) a spherical wave;
(c) a Gaussian beam.
33
Figure 2.6: Phase delay of Gaussian beam, Guoy-Phase-Shift
Phase delay of Gaussian Beam, Guoy-Phase Shift:
Guoy Phase Shift
34
Planes of const. Phase
Beam Waist
z/z R
Gaussian Beams: Summary • Solution of wave equation in paraxial approximation
• Wave confined in space and with finite amount of power
• Intensity distribution in any cross-section has the same shape (Gaussian),
only size and magnitude is scaled
• At the waist, the spot size is minimum and wave fronts are flat
• Lasers are usually built to generate Gaussian beams
Figure 2.10: Description of optical ray propagation by its distance and
inclination from the optical axis.
Z
r’1
r2
r’2r1
12
Optical
System
35
2.4 Ray Propagation
Figure 2.11: Snell’s law for paraxial rays.
Z
r’1r2
r’2
r1
1 2
n1 n2
2
1
36
Ray Matrix: Transition medium 1 to 2
M =1 0
0 1
æ
èçç
ö
ø÷÷
Figure 2.12: Free space propagation
Z
r’1
r2
r’2
r1
1 2
L
37
Ray Matrix: Propagation over Distance L
Figure 2.13: Ray propagation through a medium with refractive index n.
Z
r’1 r2
r’2
r1
1 2
L
38
Prop. in Medium with Index n over Distance L
Figure 2.14: Derivation of ABCD-matrix of a thin plano-convex lens.
Z
r’1 r2
r’2
r1
n1n2
0
R
39
Thin Plano-convex Lens
for
f
r1
z
40
Biconvex Lens and Focussing
Figure 2.15: Imaging of parallel rays through a lens with focal length f
Biconvex Lens M =
1 0
-1
f1
æ
è
ççç
ö
ø
÷÷÷
Figure 2.16: Derivation of Ray matrix for concave mirror with Radius R.
Z
r’1
r2
r’2
r1
0
-R
r’1
41
Concave Mirror with ROC R
42
Table of Ray Matrices
Figure 2.17: Gauss’ lens formula.
fr1
z
d1 d2
r2
I II
43
Application: Gauss’ Lens Formula
44
An Image is formed?
Figure 2.18: Gaussian beam transformation by ABCD law.
45
Gaussian Beam Propagation
28 CHAPTER 2. CLASSICAL OPTICS
ABCD-marix is given by
2 =1 +
1 +(2.83)
where 1 and 2 are the beam parameters at the input and the output planes
of the optical system or component, see Figure 2.18
Figure 2.18: Gaussian beam transformation by ABCD law, [6], p. 99.
To proove this law, we realize that it is true for the case of free space
propagation, i.e. pure di raction, comparing (2.83) with (2.53) and (2.70). If
we can proove that it is additionally true for a thin lens, then we are finished,
because every ABCD matrix (2x2 matrix) can be written as a product of a
lower and upper triangular matrix (LR-decomposition) like the one for free
space propagation and the thin lens. Note, the action of the lens is identical
to the action of free space propagation, but in the Fourier-domain. In the
Fourier domain the Gaussian beam parameter is replaced by its inverse (2.46)
e0( ) =
( )exp 0
µ2 + 2
2 ( )
¶¸(2.84)
e0( ) = 2 exp ( )
µ 2 + 2
2 0
¶¸(2.85)
But the inverse q-parameter transforms according to (2.83)
1
2
=
1
1+
1
1+
(2.86)
which leads for a thin lens to
1
2
=1
1
1(2.87)
28 CHAPTER 2. CLASSICAL OPTICS
ABCD-marix is given by
2 =1 +
1 +(2.83)
where 1 and 2 are the beam parameters at the input and the output planes
of the optical system or component, see Figure 2.18
Figure 2.18: Gaussian beam transformation by ABCD law, [6], p. 99.
To proove this law, we realize that it is true for the case of free space
propagation, i.e. pure di raction, comparing (2.83) with (2.53) and (2.70). If
we can proove that it is additionally true for a thin lens, then we are finished,
because every ABCD matrix (2x2 matrix) can be written as a product of a
lower and upper triangular matrix (LR-decomposition) like the one for free
space propagation and the thin lens. Note, the action of the lens is identical
to the action of free space propagation, but in the Fourier-domain. In the
Fourier domain the Gaussian beam parameter is replaced by its inverse (2.46)
e0( ) =
( )exp 0
µ2 + 2
2 ( )
¶¸(2.84)
e0( ) = 2 exp ( )
µ 2 + 2
2 0
¶¸(2.85)
But the inverse q-parameter transforms according to (2.83)
1
2
=
1
1+
1
1+
(2.86)
which leads for a thin lens to
1
2
=1
1
1(2.87)
Figure 2.19: Focussing of a Gaussian beam by a lens.
d1 d2
zR1 zR2
46
Telescope
47
Gaussian Beam Optics: Physical Optics
Ray Optics: B = 0
Physical Optics: Real part of Eq. (2.90) = 0
Gauss’ lens formula:
Figure 2.20: Fabry-Perot Resonator with finite beam size.
z1
w0
L
z2
R1R2
2.6 Optical Resonators
48
Figure 2.21: Curved-Flat Mirror
Resonator
Curved - Flat Mirror Resonator
with
49
Figure 2.22: Beam waists of the curved-flat mirror resonator as a
function of L/R1.
For given R1
50
Curved - Flat Mirror Resonator
Figure 2.20: Fabry-Perot Resonator with finite beam size.
z1
w0
L
z2
R1R2
Two Curved Mirror Resonator
51
Two Curved Mirror Resonator
by symmetry:
and:
Or:
52
Figure 2.23: Two curved
mirror resonator.
53
Figure 2.75: Stable regions (black) for the two-mirror resonator
L
0 R2 1R +R 2
R1
Resonator Stability
Or introduce cavity parameters:
54
Figure 2.24: Stability Criterion
Geometrical Interpretation
55
Figure 2.26: Stable and unstable resonators
R1R2 R1
R2
Stable Unstable
R1
R2
R2
R1
R2
R2
56
Hermite Gaussian Beams
Other solutions to the paraxial wave equation:
Hermite Polynomials:
57
Figure 2.27: Hermite Gaussians Gl(u)
Hermite Gaussian Beams
58
Figure 2.28: Intensity profile of TEMlm-beams. by ABCD law.
59
Hermite Gaussian Beams
Special Case: Confocal Resonator: L = R
Axial Mode Structure:
Roundtrip Phase = 2 p :
Resonance Frequencies:
2 1( ) ( )2
z z
60
Figure 2.29: Resonance frequencies of the confocal Fabry-Perot resonator,
d=L
Resonance Frequencies
61
