Download - 2014 02 18UFSC Lec 1 Classical Optics FXK
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
1/61
IMPRS: Ultrafast Source Technologies
1
Franz X. Krtner & 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
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
2/61
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 RadiationSources,
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
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
3/61
Schedule
Lecturer Topic Time Location
Krtner Classical Optics TuesdayFeb. 18, 9-11am
Geb. 99Raum IV, 1.OG
Uphues X-ray Optics Thursday
Feb. 20, 9-11am
Geb. 99
Raum IV, 1.OG
Krtner Ultrafast Lasers TuesdayFeb. 25, 9-11am Geb. 99Raum IV, 1.OG
Uphues Synchrotron
Radiation
Thursday
Feb. 27, 9-11am
Geb. 99
Raum IV, 1.OG
Krtner Electron Sources
and Accelerators
Tuesday
Mar. 04, 9-11am
Geb. 99
Raum IV, 1.OGUphues High Order
Harmonic Gen.
Thursday
Mar. 06, 9-11am
Geb. 99
Raum IV, 1.OG
3
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
4/61
4
2 Classical Optics2.1 Maxwells 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
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
5/61
2.1 Maxwells Equations of Isotropic Media
Maxwells Equations: Differential Form
Material Equations:
Amperes Law
Faradays Law
Gausss Law
No magnetic charge
Polarization
Magnetization
Current due to free charges
Free charge density
Bring Life into Maxwells Equations
2. Classical Optics
5
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
6/61
Vector Identity:
Vacuum speed of light:
6
Classical Optics
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
7/61
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
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
8/61
2.1.1 Helmholtz Equation
Linear, local medium
dielectric susceptibility
Medium speed of light: with
Refractive Index
8
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
9/61
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
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
10/6110
TEM-Waves
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
11/61
What about the magnetic field?
Faradays Law:
TEM-Waves
11
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
12/61
Characteristic Impedance
Vacuum Impedance:
Example: EM-Wave polarized along x-axis and propagation along z-direction:
1 ( )r
12
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
13/61
Backwards Traveling Wave
Backwards
13
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
14/61
2.1.3 Poynting Vector, Energy Density and Intensity
1r
relative permittivity:
Example: Plane Wave:
14
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
15/61
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
Maxwells 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 kzbecause once kxand ky
are fixed, kzis constrained by dispersion relation
2.2 Paraxial Wave Equation
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
16/61
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
zk
y xzx kk
zy kk
kzcan be approximated as
(paraxial approximation)
such a beam
is called aparaxial beam
Paraxial Beam
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
17/61
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
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
18/61
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
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
19/61
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
G i B
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
20/61
20
The quantity can be expressed as
)()(
11222
zwj
zRzz
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(
22 zjzwzzz
zj
jzz
j
RRR
with Rzzz /)(tan
)(
)(2)(exp
)(
1~),(
~ 2
02
2
0 zjzR
rjk
zw
r
zwzrE We get
Gaussian Beam
Fi l E i f G i B Fi ld
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
21/61
21
Finally, the field is normalized, giving
)()(2)(
exp)(
14),(
~ 202
20
0 zjzR
rjk
zw
r
zw
PZzrE F
Field of aGaussian beam
Normalization means that the total power carried by the beam is P:
Final Expression for Gaussian Beam Field
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
22/61
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
22 1)(
R
Rz
zzzw
Spot size at
z=0 is min2
0
2
)0( wzw R
2
0 1)(
Rz
zwzw
We have
Defines transversal
intensity distribution
2
0wzR Rayleigh
range
spot size as a
function of z
)(
2exp~),(
2
2
zw
rzrI )(zw
spot size
G i B Ph Di t ib ti
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
23/61
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
distributionWe have
Phase fronts (surfaces of constant phase) are parabolic, can be approximated
as spherical for smallr. R
(z) turns out to be the radius of the sphere.
Radius ofwavefront
curvature
Defines extra phase shift as compared to plane wave
G i B All E ti i O Slid
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
24/61
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)
20wzR Rayleigh range
Spot size
2
1)(z
zzzR R
Rz
z
z arctan)( Guoy phase shift
Radius of wavefront curvature
(2.56)
(2.54)
(2.55)
(2.65)
I t it Di t ib ti
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
25/61
25
Intensity Distribution
Figure 2.3: The normalized beam intensity I/I
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
S t Si F ti f
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
26/61
26
Spot Size as a Function of z2
0 1)(
Rz
zwzwSpot size w(z
R)=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
Di A l d C f l P t
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
27/61
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
angleRz
w
z
zw 0)(
Divergence Angle and Confocal Parameter
Smaller spot size
larger divergence
A i l I t it Di t ib ti
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
28/61
Figure 2.4: Normalized beam intensity I(r=0)/I
on the beam axis as a function of
propagation distance z.
r=0Intensity distribution along Z
axis (r=0)
Axial Intensity Distribution
28
P C fi t
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
29/61
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 r0from the axis?
Dependence is
exponential
Wavefront
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
30/61
30
Figure 2.8: Wavefronts of Gaussian beam
Wavefronts, i.e. surfaces of constant phase, are parabolic
constant
Wavefront
Wavefront and Radius of Curvature
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
31/61
31
Figure 2.7: Radius of curvature R(z)
Wavefront radius of curvature:
2
1)(z
zzzR R
Wavefront and Radius of Curvature
Comparison to Plane and Spherical Waves
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
32/61
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.
Guoy Phase Shift
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
33/61
33
Figure 2.6: Phase delay of Gaussian beam, Guoy-Phase-Shift
Phase delay of Gaussian Beam, Guoy-Phase Shift:
Guoy Phase Shift
Gaussian Beams: Summary
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
34/61
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
2 4 Ray Propagation
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
35/61
Figure 2.10: Description of optical ray propagation by its distance and
inclination from the optical axis.
Z
r1
r2r2r1
12
Optical
System
35
2.4 Ray Propagation
Ray Matrix: Transition medium 1 to 2
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
36/61
Figure 2.11: Snells law for paraxial rays.
Z
r1r2
r2
r1
1 2
n1 n2
2
1
36
Ray Matrix: Transition medium 1 to 2
M = 1 0
0 1
Ray Matrix: Propagation over Distance L
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
37/61
Figure 2.12: Free space propagation
Z
r1
r2
r2
r1
1 2
L
37
Ray Matrix: Propagation over Distance L
Prop in Medium with Index n over Distance L
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
38/61
Figure 2.13: Ray propagation through a medium with refractive index n.
Z
r1 r2
r2
r1
1 2
L
38
Prop. in Medium with Index n over Distance L
Thin Plano-convex Lens
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
39/61
Figure 2.14: Derivation of ABCD-matrix of a thin plano-convex lens.
Z
r1 r2
r2
r1
n1 n2
0
R
39
Thin Plano-convex Lens
for
Biconvex Lens and Focussing
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
40/61
f
r1
z
40
Biconvex Lens and Focussing
Figure 2.15: Imaging of parallel rays through a lens with focal length f
Biconvex LensM =
1 0
-
1
f 1
Concave Mirror with ROC R
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
41/61
Figure 2.16: Derivation of Ray matrix for concave mirror with Radius R.
Z
r1
r2
r2
r1
0
-R
r1
41
Concave Mirror with ROC R
Table of Ray Matrices
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
42/61
42
Table of Ray Matrices
Application: Gauss Lens Formula
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
43/61
Figure 2.17: Gausslens formula.
fr1
z
d1 d2
r2
I II
43
Application: Gauss Lens Formula
An Image is formed?
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
44/61
44
An Image is formed?
Gaussian Beam Propagation
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
45/61
Figure 2.18: Gaussian beam transformation by ABCD law.
45
Gaussian Beam Propagation
2= 1+1+
e0( ) = ( )exp 0
2 + 22 ( )
(2.84)
e0( ) = 2 exp ( )
2 + 2
2 0
(2.85)
Telescope
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
46/61
Figure 2.19: Focussing of a Gaussian beam by a lens.
d1 d2
zR1 zR2
46
Telescope
Gaussian Beam Optics: Physical Optics
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
47/61
47
Gaussian Beam Optics: Physical Optics
Ray Optics: B = 0
Physical Optics: Real part of Eq. (2.90) = 0
Gausslens formula:
2 6 Optical Resonators
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
48/61
Figure 2.20: Fabry-Perot Resonator with finite beam size.
z1
w0
L
z2
R1 R2
2.6 Optical Resonators
48
Curved - Flat Mirror Resonator
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
49/61
Figure 2.21: Curved-Flat Mirror
Resonator
Curved Flat Mirror Resonator
with
49
Curved - Flat Mirror Resonator
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
50/61
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
Two Curved Mirror Resonator
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
51/61
Figure 2.20: Fabry-Perot Resonator with finite beam size.
z1
w0
L
z2
R1 R2
Two Curved Mirror Resonator
51
Two Curved Mirror Resonator
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
52/61
Two Curved Mirror Resonator
by symmetry:
and:
Or:
52
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
53/61
Figure 2.23: Two curved
mirror resonator.
53
Resonator Stability
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
54/61
Figure 2.75: Stable regions (black) for the two-mirror resonator
L
0 R2 1R +R2R1
eso ato Stab ty
Or introduce cavity parameters:
54
Geometrical Interpretation
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
55/61
Figure 2.24: Stability Criterion
p
55
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
56/61
Figure 2.26: Stable and unstable resonators
R1 R2 R1
R2
Stable Unstable
R1
R2
R2
R1
R2
R2
56
Hermite Gaussian Beams
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
57/61
Other solutions to the paraxial wave equation:
Hermite Polynomials:
57
Hermite Gaussian Beams
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
58/61
Figure 2.27: Hermite Gaussians Gl(u)
58
Hermite Gaussian Beams
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
59/61
Figure 2.28: Intensity profile of TEMlm-beams. by ABCD law.
59
Axial Mode Structure:
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
60/61
Special Case: Confocal Resonator: L = R
Roundtrip Phase = 2 p
Resonance Frequencies:
2 1( ) ( ) 2z z
60
Resonance Frequencies
-
7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK
61/61
Figure 2.29: Resonance frequencies of the confocal Fabry-Perot resonator,
d=L
q