2014 02 18ufsc lec 1 classical optics fxk

7/23/2019 2014 02 18UFSC Lec 1 Classical Optics FXK

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IMPRS: Ultrafast Source Technologies

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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


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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

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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

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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


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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

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Vector Identity:

Vacuum speed of light:

6

Classical Optics


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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


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2.1.1 Helmholtz Equation

Linear, local medium

dielectric susceptibility

Medium speed of light: with

Refractive Index

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Real field:

Dispersion relation:

2.1.2 Plane-Wave Solutions (TEM-Waves) and Compl. Notation

Artificial, complex field:

Into wave equation (2.16):

Wavelength

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TEM-Waves


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What about the magnetic field?

Faradays Law:

TEM-Waves

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Characteristic Impedance

Vacuum Impedance:

Example: EM-Wave polarized along x-axis and propagation along z-direction:

1 ( )r

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Backwards Traveling Wave

Backwards

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2.1.3 Poynting Vector, Energy Density and Intensity

1r

relative permittivity:

Example: Plane Wave:

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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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

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P C fi t


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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


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Figure 2.8: Wavefronts of Gaussian beam

Wavefronts, i.e. surfaces of constant phase, are parabolic

constant

Wavefront

Wavefront and Radius of Curvature


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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


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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


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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


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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


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Figure 2.10: Description of optical ray propagation by its distance and

inclination from the optical axis.

Z

r1

r2r2r1

12

Optical

System

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2.4 Ray Propagation

Ray Matrix: Transition medium 1 to 2


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Figure 2.11: Snells law for paraxial rays.

Z

r1r2

r2

r1

1 2

n1 n2

2

1

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Ray Matrix: Transition medium 1 to 2

M = 1 0

0 1

Ray Matrix: Propagation over Distance L


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Figure 2.12: Free space propagation

Z

r1

r2

r2

r1

1 2

L

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Ray Matrix: Propagation over Distance L

Prop in Medium with Index n over Distance L


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Figure 2.13: Ray propagation through a medium with refractive index n.

Z

r1 r2

r2

r1

1 2

L

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Prop. in Medium with Index n over Distance L

Thin Plano-convex Lens


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Figure 2.14: Derivation of ABCD-matrix of a thin plano-convex lens.

Z

r1 r2

r2

r1

n1 n2

0

R

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Thin Plano-convex Lens

for

Biconvex Lens and Focussing


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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


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Figure 2.16: Derivation of Ray matrix for concave mirror with Radius R.

Z

r1

r2

r2

r1

0

-R

r1

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Concave Mirror with ROC R

Table of Ray Matrices


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Table of Ray Matrices

Application: Gauss Lens Formula


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Figure 2.17: Gausslens formula.

fr1

z

d1 d2

r2

I II

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Application: Gauss Lens Formula

An Image is formed?


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An Image is formed?

Gaussian Beam Propagation


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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


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Figure 2.19: Focussing of a Gaussian beam by a lens.

d1 d2

zR1 zR2

46

Telescope

Gaussian Beam Optics: Physical Optics


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Gaussian Beam Optics: Physical Optics

Ray Optics: B = 0

Physical Optics: Real part of Eq. (2.90) = 0

Gausslens formula:

2 6 Optical Resonators


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Figure 2.20: Fabry-Perot Resonator with finite beam size.

z1

w0

L

z2

R1 R2

2.6 Optical Resonators

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Curved - Flat Mirror Resonator


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Figure 2.21: Curved-Flat Mirror

Resonator

Curved Flat Mirror Resonator

with

49

Curved - Flat Mirror Resonator


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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


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Figure 2.20: Fabry-Perot Resonator with finite beam size.

z1

w0

L

z2

R1 R2


51



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by symmetry:

and:

Or:

52


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Figure 2.23: Two curved

mirror resonator.

53

Resonator Stability


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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


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Figure 2.24: Stability Criterion

p

55


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Figure 2.26: Stable and unstable resonators

R1 R2 R1

R2

Stable Unstable

R1

R2

R2

R1

R2

R2

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Hermite Gaussian Beams


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Other solutions to the paraxial wave equation:

Hermite Polynomials:

57



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Figure 2.27: Hermite Gaussians Gl(u)

58



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Figure 2.28: Intensity profile of TEMlm-beams. by ABCD law.

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Axial Mode Structure:


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Special Case: Confocal Resonator: L = R

Roundtrip Phase = 2 p

Resonance Frequencies:

2 1( ) ( ) 2z z

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Resonance Frequencies


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Figure 2.29: Resonance frequencies of the confocal Fabry-Perot resonator,

d=L

q

2014 02 18ufsc lec 1 classical optics fxk

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