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Physical Optics

Lecture Summer Semester 2018: Photon Optics

2018-05-09

Michael Kempe

Physical Optics: Content

2

No Date Subject Ref Detailed Content

1 11.04. Wave optics GComplex fields, wave equation, k-vectors, interference, light propagation,

interferometry

2 18.04. Diffraction GSlit, grating, diffraction integral, diffraction in optical systems, point spread

function, aberrations

3 25.04. Fourier optics GPlane wave expansion, resolution, image formation, transfer function,

phase imaging

4 02.05.Quality criteria and

resolutionG

Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point

resolution, criteria, contrast, axial resolution, CTF

5 09.05. Photon optics KEnergy, momentum, time-energy uncertainty, photon statistics,

fluorescence, Jablonski diagram, lifetime, quantum yield, FRET

6 16.05. Coherence KTemporal and spatial coherence, Young setup, propagation of coherence,

speckle, OCT-principle

7 23.05. Polarization GIntroduction, Jones formalism, Fresnel formulas, birefringence,

components

8 30.05. Laser KAtomic transitions, principle, resonators, modes, laser types, Q-switch,

pulses, power

9 06.06. Nonlinear optics KBasics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,

CARS microscopy, 2 photon imaging

10 13.06. PSF engineering GApodization, superresolution, extended depth of focus, particle trapping,

confocal PSF

11 20.06. Scattering LIntroduction, surface scattering in systems, volume scattering models,

calculation schemes, tissue models, Mie Scattering

12 27.06. Gaussian beams G Basic description, propagation through optical systems, aberrations

13 04.07. Generalized beams GLaguerre-Gaussian beams, phase singularities, Bessel beams, Airy

beams, applications in superresolution microscopy

14 11.07. Miscellaneous G Coatings, diffractive optics, fibers

K = Kempe G = Gross L = Lu

3

Wave-Particle Duality

• With Maxwell’s theory the description of light as electro-magnetic wave was

seemingly complete, explaining phenomena as refraction, diffraction, and interference

• Around 1900 experimental data emerged that couldn’t be explained by the wave

theory of light: black body radiation and the photo-electric effect

• Electron kinetic energy independent of intensity of light but linearly dependent on light

frequency light as particle stream with energy 𝐸 = ℎ𝑓 with Planck’s constant

(Einstein 1905)

AU

AnodePhoto-

cathode

e-

- +

electron

kinetic energy

light frequency

Evacuated tube

e-

e-

Ref.: Wikipedia

4

Photon Properties

Energy of a photon

Planck’s constant

Momentum of a photon

• stable elementary particle with no

charge, no mass

• travels at speed of light in vacuum

c0 and n-times slower in medium

with index of refraction n Particle Wave

𝑐0 = 𝜆0 ∙ 𝑓Speed of photon

𝑐 =𝑐0𝑛=𝜆0𝑛∙ 𝑓 = 𝜆 ∙ 𝑓

𝑘

f, λE,c

𝑝

𝑐0 = 299 792 458 𝑚 / 𝑠

5

mean photon flux density

mean photon flux

optical intensity

optical power

optical energy mean number of photons

Energy of photon @ λ=550nm 3.61 ∙ 10−19𝑊𝑠

mean photon flux density

phot/(cm2 s)

Photon Properties

Sun at Earth: 1000W/m² 3.61 ∙ 1018𝑝ℎ𝑜𝑡/(𝑐𝑚2𝑠)

6

Uncertainty Relations

• The wave character of the photon is also visible in the fundamental

uncertainty to measure certain quantities simultaneously

• a single-valued k-vector/momentum infinitely extended wave in space

a single-valued frequency/energy infinitely extended wave in time

Gaussian beam:𝑓 𝑥 =

1

2𝜋𝜎𝑥exp −

𝑥2

2𝜎𝑥²𝑓 𝑘 = exp −

𝑘2

2𝜎𝑘²

With 𝜎𝑥 =1

2𝜎𝑘

𝜎𝑥 ∙ 𝜎𝑘 =1

2 Smallest

possible

values𝜎𝑡 ∙ 𝜎𝜔 =1

2

Space-bandwidth product

Duration-bandwidth product

𝐸 = ℏ𝜔 𝜎𝑡 ∙ 𝜎𝐸 ≥ℏ

2

𝑝 = ℏ𝑘 𝜎𝑥 ∙ 𝜎𝑝 ≥ℏ

2

uncertainty of E during measurement time 𝜎𝑡

uncertainty of momentum at measurement

position range 𝜎𝑥

2𝜎𝑘 2𝑙𝑛2

k x

1

𝜎𝑘2𝑙𝑛2

7

polarisation properties governed by photon spin angular momentum

photon: spin ±1 (quantum number)

spin angular momentum

Polarization

By E-karimi - Own work, CC BY-SA 3.0,

https://commons.wikimedia.org/w/index.p

hp?curid=16697550

Arbitrary polarization state

Example: linear polarization

8

Experiments define the properties

• photons are indivisible (can’t be split )

• but they can still interfere!

2

1

P=50%

P=50%

2

1

P=50%

P=50%

50% detection probability at 1 and 2 0% detection probability at 1 and

100% detection probability at 2

9

Light Intensity and Probability Distribution

transmittance

reflectanceprobability intensity

Diffraction and Interference pattern

Beamsplitter:

Source: David Malkahttp://www.ing.iac.es/engineering/CCDgroup.html

10

the time of arrival and position

of a single photon is

probabilistic

probability distribution: Poisson

mean photon number

n = 0, 1, 2, … photon number

coherent light source -> arrival of photons statistically independent (at given

energy in one mode)

10

Photon Statistics: Coherent Light

n=1

n=3

n=5

n=7

n=9

𝑝 𝑛 = 𝑒− 𝑛 𝑛𝑛

𝑛!

p(n

)

n

11

the time of arrival and position

of a single photon is

probabilistic

probability distribution:

Bose-Einstein

mean photon number

thermal light source -> arrival of photons not statistically independent (governed

by energy distribution in one mode at an equilibrium temperature)

10

Photon Statistics: Thermal Light

𝑝 𝑛 =1

𝑛 + 1

𝑛

𝑛 + 1

𝑛

𝑛 = expℎ𝜈

𝑘𝐵𝑇− 1

−1

0 5 10 15 200,00

0,05

0,10

0,15

0,20

0,25

n=1

n=3

n=5

n=10

n=50

p(n

)

n

12

Photon Statistics

variance

coherent thermal

Poisson Bose-Einstein

standard deviation

mean value

signal-to-noise ratio

𝑝 𝑛 =1

𝑛 + 1

𝑛

𝑛 + 1

𝑛

𝜎𝑛2 = 𝑛 + 𝑛2

𝜎𝑛 = 𝑛 + 𝑛2

𝑆𝑁𝑅 = 𝑛

𝜎𝑛𝑆𝑁𝑅 = 𝑛 𝑆𝑁𝑅 =

𝑛

𝑛 + 𝑛2=

𝑛

𝑛 + 1

𝑝 𝑛 = 𝑒− 𝑛 𝑛𝑛

𝑛!

𝜎𝑛 = 𝑛

𝜎𝑛2 =

𝑛=0

∞

𝑛 − 𝑛 2𝑝(𝑛) 𝜎𝑛2 = 𝑛

𝐸 𝑋 =

𝑛=0

∞

𝑛 ∙ 𝑝(𝑛)

Extreme cases (single mode):

𝐸 𝑋 = 𝑛

13

Photon Statistics of Single Modes

variance

coherent partially coherent thermal

Poisson Mandel Bose-Einstein

signal-to-

noise ratio

𝑝 𝑛 =1

𝑛 + 1

𝑛

𝑛 + 1

𝑛

𝜎𝑛2 = 𝑛 + 𝑛2

𝑆𝑁𝑅 = 𝑛 𝑆𝑁𝑅 = 𝑛

𝑛 + 1

𝑝 𝑛 = 𝑒− 𝑛 𝑛𝑛

𝑛!

𝜎𝑛2 = 𝑛

𝑝 𝑛 = 0

∞

𝑒−𝒲𝒲𝑛

𝑛!𝑝 𝒲 𝑑𝒲

mean value 𝑛 𝑛 𝑛 = 𝒲

𝜎𝑛2 = 𝑛 + 𝜎𝒲

2

𝑆𝑁𝑅 = 𝑛

𝜎𝒲2

𝑛 + 1

P is constant

𝜎𝒲2 = 𝑛2

P fluctuates with distribution

𝒲 =1

ℎ𝜈 0

𝑇

𝑃 𝑡 𝑑𝑡

𝑝(𝒲)

14

energy conservation:

spontaneous emission stimulated emissionabsorption

atomic energy level differences typically lie in the optical region

Photon-Matter Interactions

𝑃𝑠𝑝 =𝑐

𝑉𝜎 𝜈𝑃𝑎𝑏𝑠 = 𝑛

𝑐

𝑉𝜎(𝜈) 𝑃𝑠𝑡 = 𝑛

𝑐

𝑉𝜎(𝜈)

Probability densities:

𝜎(𝜈): transition cross section

absorbing one photon

from a mode with n photons

emitting one photon

into a mode

emitting one photon

in a mode with n photons

𝑛𝑐

𝑉= 𝜙(𝜈) for monochromatic wave

15

molecules vibrate and rotate -> electronic, vibrational and rotational transitions

bending© Tiago Becerra Paolini

asymmetric

stretching

symmetric stretching

Energy Levels of Molecules

16

light absorption 10-15 s

internal conversion 10-12 s

fluorescence emission 10-9 s

thermal relaxation 10-12 s

Jablonski Diagram

Transition times

Not every absorbed photon results in fluorescence (fraction = quantum yield)

-> collisional quenching, fluorescence resonance energy transfer, intersystem crossing

17

luminescence

fluorescence phosphorescence

rapid emission of a photon: 108 s-1

fluorescence lifetime 10ns or shorter

light emission from excited triplet states

(spin forbidden)

slow emission rates: 103 - 1 s-1

phosphorescence lifetime ms - s or longer

Light Emission from Molecules

Spontaneous emission of light in all modes: 𝑃𝑠𝑝 = 1/𝜏 (spontaneous lifetime)

18

• fluorescent minerals

• shelly fish: Aequorea victoria

• stained:

bovine pulmonary artery endothelial cells

Fluorescence

19

energy of absorption is

larger than the energy of

emission

Absorption

Stokes

shift Δλ

Emission

Fluorescence Stokes Shift

∆𝐸

typical cross section for absorption: 𝜎 ≈ 10−16𝑐𝑚²

𝐼(𝑧) = 𝐼0 exp(−𝜎 ∙ 𝑁 ∙ 𝑧)

N: molecular concentration

20

• quantum yield Q = number of emitted photons / number of absorbed photons

Q=1 -> brightest

emissive rate (photons/s)

rate of non-radiative decay

• lifetime of excited state =

average time the molecule occupies the excited state before returning to the

ground state

single exponential decay

Fluorescence Parameters

21

=> decrease in fluorescence intensity

• non-radiative energy transfer

(FRET)

• collisional (dynamic) quenching

(chemical quenchers: oxygen, halogens,

amines)

• static quenching = contact quenching

Fluorescence Quenching

22

photoselective excitation of fluorophores by polarised light

transition dipole moment [Cm]

- defined orientation with respect to molecular axis = polarisation of

transition

electric dipole moment

Fluorescence Emission

23

Fluorescence Emission

=> rotational diffusion decreases anisotropy (less polarised fluorescent emission)

fixed moleculemolecule rotating infinitely fast

fluorescence anisotropy …intrinsic anisotropy

…fluorescence lifetime

…rotational correlation time

24

Fluorescence Lifetime Imaging (FLIM)

Scientific Reports

ISSN 2045-2322 (online)http://www.imperial.ac.uk/photonics/research/

Most often used: Time-correlated

Single-Photon-Counting (TCSPC)Can be utilized for imaging in confocal microscopy

25

Measuring diffusion of molecules via

single photon detection in confocal microscopy

fluorescence photon counts

autocorrelation function

translational

diffusion

rotational diffusion

Fluorescence Correlation Spectroscopy (FCS)

𝛿𝐼 𝑡 = 𝐼 𝑡 − 𝐼(𝑡)

26

D … diffusion coefficient

N … average number of molecules

… diffusion time

Fluorescence Correlation Spectroscopy (FCS)

27

•non-radiative energy transfer

•no intermediate photon

Förster Resonance Energy Transfer (FRET)

28

energy transfer efficiencyrate of energy transfer

... Förster distance … donor lifetime in absence of FRET

… donor-acceptor distance

Förster Resonance Energy Transfer (FRET)