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How does focusing by a lens work?


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We introduced last time• Simplest model of a focused field: Gaussian beam• Full vectorial-field model for a field focused by a lens

The tools we needed were• The angular spectrum representation• The paraxial approximation• The far-field approximation

Angular spectrum


MATH :

PHYS :

Together:

The Gaussian Beam

4

Beam waist

Wavefront radius

Phase correction (Guoy phase)

Rayleigh length

Field in focal plane z=0:

The Gaussian Beam

5

The Gaussian Beam has one free parameter. Which one?

The Gaussian Beam

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The Gaussian Beam has one free parameter. Which one?

The Gaussian Beam is an approximation!When is it a good approximation?

Does the Gaussian Beam contain evanescent field components?

Far-field


Party-Party Goggles

8

Party !

Party!Laser

Magic foil

500 µm

Party-Party

9

Party !

Party!Laser

Magic foil

500 µm

FFT

SLM technology uses Fourier optics

10

Party !

Party!Laser

Magic foil

Adaptive Version: Spatial light modulator (SLM)

Angular spectrum in terms of far-field


For kz ~ k: Fourier Optics !

From method of stationary phase:

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Back to the lens

• We can calculate the field near a focus if we just know the far-field


So what does a lens do?


Ray Continuity

(energy conservation)



Ray Continuity


Sine Condition

(aplanatic system)



Ray Continuity


Sine Condition

(aplanatic system)

What about this term?

Project plane on sphere




Ray Continuity


Sine Condition

(aplanatic system)

Field after lens


Fresnel coefficients

Angular spectrum representation


Change coordinates

Coordinates on reference sphereCoordinates in focal region NA

Simplest case: Focusing of (0,0)-Gaussian beam


:

Gaussian Beam sent into lens

Let’s skip some lengthy coordinate transformations and integrations…

If you need it, look it up in Principles of Nano Optics.

Strongly focused Gaussian beam


Mapping the field distribution in the focus


fluorescence rate ~ excitation ratex

y

contrast ~ | m .E(x,y;zo)| 2

Map of focal intensity distribution

Detector has no spatial resolution

Single molecule detection


What does the image of a point-source look like


Source Plane Image Plane

Point-spread function


On the menu today

• Motivation: Why nano-optics?

• Repetition: electromagnetism

• Optical imaging:

• Focusing by a lens

• Angular spectrum

• Paraxial approximation

• Gaussian beams

• Method of stationary phase

• The diffraction limit

• Fluorophores

• Example: Fluorescence microscopy

• Example: STED microscopy

• Example: Localization microscopy

• Example: Scanning probe microscopy


Angular spectrum


Farfield of dipole :

Paraxial approximation


On the menu today

• Motivation: Why nano-optics?

• Repetition: electromagnetism





• Gaussian beams


• The diffraction limit vs. the resolution limit

• Fluorophores






Classical resolution limit

www.photonics.ethz.ch 37E. Abbe, Arch. Mikrosk. Anat. 9, 413 (1873).

Source Plane Image Plane

4 4

Abbe’s Resolution Limit

www.photonics.ethz.ch 38E. Abbe, Arch. Mikrosk. Anat. 9, 413 (1873).

What are we actually doing here?





• Gaussian beams


• The diffraction limit: How well can we focus light?

• Optical microscopy

• Optical imaging systems

• Real-world (dipolar) sources: Fluorophores and scatterers






Fluorescent molecules – Jablonski diagram

• Stokes shift of fluorescence allows to spectrally separate (intense) pump light from (weak) fluorescence


Excitation rate ~ | m .E(x,y;zo)| 2

µ: transition dipole moment

Rhodamine 6G






• In practice, we often quantify the interaction rate between a fluorophore and a light field via a cross section s

Fluorescence microscopy: Epi-illumination

• Illuminate entire sample homogeneously

• Image sample plane onto pixelated detector

• Each fluorophore generates a signal according to the PSF

• Resolution is


Position on detector

Scanning fluorescence microscopy

• Create a pump-focus on a sample covered with fluorophores

• Move sample transversally to optical axis

• Record fluorescence photons on detector

• You can spatially separate two emitters when their distance exceeds


“bucket” detector Sample position

Confocal fluorescence microscopy



• Place pinhole in image plane

• How large should you pick the pinhole?

• What is your spatial resolution?



Confocal fluorescence microscopy



• Place pinhole in image plane

• How large should you pick the pinhole?

• pinhole suppresses out of plane signals



Single molecule detection


What are we actually doing here?





• Gaussian beams


• The diffraction limit: How well can we focus light?

• Optical microscopy

• Optical imaging systems

• Real-world (dipolar) sources: Fluorophores and scatterers

• Example: Fluorescence microscopy (diffraction limited)

• Superresolution techniques:





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