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How does focusing by a lens work?

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Boundless.com

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

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MATH :

PHYS :

Together:

The Gaussian Beam

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Beam waist

Wavefront radius

Phase correction (Guoy phase)

Rayleigh length

Field in focal plane z=0:

The Gaussian Beam

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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

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Party-Party Goggles

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Party !

Party!Laser

Magic foil

500 µm

Party-Party

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Party !

Party!Laser

Magic foil

500 µm

FFT

SLM technology uses Fourier optics

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Party !

Party!Laser

Magic foil

Adaptive Version: Spatial light modulator (SLM)

Angular spectrum in terms of far-field

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For kz ~ k: Fourier Optics !

From method of stationary phase:

Boundless.com

Back to the lens

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

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So what does a lens do?

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Ray Continuity

(energy conservation)

So what does a lens do?

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Ray Continuity

(energy conservation)

Sine Condition

(aplanatic system)

So what does a lens do?

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Ray Continuity

(energy conservation)

Sine Condition

(aplanatic system)

What about this term?

Project plane on sphere

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So what does a lens do?

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Ray Continuity

(energy conservation)

Sine Condition

(aplanatic system)

Field after lens

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Fresnel coefficients

Angular spectrum representation

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Change coordinates

Coordinates on reference sphereCoordinates in focal region NA

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

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:

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

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Strongly focused Gaussian beam

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Mapping the field distribution in the focus

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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

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What does the image of a point-source look like

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Source Plane Image Plane

Point-spread function

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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

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Point-spread function

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Angular spectrum

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Farfield of dipole :

Paraxial approximation

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Point-spread function

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Point-spread function

www.photonics.ethz.ch 35

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 vs. the resolution limit

• Fluorophores

• Example: Fluorescence microscopy

• Example: STED microscopy

• Example: Localization microscopy

• Example: Scanning probe microscopy

www.photonics.ethz.ch 36

Classical resolution limit

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

Source Plane Image Plane

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Abbe’s Resolution Limit

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

What are we actually doing here?

• Optical imaging:

• Focusing by a lens

• Angular spectrum

• Paraxial approximation

• Gaussian beams

• Method of stationary phase

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

• Optical microscopy

• Optical imaging systems

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

• Example: Fluorescence microscopy

• Example: STED microscopy

• Example: Localization microscopy

• Example: Scanning probe microscopy

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Fluorescent molecules – Jablonski diagram

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

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

µ: transition dipole moment

Rhodamine 6G

Fluorescent molecules – Jablonski diagram

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

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

µ: transition dipole moment

Fluorescent molecules – Jablonski diagram

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

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

µ: transition dipole moment

Fluorescent molecules – Jablonski diagram

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

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

µ: transition dipole moment

• 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

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Position on detector

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

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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

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“bucket” detector Sample position

Confocal fluorescence microscopy

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

• Move sample transversally to optical axis

• Place pinhole in image plane

• How large should you pick the pinhole?

• What is your spatial resolution?

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“bucket” detector Sample position

Confocal fluorescence microscopy

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

• Move sample transversally to optical axis

• Place pinhole in image plane

• How large should you pick the pinhole?

• pinhole suppresses out of plane signals

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“bucket” detector Sample position

Single molecule detection

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What are we actually doing here?

• Optical imaging:

• Focusing by a lens

• Angular spectrum

• Paraxial approximation

• Gaussian beams

• Method of stationary phase

• 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:

• Example: STED microscopy

• Example: Localization microscopy

• Example: Scanning probe microscopy

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