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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:
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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
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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 vs. the resolution limit
• Fluorophores
• Example: Fluorescence microscopy
• Example: STED microscopy
• Example: Localization microscopy
• Example: Scanning probe microscopy
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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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