Microscopy

Femius Koenderink Center for Nanophotonics FOM Institute AMOLF Amsterdam

1. Microscopy at the diffraction limit - What is the diffraction limit - Spatial frequencies and Fourier transforms - High NA imaging

2. Microscopy beyond the diffraction limit - Scanning probe to beat the diffraction limit - Example imaging molecules - Example imaing photonic structure - Artefacts

3

Ernst Abbe Cofounder of Zeiss: Zeiss, Abbe, Schott

Arbitrary source distribution

( ) ( ) i2

1, ; , , e d d4

x yk x k yx yk k z x y z x y

π − +

−∞

= ∫∫E E

Describe field as superposition of plane waves (Fourier transform):

( ) ( ) iˆ, , , ; e d dx yk x k yx y x yx y z k k z k k +

−∞

=

∫∫E E

E∞

∞

This representation is called ‘Angular spectrum representation’

Arbitrary source distribution

( ) ( ) i2

1, ; , , e d d4

x yk x k yx yk k z x y z x y

π − +

−∞

= ∫∫E E

Describe field as superposition of plane waves (Fourier transform):

( ) ( ) iˆ, , , ; e d dx yk x k yx y x yx y z k k z k k +

−∞

=

∫∫E E

Field at z=0 (object) propagates in free space as

( ) ( ) iˆ ˆ, ; , ;0 e zk zx y x yk k z k k ±=E E ( ) ( )2 2 2

0z x yk nk k k= − +

E∞

∞

Arbitrary source distribution

Field at z=0 (object) propagates in free space as

( ) ( ) iˆ ˆ, ; , ;0 e zk zx y x yk k z k k ±=E E ( ) ( )2 2 2

0z x yk nk k k= − +

The propagator is oscillating for

and exponentially decaying for

( ) ( )22 20x yk k nk+ <

( ) ( )22 20x yk k nk+ >

Near field = Exponentially confined fast spatial fluctuations Far field = Propagating fields = bound by diffraction limit

Example – Gaussian beam Suppose we have a Gaussian object

It’s spectral representation is Gaussian

Now find the field along the beam Insert the Gaussian into

Paraxial approximation If only small angles contribute

Again we are transforming a Gaussian

Gaussian beam optics

A gaussian object results in a gaussian beam as far field Diffraction: the beam widens away from the waist

Gaussian beam optics

Diffraction: the beam widens away from the waist The narrower the waist, the more the divergence Note how the law Relates spot size and numerical aperture NA=n sinθ

Diffraction optics intuition

1) Narrow beams lead to larger angular divergence 2) Larger beams can hence be more tightly focused

3) Angular far field profile to first order is just the Fourier transform of the source distribution - Gaussian beam - Also: diffraction by pinholes, and slits.

Arbitrary source distribution

Field at z=0 (object) propagates in free space as

( ) ( ) iˆ ˆ, ; , ;0 e zk zx y x yk k z k k ±=E E ( ) ( )2 2 2

0z x yk nk k k= − +

The propagator H is oscillating for

and exponentially decaying for

( ) ( )22 20x yk k nk+ <

( ) ( )22 20x yk k nk+ >

Near field = Exponentially confined fast spatial fluctuations Far field = Propagating fields = bound by diffraction limit Small objects <> wide beams and high NA’s

The diffraction limit Image of a point source in a microscope, collecting part of the angular spectrum of the source:

Rayleigh criterion: two point sources distinguishable if spaced by the distance between the maximum and the first minimum of the Airy pattern

+ θ sinNA n θ=

The diffraction limit Image of a point source in a microscope, collecting part of the angular spectrum of the source:

Rayleigh criterion: two point sources distinguishable if spaced by the distance between the maximum and the first minimum of the Airy pattern

+

0.61dNAλ

=

θ

sinNA n θ=

Rigorous non paraxial calculation gives 0.61 from Airy pattern

What’s in a rigorous calculation

Compare normal ray optics: lenses approximated as planes Abbe sine condition is `holy design rule’ for microscopes

Abbe sine condition High NA: hemispherical reference surface

What’s in a rigorous calculation

Abbe sine condition `aplanatic lens’ High NA: 1. hemispherical reference surface 2. constant power in rays upon crossing reference 3. Upon refraction, polarization vector refracts too Strategy works for illumination and collection geometry

Important consequences

Polarized as incident

Polarized along the beam

Best focus is λ/2NA in size Strong focusing adds polarization out-of-plane Focus is not quite cylindrical in shape due to polarization

High NA imaging

The ultimate smallest object is a molecule

Z-dipole In-plane Tilted 45 deg

NA=1.4 In focus

NA=0.4 In focus

Fourier microscopy

Back aperture directly maps sinθ ~k||

objective (NA=0.95)

back aperture Supercontinuum light source (Fianium) + AOTF

λ=600 nm 5 µm

Fourier microscopy

Direct evidence of k|| + G conservation

objective (NA=0.95)

back aperture Supercontinuum light source (Fianium) + AOTF

λ=600 nm

Single scatterer radiation patterns

21

Cts/pxl (0.1 s)

In phase-excited plasmon rods radiate like a line of dipoles: donut x sinc function Potential: visualize radiation pattern of any SINGLE antenna

0

1200

ky

kx

(λ=600 nm)

1um x 100 nm Au

0

500

ky

kx

(λ=600 nm)

2um x 100 nm Au

Confocal microscope

Highest resolution imaging with lenses 1. Overfill high NA objective with a parallel beam 2. Color separate output at dichroic element 3. Tube lens to focus on point detector

Resolution from confocality: (1) small laser spot (2) detection pinhole

Confocal microscope

Typical numbers 100x, NA =0.95 – 1.4 Tube lens L1/L2: f2=200 mm Objective f1 = 200/100 = 2mm Beam radius f1 NA = 3 mm Note also `conservation rule’

Resolution object plane

Gaussian optics

Why NA really helps

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

Objective opening angle (degrees)90756030 450

NA=0.4 4%

NA=0.95 35%

NA=0.9 28%

Capt

ured

frac

tion

of 4

π sr

Objective NA

NA=0.7 15%

15

NA means 1. Resolution

2. Detection efficiency

Seeing single molecules

Very dilute layer of fluorescing Molecules (DiD)

Single molecules blink one by one In: 532 nm, ~ 0.1 mW Out: ~ 600 nm

Courtesy: Martin Frimmer, AMOLF open dag

Enabling equipment tricks For room temperature experiments the universal tricks are: Dilution - λ/2 detection or excitation volume - Diluted samples to < 1 molecule per 1 µm2

-Filtering - 108 laser line rejection filters Efficient photon collection - Very high NA objective

Shot noise level detection - Silicon CCDs and APDs 60% q.efficiency

low read out & dark noise

1. Microscopy at the diffraction limit - What is the diffraction limit - Spatial frequencies and Fourier transforms - High NA imaging

2. Microscopy beyond the diffraction limit - Scanning probe to beat the diffraction limit - Example imaging molecules - Example imaing photonic structure - Artefacts

Microscopy

Why is there a barrier in optical microscopy resolution?

And how can it be broken?

Localization of a molecule Idea: if you know you have a single object, you can find its

localization to much better than the diffraction limit

Single molecule CCD images Different exposure times

Least square fitting of Gaussian Error diminishes with photon count N

Beam waist Pixel size Background noise

Biophysical Journal 82(5) 2775–2783

Note: not true diffraction barrier breaking

Cheating the diffraction limit

PALM, STORM: beat Abbe limit by seeing a single molecule at a time Using a stochastic on/off switch to keep most molecules dark

Resolution: how discernible are two objects ? If you have a single object, you can fit the center of a Gaussian with arbitrary precision (depends on noise)

Arbitrary source distribution

Field at z=0 (object) propagates in free space as

( ) ( ) iˆ ˆ, ; , ;0 e zk zx y x yk k z k k ±=E E ( ) ( )2 2 2

0z x yk nk k k= − +

The propagator H is oscillating for

and exponentially decaying for

( ) ( )22 20x yk k nk+ <

( ) ( )22 20x yk k nk+ >

Near field = Exponentially confined fast spatial fluctuations Far field = Propagating fields = bound by diffraction limit

Field of a dipole

1/r term: far field constant radiated flux

1/r3 term: near field 1/r2 term: mid field

Transition near to far at kr ~1 or r ~λ/2π

Breaking the diffraction limit in near-field microscopy

A small aperture in the near field of the source can scatter also the evanescent field of the source to a detector in the far field.

Image obtained by scanning the aperture

Alternatively, the aperture can be used to illuminate only a very small spot.

Aperture probe fibre type

Aperture probe microlever type

Metallic particle Single emitter

Probing beyond the diffraction limit

glass

aluminum

500 nm

100 nm

100 nm

λ

35 nm aperture

– well defined aperture – flat endface – isotropic polarisation – high brightness up 1 µW

Ex Ey Ez

With excitation Ex , kz, :

Focussed ion beam (FIB) etched NSOM probe

Veerman, Otter, Kuipers, van Hulst, Appl. Phys. Lett. 74, 3115 (1998)

x

y

Shear force feedback: molecular scale topography

Feedback on phase Tip -sample < 5 nm RMS ~ 0.1 nm

Feedback loop:

sample

Lateral movement, A0 ~ 0.1 nm

Tuning fork 32 kHz Q ~ 500

∆f

ω0

A0

piezo

Rensen, Ruiter, West, van Hulst, Appl. Phys. Lett. 75 1640 (1999) Ruiter, Veerman, v/d Werf, van Hulst, Appl. Phys. Lett. 71 28 (1997)

van Hulst, Garcia-Parajo, Moers, Veerman, Ruiter, J. Struct. Biol. 119, 222, (1997)

1.7 x 1.7 µm

3 x 3 µm

Steps on graphite (HOPG)

~ 0.8 nm step ~ 3 mono-atomic steps

DNA width 14 nm

height 1.4 nm

DNA on mica

Data from Kobus Kuipers and Niek van Hulst et al.

Mapping the near field of the probe

90o 0o 1 µm

100 nm

Perylene orange in PMMA

Ruiter, Veerman, Garcia-Parajo, van Hulst, J. Phys. Chem. 101 A, 7318

Two arms of the interferometer

equal in length gives temporal overlap on the

detector

Time-resolved near-field scanning tunneling microscopy

Measurement of guiding & bending

40

Sample: AIST Japan Meas: AMOLF

Artefacts

1. Topographic The tip moves over the topography Potential cross talk

1. Non-perturbative tip

Topographic artefacts

Topography: convolution of sample and tip Optical: weighted by exponential factor Tricky: topography and optical pick up are shifted sideways

43

Narrow cavity resonance

Laser: grating tunable diode laser 20 MHz linewidth around 1565 nm Detection: InGaAs APD (IdQuantique)

1565.0 1565.2 1565.40

25

50

75

100

125

Coun

ts

Wavelength (nm)

Picked up by tipFew µm above cavity Q =(1±0.5) 105

Lorentz Q =88000

November

44

Resonance shift

1565.0 1565.2 1565.40

25

50

75

100

125

Coun

ts

Wavelength (nm)

Few µm above cavity~10 nm above cavity

Line shifts by 1 linewidth

Glass tip: ∆ω/ω ∼ −1.2 10−4 (∆λ of 20 pm)

Consistent with

2 /0

2mode 0

| ( ) |max( ( ) | ( ) | )

z dE r eV E

ω αω ε

−∆= − ⋅

r r

Inserting a polarizability comparable to the mode volume shifts ω

45

Tuning vs mode intensity

1565.20

1565.25

1565.30

1565.35

1565.40

1565.45

Transverse to W1Along W1

λ of

max

. sig

nal (

nm)

-2 -1 0 1 2-4 -2 0 2 40.0

0.5

1.0

g

|E|2 (n

orm

.) in

the

slab

Position (µm)

Experiment

FDTD

In this case the ∆ω/ω and not Intensity maps |E|2

1. Microscopy at the diffraction limit - What is the diffraction limit - Spatial frequencies and Fourier transforms - High NA imaging

2. Microscopy beyond the diffraction limit - Scanning probe to beat the diffraction limit - Example imaging molecules - Example imaing photonic structure - Artefacts