# Super-resolution microscopy techniques

Notes about Super-resolution microscopy techniques.

## Course information

- Lecturer: Tony Yang
- Time: 789 (W)
- Location: MD225
- Reference books
- Bahaa Saleh and Malvin Teich, Fundamental of Photonics, 2nd ed. Wiley, New York, 2007.
- Erfle, Holger, Super-Resolution Microscopy: Methods and Protocols, Humana Press, 2017

- Grading:
- Participation in classroom discussions: 25%
- Midterm: 30%
- Term paper: 45%

- G drive: https://drive.google.com/drive/u/1/folders/1sTXB5tplkqXRCZ05VJYFFkjfPJ-hs43I

## Photonics

### Ray optics

When lenght scale of the instrument i smuch larger than that of light wavelength. Neither wave properties (diffraction, interference) nor photon ones. Optical pathlength = line integral from one point to another, with respect to refraction index (n) $$\int_A^B n(r)ds$$

#### Fermat’s principle

Light tries to tale minimal travel time Snell’s law: $$n_1sin\theta_1 = n_2sin\theta_2$$

#### Huygen’s principle

Wavefront and wavelets: explains refraction, diffraction and interference

### Total internal reflection

Dense material to loose material. With little energy loss (<0.1%) as evanescent wave, penetration depth about 100-200 nm. When incidence angle $\theta > $ the critial angle $\theta_c = sin^{-1}(\frac{n_2}{n_1})$ Used in fiber optics and qsuperresolution microscope.

### Nagative-index metamaterials

$$ n = \left( \frac{\epsilon\mu}{\epsilon_0\mu_0} \right)^{1/2} \in \mathbb{C} $$

Superlensing breaking through the diffraction limit. n is requency-dependent

### Spherical mirrors

- Approximation of the ‘perfect’ parabolic mirror at small angles
- For small angles (paraaxial) $\theta \approx sin(\theta) \approx tan(\theta)$ $$ \begin{aligned} \frac{1}{z_1} &+ \frac{1}{z_2} = \frac{1}{f} \cr f &= R/2 \cr m &= \frac{y_2}{y_1} = \frac{-z_2}{z_1} \end{aligned} $$

### Spherical boundaries of different refractive indices

$$ \begin{aligned} \frac{n_1}{z_1} &+ \frac{n_2}{z_2} = \frac{n_2 - n_1}{R} \cr y_2 &= \frac{-n_1}{n_2} \frac{z_2}{z_1} y_1 \end{aligned} $$

### Thin lens from two spherical surfaces

$$ \begin{aligned} \theta_3 &= \theta_1 - y / f \cr \frac{1}{f} &= (n_2-n_1)(\frac{1}{R_1} - \frac{1}{R_2}) \cr \frac{1}{z_1} &+ \frac{1}{z_2} = \frac{1}{f} \cr m &= \frac{-y_2}{y_1} = \frac{-z_2}{z_1} \end{aligned} $$

### Transformation in matrix forms

Light rays as 2-component vector Components as 2 by 2 matrix.

## Wave optics

### Considerations

- Diffraction (+), polarization (-), Fraunhofer (+), Fresnal (+)
- Maxwell equations: EM (E and B) vector fields
- optic
*phase*is the central quantity. - phase match at boundaries

### Wave equation

- 2nd derivative of space proprotional to that of time u: space; t: time; v: phase velocity; k: wave number; $\omega$: angular frequency; n: refractive index

$$ \begin{aligned} \nabla^2u &= \frac{1}{v^2}\frac{\partial^2u}{\partial t^2} \cr k &= \frac{2\pi}{\lambda} \cr \omega &= 2\pi v \cr v &= \frac{c}{n} \end{aligned} $$

- Linear equations => superposition possible
- Complex notation by Euler’s formula a: amplitude, ϕ(r): phase, ω: angular velocity periodic both in time and space the real part = physical quantity

$$ U (r,t) = a(r)exp(i\phi(r))exp(i\omega t) $$

### Helmholtz equations

- regardless of time $$ \begin{aligned} U (r) = a(r)exp(i\phi(r)) \cr \nabla^2U (r) + k^2U (r) = 0 \cr \end{aligned} $$

### Wavefonts

surfaces of constant phase (等相位面)

Plane waves in media with refractive index n $$ \begin{aligned} k &= k_0n \cr λ &= \frac{λ_0}{n} \end{aligned} $$

Bigger the n, higher in spatial frequency (shorter in wavelength). The same time frequency.

### Spherical waves

$$ \begin{aligned} U (r) &= \frac{A}{r}exp(-ikr) \cr r &= \sqrt{x^2 + y^2 + z^2} \end{aligned} $$

**Fresnal Approximation**: Paraaxial ($z^2 » (x^2 + y^2)$): Spherical -> paraboloidal -> planar wave
$$
\begin{aligned}
U (r) &= \frac{A}{z}exp(-ikz) exp \left( -ik \frac{x^2 + y^2}{z} \right) \cr
\nabla^2U (r) &+ k^2U (r) = 0 \cr
\end{aligned}
$$

### Reflection, Refraction

- Results are similar to ray optics at planar surfaces for planar waves
- Plane wave through thin lens -> paraboloidal waves
- Intensity = $| U(r) |^2$

### Interference

By superposition of two rays $$I = | U(r) |^2 = I_1 + I_2 + 2\sqrt{I_1I_2} cosΔϕ$$

### Paraxial waves

- Slowly varying envelope: slow change in amplitude
- Paraxial Helmholtz equation $$ ∇_T^2 A(r) = 2ik\frac{∂A}{∂z} $$

### Gaussian beam

https://en.wikipedia.org/wiki/Gaussian_beam

$$ \begin{aligned} A(r) &= \frac{A_1}{q(z)}exp \left( \frac{-ik(x^2 + y^2)}{2q(z)} \right) \cr q(z) &= z + iz_0 \end{aligned} $$

- q(z):
**q-parameter** - A solution to the paraxial Helmholtz equation
- The best we can do in real situations
- Cannot avoid spreading, but Gaussian beam’s angular divergence in minimal.
- Inside the waist (the narrowest part of the beam) is similar to planar wave
- Long wavelength and thin beam waist -> more divergence
- Depth of focus

$$ \begin{aligned} W(z) &= W_0 \sqrt{1 + (z / z_0)^2} \cr DOF &= 2z_0 = 2 \frac{W_0^2 \pi}{\lambda} \end{aligned} $$

- Calculate the divergence by the q parameter and complex distance

$$ \begin{aligned} q_2 &= q_1 + d \cr \frac{1}{q_1} &= \frac{1}{R_1} - \frac{iλ}{πW_1^2} \cr \frac{1}{q_2} &= \frac{1}{R_2} - \frac{iλ}{πW_2^2} \cr \end{aligned} $$

Beam quality: M-square factor >=1, the smaller the better.

Through thin lens

- Change in phase -> wavefront is bent
- Radius is unchanged
- Not focused on a single point like in ray optics

### Higher order modes (TEM (l,m))

- Laguerre-Gaussion beams -> important in superresolution.

## Fourier Optics

- Any wave = sum (superpositions) of plane waves
- Important properties:
*angles*and*spatial frequencies*

- Important properties:
- Optical components: linear functions with frequency response
- Impulse (with all frequencies) =>
**Impulse response function** - Inputs of various freq. =>
**Transfer function**

- Impulse (with all frequencies) =>

### Propagation of light in free space

Angles => spatial frequencies in the x-y plane

$$U(x,y,z) = A \cdot exp(-j(k_xx+k_yy+k_zz))$$

Where

- wave vector $\textbf{k} = (k_x, k_y, k_z)$
- wave length $\lambda$
- wavenumber $k = \sqrt{k_x^2 + k_y^2 + k_z^2} = \frac{2\pi}{\lambda}$

For paraxial waves

$$\theta_x = sin^{-1}(\lambda\nu_x) \approx \lambda\nu_x$$

$$\theta_y = sin^{-1}(\lambda\nu_y) \approx \lambda\nu_y$$

### Optical Fourier Transform

- Spatial frequencies at different angles
- A lens could do Fourier transform at the focal plane

### Fraunhofer Far Field Approximation

- Far field: $d \gg \frac{b^2}{\lambda} , \frac{a^2}{\lambda}$
- Near field ($d \approx \lambda$): superresolution (~nm) due to little distortion
- Far field image (diffraction pattern) is the Fourier transform of the original image
- Smaller the scale (higher spatial frequencies), larger the distortion (wider aura)

- Diffraction: is everywhere, but best demonstrated in the pinhole(aperture) experiment

#### Rectangular aperture

- expressed as cardinal sine (sinc) function
- Angular divergence (first zero value): $\theta_x = \frac{\lambda}{D_x}$

#### Circular aperture

- Bessel function, Airy pattern
- $\theta = 1.22\frac{\lambda}{D}$: angle of the Airy disk
- Focused optical beam throught an aperture: $\theta = 1.22\frac{f\lambda}{D}$:

#### 4-F imaging system

- Original image -> lens (FT) -> (spatial freqs.) -> lens(iFT) -> perfect image (in theory)
- Filtering of higher spatial freqs: less detailed image, less noise
- Spatial filtering:
*cleaning*laser beams

### Transfer function of free space

- Higher freq. => real exponent => attenuate rapidly (evanescent wave)

## Polarization

- Electric-field as a vector
- Polarization ellipse: looking at the xy plane from the z axis.
- Phase difference: $\varphi$
- Linearly polarized: $\varphi = 0 $ or $\pi$
- Circularly polarized $\varphi = \pm \pi /2 $ and $a_x = a_y$$

- Linear polarizer : only passed a certain linearly polarized light
- Wave retarder: changes $\varphi$ to change polarization pattern

### Fiber optics

- Low-loss
- Light could
*bend*inside it - Single-mode fiber (small core): Gaussain wave only
- Multimode fiber (larger core): higher order light source
- Relation to numerical aperture (NA)
- Acceptance angle of the fiber: $\theta_a = sin^{-1}(NA)$
- Larger NA: more higher order information, more noise
- Smaller NA: $V = 2\pi\frac{a}{\lambda_0}NA < 2.405$. Gaussian wave only

- Polariztion-maintaining fibers

## Quantum optics

- Quantum electrodynamics (QED)
- Energy carried by a photon: $E = h\nu = \hbar\omega$
- Typical light source: more than trillion photons per second
- $E (eV) = \frac{1.24}{\lambda_0(\mu m)}$

- Momentum carried by a photon: $p = hk$
- Probability of photon position or the squared magnitude of the SWE (indivisual behavior) is directly proportional to light intensity (group behavior)
- At smaller n : the interference pattern looks random (randomness of photon flow)
- At larger n: the interference pattern is more similar to what we see in the macroscale

- Poisson distribution (discrete ranomness with rate = photon flux)
- mean = variance
- SNR = mean^2 / variance = mean

### Schroedinger wave equation (SWE)

- Similar to solve for eigenvalues => discrete solutions => quantitized energy levels
- Particle in a well / atoms with a single electron => standing wave (discrete solutions)
- Multi-electron: no analytical solutions

## Photons and matter

- Photon absorption and release: jumping in energy levels
- Rotational : microwave to far-infrared
- Vibrational : IR e.g. CO2 laser
- Electronic : visible to UV
- Photon absorption: elcetron jump up in energy level
- Photon emmision: Spontaneous vs stimulated (laser)

## Occupation of energy levels

- Boltzmann distribution
- Pumping enegy: population inversion
- Laser stimulated emmision

## Luminescence

- Cathodo- (CRT)
- Sono- (ultrasound)
- Chemi- (lightsticks)
- Bio- (firefly)
- Electro- (LED)
- Photo- (Laser, Fluorescence, Phosphorescence)

## Photoluminescence

- In fact emitting a range of wavelengths (many sub-energy levels)
- Fluorescence (spin-allowed, shorter lifetime) vs phosphorescence (spin-forbidden, longer lifetime)

### Multiphoton

- Absorption of 2 lower energy photons => emission of 1 higher energy photon
- Multiphoton fluorescence

## Light scattering

- Photoluminescence: real excited states (resonant)
- Scattering: virtual excited states (non-resonant)
- Rayleigh: same energy (elastic)
- Particle size much smaller than the photon wavelength
- Reason behind blue sky
- vs Mie scatttering particle size comparable to photon wavelength

- Raman
- Stokes: Loss energy
- Ani-Stokes: Gain energy
- Molecular signature

- Brillouin: acoustic

- Rayleigh: same energy (elastic)

### Stimulated Raman scattering (SRS)

- Label-free microscopy

## Eyes

- 380 nm ~ 710 nm
- theshold of vision: 10 photons (a cluster of rod cells)
- Logarithmic perception: Weber-Fechner Law (like hearing)
- Single lens: spherical and chromatic aberration inevitable
- Astigmatism: directional abberation
- Pupil (Aperture)
- Small pupil: less spherical and chromatic aberration (paraxial), less brightness and more diffraction
- Large pupil: more brightness, more spherical and chromatic aberration
- Optimum: 3mm

- Viewing angle: the
*perceived*size

## Length scale of microscopes

- Resiolution limit of regular light microscope: 200nm
- Clear organnels structure: 30nm

## Geometrical optics of a thin lens

- Lens equation: $\frac{1}{f} = \frac{1}{a} + \frac{1}{b}$
- Magnification factor: $M = \frac{b}{a}$
- Virtual image: divergent rays forming a real image on the retina due to the lens
- Compound microscope: M = $M_{obj}$ * $M_{eye}$

### Infinity-corrected mircoscope

- Object on the focal plane of the objective lens
- Parallel rays from the objective is converged by the tube lens
- Magnification: reference tube length (160-200mm) divided by the focal length of the objective
- shorter focal length = larger magnification
- 1.5mm => 100x

## Microscope anatomy and design

- The most important: resolving power (distinguish between two points) = numericalaperture (NA)
- 2nd: Contrast : object v.s. background (noise) signal strength
- 3rd: Magnificaition: $M_{obj}$ * $M_{eye}$

### Anatomy

- Light source: Koehler illumination to see the sample, not the light source
- Diaphragm
- Field: field of view
- Condenser / aperture: resolution + brightness (open, larger angle) vs contrast + depth of view (closed, smaller angle)

- Condenser
- Objective
- Eyepiece / camera

### Different types of microscopic design

- Transmitted light
- Bright field
- Dark field
- Phase contrast
- DIC
- Polarization

- Reflected light: objective = condenser (most common in modern microscopes)
- Fluorescence
- Upright vs inverted

## Optical aberrations

### Spherical aberrations

- Paraxial and peripheral rays have different focal planes
- Assymetry in unfocused images
- Corrected by
- 2 plano-convex lenses facing each other
- meniscus lenses
- lenses with different radii
- doubling with another lens with opposing degree of spherical aberration

### Chromatic aberrations

- Different refractive index for different wavelengthes
- Corrected by
- Doubling with a lens with a different material and shape
- Achromat: corrected for 2 wavelengths
- Apochromat: corrected for at least 3 wavelengths
- Flunar (semi-apochromat)

### Astigmatism

- Different directional plane, different foci
- Not in perfect alignment (off-axis) / curvature of field
- Esp. in high NA lens

- Caused / corrected vy a plano-cylindrical lens

### Coma

- Comet tail
- Off-axis aberration (misalignment)

### Field Curvature

- Thin flat object -> image with edges curving towards lens
- Cause: difference of lengthes of light paths
- Esp. in high NA
- Planar view objectives correct this

### Distortion

- non-linear aberrations
- different magnification across the field of view

### Transverse chromatic aberration

- Chromatic difference of magnification

### Testing for aberrations

- Color shift between channels
- Fluorescent beads

### Anti-vibration tables

- Vibrations
- Ground (low freq. 0.1 - 5 Hz)
- Acoustic
- Direct vibration from the components (10-100 Hz)

- Solution:
- Air isolators
- Active control

### Ergonomics

- Protect scietists' eyes, neck, and shoulder

## Objective

- The most important part in a microscope

### Objective class

- More corrections, more expensive
- Achromat: 1
- Semi-apochromat: 2-3
- Apochromat: 5-10 cost

### Labels on the objective

- numerical aperture (NA): resolving power (collected photons)
- magnification (e.g. 10x): field of view
- colorcorrection: Achromat / Semi-apochromat (Neofluar / fluotar) / Apochromat
- gimmersion: air / water / oil
- free working distance
- cover slip thickness (usually 170 μm)

### Numerical aperture

NA = nsinα

#### Oil immersion

- no air gap causing total internal reflection (loss of photon information)
- NA up to 1.4

#### Abbe’s law

Lateral spatial resolution (xy):

$$ d \approx \frac{\lambda}{2} $$

Axial spatial resolution (z): usually worse (~700 nm)

### Depth of field vs depth of focus

- Depth of field: moving the object
- Depth of focus: moving the image plane

### Brightness

- More NA, brighter
- More mag, dimmer
- Best brightness: NA 1.4 and mag 40x

### Illumination (lamp)

- Tungsten: 300-1500nm (reddish), dimmer
- Tungsten-halogen lamp: stable spctrum and bright
- Mercury lamp: 5 spectral peaks, 200hrs
- Meta-halide lamp: same spectral properties as the mercury lamp, latts 2000 hrs
- Xenon lamp: more constant illumination across wavelengths, 1000 hrs
- LED: small, stable, efficient, intense, multiple colors, quick to switch, long-lasting (10000 hrs)

### Filter

- Absorption vs interference (modern)
- Neutral-density (equal) vs color filters (specific wavelengths)

### Resolution

- Rayleigh’s criterion: $d = \frac{0.61 \lambda}{NA}$
- Sparrow’s (astrophysics): $d = \frac{0.47 \lambda}{NA}$
- Abbe’s: $d = \frac{0.5 \lambda}{NA}$
- Interpreted as spatial freq. response of a transfer function (low-pass filter)

### Contrast

- Signal strength of object vs background
- Human eye limit: 2% (dynamic range = 50x, 5-6 bits)
- Improved by staining (including fluorescence) and lighting techiniques

#### Interactions with the specimen

- Absorption / transmission / reflection: produce contrast (amplitude objects)
- scattering (irregular) / diffraction : edge constrast enhancement
- Refraction: difference in refractive index (n)
- Polarization: DIC (differential interference contrast) with two coherent beam and Wollaston prisms
- Phase change: phase contrast (shifting phases)/ phase interference
- Fluorescence: achieves superresolution
- Absorption and release of photons (time scale of 1fs to 1ns)
- Great resolution, constrast, sensitivity and specificity
- Live cell imaging
- Various labels (with different wavelength)

#### Bright vs Dark field

- Bright field : darker specimen than the background, lower contrast
- Dark field (by oblique illumination): birghter specimen than the background, higher contrast
- transmitted light fall outside the objective, scattered light only

## Fluorescence microscopy

- Finally the main point of superresolution microscopy
- high-contrast (clean labeling)
- sensitive: single molecule imaging (single photon)
- specific: labeling agent dependent
- multiple labeling at once with different wavelength
- versatile
- Live imaging: cell metabolsim, protein kinetics
- Molecular interaction: FRET
- Relatively cheap and safe

### Quantum processes

- Driving photon: kick electrons to an upper electronic state
- Fluorescence: electrons falling back to the ground state
- Some relaxation by vibrational energy levels (Strokes shift), or non-photogenic energy shifts
- Absorb / emit a range of wavelengths with abs. peak
- emittedwavelength is usually longer than absorbed

- Time scale: 1fs to 1ns
- Phosphorescence: singlet -> triplet -> singlet electron (spin-forbidden), much longer time scale (in seconds)

### Fluorophore

- Conjugated pi bonds providing the electronic energy levels from UV to IR
- Fluorescence lifetime:depdens on the type of fluorophores. e.g. FLIM
- Photobleaching: irreversibly destroyed after 10000 - 100000 absorption/emission cycles
- FRAP: measuinge diffusion rate

- Quenching / blinking
- Reversible supression of emission
- PALM / STORM (single molecule microscopy)

- Emission tail: increased crostalk to others
- Efficiency (Birghtness): $\Phi\epsilon_{max}$
- Quantum yield (Φ)
- Molarextinction coef. ($\epsilon_{max}$)
- The best one: quantum dots (alos the most versatile)

### Fluorescence microscope

- epi illumination is more suitable for biology
- Object = condenser
- Increased contrast (reduced background)

- transmitted light are outside field of view (only see fluorescence photons)
- Filter sets: one for excitation + one for emission + one dichromic mirror
- May need to design excitation / emssion bands for multiple fluorophores

### Fluorophores

- Smaller = better spatial resolution
- May disrupt normal cellular function
- Lables: organic dye (1 nm), protein (3 nm), quantum dots (10 nm), gold particles (100 nm)
- Specificity molecules: Antibody (15 nm), Fab, Streptavidin, Nanobody (3 nm)
- May have secondary ones (making the entire dot even bigger)

- Absorption / emission wavelengths
- Stokes shift
- Molar extinction coefficient / quantum yield = brightness
- Toxicity
- Satuartion
- Environment (pH)

### Fluorescent protein: e.g. GFP

- Introduced by transfection: not always successful (transfection and cell viability)
- Others: CFP (cyan), mCherry, mOrange, …

#### Photoactive fluorescent protein e.g. mCherry

- State transitions by activating photons
- photoactivable
- photoconvertible
- photoswitchable

### Quantum dots

- Bright and resistant to photobleaching
- Blinking under continuous activation
- Bigger (10 nm)
- Broad excitation and narrow emission spectra

### Autofluoresence

- e.g. Tryptophan, NAD(P)(H) in the cell
- Label-free imaging
- Background

### Issues of fluorescence microscopy

- Blurring
- Bleaching
- Bleed-through

### Blurring in fluorescence microscopy

- Limited depth of field compared to specimen thickness
- Reduce the SNR (out-of-focus blurred images)
- Solution: optical sectioning

### Confocal

- Pinhole: block out-of-focus light. Aperture in Airy Units (AU), optimal is 1
- Raster scanning with mirrors and a laser: point-by-point
- Phototoxicity issues: Time-lapse possible, but even higher phototoxicity
- photon detection
- PMT: high gain, low quantum efficiency(QE) (1/8)
- CCD: higher QE (65%), higher background noise (lower SNR)
- ScMOS: QE~95%
- Avalanche photodiode (APD): QE~80%, higher SNR

- Imaging parameters: no absolute rules, always trade-offs
- Resolution: slightly better than wide field (1.4x spatial freq., by FWHM of the PSF)

#### Spinning disc

- Faster imaging (parallel scans) and lower phototoxicity
- Spinning microlens array + pinholes
- Thinner optical slice of 800nm (traditional confocal: 1000nm)

### Point spread function

- Point -> psf -> Airy disk
- After Fourier transform: Optical transfer function (OTF)

## Convolution

- Lens: finite aperture, could not capture higher spatial frequencies of the object
- A way to understand and calculate blurring. Image = object * psf
- Simplified to multiplication in the frequency domain by Fourier transform
- Optical transfer function (OTF) = F{PSF}

### Point spread function (PSF)

- Hour-glass shape (sharper xy and less z resolution) due to the orientation of the objective
- Confocal pinhole open at 1 AU: less spreading of the PSF

### Deconvolution

- Computational iterative process: deblurring, restorative
- Only makes good image better

## Total Internal Reflection Fluorescence (TIRF)

- An illumination method for bottom 200nm (extent of evanescent field)
- Improves axial resolution (up to ~100 nm) and contrast

## Colocalization

- spatial overlap between two (or more) different fluorescent labels
- Pearson correlation coefficient
- Spatial colocalization doe snot mean interaction (just the same pixel: co-occurence)
- Software analysis: ImageJ
- Mander’s Colocalization coefficients
- Noise leads to underestimation of colocalization

## Spectral Overlap

- Bleed-through
- Crossover
- Cross-talk
- Managed by tweaking light sources and filters

## Resolution limit

- Abide to physical laws
- Abbe limit: 0.5 * wavelength / numerical aperture, from Fourier optics
- Electron microscope (EM): 2nm. But cells need to be fixed and processed
- Flurorescent microscopy: 200 nm. Multiple labeling methods. Multiple strategies to enhance the resolution.

## Super-resolution light microscopy (SRLM) (precisely nanoscopy)

- Cost, specimen prep, and operational complexity are in the middle between confocal and EM.

### Near field microscopy

- Evanescent waves (before the light diffracts)
- 5-10 nm axial resolution, 30-100 nm lateral resolution
- Practically zero working distance

### 4-pi microscopy

- Two opposing objectives improves z resolution
- Techical difficulties

### PALM, STED, STROM

- Using non-linear properties of the fluorophores (turing they on / off)

### Stimulated emission depletion microscopy (STED)

- Donut-shaped induced depletion laser (high power)
- At the tail of emmision spetrum to avoid cross-talk
- Donut-shape via a vortex phase plate
- Diffraction-limited. But combining another diffraction-limited excitation laser to achieve super-resolution

- Higher labels and samples preparation requirements, and optical alignment (vibration sensitive)
- Depletion efficiency: $p_{STED} = exp(-\frac{I_{STED}}{I_{sat}})$
- Resolution by the factor of $\sqrt{1 + \frac{I_{STED}}{I_{sat}}}$
- More $I_{STED}$, more resolution, but more power (photobleaching)

- Implementation: Pulsed, continous wave, gated
- Pulsed: synchronization challenges
- continous wave (CW): high background noises
- Gated: lower background noises than CW, easier than pulsed, mainstream

- Protected STED: less photobleaching using photoswitable dyes
- Long-time observation

- STED with 4-pi: improved axial(z) resolution by another phase plate

#### Fluorescence probes

- More restricted
- Two color: Long Stoke shift + normal Stoke shift dyes

## Localization microscopy

- Tracking the particles central positions from reversing the point spread function (e.g. fittin gthe Gaussian distribution). Only possible with sparse points, thus stochastic.
- Reconstruct the whole image from a series of sparse excited dyes.
- Switching-based separation is the mainstream of sparse activation

### Photoactivated localization microscopy (PALM)

- Less convenient than dSTORM.

### Stochastic optical reconstruction microscopy (STORM)

- Direct STORM (dSTORM) currently
- Readily implemented on regular wide-field microscopes.
- Selected dye (esp. Alexa 647) and imaging buffers.
- Cameras instead of PMTs to see the whole field.
- Gaussian distributions fitting the intensity of dots to calculate the centroid point.
- Labels could have an impact on the measured length (e.g. primaryand secondary antibodies)
- Localization precision: more photons, less uncertainty (more precision, up to 5-20 nm), more frames (time) required
- Precision estimation is a statistical issue.
- FWHM = 2.35 uncertainty ($\sigma_{loc}$)

- Imaging buffer: together with activation laser, determines the state (active, vs dark) of dyes
- More fluorophores could be reactivated when the signal gets too weak by the activation laser (typically UV). But not too strong to ruin the single molecule signals.
- To avoid cross-talk (activating multiple types of dyes at once) and photobleaching by stronger activation photons, starting activating with far-red (long-wavelength) dyes
- Irradiation density
- Too high: no single molecule anymore, poor localization quality
- Too low: more time required and more background noise

- Threshold for signal detection and rejection criteria
- Too strict: wasted the real signal
- Too loose: more noise
- Too many phtons at one time indicate multiple molecules = false positive, poorly localized

- Structual averaging: reducing noise by a series of images (time info. -> spatial info.)
- Pair correlation analysis and molecular cluster analysis (not randomly distributed particles)
- Single molecule tracking
- 3D localization by encoding z information into the optic system
- Bi-plane
- Dual helix
- Astigmatism

## Structured illumination microscopy

- SIM for short
- Grating pattern for structured illumination (stripes) encoding high frequency information
- Indicated by Fourier optics (extension of optical transfer function (OTF))
- Multiple images by superimposing illimunation stripes in different angles
- Increasing resolving power by 2x
- Even more resolution improvement by non-linear optics (saturation SIM)

## Light sheet microscopy

- Orthogonal illumination
- Improved z axis and optical section
- Low laser intensity for live cell imaging, minimal phototoxicity
- Scanning beam / lattice for even illumination and more z resolution