Quantitative Phase Imaging of Cells and Tissues - PowerPoint Presentation

Slide1

Quantitative Phase Imaging of Cells and TissuesSlide2

Chaps.1-2

Introduction and groundworkSlide3

Light Microscopy

Started around 1600 during and played a central role in the Scientific Revolution

Much of the efforts in the field of microscopy has been devoted to improving

resolution

and

contrastSlide4

Resolution and Contrast

Abbe’s theoretical resolution limit for far-field imaging: half wavelength

Superresolution imaging: STED, (f)PALM, STORM, structured illumination.

Two types of contrast:

Endogenous

&

Exogenous

Endogenous contrast:

Dark field, Phase contrast, Schlerein,

Quantitative Phase Microscopy

, Confocal, Endogenous florescence

Exogenous contrast:

Staining, Florescent tagging, Beads, Nanoparticles, Quantum DotsSlide5

Quantitative Phase Imaging (QPI)

The great obstacle in generating intrinsic contrast from optically thin specimens (including live cells) is that, generally, they do not absorb or scatter light significantly, i.e., they are transparent, or

phase objects

.

Abbe

described image formation as an interference phenomenon.

In 1930s,

Zernike

developed phase-contrast microscopy (PCM), but phase is not quantitatively retrieved

In 1940s,

Gabor

proposed

holographySlide6

Quantitative Phase Imaging (QPI)

QPI combines these pioneering ideas, and the resulting image is a map of pathlength shifts associated with the specimen.

QPI

has the ability to

quantify cell

growth with

femtogram

sensitivity and without

contact.Slide7

Quantitative Phase Imaging (QPI)

Multimodal InvestigationSlide8

Quantitative Phase Imaging (QPI)

QPI and

light-scattering

data with extreme sensitivity.Slide9

Two confusions

Nanoscale sensitivity

Three-Dimensional ImagingSlide10

Light Propagation in Free Space

Helmholtz Equation

1-D Propagation

Plane wavesSlide11

Plane wave propagationSlide12

Light Propagation in Free Space

3-D Propagation

Spherical WavesSlide13

Huygen’s Principle

Each

point reached by the field becomes a

secondary source

, which emits a new spherical wavelet and so on

.

Generally this integral is hard to evaluateSlide14

Fresnel Approximation of Wave Propagation

The spherical

wavelet at far field

is now approximated bySlide15

Fresnel Approximation of Wave Propagation

For a given planar field distribution at

z

= 0,

U

(

u

,

v

), we can

calculate the

resulting propagated field at distance

z

,

U

(

x

,

y

,

z

) by

convolving with

the Fresnel

wavelet Slide16

Fourier Transform Properties of Free Space

Fraunhofer approximation

is valid when the observation plane is even farther away. And the quadratic phase becomes

The field distribution then is described as a Fourier transformSlide17

Fourier Transform Properties of Lenses

Lenses have the capability to perform Fourier Transforms, eliminating the need for large distances of propagation.

Transmission function in the form of Slide18

Fourier Transform Properties of LensesSlide19

Fourier Transform Properties of Lenses

Fourier transform of the field diffracted by a sinusoidal grating.Slide20

The First Order Born Approximation of Light Scattering in Inhomogeneous Media

light interacts with

inhomogeneous media

, a

process

generally referred to as

scattering

.

The

scattering inverse problem

can be solved analytically if we assume

weakly scattering media

, this is the

first-order Born ApproximationSlide21

The First Order Born Approximation

The Helmholtz Equation can be rearranged to reveal the scattering potential associated with the medium

the

solution for the

scattered field

is a convolution between the source term, i.e.,

F

(

r

,ω

) ⋅U(r,ω), and the Green function, g(r,ω

).

and thus the

scattering amplitudeSlide22

First-order Born approximation assumes that

the field inside

the scattering volume is constant and equal to the incident

field, assumed

to be a plain

wave

The resulting scattering amplitude:

q

is the

scattering wave vector

or the momentum transferSlide23

Inverse scattering problem

The expression for scattering amplitude can be inverted due to the Reversibility of Fourier Integral, to express the

scattering potential

Note that in order to retrieve the scattering potential

F

(

r

′,

ω

) experimentally

, two essential conditions must be met:1. The measurement has to provide the complex scattered field (i.e., amplitude and phase).2. The scattered field has to be measured over an infinite range of spatial frequencies q [i.e., the limits of integration are −∞ to ∞].Slide24

Ewald’s

limiting sphere

As we rotate the incident wave vector from

k

i

to −

k

i

, the

respective backscattering wave vector rotates from kb to −kb, such that the tip of q describes a sphere of radius 2k 0

. This is known as the

Ewald sphere

, or

Ewald limiting sphere

.Slide25

Effect of Ewald Sphere

The effect of Ewald Sphere is a bandwidth limitation in the best case scenario, i.e. a truncation in frequency. The resulting field can be expressed as

The scattering potential can be obtained via the 3D Fourier transformSlide26

Scattering by Single Particles

The field scattered in the far zone has the general form of a perturbed spherical wave

The

Differential Cross Section

and

Scattered Cross Section

associated with the particle is defined as Slide27

Scattering by Single Particles

In the case if the particle absorbs light, we can define an analogous

absorption cross section

, and the attenuation due to the combined effect,

total cross section

For particles of arbitrary shapes, sizes, and refractive indices,

deriving expression

for the scattering cross

sections is

very

difficult.However, if simplifying assumptions can be made, the problem becomes tractableSlide28

Particles Under the Born Approximation

When the refractive index of a particle is only slightly different

from that

of the surrounding medium, its scattering properties can

be derived

analytically within the framework of the Born

approximation

Rule of thumb: total phase shift of the particle is smaller than 1 rad, i.e.Slide29

Particles Under the Born Approximation—

Spherical Particle

Scattering potential:

Scattering amplitude:

Differential scattering potential:

The scattering angle is in the momentum transfer

Also known as Rayleigh-

Gans

Particles Slide30

Particles Under the Born Approximation—

Spherical Particle

In case when

qr

->0,

i.e

r->0 very small particle, q->0 very small angle:

Measurements at small angles can reveal volume of particle. This is the basis for many flow cytometry instruments.

The scattering cross section,

Indicates the Rayleigh scattering is

isotropicSlide31

Particles Under the Born Approximation—

Cubical Particle

Scattering potential:

Scattering amplitude:

Differential scattering potential:

In the case when size decreases, a~0, Rayleigh regime is recovered

For particles smaller than the wavelength, the details do not affect far-zone scattering.Slide32

Particles Under the Born Approximation—

Cylindrical Particle

Scattering potential:

Scattering amplitude:

where

J

1

is the Bessel function of first order and kind. As before,

σ

d

and σs can be easily obtainedSlide33

Particles Under the Born Approximation—

Ensemble of Particles

Scattering potential:

Scattering amplitude:

We express the scattering amplitude as the scattering amplitude of a single particle, multiplied by the structure functionSlide34

Mie Scattering

In 1908, Mie provided the full electromagnetic solutions of Maxwell’s equations

for a

spherical particle

of arbitrary size and refractive

index. The

scattering cross section has the

form

where and

are functions

of

,

This can only be solved numerically, and note that as

partical

size increase, the summation converges more slowly.

Mie scattering is sometimes used for modeling tissue scattering.

 Slide35

Chap. 5

Light MicroscopySlide36

Abbe’s Theory of Imaging

One way to describe an imaging system (e.g., a microscope) is in terms of a system of two lenses that perform two successive Fourier Transforms.

Magnification is given by

Cascading many imaging systems does not mean unlimited resolution!Slide37

Resolution Limit

‘’The microscope image

is the

interference effect of a

diffraction

phenomenon

” –Abbe, 1873

The

image field can

therefore be decomposed into sinusoids of various

frequencies and phase shiftsSlide38

Resolution Limit

the apertures present in the

microscope objective

limit the maximum angle associated with the light

scattered by

the specimen

.

The effect of the objective is that of a

low-pass

filter

, with the cut-off frequency in 1D given by Slide39

Resolution Limit

The image is truncated by the pupil function at the frequency domain

The resulting field is then the convolution of the original field and the Fourier Transform of PSlide40

Resolution Limit

function

g

is the Green’s function or PSF of the instrument, and is defined as

The Rayleigh criterion for resolution: maxima separated by at least the first root.

Resolution:Slide41

Imaging of Phase Objects

Complex transmission function:

For transparent specimen

,

is constant and so is

for an ideal imaging system.

Detector at image plane are only sensitive to intensity, therefore zero contrast for imaging transparent specimen

 Slide42

Dark Field Microscopy

One straightforward way to increase contrast is to remove the low-frequency content of the image, i.e. DC component, before the light is detected.

For coherent illumination, this high pass operation can be easily accomplished by placing an obstruct where the incident plane wave is focused on axis.Slide43

Zernike’s Phase Contrast Microscopy

Developed in the 1930s by the Dutch physicist

Frits

Zernike

Allows

label-free, noninvasive

investigation

of live cells

Interpreting the field as spatial average

and the fluctuating components,

Note that the average

must be taken inside the coherence area. Phase is not well defined outside coherence area.

 Slide44

Zernike’s Phase Contrast Microscopy

The field in Fourier Domain can be interpreted as the incident field and the scattered field.

The image field is described as the interference between these two fields.

The key to PCM: the image intensity, unlike the phase, is very sensitive to

changes around

. The Taylor expansion around zero of cosine is negligible for small x, but linear dependent for sine

 Slide45

Zernike’s Phase Contrast Microscopy

By placing a small metal film that covers the DC part in the Fourier Plane can both attenuate and shift the phase of the unscattered field.

If

is chosen,

 Slide46

Zernike’s Phase Contrast MicroscopySlide47

Chap. 6

HolographySlide48

Gabor’s (In-Line) Holography

In 1948, Dennis Gabor introduced “A new microscopic principle”,1 which he termed holography (from Greek

holos

, meaning “whole” or “entire,” and

grafe

, “writing”).

Record

amplitude

and

phase

Film records the Interference of light passing through a semitransparent object consists of the scattered (

U

1) and

unscattered

field (

U

0). Slide49

Reading the hologram essentially means illuminating it as if it is a new object (Fig. 6.2). The field scattered from the hologram is the product between the illuminating plane wave (assumed to be ) and the transmission function

The last two terms contain the scattered complex field and its back-scattered counterpart. The observer behind the hologram is able to see the image that resembles the object.

The backscattering field forms a virtual image that overlaps with the focused image.

In-line HolographySlide50

Off-Axis Holography

Emmitt Leith and

Juris

Upatnieks developed this off-axis reference hologram, the evolution from Gabor’s inline hologram.

Writing the hologram:

The field distribution across the film, i.e. the Fresnel diffraction pattern is a convolution between the transmission function of the object U, and the Fresnel kernel

The resulting transmission function associated with the hologram is proportional to the intensity, i.e.Slide51

Off-Axis Holography

Reading the hologram:

Illuminating the hologram with a reference plane wave,

, the field at the plane of the film becomes

The last two terms recover the Real image and the virtual image at an angle different than the real image.

 Slide52

Nonlinear (Real Time) Holography or Phase Conjugation

Nonlinear

four-wave mixing

can be

interpreted as

real-time

holography

The idea relies on third-order nonlinearity response of the material used as writing/reading medium

.

Two strong field

,

, that are time reverse of each other and incident on the

. An object field

is applied simultaneously, inducing the nonlinear polarization

Clearly, the field emerging from the material,

U

4, is the

time-reversed version

of

U

3, i.e. ω4 = -ω3 and

k4

= -

k3

, as indicated by

the complex

conjugation

(

U

3

∗

).Slide53

Digital Hologram

idea is to calculate the cross-correlation between the known signal

of interest

and an unknown signal which, as a result, determines (i.e

., recognizes

) the presence of the first in the

second

The result field on the image plane is characterized by the cross correlation between the image in interest and the image being comparedSlide54
Slide55

The transparency containing the signal of interest is

illuminating by a plane wave. The emerging field,

U

0

, is Fourier transformed

by the

lens at its back focal plane, where the 2D

detector array

is positioned.

The off-axis

reference field Ur is incident on the detector at an angle θ.Fourier Transforms are numerically processed with FFT algorithmDigital HolographySlide56
Slide57

Chap. 8

Principles of Full-Field QPISlide58

Interferometric Imaging

The image field can be expressed in space-time as

Detector is only sensitive to intensity, phase information is lost in the modulus square of field.

Mixing with a reference field

 Slide59

Temporal Phase Modulation:

Phase-Shifting Interferometry

The idea is to introduce a control over the phase

difference between

two interfering fields, such that the intensity of

the resulting

signal has the formSlide60

Temporal Phase Modulation:

Phase-Shifting Interferometry

Three unknown variables:

,

and the phase difference

Minimum three measurements is needed

However, three measurements only provide

over half of the trigonometric circle, since sine and cosine are only

bijective

over half circle, i.e.

and

respectively

Four measurements with phase shift in

increment.

 Slide61

Temporal Phase Modulation:

Phase-Shifting InterferometrySlide62

Spatial Phase Modulation: Off-Axis Interferometry

Off-axis interferometry takes advantage of the

spatial phase

modulation

introduced

by the

angularly shifted

reference plane

waveSlide63

Spatial Phase Modulation: Off-Axis Interferometry

The goal is to isolate

and calculate the imaginary counterpart through a Hilbert Transform

Finally the argument is obtained uniquely as

the frequency of modulation,

Δ

k

, sets an upper limit on the

highest spatial

frequency resolvable in an image.

 Slide64

Phase Unwrapping

Phase measurements yields value within

interval and modulo(2

. In other words, the phase measurements cannot distinguish between

, and

Unwrapping operation searches for

jumps in the signal and corrects them by adding

back to the signal.

 Slide65

Figures of Merit in QPI

Temporal Sampling: Acquisition Rate

Must be at least twice the frequency of the signal of interest, according to

Nyquist

sampling theorem.

In QPI acquisition rate vary from application: from >100Hz in the case of membrane fluctuations to <1mHz when studying the cell cycle.

Trade-off between acquisition rate and sensitivity.

Off-axis has the advantage of “single shot” over phase-shifting techniques, which acquires at best four time slower than that of the camera.Slide66

Figures of Merit in QPI

Spatial Sampling: Transverse Resolution

QPI offer new opportunities in terms of transverse resolution, not clear-cut in the case of coherent imaging.

The phase difference between the two points has a significant effect on the intensity distribution and resolution.

Phase shifting methods are more likely than phase shifting method to preserve the diffraction limited resolutionSlide67

Figures of Merit in QPI

Temporal Stability: Temporal-Phase Sensitivity

Assess phase stability experimentally: perform successive measurements of a stable sample and describe the phase fluctuation of one point by its standard deviationSlide68

Figures of Merit in QPI

Temporal Stability: Temporal-Phase Sensitivity

Reducing Noise Level:

Simple yet effective means is to reference the phase image to a point in the field of view that is known to be stable. This reduces the common mode noise, i.e. phase fluctuations that are common to the entire field of view.

A fuller descriptor of the temporal phase noise is obtained by computing numerically the power spectrum of the measured signal. The area of the normalized spectrum gives the variance of the signalSlide69

Figures of Merit in QPI

Temporal Stability: Temporal-Phase Sensitivity

Reducing

Noise Level:

Function

is the analog of the

noise equivalent power (NEP)

commonly used a s figure of merit for photodetectors.

NEP

represents the smallest phase change (in rad) that can be measured (SNR =1) at a frequency bandwidth of 1 rad/s.

High sensitivity thus can be achieved by locking the measurement onto a narrow band of frequency.

 Slide70

Figures of Merit in QPI

Temporal Stability: Temporal-Phase Sensitivity

Passive stabilization

Active stabilization

Differential measurements

Common path interferometrySlide71

Figures of Merit in QPI

Spatial Uniformity: Spatial Phase Sensitivity

Analog to the “frame-to-frame” phase noise, there is a “point-to-point” (spatial) phase noise affects measurements.Slide72

Figures of Merit in QPI

Spatial Uniformity: Spatial Phase Sensitivity

The standard deviation for the entire field of view, following the time domain definition:

The normalized spectrum density

Thus variance defined as Slide73

Figures of Merit in QPI

Spatial Uniformity: Spatial Phase Sensitivity

Again, phase sensitivity can be increased significantly if the

measurement is

band-passed around a certain spatial

frequency

Spatial and Temporal power spectrum Slide74

Summary of QPI Approaches and Figures of Merit

T

here

is no technique that performs

optimally with

respect to all figures of merit identifiedSlide75

Summary of QPI Approaches and Figures of Merit

Thus, there are

possible combinations of

two

methods,

as follows

 Slide76

Summary of QPI Approaches and Figures of Merit

Thus, there are

possible combinations of three methods,

as follows