Transcript Quantitative Phase Imaging of Cells and Tissues

Quantitative Phase Imaging of
Cells and Tissues
Chaps.1-2
Introduction and groundwork
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
contrast
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
Dots
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 holography
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.
Quantitative Phase Imaging (QPI)
• Multimodal Investigation
Quantitative Phase Imaging (QPI)
• QPI and light-scattering data with extreme
sensitivity.
Two confusions
Nanoscale sensitivity
Three-Dimensional Imaging
Light Propagation in Free Space
• Helmholtz Equation
• 1-D Propagation
• Plane waves
Plane wave propagation
Light Propagation in Free Space
• 3-D Propagation
• Spherical Waves
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 evaluate
Fresnel Approximation of Wave
Propagation
• The spherical wavelet at far field is now
approximated by
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
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 transform
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
Fourier Transform Properties of Lenses
Fourier Transform Properties of Lenses
• Fourier transform of the field diffracted by a
sinusoidal grating.
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 Approximation
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 amplitude
• 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 transfer
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
∞].
Ewald’s limiting sphere
• As we rotate the incident
wave vector from ki to −ki, 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.
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
transform
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
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 tractable
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.
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
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 isotropic
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.
Particles Under the Born
Approximation—Cylindrical Particle
• Scattering potential:
• Scattering amplitude:
• where J1 is the Bessel function of
first order and kind. As before,
• σd and σs can be easily obtained
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
function
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 𝑎0 and 𝑏0 are functions of 𝑘 = 𝑘0 𝑎,𝛽 = 𝑘0 𝑛𝑎/𝑛0
• 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.
Chap. 5
Light Microscopy
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!
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 shifts
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
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 P
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:
Imaging of Phase Objects
• Complex transmission function:
• For transparent specimen, 𝐴𝑠 is
constant and so is 𝐴𝑖 for an ideal
imaging system.
• Detector at image plane are only
sensitive to intensity, therefore zero
contrast for imaging transparent
specimen
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.
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 𝑈0 and the
fluctuating components, 𝑈1 𝑥, 𝑦
• Note that the average 𝑈0 must be taken inside the
coherence area. Phase is not well defined outside
coherence area.
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
Δ𝜙 changes around 𝜙/2. The Taylor expansion around zero of cosine is
negligible for small x, but linear dependent for sine
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.
𝜋
2
• If 𝜙 = is chosen,
Zernike’s Phase Contrast Microscopy
Chap. 6
Holography
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 (U1) and unscattered field (U0).
In-line Holography
• 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 backscattered 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.
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.
Off-Axis Holography
• Reading the hologram:
– Illuminating the hologram with a
reference plane wave, 𝑈𝑟 , 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.
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 𝑈1 , 𝑈2 , that are time reverse
of each other and incident on the 𝜒 (3) . An
object field U3 is applied simultaneously,
inducing the nonlinear polarization
Clearly, the field emerging from
the material, U4, is the timereversed version of U3, i.e. ω4
= -ω3 and k4 = -k3, as indicated
by the complex conjugation
(U3∗).
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 compared
Digital Holography
• The transparency containing the signal of interest is illuminating by a
plane wave. The emerging field, U0, 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 algorithm
Chap. 8
Principles of Full-Field QPI
Interferometric Imaging
• The image field can be expressed in spacetime as
• Detector is only sensitive to intensity, phase
information is lost in the modulus square of
field. |𝑈 𝑥, 𝑦, 𝑡 |2
• Mixing with a reference field 𝑈𝑟
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 form
Temporal Phase Modulation:
Phase-Shifting Interferometry
• Three unknown variables: 𝐼1 , 𝐼2 and the phase difference 𝜙
• Minimum three measurements is needed
• However, three measurements only provide 𝜙 over half of the
trigonometric circle, since sine and cosine are only bijective
𝜋 𝜋
over half circle, i.e. (− , ) and 0, 𝜋 respectively
2
2
• Four measurements with phase shift
𝜋
in
2
increment.
Temporal Phase Modulation:
Phase-Shifting Interferometry
Spatial Phase Modulation: Off-Axis
Interferometry
• Off-axis interferometry takes advantage of the spatial phase
modulation introduced by the angularly shifted reference
plane wave
Spatial Phase Modulation: Off-Axis
Interferometry
• The goal is to isolate cos[Δ𝑘 𝑥 ′ + 𝜙(𝑥 ′ , 𝑦)] and
calculate the imaginary counterpart through a Hilbert
Transform
• Finally the argument is obtained uniquely as
• the frequency of modulation, Δk, sets an upper limit on
the highest spatial frequency resolvable in an image.
Phase Unwrapping
• Phase measurements
yields value within
(−𝜋, 𝜋] interval and
modulo(2𝜋) . In
other words, the
phase measurements
cannot distinguish
between 𝜙0 , and
𝜋
𝜙0 +
2
• Unwrapping
operation searches
for 2𝜋 jumps in the
signal and corrects
them by adding 2𝜋
back to the signal.
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.
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 resolution
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 deviation
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 signal
Figures of Merit in QPI
• Temporal Stability: Temporal-Phase
Sensitivity
Reducing Noise Level:
– Function Φ 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.
Figures of Merit in QPI
• Temporal Stability: Temporal-Phase Sensitivity
– Passive stabilization
– Active stabilization
– Differential measurements
– Common path interferometry
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.
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
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
Summary of QPI Approaches and
Figures of Merit
• There is no technique that performs optimally
with respect to all figures of merit identified
Summary of QPI Approaches and
Figures of Merit
2
are 𝐶4
• Thus, there
= 6 possible combinations
of two methods, as follows
Summary of QPI Approaches and
Figures of Merit
3
are 𝐶4
• Thus, there
= 4 possible combinations
of three methods, as follows