Quantitative Phase Imaging of Cells and Tissues

Introduction and groundwork 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 (a) QPI of a neuron (colorbar shows pathlength in nanometer). (b) Synthetic DIC image obtained by taking a 1D derivative of a. (c) Image obtained by taking the modulus squared of the gradient associated with a. (d) Image obtained by taking the Laplacian of a.

Quantitative Phase Imaging (QPI) QPI and light-scattering data with extreme sensitivity. Fourier transform light scattering. (a) QPI of dendrites; (b) scattering map from dendrites in a.

Two confusions Three-Dimensional Imaging Nanoscale sensitivity

Light Propagation in Free Space Helmholtz Equation 1-D Propagation Plane waves One-dimensional propagation of the field from a planar source s(x): (a) source; (b) real part of the field propagating along x; k = k0xˆ is the wave vector associate with the plane wave.

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 𝑎 0 and 𝑏 0 are functions of 𝑘= 𝑘 0 𝑎, 𝛽= 𝑘 0 𝑛𝑎/ 𝑛 0 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, 𝐴 𝑠 is constant and so is 𝐴 𝑖 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 𝑈 0 and the fluctuating components, 𝑈 1 𝑥,𝑦 Note that the average 𝑈 0 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. If 𝜙= 𝜋 2 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 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.

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, 𝑈 𝑟 , 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 𝑈 1 , 𝑈 2 , that are time reverse of each other and incident on the 𝜒 (3) . An object field U 3 is applied simultaneously, inducing the nonlinear polarization Clearly, the field emerging from the material, U4, is the time-reversed 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

Principles of Full-Field QPI Chap. 8 Principles of Full-Field QPI

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. | 𝑈 𝑥,𝑦,𝑡 | 2 Mixing with a reference field 𝑈 𝑟

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: 𝐼 1 , 𝐼 2 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 over half circle, i.e. (− 𝜋 2 , 𝜋 2 ) and 0, 𝜋 respectively Four measurements with phase shift in 𝜋 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 𝜙 0 , and 𝜙 0 + 𝜋 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 and Temporal power spectrum 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 Thus, there are 𝐶 4 2 =6 possible combinations of two methods, as follows

Summary of QPI Approaches and Figures of Merit Thus, there are 𝐶 4 3 =4 possible combinations of three methods, as follows