Difference between revisions of "Physical optics and resolution"

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The performance of an imaging system is limited by both fundamental and technical constraints.
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In the seventeenth century, Antonie van Leeuwenhoek and Robert Hook made dramatic improvements in microscopes. Their work brought about many fundamental discoveries. Both men documented the existence of cells, one of the most significant observations in the history of science. But Hooke and van Leeuwenhoek didn't get much further than that. Why didn't they discover organelles, proteins, or atoms?
As we reviewed the basic principles of [[Optical Microscopy Part 1b: Geometrical Optics and Ray Tracing|geometrical optics and ray tracing]], treating light as a particle, we learned how aberrations (inherent to the polychromatic spectrum of light, to the nominal curvature of lenses, or introduced by human imperfection) could deform results. In this section, adding the descriptive framework of light as a wave, we'll study other factors that contribute to measurement uncertainty.
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Limits of detection in microscopy can be understood as the compound of:
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Two things are required to visualize something with a microscope: resolution and contrast. Resolution is a measure of the finest detail that can be seen through an optical system. Alone, resolution is insufficient to make something visible. There must also be contrast — some kind of difference between the thing you want to see and whatever else makes up the rest of the sample.
* aberrations
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* resolution
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* contrast
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* detector construction
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* noise.
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Hooke and van Leeuwenhoek couldn't see things much smaller than cells because their microscopes had very bad [[http://measure.mit.edu/~20.309/wiki/index.php?title=Optical_Microscopy_Part_1b:_Geometrical_Optics_and_Ray_Tracing|optical aberrations]]. Aberrations are technical shortcomings that distort images made by optical systems. Over the intervening three and a half centuries, engineers developed clever designs that reduce optical aberrations to arbitrarily small levels (in some cases, at arbitrarily large price points). Even assuming that all the aberrations had been eliminated from their microscopes, though, the early microscopists still would not have glimpsed a protein molecule or an atom. In the ray tracing model, rays passing through a lens come together at a point in the image plane. In actuality, light passing through an aperture undergoes a modification called '''diffraction''' that spreads the point predicted by the ray model into a disk of light. This spreading causes points of light to intrude on their neighbors, blurring the image.
  
 
==Diffraction and resolution==
 
==Diffraction and resolution==

Revision as of 04:24, 24 August 2013

20.309: Biological Instrumentation and Measurement

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In the seventeenth century, Antonie van Leeuwenhoek and Robert Hook made dramatic improvements in microscopes. Their work brought about many fundamental discoveries. Both men documented the existence of cells, one of the most significant observations in the history of science. But Hooke and van Leeuwenhoek didn't get much further than that. Why didn't they discover organelles, proteins, or atoms?

Two things are required to visualize something with a microscope: resolution and contrast. Resolution is a measure of the finest detail that can be seen through an optical system. Alone, resolution is insufficient to make something visible. There must also be contrast — some kind of difference between the thing you want to see and whatever else makes up the rest of the sample.

Hooke and van Leeuwenhoek couldn't see things much smaller than cells because their microscopes had very bad [aberrations]. Aberrations are technical shortcomings that distort images made by optical systems. Over the intervening three and a half centuries, engineers developed clever designs that reduce optical aberrations to arbitrarily small levels (in some cases, at arbitrarily large price points). Even assuming that all the aberrations had been eliminated from their microscopes, though, the early microscopists still would not have glimpsed a protein molecule or an atom. In the ray tracing model, rays passing through a lens come together at a point in the image plane. In actuality, light passing through an aperture undergoes a modification called diffraction that spreads the point predicted by the ray model into a disk of light. This spreading causes points of light to intrude on their neighbors, blurring the image.

Diffraction and resolution

Note: Figures in this section are from spie.org [1]. To ray optics, or geometrical optics, that provided intuition and equations to account for reflection and refraction and for imaging with mirrors and lenses, we can add the concepts of wave optics, also dubbed physical optics, and thereby grasp phenomena including interferences, diffraction, and polarization.

Maxwell's equations

Profiles of the transverse electric and magnetic fields of a light wave at an instant of time.
  • The set of partial differential equations unified under the term 'Maxwell's equations' describes how electric $ \vec E $ and magnetic $ \vec {B} $ fields are generated and altered by each other and by charges and currents.
$ \nabla \cdot \vec E = {\rho \over \varepsilon_0} $
$ \nabla \cdot \vec B = 0 $
$ \nabla \times \vec E = - {\partial \vec B \over \partial t} $
$ \nabla \times \vec B = \mu_0 \left ( \vec J + \varepsilon_0 {\partial \vec E \over \partial t} \right ) $
where ρ and $ \vec J $ are the charge density and current density of a region of space, and the universal constants $ \varepsilon_0 $ and $ \mu_0 $ are the permittivity and permeability of free space. The nabla symbol $ \nabla $ denotes the three-dimensional gradient operator, $ \nabla \cdot $ the divergence operator, and $ \nabla \times $ the curl operator.
  • In vacuum where there are no charges (ρ = 0) and no currents ($ \vec J = \vec 0 $), Maxwell's equations reduce to
$ \nabla^2 \vec E = {1 \over c^2}{\partial^2 \vec E \over \partial t^2} $
$ \nabla^2 \vec B = {1 \over c^2}{\partial^2 \vec B \over \partial t^2} $
  • At large distances from the source, a spherical wave may be approximated by a plane wave, of direction of propagation $ \vec E\ \times\ \vec B $. In space and time, the electric and magnetic fields vary sinusoidally:
$ \vec E (\vec r, t) = \vec E_0 \cos ({\vec k \cdot \vec r} - \omega t + \phi_0) $
$ \vec B (\vec r, t) = \vec B_0 \cos ({\vec k \cdot \vec r} - \omega t + \phi_0) $
where $ t $ is time (in seconds), $ \omega $ is the angular frequency (in radians per second), $ \vec k = (k_x,\ k_y,\ k_z) $ is the wave vector (in radians per meter), and $ \phi_0 $ is the phase angle (in radians). The wave vector is related to the angular frequency by $ k = \left\vert \vec k \right\vert = { \omega \over c } = { 2 \pi \over \lambda } $, where k is the wavenumber and λ is the wavelength.

Interferences

Key results from the theory of interactions of wave lights are:

Principle of linear superposition

The (vector) electric and magnetic fields from each source of an electromagnetic wave add.

20.309 130823 LinearSuperpositionWaves.png
  • In-phase waves reinforce each other.
  • Out of phase waves cancel.
  • Different path length causes relative phase shift.

Huygens-Fresnel principle

As a wavefront propagates, each point on the wavefront acts as a point source of secondary spherical light waves.

Fraunhofer and Fresnel diffractions

The Huygens-Fresnel principle can be used to solve diffraction of a plane wave as it passes through a slit by putting many sources along the wavefront. Fraunhofer diffraction refers to the pattern when observed far from the slit (far-field diffraction) or through a lens, while Fresnel diffraction refers to the near-field counterpart.

Diffraction pattern through a single slit
20.309 130823 DiffractionSlit.png 20.309 130823 SlitDiffraction.png

Resolution

20.309 130823 PinholeDiffraction.png
20.309 130823 SurfacePlotAiry.png

Airy discs

  • Diffraction through a circular pinhole creates Airy discs, whose intensity on the imaging screen follows the equation
$ I ( \theta ) = I_0 \left ( {2 J_1 {(ka \sin \theta)} \over ka \sin \theta}\right ) ^2 $
where J1 is the Bessel function of first kind, k = 2 π / λ, and θ is the angle from the direction of light propagation.
  • The half-angle beam spread to first minimum, θ, occurs at
$ \sin \theta = 1.22 {\lambda \over D} $
with λ the wavelength of the light and D the diameter of the round aperture (pinhole).

Rayleigh resolution

  • Rayleigh criterion (a somewhat arbitrary definition of resolution!) claims that two points are resolved if the maximum of one Airy disk lies at the first zero of the other.
Diffraction patterns for various angular separations of the two point sources. (a) The sources are far apart, and the patterns are well resolved. (b) The sources are at an angular separation just satisfying the Rayleigh criterion, and the patterns are just resolved. (c) The sources are so close together that the patterns are not resolved. [2]
  • For small angles,
$ R \approx 1.22 {\lambda f \over D} \approx 0.61 {\lambda \over NA} $
where $ R $ is the separation of the images of the two point objects on the film, $ f $ is the distance from the lens to the film, and NA = n sin α is the numerical aperture of the imaging lens.
  • It ensues that it is very difficult to achieve a resolution R less than ~ 160 nm with a light microscope (even with a big lens of small focal length f and sin α ~ 1, even with oil immersion granting n ~ 1.5, and blue visible light of &lambda ~ 400 nm)!
  • Be aware that since real-life lenses are of finite dimensions, diffraction through a circular aperture is unavoidable and does take place in your microscope. Diffraction through a circular aperture can be simulated by convolution of the original image with a simulated Airy disk.


Enhancing image contrast

Optical microscopy, involving visible light either transmitted through or reflected by the sample, is fettered by three categories of limitations:

  • Diffraction limits resolution to ~ 0.2 μm (see Diffraction above).
  • Only strongly absorbing (dark) or strongly refracting objects can be successfully imaged.
  • Background signal from points outside of the focal plane undermine image contrast.

To overcome these limitations, several specific microscopy techniques have been developed. Good overviews can be found at zeiss.com [3] and fsu.edu [4] (where the figures in this section were taken from) if you're interested in learning more.

20.309 130823 BrightField.png
  • Bright-field microscopy, which relies of transmitted white light illumination of the sample, can benefit from color dying of specimens, and its images' contrast can increase if the degree of absorption depends on the light wavelength and varies from point to point within the specimen.
20.309 130823 DarkField.png
  • Dark-field microscopy capitalizes on oblique illumination: an opaque stop blocks direct light toward the sample, and only faint rays diffracted, reflected, and/or refracted by optical discontinuities in the sample enter the objective lens.
  • Rheinberg illumination also derive from oblique ray visualization.
20.309 130823 PhaseContrastColor.png
  • Phase contrast microscopy takes advantage from the phase shift that arises when in-phase incident light interacts with specimen regions of varying indices of refraction. A specialized condenser containing one or several annuli, matched to phase rings in the rear plane of the objective lens amplify the phase contrast generated by distinct optical path lengths.
20.309 130823 DIC.png
  • Differential interference contrast (DIC) microscopy relies on beam sharing to create interferences between the reference beam and a minuscule sheared amount of its light, and uses the detection of gradients in the optical phase to create sample contrast.
  • Hoffman modulation contrast is also based on optical phase gradient detection.
20.309 130823 Polarization.png
  • Polarization microscopes are equipped with a polarizer before the sample and an analyzer (i.e. a second polarizer of vibration azimuth positioned at right angle to that of the first polarizer) after the sample. The two individual wave components, polarized in perpendicular planes, become out of phase if they pass through a bi-refringent specimen, and contrast arises as they are recombined with constructive and destructive interferences.
20.309 130823 Fluo.png
  • Fluorescence microscopy uses light (absorbing and) emitting labels to identify and distinguish molecular and cellular structures of interest. More on the topic below!

Fluorescence microscopy

Noise sources in optical detectors

  • The maximum information you can extract from an image is limited by its signal-to-noise ratio (SNR).
  • Examining the sources of noise in your optical detector, and understanding which contributions have the greatest impact on the uncertainty of your measurement, will be central to optimizing the quality of your results and conclusions.
  • Let's first look at the camera used in 20.309. It is subjected to
    • Optical shot noise (Ns): inherent noise in counting a finite number of photons per unit time
    • Dark current noise (Nd): thermally induced “firing” of the detector
    • Johnson noise (NJ): thermally induced current fluctuation in the load resistor
    • Technical noise due to various imperfections.
  • Uncorrelated sources of noise N
$ \lim_{T \to {+ \infty}} {{1 \over T} \int_{-T/2}^{T/2} (n(t+\tau) - \bar{n})(n(t) - \bar{n})^*\, d\tau} = \left \langle \Delta {n(t+\tau)} \Delta {n^* (t)} \right \rangle = 0 $
(with $ \tau \ne 0 $ and * denoting the complex conjugate)
will add in quadrature:
$ N^2 \propto N_s^2 + N_d^2 + N_J^2 $


References

  1. http://spie.org/Documents/Publications/00%20STEP%20Module%2004.pdf
  2. www.kshitij-school.com
  3. http://zeiss-campus.magnet.fsu.edu/articles/basics/contrast.html
  4. http://micro.magnet.fsu.edu/primer/techniques/index.html