Difference between revisions of "Physical optics and resolution"

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|[[Image: 20.309 130823 PinholeDiffraction.png|frameless|200 px]]
 
|[[Image: 20.309 130823 PinholeDiffraction.png|frameless|200 px]]
 
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===Resolution===
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* Diffraction through a circular pinhole creates Airy discs, whose intensity on the imaging screen follows the equation
 +
:<math> I ( \theta ) = I_0 \left ( {2 J_1 {(ka \sin \theta)} \over ka \sin \theta}\right ) ^2 </math>
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:where ''J<sub>1</sub>'' is the Bessel function of first kind, ''k'' = 2 &pi; / &lambda;, and ''&theta;'' is the angle from the direction of light propagation.
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* The half-angle beam spread to first minimum, ''&theta;'', occurs at
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:<math> \sin \theta = 1.22 {\lambda \over D}</math>
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:with &lambda; the wavelength of the light and ''D'' the diameter of the round aperture (pinhole).
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* For small angles, <math>\sin \theta \approx \theta </math>, so the first dark coordinate is
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:<math>x = 1.22 {\lambda f \over D}</math>
  
  

Revision as of 15:34, 23 August 2013

20.309: Biological Instrumentation and Measurement

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The performance of an imaging system is limited by both fundamental and technical constraints. As we reviewed the basic principles of 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.

Limits of detection in microscopy can be understood as the compound of:

  • aberrations
  • resolution
  • contrast
  • detector construction
  • noise.


Diffraction

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 Pinhole diffraction
20.309 130823 DiffractionSlit.png 20.309 130823 PinholeDiffraction.png

Resolution

  • 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).
  • For small angles, $ \sin \theta \approx \theta $, so the first dark coordinate is
$ x = 1.22 {\lambda f \over D} $


References

  1. http://spie.org/Documents/Publications/00%20STEP%20Module%2004.pdf