20.309: Exam 1 Topics

Measurement error

• Physical measurements include observational error.
• A simple mathematical model for observational error is: $ M = Q + E $, where $ Q $ is the true value of the physical quantity, $ M $ is the measured value, and $ E $ is the error.
• Useful probability distributions for modeling random measurement errors include the Gaussian distribution and the Poisson distribution.
  • The Gaussian distribution has two parameters: the mean value $ \mu $, and the standard deviation $ \sigma $.
    • About two thirds of the time, the value of a Gaussian random variable falls in the interval $ \mu\plusmi\sigma $. About 95% of the time, it falls in the interval $ \mu\plusmi\sigma $

o Gaussian distribution o Central limit theorem o Variances add o Poisson distribution o Types of errors o Effect of averaging • Models of light o Particle (ray tracing) o Wave (diffraction and interference; resolution) • Spherical and plane wave solutions to Maxwell’s equations o Quantum theory (photoelectric effect; photovoltaic effect; fluorescence) • Law of reflection • Law of refraction o n = speed of light in material/speed in vacuum • Refraction at a spherical interface o Paraxial approximation o Thin lens approximation o Lens maker’s formula o Thin lens equation o Lens underwater • Ray tracing o What is a ray? o Rules o Locating image plane, computing magnification • Good Questions about optical systems o Where is the image? o What type of image? o What is the magnification? o How much light does the system gather? o What is the field of view? o What is the resolution? • Optical aberrations o Spherical o Chromatic o Coma o Field curvature o Lens types o Minimizing aberration • Microscope design o 4f microscope o Magnification o Objective lenses • Working distance • Effective focal length • Infinity correction • Oil/water immersion o Tube lens o Epifluorescence • Fluorescence/Jablonski diagram • Excitation filter • Dichroic mirror • Barrier filters • Beam expander o Lab 1 microscope design • Optical resolution o Airy disk function o Rayleigh resolution o Equivalent Gaussian • Noise in images o Shot noise o Dark current noise o Readout noise o Pixel size • Image processing o Image representation o Segmentation/threshold o Centroid finding o Estimating resolution by nonlinear regression