Difference between revisions of "20.309: Exam 1 Topics"

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Measurement error

• Physical measurements include observational error (also called measurement 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.
  • The total error $ E $ in a measurement is equal to the sum of contributions from each error source $ E=\sum{\epsilon_i} $.
• Errors sources can cause random or systematic errors (or both).
  • The magnitude of a random error decreases when identical measurements are averaged; the magnitude of a systematic error does not.
• The Gaussian and Poisson distributions are useful models for many kinds of random measurement errors.
  • The Gaussian distribution has two parameters: a mean value $ \mu $, and a 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\plusmi2\sigma $. 98% of the time, the value is between $ \mu\plusmi3\sigma $
  • The Poisson distribution has one parameter, it's mean value $ \mu $. The variance of a Poisson distributed random variable is equal to its mean.
• When you add two random variables, their variances add.
  • Thus, if you average '$ N $ identically distributed random values with standard deviation $ \sigma $, the standard deviation of the average is $ \frac{\sigma}{\sqrt{n}} ==Models of light== * Three useful models of light are: particle, wave, and quantum. ** The particle model is the most intuitive, but it neglects important behaviors of light such as diffraction and interference. The particle model doesn't do a good job of explaining resolution, for example. ** The wave model is less intuitive than the particle model, but it is more accurate. The wave model does a good job of explaining resolution, but it falls apart when you try to explain fluorescence or noise in images. ** The quantum model is incredibly accurate, but it is just about impossible to do in your head. (The quantum model is so nuts that some people think it implies that there are multiple universes.) The quantum model predicts all the behaviors of light. * A useful way to model a light field is as a set of rays that point in the direction of propagation. ==Reflection and refraction== * Transparent materials have a property called the ''index of refraction'', <math>n $. $ n $ is the ratio of the speed of light in a vacuum to the speed of light propagating in the material.
• Snell's law says that light bends when it encounters an interface between dissimilar indices of refraction according to the equation $ n_1 \sin(\theta_1) = n_2 \sin(\theta_2). This formula is on the cheat sheet. ** Angles are measured from the normal. ==Systems of lenses and mirrors== * ''Ray tracing'' (also called''geometrical optics'', ''geometric optics'', or ''Gaussian optics'') is a simple set or rules that allows you to determine the location of images in a system of lenses and mirrors. ** Ray tracing relies on two assumptions: that the lenses are thin and angles are small. * [http://www.colorado.edu/physics/phys1230/phys1230_fa01/topic23.html Here is a page that summarizes the ray tracing rules] * The image is located where two rays that begin at the same point cross. ** The image can be ''real'' or ''virtual''; ''upright'' or ''inverted''; and ''magnified'', ''shrunk'', or ''the same size''. ** Magnification is equal to the height of the image divided by the height of the object. ** Use three adjectives to describe an image. * You can also use the thin lens equation to determine the location of an image: <math>\frac{1}{f}=\frac{1}{S_o}+\frac{1}{S_i} $
• $ f $ is the focal length of a lens. It depends on the shape of the lens, the refractive index of the material the lens is made out of, and the refractive index of whatever medium the lens is in (frequently air). $ \frac{1}{f}=\frac{n_{lens}-n_{medium}}{n_{medium}}{\frac{1}{R_1}+\frac{1}{R_2} $. Note the sign convention on the lens radius.
• if you put a lens under water (or oil, or whatever with $ n>1 $, its focal length gets longer'. The $ \frac{n_{lens}-n_{medium}}{n_{medium}} $ term gets smaller.
• Field of view is the size of the largest object that could be imaged with an optical system.
• Deviations from the ideal lens assumptions result in optical aberrations that reduce the performance of optical systems.
  • Various technical solutions exist to reduce optical aberrations. Optical systems with very low aberrations usually cost lots of money.
  • One simple thing you can do to minimize aberrations is to put cylindrical lenses in the correct orientation.

Microscope

• You should understand every component in your microscope and why it is located where it is.
• Objective lenses can be modeled as a simple lens with a focal length $ f=\frac{200}{m} $

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