20.309: Exam 1 Topics

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 random errors 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\pm\sigma $. About 95% of the time, it falls in the interval $ \mu\pm2\sigma $. 98% of the time, the value is between $ \mu\pm3\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 random variables, their variances add. (Formula on cheat sheet.)
  • 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.

Interactions of light and matter

Reflection and refraction

• Light goes slower when it propagates inside transparent material than in a vacuum.
  • Transparent materials have a property called the index of refraction, $ 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.

Absorption

• Beer's law models the absorption of light: $ \log{\frac{I}{I_0}}=-\epsilon l c $. (Equation on cheat sheet.)

Systems of lenses and mirrors

• Ray tracing (also calledgeometrical 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: the lenses are thin and the angles are small.
• 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. For example, a magnifying glass produces an image that is upright, magnified, and virtual.
• You can also use the thin lens equation to determine the location of an image: $ \frac{1}{f}=\frac{1}{S_o}+\frac{1}{S_i} $ (This formula is on the cheat sheet.)
• $ f $ is the focal length of a lens.
  • $ f $ 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}}\left\{\frac{1}{R_1}+\frac{1}{R_2}\right\} $. Note the sign convention on the lens radius. (This formula is on the cheat sheet.)
• 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} $, called the effective focal length. (The 200 number is manufacturer specific for Nikon.)
• In the infinity corrected microscopes we built, the sample is placed at the front focal point of the objective.
  • The image created by the objective is at infinity.
  • A second lens called a tube lens is required to create an image on the detector.
• This kind of microscope is sometimes called a "4f" microscope because it is $ 2f_{objective}+2f_{tube lens} $ long. (I think it's kind of a dumb name.)
• A dichroic mirror directs the (green) epi illumination toward the objective.
  • The excitation tube lens focuses light at the back focus of the objected so that the sample is bathed in collimated light.
  • The beam expander ensures that the illumination covers the entire field of view.
• Our microscopes do not have an excitation filter because the laser has a narrow emission bandwidth.
• The barrier or emission filter removes that vast majority of excitation (green) light from the image.

Resolution

• Diffraction causes light propagating through a circular aperture to blur out into an Airy disk, which looks like a bright blob surrounded by rings.
  • The blurring limits your ability to resolve fine details in an image.
• The most common definition of resolution is the Rayleigh criterion, which states that one point can be just resolved from another if the second point lies on the first zero of the first point.
• Rayleigh resolution depends on the light gathering capacity of an optical system and is given by the formula $ R=0.61\frac{\lambda}{\text{NA}} $. (Formula on cheat sheet.)
• $ \text{NA}=n\sin(\theta) $, where $ \theta $ is the angle of the tippiest ray from the object (at the optical axis) that enters the makes it through the optical system. $ n $ is the refractive index between the object and the objective lens — 1 for air, about 1.33 for water, and about 1.51 for microscope oil. (Formula on cheat sheet.)

Fluorescence

• Fluorescence is a chain of energy level transitions of specific molecules called fluorophores.
• The Jablonski diagram illustrates the transitions. (There is a Jablonski diagram on the cheat sheet.)
• Fluorophores are excited by photons in a certain wavelength range and emit photons with a longer wavelength.
• Fluorophores may also return to the ground state without emitting a photon.
• The ratio of radiative decays to total excitations is called the quantum yield of a fluorophore.

Noise in images

• Since photon emission is stochastic, all images include shot noise (also called photon noise)
  • Shot noise follows a Poisson distribution, so even a perfect image has noise
  • An ideal image includes only shot noise and has a signal to noise ratio of $ \frac{N}{\sqrt{N}}=\sqrt{N} $
• CCD and CMOS cameras have other (technical) noise sources, principally dark current noise and read noise.
• Dark current results from thermal electrons falling into the rain buckets.
  • Dark current is stochastic, with an average value that is proportional to the exposure time.
  • The distribution of dark electrons is well modeled by the Poisson distribution.
  • Thus, dark current noise is proportion to the square root of the exposure time.
  • You can reduce the magnitude of dark current by cooling the detector.
• Read noise results from various noise sources that come up in the process of counting the electrons.

Good images

• The amount of information you can extract from an image is determined by the signal to noise ratio.
• Choose the right size pixels.
  • Pixels that are small receive fewer photons and thus are noise.
  • Pixels that are large have low noise but cannot resolve fine details.
  • The pixels size should be small enough to capture all of the information available from the optical system (as determined by its resolution) and no smaller
  • A good rule of thumb is to use a pixel size of $ \frac{MR}{2} $

Image processing

• You can estimate the location of an object in an image by computing its center of mass or centroid.
• Noise in images limits the accuracy of centroids.