Summary of Cryo-Electron Microscopy for Structural Biology

## Introduction Microscope optics explains how lenses and imaging systems let us see objects too small for the naked eye by forming enlarged images and how physical limits and digital sampling affect what we can observe. This material breaks down image formation, resolution limits, and digitization into clear steps with examples and practical notes. ## Basic image formation by a lens ### Ray diagram and key distances A thin lens forms an image of an object according to the lens formula. Important distances are: - $u$: distance from the object to the lens - $v$: distance from the image to the lens - $f$: focal length of the lens > Definition: The focal length $f$ is the distance from the lens at which parallel rays converge to a point (for a converging lens) or appear to diverge from (for a diverging lens). The thin lens equation relates these distances: $$\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$$ Magnification $M$ is given by the ratio of image size to object size and also by distances: $$M = \frac{\text{image height}}{\text{object height}} = -\frac{v}{u}$$ Negative $M$ indicates an inverted image. ### Practical example - If an object is at $u = 30\,\text{cm}$ and the lens focal length is $f = 10\,\text{cm}$, then $$\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{10} - \frac{1}{30} = \frac{2}{30}$$ so $$v = 15\,\text{cm}$$ and magnification is $$M = -\frac{v}{u} = -\frac{15}{30} = -0.5\,.$$ This gives an inverted image half the size of the object at $15\,\text{cm}$ from the lens. ## Optical resolution and diffraction limits ### What is resolution? > Definition: Resolution is the smallest distance between two points on an object that can be distinguished as separate in an image. Even with perfect lenses, diffraction by the aperture limits the resolution. A point source produces an Airy pattern rather than a point. The central bright spot of that pattern is the Airy disk, which contains about 84% of the light energy. > Definition: The Airy disk is the central maximum of the diffraction pattern produced by a circular aperture; it is enclosed by the first minimum. ### Rayleigh criterion Two point sources are just resolvable if the intensity maximum of one Airy disk falls on the first minimum of the other. The Rayleigh criterion gives the minimum resolvable distance $r$ in the object plane: $$r = \frac{d}{2} = \frac{0.61\lambda}{\mu \sin\alpha}$$ where: - $\lambda$ is the wavelength of light used - $\mu$ is the refractive index of the medium between the object and the objective - $\alpha$ is the semi-angle of the light cone collected by the objective (see figure) The quantity $\text{NA} = \mu\sin\alpha$ is called the numerical aperture, so the formula is often written as $$r = \frac{0.61\lambda}{\text{NA}}\,.$$ ### Table: Factors affecting resolution | Factor | Effect on resolution | Practical implication | |---|---:|---| | Wavelength $\lambda$ | Larger $\lambda$ increases $r$ (worse resolution) | Use shorter wavelengths (e.g., blue light) to improve resolution | | Numerical aperture $\text{NA}$ | Larger NA decreases $r$ (better resolution) | Use higher-NA objectives and immersion media | | Refractive index $\mu$ | Higher $\mu$ decreases $r$ (better resolution) | Use immersion oil ($\mu\approx1.52$) between slide and objective | ### Practical application - In light microscopy, using blue light ($\lambda \approx 450\,\text{nm}$) and an oil-immersion lens with NA $\approx 1.4$ gives: $$r \approx \frac{0.61\times 450\times10^{-9}}{1.4} \approx 196\,\text{nm}\,.$$ - This means two points closer than about $200\,\text{nm}$ cannot be resolved by conventional optical microscopy under these conditions. Fun fact: The Rayleigh criterion is a convention based on the shape of the diffraction pattern and provides a useful practical limit, but modern techniques like super-resolution microscopy can bypass this limit by exploiting fluorescence and computational methods. ## Image formation on detectors and