Wavelength and precision of observation

[annie122]

I don't understand why when you make an observation, you need a wavelength shorter than the wanted precision.
i.e. you can't make clear pictures of golf balls with radio waves, you can't observe things smaller than a photon with an optical microscope.

 

[Claude Bile]

It is due to the diffraction limit, first formulated by Abbe. The essence of the diffraction limit is as follows;

- The resolution of an optical field is given by the range of spatial frequencies (or wave-vectors) present. Sharper details require more spatial frequencies to be present, i.e. typical Fourier transform principles apply.

- A focusing system can only transmit a limited range of spatial frequencies. For a monochromatic wave (fixed wave-vector magnitude), the size of the transverse wave-vector components depends on the angle made with the optic axis. Hence systems with a higher numerical aperture (or acceptance angle) are capable of higher resolution in the transverse (image) plane because they can transmit a higher range of spatial frequencies (parallel to that plane).

- Shorter wavelengths are capable of higher resolution because the overall magnitude of the wave-vector is larger (k = 2*pi/wavelength).

- Even free-space limits resolution. To obtain a transverse frequency component greater than the magnitude of the wave-vector, one of the components must become imaginary. This results in wave components that are evanescent. Evanescent waves are bound the surface (or source) that generates them and do not propagate through free-space. The decay length of these waves is on the scale of one wavelength or less.

- The diffraction limit can therefore be circumvented if the evanescent waves can somehow be detected. This is the basic principle behind most near-field imaging and super-resolution imaging methods.

- For most definitions of resolution, the diffraction limit is around half the wavelength, i.e. for a wavelength of 590 nm, the smallest detail that can be transmitted to the far-field (i.e. beyond a few wavelengths) is around 295 nm.

Claude.

 

[johnbbahm]

X2 on all Claude said, It helps me to think about it in terms of waves and phase.
When you only have one detector, you can tell when a reflection occurred only within one wavelength. The only way to get better is to bring in some phase info, which cannot be done with a single detector. Think of the wavelengths as integer measuring sticks, you can only measure something in whole stick units, The smaller the stick unit, the more accurate the measurement.