It is also important to understand that a waveplate is designed at a specific thickness for a defined range of wavelengh at a specific angular position.

Before giving an answer to that question it is important to understand that light signal, which interact like a wave, can be resolved in to orthogonal polarisations usually called p and s.  To simplify with an image in a classic x,y,z orthogonal projection, a light wave going into a z direction will oscillate along the z axis in x and y directions.

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Quarter waveplates will be used to delay polarisation of π/2. Quarter waveplate can change linearly polarized light to circular polarization at its output.

The real retardation of one polarization to an other is serval times the wavelength plus the expected nominal retardation. It can be seen as the position a car wheel it would be the same if it turns only 45° or 10 times on itself plus 45°.

Manufacturers of waveplates usually also provide other precision optics components but not all precision optics supplier can provide waveplate. Very precise calculation and positioning along the optical axis of the crystals should be done. It is therefore important to work only with suppliers experiences in waveplates.

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Waveplates will modify the relative phase of p & s polarisations. Practically it will delay a polarisation from an other, hence waveplates are also called retarders.

Multi-order wavepates are thicker than true zero-orders ones, so more mechanically resistant and with higher LIDT.  But they are less precise especially when there are temperature variations as the birefringence of the materials change with the temperature.

I am solving an optics exercise that requires to design a Keplerian telescope with a given magnification M (negative), a certain object size L and the restriction of not having lenses faster than F/1 (wavelength of the signal and resolution are also given, but they are used for solve successive questions). In particular, the exercise requires to retrieve the NA of the objective lens from the optical invariant of the system.

They can be manufactured either with very thin material film (usually sandwiched between two optically neutral windows for more mechanical stability), or several multi-orders waveplates positions in different angles in order to eliminate the retardation shifts (extra turns of the wheel).

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Waveplates manufacturing rely on the birefringence of some optical material. Birefringence being the characteristic that bears some materials having different refractive index in different orthogonal directions. We will then speak about a fast axis and a slow axis making the delay between polarisation while the light is going through the waveplate.

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Understanding light, how it is made of photons how it behaves like an electromagnetic wave has been a long and difficult journey. Optical waveplates are one of the tool that came from this research and that enable us to make the most of light signals and so to enhance our comprehension of the world around us. Below you’ll find complete explanation about optical waveplates :

A zero-order waveplate only retard the signal of the nominal value. (whitout extra turn of the wheel in our car wheel example).

Given these information, I would say that, using the fastest lenses possible, I would need a first lens with radius R1=L and a focal length of F1=2L (so that the F/1 requirement is satisfied), while the second lens should have a focal length of F2=|M|F1, a distance from the first of d=F1+F2 and a radius of R2=|M|L. However, I am not sure how to retrieve the Lagrange invariant (https://en.wikipedia.org/wiki/Lagrange_invariant) from this theoretical design: for once, since the system is afocal, the object should be at infinity, thus how can the marginal ray be defined? Moreover, in this case both the first and the second lens limit the marginal rays of the system, and being the object at infinity I am not sure how should it be tilted (shouldn't it be parallel to the optical axis? But then wouldn't it coincide with the chief ray?).

As the retardation is in the order of the wavelength, for both waveplates types it is important to have a high quality of the optical part with surface quality as high as λ/10.

I can't fully answer this question, but I did find this helpful diagram which answers part of it...(on p 28 of this PDF)