Polarizer of entry of matrix _{ } delivering at exit a vecteuR _{ } Birefringent blade directed with 45° of the polarizer, centers slow on the first bisectrix of the starting reference mark giving a dephasing of _{ } .
with DT the temporal delay of the slow vibration on the fast vibration at the time of the crossing of the thinnest blade of the filter thickness D and the central wavelength L of the filter is the speed of the light in the vacuum, finally is the ordinary index (fast axis) of the crystalline blade and is the extraordinary index (slow axis) and and thus one thus has for the first stages the matrix of Jones, _{ } _{}One reconnait in the member of left of the equality above, successively of righthand side on the left, the matrix of the rotation of angle then the matrix of the blade, and finally the matrix of the rotation of angle . These two matrices of rotation make it possible to work in the reference mark consisted the two axes of the crystalline blade, the fast axis being the xaxis and the slow axis the yaxis. In these axes the matrix of Jones of the blade has the simple form mentioned above. The member of righthand side is thus the matrix of Jones of the blade of the first stage of the filter of Lyot in the reference mark whose axes have the direction of the polarizer of exit for xaxis and the direction of the polarizer of entry for yaxis (it is pointed out that these two polarizers are cross). The vector of Jones at exit east thus and after the polarizer of exit of matrix the vector of Jones of the vibration of exit east _{ } of complex amplitude _{ } The intensity at exit of the first stage is equal to the square of the module of the vector of exit which we have just obtained, that is to say _{ } 
_{}One reconnait on the figure the two polarizers of entry and exit. The blade has its axes with 45° common direction of the two polarizers. The slow axis is directed by OY and the fast axis by OX. The rectilinear vibration of entry delivered by the first polarizer breaks up into two rectilinear vibrations oH on the fast axis and oK on the slow axis of the blade. At the exit of the blade, the vibration oK is delayed compared to oH proportionally with the length oPde the blade. It results from it a vibration in general elliptic which can become circular left if the delay is equal to a quarter of wave, rectilinear orthogonal with the vibration of entry if the delay is equal to a halfwave, circular right for a delay of 3/4 of wave and becomes again rectilinear identical to the vibration of entry for a delay of a whole wave (the crossing of the blade is then without effect on nature of the vibration). 

Action on the figure: Dice which the figure is active (click on the left) one can activate time (double click to on the left open the dropdown menu then to control then active time). One can regulate with the mouse the thickness of the blade by trainant the point P. Rechercher the thickness of the blade which returns this one quarter of wave, halfwave or wave. The "value" of the blade in "waves is posted". One fear also of drawing aside the polarizers for better seeing the blade (to however maintain them outside the blade!). 
At the exit of the first two stages, the second blade being twice thicker than the first, At the exit of N stages, each blade being twice thicker than the preceding one _{ } _{}the amplitude complexes at exit of the nth stage is written the intensity at exit of the nth stage is written Let us represent the variation of this intensity on the outlet side of the filter according to the phase (in degrees) according to the number of stages of the filter. 
Transmission after N stages of a filter of Lyot on 6 floors. The red curve thus represents the transmission of the complete filter. 
Action on the figure: Dice which the figure is active (click on the left) one can activate time (double click to on the left open the dropdown menu then to control then active time). One can vary the point of view with the mouse by moving it under click maintained right. One can vary the thicknesses of the blades to the keyboard with the keys haut/bas.On can move the point O and thus all the filter by Ctl and click maintained right. 
Action on the figure: Dice which the figure is active (click on the left) one can vary the direction of the incidental vibration to the mouse by moving the point V One can vary number N of blades of the filter of Lyot between 1 and 10 by the keys haut/bas. The graph represents the spectral transmission of the filter. 