Synthesis mirror-diopter-lens

To position in Optical geometrical the image provided by a mirror, a diopter, a lens or a dioptric centered system, there exist two methods strictly mathematically equivalent: algebraic method and geometrical method

  • homographic algebraic method

The position of the image results from the position of the object by formulas known as of conjugation: The French-speaking people use the formulas known as of Descartes and the english-speaking use the formulas known as of Newton which are much simpler formally and much easier of employment. These two formalisms are strictly equivalent and of only one formula usable in geometrical Optique is only mathematical particular case for a Miroir, a Dioptre and a lens or even any centered system characterized by a cylindrical symmetry having an axis of revolution or optical axis; the mathematicians use the homographic term of function which with the following exeptionnelle property: the composition of two homographic transformations is a homographic transformation .
  • the geometrical method is simply the transcription in a plan containing the optical axis of these transformations.

Note:: This is valid only in the approximation of Gauss also said approximation of the small angles

General information

The formula known as of conjugation

ƒ O / xo + ƒ I / xi = 1
It is the relation of Descartes which functions for the mirrors, diopters and lenses;
It is written after two lines of calculus:
ƒ I / xi = 1 - ƒ O / xo is xi = ƒ I /(1 - ƒ O / xo ) and finally:
xi = ƒ I xo /( X - ƒ O )
  • where x O is the algebraic position of the object or homographic transformation,
  • x I is the algebraic position of the image, the instrument being in x=0;
  • f O is the algebraic position of the hearth object
  • and f I is the algebraic position of the hearth image

or by cutting off ƒ I on each side

( xi - ƒ I ) ( X 0 - ƒ O ) = ƒ I ƒ O
C' is the relation known as of Newton.

growth

the growth
\ gamma= \ frac {y_i} {y_o} = \ frac {- f_o} {(x_o-f_o)}= - \ frac {(x_i-f_i)}{f_i}

to note that it is the only formula to retain finally, The others result some easily.

  • ƒ O = ƒ I for a mirror

  • ƒ O = - ƒ I for a lens
  • ƒ O = - ƒ I / N for a diopter

Note: one can apply the formula for the diopter in all the cases, by considering

  • N = -1 for a mirror, ƒ O = - ƒ I /-1;
  • N = 1 for a lens, ƒ O = - ƒ I /1;
C has as coordinates (ƒ I + ƒ O , 0).

The growth γ is worth: γ = yi / yo = - ƒ O /( xo - ƒ O ) = - ( xi - ƒ I )/ƒ I this makes it possible to also note that the place of the image yi is the line of equation: there = ( there o /ƒ I )·( xi - ƒ I ) who passes by the point (ƒ I , 0) and from slope - yo /ƒ I .

geometrical construction

Animation Ci below watch the way in which one can carry out the composition of two homographic transformations which is itself a homographic transformation making it possible to position the image produced by a Doublet
  • an incidental ray parallel with the axis is deviated out of I by the first lens and passes by the hearth image F' 1 of the first lens; it meets the second lens in J.
  • One then traces a ray parallel with IJ and passing by O2: it is not deviated and it passes obligatorily by F' 2s i.e. the secondary hearth image of the second lens located at the perpendicular of F' 2;
  • One then plots the straight line JF' 2s which is the place of the final image if one moves 'object B on the parallel incidental ray (in order to have a fixed object of size.
  • by tracing BO1 which is not deviated by the first lens one finds on the line passing by I and J, the image B' 1 provided by the first lens; this image which is used as object with the second lens has by the second lens an image B' 2 which is the final image.
  • by plotting the straight line passing by B' 1 and O2 one finds B' to 2 the final image with the intersection with the place of B' 2: this point is the image of B by the doublet
  • if one projects on the optical axis B in has, B' 1 in A' 1, B' 2 in A' 2 one can say that A' 2B' 2 is the image of AB by the doublet.

formulas of conjuguaison

Since the composition of two homographies is an homography, it is shown algebraically that the position of the image is obtained by the formula F. F' A' = constant and the constant can be calculated for two unspecified combined points; one takes usually the points for which the growth is worth +1 because they are easy to position and are called principal points.

See too

References

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