Thick model, for two different homogeneous fluids

Most vision applications deal with a camera immersed in an homogeneous fluid, namely air. Under such an hypothesis some simplifications arise and it can be shown [LRD93] that nodal points and principal points coincide. The use of a pin-hole model consists in merging the two principal planes in order to only retain rays through the equivalent optical center.

Paraxial formulas for a lens located between two distinct homogeneous fluids are found in most handbooks of geometrical optics [Per88]. We recall the expressions that will be used hereafter.

These formulas extend the properties of the lenses to the arbitrary refractive index of the object ($n_1$) and of the image ($n_2$) media, also involving mechanical specification of the lenses in which glass has an index equals to $n$:

For opticians, refractive index is the ratio of the speed of light in air and the speed of light in the considered medium. When the situation involves two different extremal indices, the focal length $f$ has two distinct values $f_o$ for the object medium and $f_i$ for the image medium. Moreover nodal and principal points are now distinct.

Figure 2: Optical ways in fluids of different indices

The following relations hold between the different optical variables (C.f. Figure 2.2).

1. Lens constant

   $$k=\frac{n-n_1}{r_1} +\frac{n_2-n}{r_2} - \frac{tc(n-n_1)(n_2-n)}{n.r_1.r_2}$$ (1)

   
   

2. Focal lengths:

   $$f_o=\frac{n_1}{k}, f_i=\frac{n_2}{k}$$ (2)

   
   

3. Gauss relation

   $$\frac{n_1}{p_1} + \frac{n_2}{p_2} = k$$ (3)

   
   

4. Principal points locations

   $$EH_o= \frac{n_1.tc}{k} \frac{(n_2-n)}{n.r_2}$$ (4)

   
   

   $$SH_i= \frac{-n_2.tc}{k} \frac{(n-n_1)}{n.r_1}$$ (5)

   
   

5. Nodal points locations

   $$EN_o=EH_o + H_oN_o$$ (6)

   
   

   $$SN_i=SH_i + H_iN_i$$ (7)

   
   

   with

   $$H_oN_o=H_iN_i=\frac{(n2-n1)}{k}$$ (8)