1. PRISMS – one of the e.g. of optical instrumentation
Prism plays many different roles (beamsplitter, polarizing device, interferometer) in Optics, but most applications make use of 2 main prism functions:
as a dispersive device – separates constituent frequency components in polychromatic light
to effect a change in orientation of an image or in direction of propagation of a beam
Dispersing Prism
(The variation of refractive index and light speed with wavelength is dispersion) A   D B n
i1
t1
i2
t2
(i1 – t1)
(t2 – i2) C
Apex angle E

2. emerging ray is deflected from original direction by angular deviation 
Refraction at surface 1; ray deflected by (i1  t1)
Refraction at surface 2; ray further deflected by (t2  i2)
Total deviation is
Polygon ABCD has two right angles, BCD must be (180  ); BCE =  = exterior angle to BCD, thus,
and
To write  in terms of i1 and  (assume prism index = n and it is immersed in air, na  1) 
from Snell’s law:  

3. From above,  increases with n
but
and  
From above,  increases with n
But we know that n = f(); thus  = f()
n() decreases when  increases
Thus, () decreases when  increases

4. For fixed n and ; variation of  with i1 shows a minimum
Deviation is less for red than for violet light
white
red
green
violet
For a monochromatic ( fixed) light source incident on a prism (n and  fixed), a minimum deviation m exists as incident angle i1 varies.
For fixed n and ; variation of  with i1 shows a minimum
To obtain the minimum, we differentiate equation for  with respect to i1 and equate to zero, i.e.
Indirect differentiation is easier:
Differentiate  Or

5. Applying Snell’s law at interfaces 1 and 2:
Taking the derivatives:
(1)
(1a)
(2)
(2a)
Take derivative of
( is fixed); or
Dividing equation (1a) by (2a): 

6. Using identity 
We have:
But from Snell’s law again in (1) and (2): ;
thus,
The value of i1 for which the above is satisfied is the one for which
. We know that n  1, it follows that
and therefore,

7. From Snell’s law at the 1st interface:
it means that the ray for which the deviation is a minimum traverses the prism symmetrically, i.e., parallel to the base
From A  m n
i1
t1
i2
t2
From 
From Snell’s law at the 1st interface:
this forms the basis of one of the most accurate techniques for determining the refractive index of a transparent material

8. Dispersion
The refractive index depends on the wavelength n
thus total deviation is wavelength dependent, various wavelength components of incident light can be separated on refraction through prism
Shorter wavelengths have larger refractive indices, thus smaller speeds in prism
The normal dispersion (excludes absorption) curve is given empirically by Augustin Cauchy (Cauchy’s formula):
where A, B, C, … are empirical constants fitted to the dispersion data of a particular material.
The dispersion defined as dn/d is approximated by:

9. Dispersion versus deviation: D 
Large deviation , small deviation ,
small dispersion D large dispersion D
Dispersion is characterized by using 3 wavelengths of light near middle and exremes of the VIS spectrum, called the Fraunhofer lines (observed in the solar spectrum)
(nm)
Characterisation n
Crown glass
Flint glass
486.1
F, blue
1.5286
1.7328
589.2
D, yellow
1.5230
1.7205
656.3
C, red
1.5205
1.7076
F & C dark lines due to absorption by hydrogen atoms, D dark line due to absorption by sodium atoms in sun’s outer atmosphere

10. Using the thin prism at minimum deviation for the D line, the ratio of angular spread of the F and C wavelengths to the deviation D of D wavelength, is
This ratio is also known as the dispersive power  = D/
Reciprocal of dispersive power is called the Abbe number (e.g.  = 1/65 in crown glass;  = 1/29 in flint glass)
Prism spectrometer:

11. CHROMATIC RESOLVING POWER
Chromatic resolving power of a prism is its ability to resolve two wavelengths which are very close.
Consider a monochromatic parallel beam incident on a prism, such that it fills the prism face.
Using Fermat’s principle, ray FTW is isochronous with ray GX as they begin and end on the same plane wavefronts, GF & XW respectively.
Their optical paths are equal, i.e.,
FT + TW = nb …(i)
(n = refractive index of the prism associated with wavelength )

12. A 2nd neighbouring wavelength ’ is now present, where ’   = ;
and ’ is associated with a different refractive index n’ = n  n
The emerging wavefronts for the two components are separated by a small angular difference 
Applying Fermat’s principle to the 2nd wavelength:
FT + TW’ = (n  n) b
FT + TW  s = (n  n) b …(ii)

13. Subtracting (ii) from (i),
Introducing the dispersion term,
The angular difference is:
… (iii)
where d = beam width
From Rayleigh’s criterion (to be discussed in diffraction topic later) which determines the limit of resolution, it gives the minimum separation of the two wavefronts, such that images formed are just barely resolvable as:
… (iv)
Equating (iii) & (iv):
… (v)

14. Or the minimum wavelength separation permitted so that images can be resolved is:
… (vi)
The resolving power of the prism(an alternate way of describing the resolution limit of the instrument) is defined as:
… (vii)

15. Quantitative example: Find the resolving power and minimum resolvable wavelength difference for a prism made from flint glass with base 5 cm.
Choosing mean wavelength  = 550 nm, the approximate average value of the dispersion is
Thus resolving power is:
The minimum resolvable wavelength difference in region ~ 550 nm is
1 Å

16. Reflecting prisms:
beam introduced in such a way that at least one internal reflection occurs – to either change direction of propagation or change orientation of image, or both
advantage of reflecting prisms over mirrors:
- prism’s reflecting faces easier to keep free of contamination
- total internal reflection capable of higher reflectivity
Dove prism
Right-angle prism

17. Penta prism; pentagonal cross section

18. Many optical instruments use prisms to reflect light (by total internal reflection). Cameras and binoculars are two common examples.
When light is reflected internally by a 45° prism as shown below, inversion takes place.

19. Two such prisms are used in "prismatic binoculars" in order to produce an image which is "the same way round" and "the same way up" as the object.
This system also has the advantage of including a longer optical path in a given length of instrument. This allows for greater magnification.