COMPARISON OF OPTICAL CONTOURING METHODS 117 
COLLIMATED J LENS 
TRANPARENCY 
| FINAL IMAGE 
SPATIAL 
FILTER 
Fic. 2. Optical system for eliminating noise. 
t(x,y) = e eJ exp i Shin + h(x) tano) (4) 
n:1 
where], (a,) represents the Bessel function of 
order n. If the transparency is illuminated by 
a plane wave, the diffracted orders are rep- 
resented by the corresponding terms of the 
series in Equation 4. The order of diffraction 
is represented by n. If two collimated beams 
illuminate the transparency in such a way 
that the nth order and the -nth order beams 
interfere with each other, the resulting fringe 
pattern can be shown to represent elevation 
contours. The contour spacing is given by 
ied d 
h(x) 2n tan0 
When 0 = 45°, the expression for the contour 
spacing becomes d/2n instead of d as in the 
conventional case. Thus when n = 5 the sen- 
sitivity is increased by a factor of 10. 
This method of contour generation offers 
many advantages over conventional 
techniques. Since the first recording is done 
with low spatial frequencies, the imaging sys- 
tem can operate at large f-numbers, thus in- 
creasing the depth-of-field. While the resolu- 
tion is proportional to the f-number, the 
depth-of-field is proportional to the square of 
the f-number. There is no compromise on 
resolution of contour spacing since it can be 
increased independently at the processing 
stage. 
CONTOURING WITH HOLOGRAPH 
INTERFEROMETRY 
Elevation contour generation using holo- 
graphic interferometry has been accomplished 
by the superposition of two holographic re- 
cordings on a single photographic plate. The 
holographic recordings differ in phase in 
such a way that the phase difference corres- 
ponding to every point on the object is pro- 
portional to the distance of the object point 
from the hologram. The interference be- 
tween the two reconstructed images of the 
hologram produces a fringe pattern repre- 
senting the elevation contours of the object. 
The different holographic contouring 
methods arise because of the various options 
available to change the phase between expo- 
sures as a function of elevation. The three 
significant methods that have been proposed 
and demonstrated are the multiple wave- 
length method, the multiple source method, 
and the multiple index method. 
In the multiple wave-length method, the 
two exposures are made with two different 
wave lengths of light. The angle of the refer- 
ence beam in each case is adjusted so as to 
make the two spatial carrier frequencies the 
same. With a collimated reference beam 
there is only a longitudional magnification 
due to change in wave length. The longitudi- 
nal magnification at every point in turn is 
proportional to its distance from the holo- 
gram. Hence upon reconstruction with a single 
wave length reference beam, the images have 
phase differences proportional to the differ- 
ential longitudinal magnification. This 
phase difference gives rise to a contour fringe 
pattern. The contour interval is given by dh 
-X Ay|X - X. 
In the multiple source method, the holo- 
gram is recorded using a single wave length. 
Multiple object beams are used to project a 
set of fringe surfaces whose normals are 
roughly parallel to the direction of observa- 
tion. The contour interval is given by dh 
= M2 sin 6/2 where 0 is the angle between the 
two object beams. 
In the multiple index method, the index- 
of-refraction of the medium surrounding the 
object is changed to provide the phase differ- 
ence between exposures. This may be ac- 
complished by immersing the object in 
liquids or gases of differing indices. The ex- 
pression for the contour interval is given by 
dh = M2 dn where dn is the change in re- 
fractive index. 
In all the holographic techniques, it is pos- 
sible to make appropriate changes to make 
contour generation a real-time process. Some 
form of photographic recording technique 
must be employed to record the contour