1350 
which energy and information are propa- 
gated, and that it provides the key to a quan- 
titative understanding of resolving power 
and accuracy of locating images. Current 
theories are developed from the calculation 
of the image of a small source by a perfect 
optical system of given aperture, as 
exemplified by Airy’s calculation of the dif- 
fraction pattern, and by the Rayleigh criterion 
for the resolution of the images of two incoher- 
ent point sources!?. 
One of the basic advantages of coherent 
radiation is the narrow angular distribution of 
emission that is possible, and this im- 
mediately suggests the possibility of using 
such a coherent beam for defining an axis, or 
for use in measuring transverse offsets. Most 
lasers will emit with a transverse distribution 
which approximates to a fundamental mode 
distribution. The distribution of intensity, I, 
as a function of the radial distance r from the 
axis, may be represented? as 
I(r) = 1, exp (-2 r2/w?) 
where I, is the maximum intensity at the 
center r = 0. I falls to a value of (1/e?) I, at a 
distance w from the center (Figure 1). As the 
lightbeam propagates, the Gaussian distribu- 
tion is unchanged, except in scale, and this is 
shown by the change in w as a function of 
axial distance z. 
If the wavefront is made to converge by a 
lens of sufficient aperture and of focal length f, 
the light patch converges to a minimum size 
in which w has a minimum value 
09, — A/n0O (0 small) 
where 6 is the angle subtended by w at the 
distance z when both w and z tend to infinity 
6 = w'/f where w' is the value 
of w at the lens. 
For 0 = 0.1 and à = 633 nm 
2w, = 4 um 
  
| 
Lens 
  
Fic. 1. Gaussian distribution of intensity for 
coherent radiation. 
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975 
In some of our past work we have found 
that transverse settings can be made to about 
one-thousandth of this minimum size, that is, 
to about * 4 nm, using a split-field 
photocell-pair*. 
The Gaussian distribution, strictly, should 
be permitted to extend to a very large radius 
in the aperture ofthe lens in orderto preserve 
the Gaussian distribution at the minimum or 
“waist.” In practice, the lens semi-aperture a 
is fixed by other considerations, and the scale 
of the Gaussian distribution must be chosen as 
the best compromise between maximizing 
the energy transmitted by the lens and 
minimizing the size of the central concentra- 
tion in the image. 
As an example, it has been calculated5 that 
overfilling the lens aperture with a Gaussian 
distribution to the extent that w — 1.4 a will 
result in a pattern showing rings like the Airy 
pattern, and with a central maximum of 
nearly the same size as the Airy pattern, in 
which the diameter of the first dark ring is 
1.22 4/0 — 7.7 um for the value of 0 
previously taken. 
INvERTING ÍNTERFEROMETERS 
With coherent light, the problem of making 
a measurement of the position of a point 
source is capable of treatment in new ways. 
As a simple example, we may consider an 
optical system which splits an incident 
wavefront into two identical parts (by divid- 
ing the amplitude at each point at a suitable 
beamsplitter) and then causes the two halves 
to be mutually inverted before they are re- 
combined. Such a system (Figure 2) defines a 
geometrical axis of symmetry with respect to 
the emerging beams. A ray incident on the 
system which does not coincide with this axis 
on emerging will appear as two separate rays. 
Depending on the focal power in the system, 
a number of different possibilities now exist 
to make use of interference between the two 
separated halves of the original incident 
Inverting Interferometer —; 
Á Mutually inverted 
yf wavefronts —— 
"s 
/ 7 
X d. m 
> 
  
  
  
med 
  
N Di 
Incident wavefront — > 
Axis of symmetry 
Fıc. 2. - An inverting interferometer.