The theoretical analysis presented in the paper below has been conducted by R. Al Jishi, D. B. Chang, and D. Webb demonstrating the feasibility of the analog image transmission through a multi-mode optical fiber, provided tight tolerances can be maintained on several important parameters. The ability to maintain the required tolerances is mainly a technological issue. The experimental results obtained indicate that several other parameters of importance must also be included in the analysis (please see the discussion under the experiment button of the web page).
This result is valid when the transformed radiation field is measured
in the Fraunhofer diffraction zone. A derivation is given in reference
[2] and in various other places. The pupil function acts as a low-pass
spatial filter, and if only paraxial rays are important, it can be treated
as a constant. Omitting 
and setting d = f, one easily sees that 
is the two-dimensional Fourier transform of 
.
Strictly speaking, 
should be interpreted as the wavelength in air (if that is the medium between
the lens and its focal plane), but this slight inaccuracy will be canceled
out when a second OFT is performed. Coordinates 
of points in the Fourier transform plane exist in ordinary space, and 
therefore having the dimension of length. Points in the Fourier transform
plane constitute a one-to-one mapping of spatial frequencies 
occurring in the input image, such that 
,
and 
.
Hence, the largest spatial frequency that can couple into a waveguide of
core radius R is determined by
and 
.
For example: If 
,
and 
,
the linear resolution is 20 lines/centimeter. For an input image of I cm
x I cm, this implies a transformed image consisting of about 400 pixels.
If the core radius of the fiber is increased to 
and the lens is capable of utilizing the full capacity of the larger fiber,
about 40000 pixels can be transmitted. Alternatively, the number of pixels
can be increased, without changing the core radius, by using magnifying
optics to enlarge the 
image.
The radial wave number 
is the n-th positive root of 
,
where R is the radius of the core. Note that m can be positive
or negative, and so the radial wave number 
and axial wave number 
characterize two linearly independent normal modes unless m = 0.
The radial E field is continuous across the boundary between the
two dielectrics, although it does not exactly vanish there as implied by
the above approximation. The H field is discontinuous at the boundary,
due to surface currents which are created there. Thus the tight-trapping
condition is satisfied. If, as is usually the case, the operating frequency 
is not subject to control by the designer, then the axial wave number 
is fixed by the constraint 
where 
means the refractive index of the core. The 
mode is allowed if 
.
In the proposed optical system, when the radiation field arrives at the
entrance plane of the waveguide, it has been Fourier transformed by the
lens. To analyze what happens next consider an input consisting of a single
pixel treated as a point source and located somewhere in the input plane.
The required Fourier transform is trivial. Moreover, the equations determining
eigenvalues and field configurations in a step-index cylindrical fiber
are known In this way one can obtain numerical examples which illustrate
this key fact: single pixel elements in the input image are selective in
exciting particular eigenmodes. Most of the radiation which originally
emanated from a pixel excites a subset of normal modes that have the same
phase velocity, so that the pixel becomes confined to a diagonal band in
"modal index" or 
space. Some illustrative calculations are shown in table 1. Each allowed
mode propagates through the fiber in the axial (z) direction, and
therefore it acquires a phase shift, due to the factor 
.
On exit from the waveguide, the modes are recombined into a spatial field.
Finally, the resolving lens performs an optical Fourier transform on the
reconstituted field, resulting in a superposition of Fourier transforms
of the eigemodes. The latter are bright spots with secondary maxima, one
example of which is shown in figure 2. At this point, the amplitude of
the radiation field exists in image (not spatial frequency) coordinates.
The form of the final image differs in three ways from that of the input:
relative to the input; the presentation of the image is unimportant for
most applications.It is convenient to work with eigenfunctions obtained by separation
of the Helmholtz equation in cylindrical coordinates, 
and 
.
The optical Fourier transform of 
,
the input image function, is 
, the wave function at the entrance to the waveguide, because the waveguide's
nearer end is at the back focal plane of the first lens. For the same reason, 
,
the output at the farther end, is related to 
,
the final image, by an optical Fourier transform. When the field described
by 
is
presented to the entry surface of the waveguide, it is expanded in terms
of a set of normal modes. In the case of TM-like modes, the expansion is
where 
is determined such that 
.
Generalization to the case where both TE-like and TM-like modes are involved
in the expansion is straightforward. Each allowed normal mode propagates
through the waveguide with a wave vector directed along the z-axis. The
wave function at the exit from the waveguide, of length L, is given
by:
The wave function, 
,
following the Fourier transform on 
,
is thus given by:
where 
.
The factor C alters the scale and phase of the output, but is otherwise
unimportant, since it is the same for all modes. Making use of the Jacobi-Anger
relationship
where
and 
and carrying out the integration over 
we obtain:
Now note that, for values of p which are not too small relative to f,
we can assume that 
since 
.
In this case, it follows that:
The original expansion coefficients (Am,n) are given explicitly by:
Finally, we define the position-dependent phase shift, 
,
which is independent of m and n if 
;
then it follows that
Equation 10 is the sought result: The intensity
of the reconstituted image, which is the absolute square of U, is
the same as that of the input image (
),
except for a spatial inversion and a proportional constant.
is not exactly zero, but is much smaller that 
or 
; the
magnetic field in the TE-like normal mode behaves similarly. Modes of either
type can be characterized by a wave function of the form shown in equation
(2). Allowed modes are those for which 
.
Forbidden modes (those for which this condition is violated) also participate
in the modal resolution, but forbidden modes cannot propagate through the
fiber, and therefore any information carried by them is lost. The capacity
of a fiber of any size can be estimated by noting that, in the limit of
large M, the number of allowed tight-trapping modes tends asymptotically
to 
. To
illustrate this approximation, consider the simple case of M = 100. The
estimated number of allowed modes is 5000; the exact number (as verified
by a computer count) is 5002, of which 2456 are TM-like and 2556 are TE-like.
In a real-world application, of course, the cutoff parameter M will be
much larger than 100. Suppose 
, 
and 
;
then 
and the number of allowed modes is about 3.26x 
.
Consider a test case, consisting of a point source located at a distance
g
from the origin, on the +x axis in the input plane: 
.
The optical Fourier transform of 
is 
.
The expansion coefficients for this special case can be derived analytically,
adjusted for propagation, and then summed numerically, giving 
.
The final Fourier transform to 
can then be performed analytically. The following dataset was used for
a sample calculation, using TM-like eigenfunctions only:
The delta function in 2-space carries the dimension of amplitude (of
some field-related quantity) per square millimeter. Processing by the system
does not change the dimension of the signal, but as pointed out previously,
the output of interest is proportional to the amplitude squared, to which
we shall therefore assign the dimension of 
. As expected, the squared output (see figures 4, 5 and 6) resembles the
diffraction image of a point source viewed through a circular aperture.
The discrepancies between results at three different cutoff levels are
so minute that they are hard to exhibit. Actually, M = 100 is too
small by an order of magnitude for the fiber described in the above dataset.
Therefore, we believe there will be no significant loss of fidelity due
to the fact that the modal expansion is limited to a finite set of modes
supported by the fiber. Similar results (not shown here) can also be obtained
using the TE-like modes or a combination of both. This observation is important
because we do not know, at this time, which type of modes are more efficiently
coupled to the input signal.
where 
,
and where
is the ordinary Fourier transform of 
in rectangular coordinates, it is required that the variation in 
be small relative to the free-space wavelength (
).
Here do, is the distance of the object from the input lens, and 
is the distance from the lens to the fiber at transverse position coordinates 
.
The same consideration also applies, by symmetry, at the output end of
the fiber. Conclusion: Both ends of the fiber must be flat within a
small fraction of 
.
This degree of flatness is readily achievable by polishing. On the other
hand, it may be more cost-effective to satisfy the flatness requirement
with the aid of an index-matching fluid than to do so by polishing alone.
defined as before.
Setting 
,
this implies 
.
If 
with 
.
the tolerable error condition is: 
A similar expression applies at the output end of the fiber. Conclusion:
A close tolerance on the entrance distance is also required. This can
be achieved with a micrometer table.
Setting 
for 
this implies 
.
With 
,
the tolerable error condition is 
meters. The same condition also applies at the output end of the fiber.
Conclusion: The two object planes can be located almost anywhere.
to 
over
current digital techniques;
digital
processing
analog;
) and
output (
)
image spaces, is given by
In the above result, the approximate equality acknowledges the
existence of a truncation error: i.e., replacement of the complete set
of eigenfunctions with a finite subset consisting of allowed modes. Spatial
inversion takes place because two Fourier transforms were performed with
the same sign of j in the exponent. The position-dependent phase shift
does not affect the intensity of the radiation field.
Figure 1: Processing proceeds from left to right
Figure 2: This plot shows the amplitude (square root of intensity),
observed at distance x from the origin, of the Fourier transform
of the radial E field attributable to a TM-like mode. More exactly,
the amplitude is a complex number consisteing of the plotted function times 
,
where 
is the azimuthal position of the observer and z measures distance
along the axis of the fiber. The phase shift kz is about the same
for all modes whise central maximum occurs at the same abscissa. parameters
used in these computation are: 
,
,
.
Figure 4: Contour plot showing intensity (i.e 
of the output due to a point input at 
,
.
Cutoff parameter 
:
other parameters are listed in section 6
Figure 5: Intensity profile sliced along the x axis, for the test case shown in figure 4. Cutoff parameter M is 100.0 for this plot; plots with M=50.0,100.0,150.0 are indistinguishable when plotted at the scale of this curve.
Figure 6: Enlarged plot of part of the intensity profile from figure 5, also showing results for M=50 and M=150 for comparison. Differences between the curves are at maximum near the peak.
Table 1: Excitation of the E field (assuming tight trapping
and TM-like modes) due to a point source at a distance x from the
origin of the object plane, where x is defined such that 
zero of 
.
The expansion coefficient 
is a complex number, consisting of the tabulated value, multiplied by 
where 
is the azimuthal coordinate of the point source. All large amplitudes lie
along a diagonal band in 
coordinates, and correspond to modes having about the same phase velocity.
