3.1. Theoretical Analysis
The source of the observed patterns is the variations of backscattered Rayleigh radiation. It is of interest how the backscattered light implies information about the internal properties of fiber media. The signal is formed by local scattering acts, which are determined by local properties of the fiber: the refractive index, attenuation constant and backscattering factor, which may vary with distance. These value variations over fiber length should be described and connected with detected power. Following the standard approach [
21,
22], we derive these variations.
To obtain the backscattered power from a local fiber section, let us consider electric field
of the initial linearly polarized speculative electric pulse propagating in the fiber in the
z direction as
where
denotes the pulse’s magnitude,
denotes the field distribution of the fundamental mode,
denotes the group speed of light,
denotes the propagation constant,
denotes the light frequency and
denotes the attenuation constant.
is a rectangular pulse envelope function with width
l:
The usual formalism used in optical communication theory expresses the pulse shape as a function of exclusively time. Here, the travelling wave formalism for the pulse shape was chosen in order to describe the interaction between the fiber medium and the pulse.
The differential amplitude (at the moment
t in
) of the electric field scattered from section
at point
:
where
denotes the local inhomogeneities of electric susceptibility which cause the scattering,
denotes averaged value of
over the distance
,
denotes the mode field diameter while the field distribution is assumed to have a Gaussian form:
. Mode field distribution, in general, may slightly vary on
z, but this dependence is insignificant since these fluctuations are averaged.
Integration over the pulse length gives the following:
Now, we assume that the exponential decay
does not change on the scales of the wavelength. That is true for the typical attenuation constant
which is about 0.18–0.24 dB/km. Moreover, this is also justified by the fact that
makes sense only on the scales of several wavelengths. This makes it possible to calculate backscattered power:
where the initial power is
and
For further simplicity, we denote the probing pulse center coordinate by
. The backscattered power at the input of the fiber at the moment
t corresponds to backscattering from the section of
length at the point
. Assuming that
l is at least several times larger than both atomic scales and wavelength, we rewrite
as
where
denotes a weighted average of
with the squared distribution of the fundamental mode
.
in Equation (
7) can be treated as the squared absolute value of spatial Fourier transform of electric susceptibility with spatial vector
of the fiber section in the vicinity of the point
. Equation (
5) can be rewritten as
Equations (
7) and (
8) completely describe the scattering process in single-mode optical fibers, showing how the reflected light brings the information about internal structure of the fiber. The power scattered in the vicinity of the auxiliary point of the fiber is defined by the local electric susceptibility distribution, refraction coefficient and mode field distribution. The mode field distribution may also vary in different fiber sections due to geometry defects and refractive index variability. For convenience,
can be replaced with
z. Following the results obtained in previous works [
21,
22],
can be linked with observable value
:
In Equation (
9),
is the coefficient of loss due to the scattering. Similar to
in Equation (
7),
in Equation (
9) can be defined only at a fiber section the size of at least several wavelengths. Otherwise, it loses its physical meaning. Preferably, this fiber section length
should be more than several dozens of wavelengths. Together with the backscattering factor, it determines the power scattered in the backward direction. Thus, from Equation (
9), we can see that apart from the refractive index and the mode field diameter, possible variations of the backscattered radiation are due to possible oscillations of
along the fiber.
To derive the power of the real pulse backscattered from the length
, let us substitute Equation (
9) into Equation (
8) and obtain:
Here, denotes the backscattering factor with and depending on distance in general case. denotes a pulse energy.
from Equation (
10) may be considered as backscattered power from a delta-like probing pulse. This assumption is justified for a narrow pulse. As discussed above, the pulse length should be no less than several wavelengths. This length should be more than the coherence length to avoid interference effects. In consideration of this,
can be treated as value at the point.
W corresponding to delta-pulses should be replaced with
for the general case of extended pulses. Here, a normalized power envelope function
describes the pulse shape. The backscattered power for extended pulses may be expressed as
where
L denotes the total length of the fiber. Thus, the backscattered power is a convolution of the pulse with
and
, which describe the unique scattering properties of the optical fiber, e.g., for the rectangular optical pulses, which are relatively short compared to an exponential decay, Equation (
11) takes the form of a window-averaged product of
and
B:
where
corresponds to the pulse length. This equation demonstrates that individual scattering properties may be observed through conventional OTDR technology where the typical pulse duration is ∼1 ns–10
s (∼0.2 m–2 km).
Rayleigh scattering is due to fluctuations of on the scales less than optical signal wavelength. These small-scale fluctuations, often regarded as white spatial noise, determine the value of , which causes the optical signal backscattering. Otherwise, the inhomogeneities in optical fibers are in an extensive range of spatial scales. Therefore, they will define variations of on scales larger than the optical signal wavelength. As these variations in an optical fiber may have an extensive spatial range on the larger scales, they can also be described, at first approximation, as the white spatial noise. These large-scale variations can be observed with, e.g., a conventional OTDR technique.
3.2. Experimental Verification of Fiber Key Uniqueness and Reproducibility
As mentioned above, variations remaining on the reflectogram (i.e., patterns) are due to different chaotically distributed inhomogeneities of the fiber waveguide. Furthermore, since it is technologically impossible to reproduce the exact locations of such inhomogeneities, the fiber sections can be considered in terms of PUF in the language of challenge–response pairs [
2]. As a result, they, as physically unique objects, can be considered as keys for authentication after conducting reflectometric measurements (in our case, OTDR measurements) with light pulses having predetermined parameters.
As was stated earlier, the OTDR technique may demonstrate several kinds of instability, particularly in laser pulse intensity or shape impermanence, photodetector noises, delay mismatching, etc. This instability may not be eliminated sufficiently by proper averaging to achieve the required properties of uniqueness and reproducibility. Thus, these properties should be clearly verified experimentally.
The successive measurements were carried out with the same pulse parameters for two independent 400 m long sections of the same fiber from the 50 km long fiber span in the spool. Namely, the first section was 20.0–20.4 km from the OTDR device and the second was 20.5–20.9 km. Five consecutive measurements of the span were made, and after that, patterns corresponding to each fiber section were obtained by subtracting the linear contribution from the reflectograms. Based on the obtained patterns, the joint correlation matrix was constructed. This matrix is presented in
Figure 3a. The patterns of fiber sections are individually well reproduced (red blocks), as the values of the standard correlation coefficient are not less than 0.93. On the other hand, patterns of independent sections of the fiber are entirely different (blue blocks), since the absolute values of the correlation coefficient do not exceed 0.26. One can see that even sections of the same fiber may be distinguished with a high degree of accuracy.
Backscattered power is conjointly determined by the probing pulse and fiber waveguide properties. Therefore, if the parameters of the pulse vary, the resulting patterns may also change. An increase in the duration of pulses leads to deterioration in the effective spatial resolution of the OTDR device, thereby eliminating the variations with high spatial frequencies and changing the appearance of patterns. Thus, the same fiber affected by pulses with distinct shapes should produce essentially different patterns. To experimentally confirm this fact, we measured a fixed section of the fiber, applying probing pulses of various duration: 500 ns and 1000 ns. In
Figure 3b, a similar matrix is shown. Again, the section which is 20.0–20.4 km from the OTDR device was chosen. The patterns for each series of individual measurements are well reproduced again since the correlation coefficient values are at least 0.90. Still, when the pulse parameters change, the patterns become dissimilar.
To prove the uniqueness and reproducibility of patterns, the statistics were obtained for the inter- and intra- distances between them.
Figure 3c shows the results of the tests. In total, 104 different 1 km long fiber sections at various distances up to 27 km from the reflectometer were processed, including even significantly remote fiber sections with worse reproducibility due to higher relative noise. The statistical distribution for the
distance between successive measurements of these sections corresponding to 5700 pairs of patterns is presented by the pink histogram. This distribution demonstrates the Poisson-like probability with a mean value of
.
Statistics for inter-distance corresponding to 137,500 pairs of patterns are presented by the blue histogram. These statistics demonstrate the Gaussian distribution with a mean value of and a variance of . The conducted tests show that the overlap of histograms is insignificant. As a result, the distance threshold appropriate for the successful 1 km long fiber section identification can be chosen as 0.25.
According to Equation (
12), the backscattering signal value is defined via window averaged product of
and B on the pulse length. It means that the quality of reproducibility and uniqueness depends on the fiber section and the pulse length ratio. In the above experiments, this ratio value was 10, which seems close to the lower limit. However, even for this case, the uniqueness and reproducibility were established at a fairly high level.
Additionally, the stability of correlation coefficient values over time was checked. The patterns were again processed for the same 400 m long fiber section, which is 20 km away from the reflectometer. Twenty five measurements were sequentially made on one day and the same number of measurements a week later.
Figure 4 shows the corresponding joint correlation matrix. All matrix elements are greater than 0.80, which indicates good reproducibility of patterns with time. It is also worth noting that an increase in the fiber section length leads to additional reproducibility improvement.
The obtained patterns indeed demonstrate uniqueness and reproducibility, which are required for identification purposes. Therefore, our experimental results prove that an optical fiber section may be recognized in the authentication procedure.
3.3. Spatial Frequencies Spectrum Investigation
On the one hand, we experimentally observed unique patterns on the scales of tens of kilometers with the resolution from tens to hundreds of meters. On the other hand, patterns may also be observed with the OFDR technique on the scales of meters with submillimeter resolution [
10,
11,
12,
13]. These patterns correspond to independent variations of the fiber media. As unique patterns can be observed in a wide range of spatial scales, it is of interest to investigate their spatial spectrum.
To determine characteristic spatial frequencies of observed variations of patterns, the discrete Fourier transform (DFT) of patterns has been carried out (
Figure 5a). For convenience, the spatial vector k was used, related to the spatial period
as
. The patterns were obtained in OTDR measurements of the first 25 km long fiber section of 50 km long fiber span in the spool with pulse durations of 200 ns and 1000 ns (
Figure 5b).
According to the results, the spatial spectrum is dense with frequencies from ∼
m
to ∼
m
. Moreover, for both pulse durations, the amplitudes of the lowest spatial frequencies have essentially non-zero values (
Figure 5a, subplot). Thus, spatial periods of variations of patterns may reach scales of the length of fiber line and are limited only by OTDR distance range or the fiber length.
In a high spatial frequency region, the spectrum is limited by the pulse length, which determines the spatial resolution of a device. As an estimate of the boundary for high spatial frequencies, we considered the point at which the Fourier transform value is halved compared to the maximum. For convenience, a polynomial approximation was applied to the data (dashed lines in
Figure 5a). For the selected pulses, these boundaries correspond to a spatial period of about fifty meters for a pulse duration of 200 ns and about two hundred meters for a pulse duration of 1000 ns. These values are of the order of corresponding pulse lengths in the fiber waveguide which are
, where
m/s denotes the speed of light in optical fiber, and
denotes the duration of the pulses.
Figure 5a also includes the spatial spectrum of the OTDR device’s probing pulse. The length of the pulse is 200 m which corresponds to its duration of 1000 ns. It can be seen that this spectrum looks like the envelope function for corresponding spatial frequency spectrum. That is also in agreement with the theoretical predictions. In accordance with Equation (
11), since the backscattered signal is a convolution of the probing pulse shape with the scattering function of the fiber, the resulting spatial spectrum should be their product in Fourier space:
. Wherein, zero values of the spatial spectrum should coincide with corresponding zero values of the rectangular pulse spectrum, what is indeed observed. Moreover, since the pulse spectrum value is substantially constant in the vicinity of
, the measured spatial spectra in this range have fairly close values for different pulse durations (
Figure 5a, subplot).
The above results indicate that OTDR-processed pattern variations can be observed in a wide range of spatial periods from tens of meters to tens of kilometers. Considering the results presented in OFDR experiments, the overall spectrum extends from submillimeter scales to tens of kilometers and possibly even greater values. Since the spectrum is so broad and dense, the patterns’ oscillations’ structure is similar to that of white noise. Furthermore, the observations mentioned above provide extremely high robustness against altering the fiber section by the intruder. These results additionally confirm the possibility of using these patterns to authenticate remote users or fiber sections of various lengths.