Method and apparatus for measuring temperature profile with a single optical fiber

A method and apparatus for determining a temperature profile along a single optical fiber. The temperature profile is determined by measuring the output power spectral distribution. By measuring n points along the spectral power distribution curve, n "resolution elements" of temperature can be determined.

BACKGROUND OF THE INVENTION 
The invention is generally related to the use of fiber optic technology for 
thermal radiation detection. More particularly, the invention provides a 
method and an apparatus for determining a temperature profile along a 
single optical fiber. 
It is becoming widely recognized that various types of sensor devices can 
be advantageously made by using optical fibers either as the medium for 
data transmission, or as the sensor transducer, or both. Fiber optics for 
sensor systems may be useful in the presence of a high electromagnetic 
noise background or in environments where electrical signals cannot be 
used, such as in the presence of explosive atmospheres. Fiber optic sensor 
systems can be implemented in the measurement of a variety of parameters, 
such as current, pressure, moisture and temperature. 
Temperature may be measured with optical fibers in several known ways. For 
example, optical fibers have been developed in which the light propagation 
characteristics of the fiber are dependent on temperature. In U.S. Pat. 
No. 4,151,747, an optical fiber cooperates with a light source and a 
detector for sensing changes in the temperature being monitored. The 
amount of light which passes through the fiber varies with changes in the 
temperature of the fiber. The disadvantage of this sensor is that the 
optical fiber is very difficult to manufacture because extremely accurate 
control of the optical and thermal properties of the fiber core and 
cladding is required. Another type of optical fiber temperature sensor 
measures the internally generated, black body radiation emitted from the 
fiber when it is heated. This sensor is taught in "Fiber-optic Temperature 
Sensor Based on Internally Generated Thermal Radiation", M. Gottlieb and 
G. B. Brandt, Applied Optics, Vol. 20, No. 19, October 1981. Both of the 
temperature sensors described therein only provide an indication of 
temperature along the hottest region of the fiber and thus function only 
as a "hot-spot" probe. Although such fiber optic "hot spot" probes are of 
potential value in a variety of applications, the existence of multiple 
hot spots along the fiber can, in some cases, significantly degrade the 
accuracy of the sensor. 
It is an object of this invention to provide a method and an apparatus for 
determining a temperature profile along a single optical fiber by 
evaluating the spectral power density of the internally generated thermal 
radiation of an optical fiber. 
Infrared radiation is emitted by all solid objects which are not perfectly 
transparent. This emission is often referred to as "black body radiation". 
According to the Stefan-Boltzmann law, the total rate per unit area of 
emission of energy of all wavelengths is directly proportional to the 
fourth power of the absolute temperature. And, according to Wien's law, 
the wavelength of maximum intensity is inversely proportional to the 
absolute temperature of the emitting body. All solid objects have an 
emissivity between zero and one; a perfectly transparent object which 
absorbs nothing and emits nothing, has an emissivity of zero, while an 
ideal, perfectly black object has an emissivity of one. Fiber optic 
material is a semi-transparent material with an emissivity value between 
zero and one. 
The concept of monitoring temperature with an optical fiber by measuring 
the internally generated radiation of the fiber itself is disclosed in the 
article "Fiber-optic Temperature Sensor Based on Internally Generated 
Thermal Radiation" by M. Gottlieb and G. B. Brandt; Applied Optics; Vol. 
20, No. 19; Oct. 1, 1981, the contents of which are incorporated herein by 
reference. In this article, a method of determining the hottest spot along 
a fiber is described. This method assumes that only one hot spot exists 
along the fiber and that this single hot spot generates nearly all the 
thermal radiation in the fiber. This radiation propagates through the 
fiber and is measured. A single number, the output voltage of the detector 
at the end of the fiber is converted into a temperature. 
There are several assumptions made in the above-identified article. One hot 
spot exists along the fiber which has a constant temperature `T` across 
its length `l`. The hot spot is a distace `L` away from the detector. The 
length of the fiber, not including the `hot spot` is cold and contributes 
no thermal radiation. The distance `L` of cold fiber, attenuates the 
radiation emanating from the hot spot by a total amount of 
e.sup.-.alpha.L, where .alpha. is the absorptivity of the fiber and e is a 
constant (e=2.732), the base of the natural log. It is also assumed that 
the fiber is not perfectly clear and that some radiation passing through 
the fiber is lost because the fiber absorbs it. The clearness of the 
fiber, its absorptivity, is indicated by .alpha.. The lower the .alpha. 
the clearer or more perfect the fiber with .alpha.=0 being a perfect 
fiber. Accordingly, in a perfect fiber, if one watt of light enters the 
fiber (P.sub.in) and travels the length of the fiber, one watt will come 
out of the fiber (P.sub.out): P.sub.out =P in e.sup.-.alpha.L. Thus, if 
.alpha.=0: P.sub.out =P.sub.in e.sup.o ; or Pout=Pin. However, fibers are 
not perfectly clear and .alpha.&gt;0. It is also assumed by the Gottlieb et 
al. disclosure that the clearness, or the ability of the fiber to pass 
light, .alpha., is not dependent upon the color of light selected to 
measure fiber clearness. Accordingly, the absorptivity of the fiber is 
assumed to be a constant. Every part of the fiber is exactly the same as 
any other part, i.e., it has the same diameter, absorption constant, etc. 
It is also assumed that the fiber is in good, uniform, thermal contact 
with the object being monitored. Finally, the detector at the end of the 
fiber has a flat response and measures integrated power between two 
selected wavelengths. 
The single hot spot sensor of Gottlieb et al. only provides an estimate of 
the hottest spot along a fiber's entire length. If in fact, two or more 
hot spots of similar temperature exist along the fiber, only one estimate 
of the hottest spot along the fiber is obtained, and in this situation, 
that value is obtained through a weighted sum of both hot spots. 
Accordingly, the weighted sum could yield a totally erroneous result. 
In order to determine the temperature of an object at every point along the 
length of the optical fiber in contact therewidth, a temperature 
distribution monitor was developed and disclosed by the inventor of the 
instant invention is U.S. patent application Ser. No. 304,761, Distributed 
Fiber Optic Temperature Monitoring Apparatus and Method, filed Sept. 9, 
1981, assigned to the assignee of the present invention and incorporated 
herein by reference. 
U.S. patent application Ser. No. 304,761 discloses a multifiber bundle 
which monitors an entire temperature distribution. Each fiber of the 
bundle has a predetermined absorption constant .alpha. which is measurably 
distinct from that of every other fiber. This disclosure assumes that `N` 
hot spots exist along the fiber which has a temperature T.sub.1, T.sub.2 . 
. . , T.sub.N across its length and each hot spot temperature is a 
constant value across its length. It is further assumed that all parts of 
each fiber generate radiation which is transmitted to the detector as a 
weighted sum. Thus for a bundle containing N fibers, N values of 
integrated output power are obtained. Because each fiber in the bundle has 
a distinct absorptive constant .alpha., the attenuation and generating 
characteristics of each fiber are different, thus N values of integrated 
power are measured by N detectors. It is also assumed that absorbtivity 
.alpha. is constant with wavelength and that the total length and diameter 
of each fiber are homogeneous. 
This multifiber bundle optical device provides an estimation of the actual 
thermal distribution, the accuracy of which is reflected in the number of 
fibers in the bundle. However, the device does utilize an integrated power 
output which is applied to a linear system of equations. 
SUMMARY OF THE INVENTION 
A method and apparatus for analyzing the thermal radiation output of an 
optical fiber capable of the self-generation of a thermal radiation output 
in response to the temperature of the environment proximate the optical 
fiber. The optical fiber is characterized by a predetermined discrete 
value for each wavelength. The method includes the steps of separating the 
spectral power density of the thermal radiation output into a determinable 
number of bandwidths and measuring the output spectral power of each 
bandwidth to provide an estimation of the temperature profile along at 
least a portion of the optical fiber. 
The invention also provides a temperature monitoring apparatus which 
utilizes a single optical fiber and means for measuring the spectral 
distribution of the self generated thermal radiation. The apparatus also 
includes means for selectively identifying a plurality of predetermined 
wavelengths which are analyzed for spectral output power content.

DETAILED DESCRIPTION OF THE INVENTION 
This invention provides a method and an apparatus for determining a 
temperature profile along a single optical fiber. Temperature profiling 
with a single fiber is possible by assuming that the absorption value in a 
single fiber is different for a given number of discrete wavelengths and 
therefore, a given number of equations of spectral power distribution can 
be derived. As will be more fully described below, the spectral power 
distribution measurement is analyzed by a given number of non-linear 
equations. 
Considering the schematic illustration of the apparatus of this invention, 
a single optical fiber temperature sensor is generally indicated by the 
reference character 11. The system 11 includes a single optical fiber 13 
which is capable of the self-generating of thermal radiation in response 
to the temperature of the environment or thermal source 15 proximate 
thereto. The optical fiber 13 is characterized by having a discrete 
absorption value for each wavelength of a spectral distribution of the 
self-generated thermal radiation. A means for measuring the spectral 
distribution of the thermal radiation output is operably associated with 
the optical fiber 13 and includes a monochromator or similar device which 
isolates narrow portions of the spectrum through the dispersion of light. 
The accompanying illustration depicts a lens 17 which focuses the spectral 
output of the fiber 13 onto a diffraction grating schematically indicated 
at 19. A detector array means 21 measures the spectral output power of N 
bandwidth as diffracted by the grating 19 or, for example, isolated by a 
monochromator. Alternatively, an acousto-optic tunable filter can be 
employed to select predetermined bandwidths in the spectral output of the 
optical fiber 13 for analysis. The analysis of the spectral power output 
requires an evaluation through a somewhat cumbersome series of 
simultaneous equations. Accordingly, the evaluation can be affected in the 
spectral distribution measuring means by connecting the detector array to 
a computer system 23 which can include a multiplexer 25, a high gain 
amplifier 27, an analog to digital converter 29 and a computer 31. The 
spectral power analysis which consists of n number of thermal radiation 
measurements, can be sent to the computer from the detector array and 
mathematically associated with the absorption constant, .alpha. for the 
specific bandwidth being analyzed. Following the mathematical solution of 
the n number of simultaneous equations, as will be outlined below, the 
results can be automatically displayed on a readout device 33. 
The use of a single optical fiber to measure a temperature distribution is 
based on the assumption that the value of the absorption constant in one 
fiber is different at n wavelengths and that therefore N equations of 
spectral power distribution can be derived. However, as indicated above, 
these equations are nonlinear. Moreover, because spectral power 
distribution is being measured, a diffraction grating or similarly 
functioning device is used to define the radiation output into N 
wavelength bands. The strength of each of the N wavelength bands is then 
measured to provide an estimate of the spectral power density. 
Certain assumptions on the behavioral characteristics of an optical fiber 
are relied upon in the method and apparatus of this invention. There 
exists along the length of the fiber N hotspots, each of which has a 
temperature T.sub.1, T.sub.2 . . . , T.sub.N. All parts of a single fiber 
generate radiation which is transmitted to a diffraction grating at the 
end of the fiber and which is broken into N wavelength bands. The strength 
of each wavelength band is measured with N detectors and N values if 
spectral power is obtained. The absorption constant of a single optical 
fiber has N values at N different wavelengths. The attenuation and 
spectral output power generation characteristics of the fiber at N 
wavelengths is different, thus N values of spectral power distribution are 
measured. The absorptivity of the fiber varies with wavelengths but does 
not vary throughout the fiber. In other words, the characteristics of the 
fiber described above remain constant along the entire length of the 
fiber. 
In order to fully appreciate the derivation of the equations by which the 
spectral power output of a single optical fiber sensor of this invention 
is measured, it is necessary to build on some of the basic concepts 
described in the background portion of this specification. 
The per unit length ability of a material to absorb external thermal 
radiation is given by its absorptivity .alpha.. To measure .alpha., which 
herein is examined as a function of wavelength, .lambda.. If a light of 
wavelength .lambda..sub.1 provides a known input power, P.sub.in, into a 
fiber of known length, L, the power output, P.sub.out, is defined by the 
equation: 
EQU P.sub.out =P.sub.in e.sup..alpha.(.lambda.)L (1) 
Based on this equation, the absorptivity .alpha. of a fiber at N wavelength 
.lambda. can be calculated. 
An analysis of a temperature sensing fiber can be made on the basis of the 
well-known theory of blackbody radiation and the light guiding properties 
of optical fibers. The total power density radiated from the surface of a 
heated body, between the wavelengths .lambda..sub.S and .lambda..sub.L, 
is: 
##EQU1## 
where E is the emissivity or absorptance of the solid object and is in 
general a function of temperature and wavelength and P is the total 
integrated power emanating from the surface. Building on equation 2 above, 
the emissivity of a slab having a thickness dx on the surface of the 
object is given by the statement: 
EQU E=absorptance=.alpha.dx (3) 
Combining equations 2 and 3 yields: 
##EQU2## 
where T is the temperature of the slab and .alpha. is absorptivity. Now, 
radiation is attenuated as it travels through the semitransparent object 
to the surface thereof by: 
EQU e.sup.-.alpha.x (5) 
where x is the distance to the surface and .alpha. is the absorptivity. 
Summing the radiation emanating from within the semitransparent object 
renders an output radiation according to this statement: 
##EQU3## 
where T(x) is the temperature of the object as a function of x. In an 
optical fiber, only free radiation along the axis of the fiber is going to 
reach the detector at one end thereof. Thus, a geometric constant must be 
placed in front of this equation. For an optical fiber with diameter D, 
cladding having an index of refraction n.sub.clad, and a core having an 
index of refraction n.sub.core, the geometric constant is: 
##EQU4## 
The final expression for integrated power emanating from a fiber of 
absorption .alpha.(.lambda.), with a temperature distribution T(x) across 
its length is: 
##EQU5## 
where a is equal to equation 7, D is diameter of the fiber in centimeters, 
C.sub.1 is 3.74.times.10.sup.-12 (watts/cm.sup.2); C.sub.2 is 1.43 
(cm-.degree. K.), .alpha.(.lambda.) is absorption as a function of 
wavelength (cm.sup.-1), T(x) is temperature as a function of distance 
along the fiber (cm), L is the total length of the fiber, and .lambda.a 
and .lambda.b are detector cutoff wavelengths (.mu.m). 
Hot spot sensors described above, merely utilize a detector with cutoff 
wavelengths .lambda.a and .lambda.b to evaluate equation (8). Then by 
assuming a temperature profile consisting of one hot spot with a typical 
width (determined by the application) an accurate estimate of the hottest 
point along the profile can be determined. In contrast, the present 
invention examines the spectral content of the output radiation. Using the 
fundamental theorm of calculus .differential.P/.differential..lambda. can 
be shown to be: 
##EQU6## 
The temperature distribution T(x) is treated as a step-like function along 
the fiber where each step is designated by T.sub.1, T.sub.2, . . . 
T.sub.N. Additionally, .alpha.(.lambda.) is assumed to be known at N 
points in wavelength and is designated by .alpha..sub.1, .alpha..sub.2, . 
. . .alpha..sub.N. P(.lambda.) is then measured at each of the wavelengths 
yielding: 
##EQU7## 
This formulation yields N non-linear equations in N unknowns as follows: 
##EQU8## 
These equations can be solved by using various numerical techniques, for 
example, Newtons Method or the method of successive iterations. They would 
be most appropriately used knowing P(.lambda.). 
Using these calculations, an estimate of the actual power distribution can 
be obtained, with the quality of the estimate being enhanced according to 
the number of wavelength bands evaluated. This system has the advantage of 
being very flexible. The more wavelengths of spectral power density 
measured, the greater the accuracy to which the actual temperature 
distribution is measured. 
The temperature range within which an optical fiber temperature sensor 
according to this invention can operate is a function of the heat 
resistant characteristics of the optical fiber's cladding and the 
impurities within the optical fiber itself which generate the thermal 
radiation spectral power output. 
What has been described is a method and apparatus for measuring a 
temperature profile with a sensor utilizing a single optical fiber by 
examining the spectral power output of the optical fiber.