Vibration registration system

Introduction

The bend of the fiber–optic wire (photos 1 and 2) begins to glow due to losses in the fiber. This is due to microscopic light leaks, when the total internal reflection is disrupted when the optical fiber is deformed when it is bent.

Thanks to this phenomenon, we can identify the place where the bend occurs, which indicates the localization of unwanted vibrations. To enhance this effect, it would be useful to use multimode fiber.

The bending signal can then be transmitted to a single-mode fiber spiraled around a multimode fiber: photon beams from losses in the multimode fiber will end up in the optical trap of the single-mode fiber. The signal will propagate through the fiber in one of two directions.

Since the length of a direct multimode fiber is less than the length of the fiber wound on it in a spiral, the fixed time between signal registrations allows you to calculate the bending coordinate of the cable.

Construction

Vibration of an optical fiber stretched along any structure will lead to a change in the length of the laser beam path and the angle of reflection of the beam inside the fiber (see Figure 1).

Along the structure, two optical fibers must be stretched in a common braid. The first fiber is a single-mode, while the second is a multimode without an insulating coating. A single-mode fiber is wound around the multimode in a spiral (see Figure 2).

The emitter is connected to a multimode fiber at one end of the circuit; the other end of the multimode fiber is connected to receiver 1. The spiral fiber is connected to a photodetector with a translucent mirror from the same end where the emitter is connected at both ends and connected to receiver 2 with the other end, respectively, as shown in the diagram (Figure 3). These receivers and a photodetector convert an optical signal into an electrical one.

A receiver in a single-mode fiber consists of a receiver and an optical element (mirror or a membrane of metamaterial) that reflects rays falling on its surface not at right angle (Figure 4).

After installing the system on the structure, the optical element is adjusted so that the signal coming out of the fiber hits it at a right angle and is absorbed. Receiver 1 does not receive any signal. Receiver 2 receives a constant signal, since the installed fiber-optic cable has only bent that simulate structural elements.

When the engineering structure on which this system is installed passes a test run and operates without disruption, receivers 1 and 2 begins to receive a photon signal of a certain modulation. The full period of this signal is recorded using a controller connected to the receiver.

When atypical vibrations appear in the structure, signal of receivers differ from their recorded values.

The controller, which communicates with the receivers, receives a signal indicating the presence of undesired vibrations. Due to the fact that the length of the single-mode winding in a spiral is different from the length of the multi-mode fiber, the signal from the vibrations at one point in the structure reaches the receivers 1 and 2 at the different times.

The signal of the single-mode fiber on receiver 2 from vibrations caused by abnormal system operation will complement the signal recorded in normal mode from the same receiver. The multimode fiber signal on receiver 1 from abnormal vibrations will have a lower intensity than on the recording of the signal from receiver 1. The time difference between signal anomalies makes it possible to calculate the coordinate of abnormal vibration.

A program for the controller to compare recordings of normal vibrations with the current signal and calculate the coordinates of abnormal vibrations based on the time difference between anomalies in the current signals of receiver 1 and receiver 2.

If the signal in the spiral fiber propagates in a direction opposite to the direction of propagation of the signal in the multimode fiber, the signal will be received by a translucent mirror with a photodetector. The photodetector detects the presence of a signal and sends a message to the controller, the mirror reflects most of the pulse back and it arrives at the main receiver, where its delay time is recorded. When registering an event in this way, a different coordinate calculation algorithm will be applied. The formula for this case will be given below.

A photodetector is needed to determine the correspondence between pairs of signals when several vibrations occur at different points. Since these vibrations can be of the same or very close frequencies, modern methods of establishing a correlation between signals by analyzing their difference in phase, frequency, and intensity can be inefficient and cumbersome in computing power.

If the signals from previous vibrations are received at a greater distance than those from later vibrations, there is a risk that their order may be mixed up.

The light scattered on the bend in multi-mode fiber goes both in the forward and in the opposite direction in the single-mode fiber. The exact distribution depends on:

·       the bending geometry,

·       the mode profile,

·       the cladding roughness,

·       the numerical aperture coefficient.

But in statistics and in real experiments, the scattered signal is divided approximately 50/50.

During the continued registration of the confused signals, the controller tries to calculate the coordinate, sees that there are two or more different options for the same coordinates, and waits, receiving statistics for several seconds. Then, it calculates the location along the forward and reverse paths of the spiral signal - this gives the second equation for the second unknown and the algebraic system ceases to be degenerate. A system of two equations is compiled for each pair from an array of mixed signals.

This method can be combined into an adaptive algorithm inside the controller:

Due to this design feature, the system can only be manufactured in a standard cable length range. Each length will have its own constant in the line of the controller program code, as well as constants with the values of the speed of light propagation in the fiber optic material, the pitch of the spiral and the refractive index of optic fiber.

Since the minimum bending radius of a single-mode fiber is much larger than that of a multimode fiber, it is necessary to slightly modify the cable design. There should be a soft core inside, which will not increase the rigidity of the entire cable, so that it can easily bend under the influence of vibrations. The radius of the core should be approximately equal to the minimum bending radius of a single-mode optical fiber. Different classes of single-mode fibers within G.657 have varying minimum bend radii: G.657.A1 has a 10.0 mm minimum, G.657.A2 and G.657.B1 have a 7.5 mm minimum, and G.657.B2 has a 5.0 mm minimum.

Direct multimode optical fibers are arranged around the core, covering the entire surface of the core with a single layer. All multimode fibers are connected to a common emitter and receiver. It is also necessary to cover the entire surface of multimode fibers with a single layer using a single-mode fiber. All single-mode fibers are connected to common receivers. The cable insulation must perform the function of optical insulation and protect the cable from minor damage to the fiber, both mechanical and chemical. Insulation also cannot significantly increase the rigidity of the cable to maintain vibration sensitivity. The core and insulation layer must be made of tension-resistant materials: this is necessary to preserve the integrity of the cable structure, since a significant change in the pitch of the spiral can affect the accuracy of detecting the vibration site.

Calculation

Based on the minimum bending radius of a single-mode fiber, we assume for preliminary calculation the radius of the core with an environment of multimode fiber equal to 5 millimeters. Since the outer diameter of a standard optical fiber is 125 micrometers, the diameter of the core itself will be 9.750 millimeters.

Since the single-mode fiber is wound on a core with a multimode without gaps, the spiral pitch will be a multiple of the outer diameter of the fiber, i.e. 125 micrometers. Let's take the number of single-mode fibers equal to 48. Then the spiral pitch is 48*0.125 = 6.0 millimeters.

Spiral pitch (along the multimode fiber axis): 6 mm;

Multimode fiber diameter (without insulation): 125 μm;

The formula for the length of a cylindrical spiral in one step:

l = √((2πr)^2 + h^2),

Where: h = 6.0 mm – pitch of spiral;

r = 5.0 mm - radius;

l = √((2π⋅0.005)^2 +(0.006)^2) ≈ 0,032 m;

This means that for every 6 mm along the axis (multi-mode fiber), there is ≈ 32 mm of single-mode fiber.

x/x' = h/l,

x/x' = 3/16 = 0.1875,

Where x/x' - the ratio of the lengths of a multi-mode fiber to a single-mode fiber in a cable section from the coordinate of the detected vibration to one of the cable ends.

Now let's calculate the signal travel time over a 20-meter-long cable. The speed of light in a fiber:

V = c/n,

Where: n - refractive indices of the fiber core;

c – speed of light in absolute vacuum.

Time of passage over multimode optical fiber, taking into account pulse dispersion:

t = (L⋅nm)/c + ∆tmodal,

∆tmodal ≈ (L⋅nm)/c ⋅ ∆/2,

Where: nm - refractive index of multi-mode fiber: ≈ 1.48 - typical for quartz fiber;

∆ - the relative difference in the refractive indices of the fiber core and cladding, let's take the value for standard multimode fibers ≈ 0.01. The formula for relative difference in the refractive indices: ∆ = (n2core – n2clad)/2n2core .

t1 = (20⋅1.48)/3⋅108 + (20⋅1.48⋅0.01)/2⋅3⋅108 ≈ 9.916⋅10-8 s = 99.16 ns;

Time of passage over spiral single-mode optical fiber:

t = (L⋅l⋅ns)/(c⋅h),

Where ns - refractive index of single-mode fiber: ≈ 1.468 (typical for quartz fiber at 1.550):

t2 = (20⋅0.032⋅1.468)/(0.006⋅3⋅108) ≈ 52.196⋅10-8 s = 521.96 ns;

Time difference (delay):

Δt ≈ 422.80 ns.

Now we will derive the equation for finding the coordinate from the time difference between the movement of the signal along different paths, taking into account the main corrections:

t1 = x⋅nm/c + x⋅nm⋅Δ/2c; t2 = 16x⋅ns/3c,

Δt = 16x⋅ns/3c – (x⋅nm/c + x⋅nm⋅Δ/2c),

Δt = (32x⋅ns - 6x⋅nm - 3x⋅nm⋅Δ)/6c,

x = (Δt⋅6c)/(32ns - 6nm - 3nm⋅Δ);

The same equation is corrected for the case when the signal propagates in the opposite direction along the spiral fiber:

t2 = (2L-x)/Vs,

t1 = x⋅nm/c + x⋅nm⋅Δ/2c; t2 = 16(2L-x)⋅ns/3c,

Δt = 32L⋅ns/3c - 16x⋅ns/3c – (x⋅nm/c + x⋅nm⋅Δ/2c),

x = (32L⋅ns/3c - Δt)⋅6c/(6nm + 3nm⋅Δ + 32ns);

In the case of mixed pairs of signals from different vibration points, the controller solves both equations in one system for all possible combinations of pairs:

x = (Δt⋅6c)/(32ns - 6nm - 3nm⋅Δ);

x = (32L⋅ns/3c - Δt)⋅6c/(6nm + 3nm⋅Δ + 32ns).

Only those pairs out of all possible pairs for which this system is solvable are a pair of signals from one vibration point. In this case, having a solution to a system of equations means that both equations individually have the same solutions for the same pair of signals.

Now, let's calculate the accuracy of the localization of the vibration point based on the difference in the signal travel time. Let's say we have the ability to capture the delay with an accuracy of 1 nanosecond (1·10⁻⁹ s, realistic for modern photonic):

x = (Δt⋅6c)/(32ns - 6nm - 3nm⋅Δ),

x = (10-9 ⋅6⋅3⋅108)/(32⋅1.468 - 6⋅1.48 - 3⋅1.48⋅0.01) = 0,047 m;

Calculation of accuracy according to the signal of a single-mode fiber going in the opposite direction:

x = (32L⋅ns/3c - Δt)⋅6c/(6nm + 3nm⋅Δ + 32ns),

x = (32⋅20⋅1.468/3⋅3⋅108 - 10-9)⋅6⋅3⋅108/(6⋅1.48+3⋅1.48⋅0.01+32⋅1.468) = 33.58 m;

As can be seen from the calculations, the method of recording simultaneous signals from different locations may be ineffective due to the large error. If you register a signal going in the opposite direction at a point with an emitter, the error will decrease by about half.

Instead, you can use two such systems connected to a common controller in parallel. The second cable should have a spiral pitch of a single-mode fiber several times larger. Both cables must be tightly clamped in a common braid with optical fibers of the same standard.

Then, in the case of several vibration locations, the controller program finds a solution to two equations of the same type for two pairs of signals, sorting through all possible pairs of signals. The pair of signals for which the solutions of the two equations coincide is a pair of signals from the same location of extraneous vibrations. Since pairs of signals are duplicated at different time intervals between the signals in the pair, the correspondence is uniquely identified.

The controller calculates only those signals that go in the forward direction, the photodetector in the opposite direction sends messages to the controller about the reverse signals and they are not processed.

let's calculate the parameters of the second cable by taking the spiral pitch h' value of 60 millimeters:

l' = √((2πr)^2 + h'^2),

l' = √((2π⋅0.005)^2 +(0.06)^2) ≈ 0.068 m,

This means that for every 6 mm along the axis (multi-mode fiber), there is ≈ 68 mm of single-mode fiber:

h'/l' ≈ 0.0882;

Time intervals and equation for coordinate:

t1 = t'1 = x⋅nm/c + x⋅nm⋅Δ/2c; t2 = 16x⋅ns/3c; t'2 = 34x⋅ns/3c,

Δt' = 34x⋅ns/3c – (x⋅nm/c + x⋅nm⋅Δ/2c),

Δt' = (68x⋅ns - 6x⋅nm - 3x⋅nm⋅Δ)/6c,

x = (Δt'⋅6c)/(68ns - 6nm - 3nm⋅Δ);

Calculation of accuracy for second line:

x = (10-9 ⋅6⋅3⋅108)/(68⋅1.468 - 6⋅1.48 - 3⋅1.48⋅0.01) = 0.0198 m;

For the case of mixed pairs of signals from different vibration points:

x = (Δt⋅6c)/(32ns - 6nm - 3nm⋅Δ),

x = (Δt'⋅6c)/(68ns - 6nm - 3nm⋅Δ).

Conclusion

The second method of determining the coordinates of simultaneous signals from different locations provides an advantage both in accuracy and in the absence of the need to accumulate statistics on incoming signals. Nevertheless, it makes sense to test both methods, perhaps the first method will remain relevant as an emergency in case of cable damage.

For efficient operation of the system, it makes sense to use more powerful radiators for optical fiber - this way the signal captured on bends by a single-mode fiber will be brighter and more noticeable. Due to the design features of the system, the error will be higher over short distances due to the smaller difference in signal travel time along different paths. At long distances, the error may increase due to the large signal scattering in the multimode fiber.

It is also possible to draw conclusions about the effect of the spiral pitch of a single-mode fiber on the accuracy of coordinate measurement: the larger the pitch, the more accurately the coordinate is calculated. It may make sense to leave one of the cables with a direct single mode fiber and record the difference in signal registration time due to the dispersion in the multi-mode fiber. On the other hand, a straight single-mode fiber will be less able to detect scattering at the bends of a multimode fiber.

Instead of winding a single-mode fiber on a multimode fiber, a multimode fiber can be coated with a material that converts photons incident on it into electrical signals. The exact location of the vibration can be determined by measuring the difference in time between the arrival of photonic and electrical signals at the receiver.

This technology is a photonic electronic controller. With a large number of fiber-optic cables installed in complex systems, the optical cable can be connected not to electronic controllers, but directly to the quantum CPU that controls the entire system. In addition, the same emitter emitting single photons can be connected to spiral optical fibers and straight ones - this will make it possible to use the effect of entangled photons for a number of computational and sensory tasks.

This technology can be used to control vibrations in various engineering systems: factory equipment, seismographs, and the aerospace industry. The system is able to instantly register emerging vibrations and filter them out from the noise of the structure itself; recognize the frequencies of extraneous vibrations.

©Daniel Mervel

Vibration registration system

Introduction

The bend of the fiber–optic wire (photos 1 and 2) begins to glow due to losses in the fiber. This is due to microscopic light leaks, when the total internal reflection is disrupted when the optical fiber is deformed when it is bent.

Thanks to this phenomenon, we can identify the place where the bend occurs, which indicates the localization of unwanted vibrations. To enhance this effect, it would be useful to use multimode fiber.

The bending signal can then be transmitted to a single-mode fiber spiraled around a multimode fiber: photon beams from losses in the multimode fiber will end up in the optical trap of the single-mode fiber. The signal will propagate through the fiber in one of two directions.

Since the length of a direct multimode fiber is less than the length of the fiber wound on it in a spiral, the fixed time between signal registrations allows you to calculate the bending coordinate of the cable.

Construction

Vibration of an optical fiber stretched along any structure will lead to a change in the length of the laser beam path and the angle of reflection of the beam inside the fiber (see Figure 1).

Along the structure, two optical fibers must be stretched in a common braid. The first fiber is a single-mode, while the second is a multimode without an insulating coating. A single-mode fiber is wound around the multimode in a spiral (see Figure 2).

The emitter is connected to a multimode fiber at one end of the circuit; the other end of the multimode fiber is connected to receiver 1. The spiral fiber is connected to a photodetector with a translucent mirror from the same end where the emitter is connected at both ends and connected to receiver 2 with the other end, respectively, as shown in the diagram (Figure 3). These receivers and a photodetector convert an optical signal into an electrical one.

A receiver in a single-mode fiber consists of a receiver and an optical element (mirror or a membrane of metamaterial) that reflects rays falling on its surface not at right angle (Figure 4).

After installing the system on the structure, the optical element is adjusted so that the signal coming out of the fiber hits it at a right angle and is absorbed. Receiver 1 does not receive any signal. Receiver 2 receives a constant signal, since the installed fiber-optic cable has only bent that simulate structural elements.

When the engineering structure on which this system is installed passes a test run and operates without disruption, receivers 1 and 2 begins to receive a photon signal of a certain modulation. The full period of this signal is recorded using a controller connected to the receiver.

When atypical vibrations appear in the structure, signal of receivers differ from their recorded values.

The controller, which communicates with the receivers, receives a signal indicating the presence of undesired vibrations. Due to the fact that the length of the single-mode winding in a spiral is different from the length of the multi-mode fiber, the signal from the vibrations at one point in the structure reaches the receivers 1 and 2 at the different times.

The signal of the single-mode fiber on receiver 2 from vibrations caused by abnormal system operation will complement the signal recorded in normal mode from the same receiver. The multimode fiber signal on receiver 1 from abnormal vibrations will have a lower intensity than on the recording of the signal from receiver 1. The time difference between signal anomalies makes it possible to calculate the coordinate of abnormal vibration.

A program for the controller to compare recordings of normal vibrations with the current signal and calculate the coordinates of abnormal vibrations based on the time difference between anomalies in the current signals of receiver 1 and receiver 2.

If the signal in the spiral fiber propagates in a direction opposite to the direction of propagation of the signal in the multimode fiber, the signal will be received by a translucent mirror with a photodetector. The photodetector detects the presence of a signal and sends a message to the controller, the mirror reflects most of the pulse back and it arrives at the main receiver, where its delay time is recorded. When registering an event in this way, a different coordinate calculation algorithm will be applied. The formula for this case will be given below.

A photodetector is needed to determine the correspondence between pairs of signals when several vibrations occur at different points. Since these vibrations can be of the same or very close frequencies, modern methods of establishing a correlation between signals by analyzing their difference in phase, frequency, and intensity can be inefficient and cumbersome in computing power.

If the signals from previous vibrations are received at a greater distance than those from later vibrations, there is a risk that their order may be mixed up.

The light scattered on the bend in multi-mode fiber goes both in the forward and in the opposite direction in the single-mode fiber. The exact distribution depends on:

·       the bending geometry,

·       the mode profile,

·       the cladding roughness,

·       the numerical aperture coefficient.

But in statistics and in real experiments, the scattered signal is divided approximately 50/50.

During the continued registration of the confused signals, the controller tries to calculate the coordinate, sees that there are two or more different options for the same coordinates, and waits, receiving statistics for several seconds. Then, it calculates the location along the forward and reverse paths of the spiral signal - this gives the second equation for the second unknown and the algebraic system ceases to be degenerate. A system of two equations is compiled for each pair from an array of mixed signals.

This method can be combined into an adaptive algorithm inside the controller:

Due to this design feature, the system can only be manufactured in a standard cable length range. Each length will have its own constant in the line of the controller program code, as well as constants with the values of the speed of light propagation in the fiber optic material, the pitch of the spiral and the refractive index of optic fiber.

Since the minimum bending radius of a single-mode fiber is much larger than that of a multimode fiber, it is necessary to slightly modify the cable design. There should be a soft core inside, which will not increase the rigidity of the entire cable, so that it can easily bend under the influence of vibrations. The radius of the core should be approximately equal to the minimum bending radius of a single-mode optical fiber. Different classes of single-mode fibers within G.657 have varying minimum bend radii: G.657.A1 has a 10.0 mm minimum, G.657.A2 and G.657.B1 have a 7.5 mm minimum, and G.657.B2 has a 5.0 mm minimum.

Direct multimode optical fibers are arranged around the core, covering the entire surface of the core with a single layer. All multimode fibers are connected to a common emitter and receiver. It is also necessary to cover the entire surface of multimode fibers with a single layer using a single-mode fiber. All single-mode fibers are connected to common receivers. The cable insulation must perform the function of optical insulation and protect the cable from minor damage to the fiber, both mechanical and chemical. Insulation also cannot significantly increase the rigidity of the cable to maintain vibration sensitivity. The core and insulation layer must be made of tension-resistant materials: this is necessary to preserve the integrity of the cable structure, since a significant change in the pitch of the spiral can affect the accuracy of detecting the vibration site.

Calculation

Based on the minimum bending radius of a single-mode fiber, we assume for preliminary calculation the radius of the core with an environment of multimode fiber equal to 5 millimeters. Since the outer diameter of a standard optical fiber is 125 micrometers, the diameter of the core itself will be 9.750 millimeters.

Since the single-mode fiber is wound on a core with a multimode without gaps, the spiral pitch will be a multiple of the outer diameter of the fiber, i.e. 125 micrometers. Let's take the number of single-mode fibers equal to 48. Then the spiral pitch is 48*0.125 = 6.0 millimeters.

Spiral pitch (along the multimode fiber axis): 6 mm;

Multimode fiber diameter (without insulation): 125 μm;

The formula for the length of a cylindrical spiral in one step:

l = √((2πr)^2 + h^2),

Where: h = 6.0 mm – pitch of spiral;

r = 5.0 mm - radius;

l = √((2π⋅0.005)^2 +(0.006)^2) ≈ 0,032 m;

This means that for every 6 mm along the axis (multi-mode fiber), there is ≈ 32 mm of single-mode fiber.

x/x' = h/l,

x/x' = 3/16 = 0.1875,

Where x/x' - the ratio of the lengths of a multi-mode fiber to a single-mode fiber in a cable section from the coordinate of the detected vibration to one of the cable ends.

Now let's calculate the signal travel time over a 20-meter-long cable. The speed of light in a fiber:

V = c/n,

Where: n - refractive indices of the fiber core;

c – speed of light in absolute vacuum.

Time of passage over multimode optical fiber, taking into account pulse dispersion:

t = (L⋅nm)/c + ∆tmodal,

∆tmodal ≈ (L⋅nm)/c ⋅ ∆/2,

Where: nm - refractive index of multi-mode fiber: ≈ 1.48 - typical for quartz fiber;

∆ - the relative difference in the refractive indices of the fiber core and cladding, let's take the value for standard multimode fibers ≈ 0.01. The formula for relative difference in the refractive indices: ∆ = (n2core – n2clad)/2n2core .

t1 = (20⋅1.48)/3⋅108 + (20⋅1.48⋅0.01)/2⋅3⋅108 ≈ 9.916⋅10-8 s = 99.16 ns;

Time of passage over spiral single-mode optical fiber:

t = (L⋅l⋅ns)/(c⋅h),

Where ns - refractive index of single-mode fiber: ≈ 1.468 (typical for quartz fiber at 1.550):

t2 = (20⋅0.032⋅1.468)/(0.006⋅3⋅108) ≈ 52.196⋅10-8 s = 521.96 ns;

Time difference (delay):

Δt ≈ 422.80 ns.

Now we will derive the equation for finding the coordinate from the time difference between the movement of the signal along different paths, taking into account the main corrections:

t1 = x⋅nm/c + x⋅nm⋅Δ/2c; t2 = 16x⋅ns/3c,

Δt = 16x⋅ns/3c – (x⋅nm/c + x⋅nm⋅Δ/2c),

Δt = (32x⋅ns - 6x⋅nm - 3x⋅nm⋅Δ)/6c,

x = (Δt⋅6c)/(32ns - 6nm - 3nm⋅Δ);

The same equation is corrected for the case when the signal propagates in the opposite direction along the spiral fiber:

t2 = (2L-x)/Vs,

t1 = x⋅nm/c + x⋅nm⋅Δ/2c; t2 = 16(2L-x)⋅ns/3c,

Δt = 32L⋅ns/3c - 16x⋅ns/3c – (x⋅nm/c + x⋅nm⋅Δ/2c),

x = (32L⋅ns/3c - Δt)⋅6c/(6nm + 3nm⋅Δ + 32ns);

In the case of mixed pairs of signals from different vibration points, the controller solves both equations in one system for all possible combinations of pairs:

x = (Δt⋅6c)/(32ns - 6nm - 3nm⋅Δ);

x = (32L⋅ns/3c - Δt)⋅6c/(6nm + 3nm⋅Δ + 32ns).

Only those pairs out of all possible pairs for which this system is solvable are a pair of signals from one vibration point. In this case, having a solution to a system of equations means that both equations individually have the same solutions for the same pair of signals.

Now, let's calculate the accuracy of the localization of the vibration point based on the difference in the signal travel time. Let's say we have the ability to capture the delay with an accuracy of 1 nanosecond (1·10⁻⁹ s, realistic for modern photonic):

x = (Δt⋅6c)/(32ns - 6nm - 3nm⋅Δ),

x = (10-9 ⋅6⋅3⋅108)/(32⋅1.468 - 6⋅1.48 - 3⋅1.48⋅0.01) = 0,047 m;

Calculation of accuracy according to the signal of a single-mode fiber going in the opposite direction:

x = (32L⋅ns/3c - Δt)⋅6c/(6nm + 3nm⋅Δ + 32ns),

x = (32⋅20⋅1.468/3⋅3⋅108 - 10-9)⋅6⋅3⋅108/(6⋅1.48+3⋅1.48⋅0.01+32⋅1.468) = 33.58 m;

As can be seen from the calculations, the method of recording simultaneous signals from different locations may be ineffective due to the large error. If you register a signal going in the opposite direction at a point with an emitter, the error will decrease by about half.

Instead, you can use two such systems connected to a common controller in parallel. The second cable should have a spiral pitch of a single-mode fiber several times larger. Both cables must be tightly clamped in a common braid with optical fibers of the same standard.

Then, in the case of several vibration locations, the controller program finds a solution to two equations of the same type for two pairs of signals, sorting through all possible pairs of signals. The pair of signals for which the solutions of the two equations coincide is a pair of signals from the same location of extraneous vibrations. Since pairs of signals are duplicated at different time intervals between the signals in the pair, the correspondence is uniquely identified.

The controller calculates only those signals that go in the forward direction, the photodetector in the opposite direction sends messages to the controller about the reverse signals and they are not processed.

let's calculate the parameters of the second cable by taking the spiral pitch h' value of 60 millimeters:

l' = √((2πr)^2 + h'^2),

l' = √((2π⋅0.005)^2 +(0.06)^2) ≈ 0.068 m,

This means that for every 6 mm along the axis (multi-mode fiber), there is ≈ 68 mm of single-mode fiber:

h'/l' ≈ 0.0882;

Time intervals and equation for coordinate:

t1 = t'1 = x⋅nm/c + x⋅nm⋅Δ/2c; t2 = 16x⋅ns/3c; t'2 = 34x⋅ns/3c,

Δt' = 34x⋅ns/3c – (x⋅nm/c + x⋅nm⋅Δ/2c),

Δt' = (68x⋅ns - 6x⋅nm - 3x⋅nm⋅Δ)/6c,

x = (Δt'⋅6c)/(68ns - 6nm - 3nm⋅Δ);

Calculation of accuracy for second line:

x = (10-9 ⋅6⋅3⋅108)/(68⋅1.468 - 6⋅1.48 - 3⋅1.48⋅0.01) = 0.0198 m;

For the case of mixed pairs of signals from different vibration points:

x = (Δt⋅6c)/(32ns - 6nm - 3nm⋅Δ),

x = (Δt'⋅6c)/(68ns - 6nm - 3nm⋅Δ).

Conclusion

The second method of determining the coordinates of simultaneous signals from different locations provides an advantage both in accuracy and in the absence of the need to accumulate statistics on incoming signals. Nevertheless, it makes sense to test both methods, perhaps the first method will remain relevant as an emergency in case of cable damage.

For efficient operation of the system, it makes sense to use more powerful radiators for optical fiber - this way the signal captured on bends by a single-mode fiber will be brighter and more noticeable. Due to the design features of the system, the error will be higher over short distances due to the smaller difference in signal travel time along different paths. At long distances, the error may increase due to the large signal scattering in the multimode fiber.

It is also possible to draw conclusions about the effect of the spiral pitch of a single-mode fiber on the accuracy of coordinate measurement: the larger the pitch, the more accurately the coordinate is calculated. It may make sense to leave one of the cables with a direct single mode fiber and record the difference in signal registration time due to the dispersion in the multi-mode fiber. On the other hand, a straight single-mode fiber will be less able to detect scattering at the bends of a multimode fiber.

Instead of winding a single-mode fiber on a multimode fiber, a multimode fiber can be coated with a material that converts photons incident on it into electrical signals. The exact location of the vibration can be determined by measuring the difference in time between the arrival of photonic and electrical signals at the receiver.

This technology is a photonic electronic controller. With a large number of fiber-optic cables installed in complex systems, the optical cable can be connected not to electronic controllers, but directly to the quantum CPU that controls the entire system. In addition, the same emitter emitting single photons can be connected to spiral optical fibers and straight ones - this will make it possible to use the effect of entangled photons for a number of computational and sensory tasks.

This technology can be used to control vibrations in various engineering systems: factory equipment, seismographs, and the aerospace industry. The system is able to instantly register emerging vibrations and filter them out from the noise of the structure itself; recognize the frequencies of extraneous vibrations.

©Daniel Mervel