Parametric Model of a Laser with External Distributed Feedback in the System of Remote Measurement of Nanovibrations

Abstract

The paper studies the actual problem of building an adequate model of remote measurement of the characteristics of barely noticeable vibrations (nano vibrations) of the object of observation, which may be caused by some processes inside the object or external influences on it. Solving such problems is of key importance both for engineering sciences and for ensuring the safety of information technologies. The paper proposes a model that otherwise explains the interaction of individual feedback energy components in the formation of a useful signal. This reveals the reasons and makes it possible to determine the quantitative difference in estimates of the useful signal between the models: generally accepted and proposed. The proposed model, based on the photon lifetime and the equivalent length of the composite resonator, is in good agreement with the results of practical experiments and opens up new opportunities for improving the efficiency of the Laser Feedback Interferometry (LFI) model.

Keywords

• 1 Interferometry
• laser
• vibrometer
• nano vibration
• LFI model

1. Introduction

The solution to many problems in engineering and information technology significantly depends on the success or failure of remote measurement of the characteristics of barely noticeable vibrations (nano vibrations) of the object of observation, which are due to some processes inside the object or external influences on it [1, 2].

The use of devices for measuring nano vibrations is an element of information warfare [3–4]. In particular, when designing information security systems, it is important to assess the possibility of unauthorized access to critical information owing to the artificial or natural occurrence of technical channels for information leakage [5]. Thus, in [6–8], an analysis was made of the principle of operation of laser acoustic reconnaissance systems and passive methods of protection against reading acoustic information.

The laser recorders and nano vibration meters have very high characteristics compared to other technical means of remote information acquisition. This affects the development of a threat model and the practical construction of an information security system in an enterprise [9– 11].

During development and practical experiments in the field of constructing laser nano vibration meters, results were obtained [12–14], which do not agree very well with the existing theoretical model for constructing a laser vibrometer, which became the basis for a detailed analysis of this model and its refinement [15–17].

2. Physical and Mathematical

Principles of Building a Model

main resonator resonator of the target To maximize the intensity of the received radiation, a scheme with wavefront division was used, since the intensity distribution over the aperture when the beam moves in opposite directions is different. (From the laser—Gaussian radiation, with a maximum in the center, into the laser—uniform).

The ratio of the area of the main mirror—3 resonators 1–3 and the beam expanded by the collimator—2, is approximately 1⁄1000. The second resonator, 1–5, is formed by the main mirror and the surface of the object. The third resonator of the circuit, formed by mirrors 3–5, has a lower quality factor by several orders of magnitude compared to resonators 1–5, and even more so with resonators 1–3, that’s why we will not consider its influence.

Further considerations are also applicable to the amplitude division scheme and are based (Fig. 1) on simple, obvious assumptions: • A laser—conservative system • An object surface—3rd laser mirror • An active medium and a deaf mirror—a common part of a composite resonator.

A consequence of our assumptions is the

conclusion that the feedback interferometry (Laser Feedback Interferometry—LFI) is due to the evolution of the photon lifetime in the composite resonator.

Since the lifetime is an energy-intensive

parameter, the physical processes in a composite resonator are parametric. The model based on this approach will be called parametric (Parametric or

P-model).

Doppler modulation is a kind of frequency

modulation that has an asymmetric spectrum [18].

In the absence of a frequency shift of the

probing beam relative to the reference beam, at least by an amount equal to half the useful signal spectrum width, the negative and positive parts of the spectrum overlap.

As a result, the task of restoring the law of

change of any complex useful signal becomes practically unsolvable. To exclude such a situation, to shift the frequency, an Acousto-Optic Modulator (AOM)—4 was introduced into the optical scheme (Fig. 1). Modulation frequency is ω_b=30〖∙10〗^6 Hz.

In addition, AOM performs another important function. It shifts the center of the spectrum of the useful signal much further, to the region of high frequencies, which makes the received signal relatively narrow-band and allows to filter and demodulate it with conventional radio engineering methods.

The processes that cause a change in the laser

parameters are associated with a change in the current by [19, 20], which causes a change in the temperature of the laser diode, which, in turn, is due to a change in the value of the feedback from the mirror of the object—5 (Fig. 1).

In our scheme, the frequency of the feedback signal from the object, because of its narrow bandwidth, is determined not by the frequency 285

−1( )= −1( )+ ∆ −1( ).

From ( 2 ) it follows that as a result, an increment can be obtained as the pump current: = ( + ∆ ) and (with the opposite sign) the radiation power: = ( − ∆ ). In the general case, we get both increments together. Let’s assume the general case.

We subtract the original equation (1) from the

same equation, but with increments—∆ и ∆ : = ( )( ( )− −ℎ1),

Dividing the left and right sides of ( 3 ) into ( 3 ) − ∆ −1( ), we get:

= ( )( ( )− −ℎ1), = ( −1),

∆ ∙ ∙∆ −1( ) = ( −1), range of the useful signal, but by the frequency of the AOM, which is chosen high enough so that the temperature of the laser active medium does not have time to track it [19]. This eliminates the inevitable (in the absence of AOM) change in the laser parameters during the reception of a relatively low-frequency useful signal and the distortions associated with their change.

Let us consider the behavior of the radiation

power and the pump current caused by the change in the photon lifetime in the composite resonator, which is determined by expression ( 2 ).

A simplified photon balance equation, without considering insignificant losses due to transitions without radiation [20], has the form: where: ( ) is the number of induced photons; ( )= ( ) electrons; ; ( ) is the power of injected is the power of photons induced over time ; ( )is amplification per unit of time; is the number of injected electrons; −ℎ1 is a reciprocal lifetime of photons. ℎ = where is resonator length, reflectance, с is the speed of light.

2 ln −1

, is resonator

In the stationary regime, the pump rate always exceeds the loss rate:

( )> −ℎ1.

Let us multiply the left and right parts of ( 1 ) by the photon energy—ℎ . Then ( 1 ) will represent the ratio between the power of stimulated emission , on the one hand, and pump power ℎ, on the other: = expression can be represented as: and threshold − ℎ. The last = ( ( )− ℎ( )). We have obtained the dependence of the radiation power on the pump current—the output or watt-ampere characteristic of the laser, which can otherwise be represented as: = ( − −1( )), ( 2 ) ( = photon lifetime: where: is the power of radiation (W); is pump current (А); voltage (V); is quantum efficiency; is laser is the electron charge ( ∙ ); ∙ ∙ −1( )= ℎ is a threshold exceed the threshold. current (А). Generation condition: I ≥ Ith ≥ 0—the current must exceed the threshold. Generation condition: ≥ ℎ ≥ 0 is the current that must Let the laser be powered by a voltage generator ). We give a small increment of the ( 3 ) ( 4 ) ( 5 ) ( ) 0 ≤ = ( )( ( )− −ℎ1), ( −1),

( −1)≤ 1.

Nothing prevents us from interpreting expressions ( 4 ) and ( 5 ) as the density of a twopoint probability distribution—{∆P, ∆I}, where the terms are the probabilities of these values. So, δP(τ_p^(-1)) is the probability that an increase in the threshold power will cause an increase in the radiation power ∆P, and δI(τ_p^(-1)) is the probability that an increase in the threshold power will cause an increase in the pump power ∆I. Indeed, each of its elements is less than 1, positive, dimensionless, their sum is equal to 1.

The change in the variables in equation ( 2 )— radiation power ∆P and pump current ∆I with a change in the parameter (photon lifetime) occurs as a result of the evolution of the energyintensive parameter of the operating laser, i.e. we are talking about the behavior of a laser as a nonequilibrium system in a transitional mode.

Relation (2) and the resulting probability

distribution ( 5 ) do not allow us to find the desired dependence. An additional condition is required. It can be obtained by relying on one of the appropriate universal principles, such as variational ones, corresponding to the conditions of use. This possibility is provided by the of maximum

entropy

Entropy

Production

production

Principle—

principle (Maximum

MEPP).

MEPP is formulated as follows: a nonequilibrium system, developing in natural conditions, tends to the state corresponding to the maximum entropy with the maximum possible ( , )→

Considering that

( −1)= 1 − ( −1), the MPP compliance condition for distribution ( 4 ) will look like this: ( 6 ) ( 7 ) coefficients = 1, with 2. Then, if 2 = 0, then = 2.

We notice that the virtual is closer to the mirror, the reflectivity Really:

the normalized “speed” [21]. By “natural”, we mean the conditions under which there is no “targeted external influence” [22]. The latter corresponds to the first of the Assumptions we have accepted.

Formulating

MEPP, Jaynes [21] adopted the

Shannon form of information entropy [22].

For the distribution entropy ( 5 ), the principle in the form [22] can be written as:

( ) ( ( −1)+ ( −1))≡ − ( ) coefficients formally correspond to the definition mathematical expectation of the length distributed according to it is as follows: ( )= ∑ = 1 1 + 2 2, = 1,2.

In statistics, various estimates of mean values are used—harmonic, quadratic, cubic, in addition, mode, and median, but only mathematical expectation according to Cramer-Rao inequality, is optimal in the mean square sense. From this, it follows that the optimal estimate of the mean should have a minimum dispersion. value

Let’s check this:

1 1 = 0 position of which is higher. obtain a two-point distribution.

Such conditions will satisfy any of the many possible estimates of the average length. The mathematical expectation is one of them.

Representing the reflection coefficients 1, 2, normalized to the sum 1 + 2 as probabilities, we ( ln + (1 − )ln(1 − ))= 0, where (∗) is information entropy in Shannon form.

ln obvious that it is located between the mirrors 1, 2, that is.: 1 ≤ ≤ 2, 1 < 2 and its ( )= ∑ ( − ( )) = ( 1 − We find at what

a minimum dispersion is achieved: ∆ = −

2∆ ( ) , ∆ =

2∆ ( ) uniform distribution has the maximum entropy.

From ( 8 ) it follows that due to the evolution of the photon lifetime— , the radiation power and the pump current undergo equal value and opposite change in sign. An increase in the value of τ_p causes a simultaneous decrease in the pump current and, equal in absolute value—an increase in the radiation power. Accordingly, a decrease in the value of τ_p will give an increase in the pump proportional decrease in the reflection resonators.

Formulas (8) make it possible to numerically

estimate ∆ и ∆ as functions .

Previously, in (3) we specified an increment

∆ . Let us establish the dependence of ∆ on the reflection coefficient of the object’s mirror.

Equivalent reflection coefficient of a composite resonator:

= 1 + 2 is here and below, r is the intensity reflection coefficient, 1, 2 are coefficients of mirrors of partial

The distances from the common rear mirror to the mirrors 1, 2 are equal, respectively 1, 2, here 1, 2, —optical lengths of and

below resonators.

( ( )) 2 = (−2 1 + 2 = −2 ( 1

+ 2 1 ) 1 2 )+ 2 + (−2 2 + 2 ( 1 + ) 2 ) = 0 − 2 = considering that 1 + 2 = 1,

We made sure that the accepted estimate of the

equivalent length of the composite resonator (in the form of the mathematical expectation of the lengths of partial resonators) is optimal.

The above derivation of the equivalent length

estimate is of a “statistical” nature.

Understanding the importance of correctly determining ℎ is the equivalent length of a composite resonator, we will give, in our opinion, a more “physical” way of determining it, which also leads to a similar result (Section 3). value

Knowing the value

, let’s find the is lifetime of photons in a composite resonator. To do this, we use two equations. We obtain the first equation by assuming in equation ( 1 ) ( )= 0, which is equivalent to a virtual “turn off” of gain in the generating laser. Then ( 1 ) is converted to the form:

We obtain the second equation using the definition of m is a multiple reflection coefficient: where (0)is energy in the resonator at the time of switching off the pump current; ( )= is into account: the number of re-reflections in the resonator during the time .

Let us separate the variables and integrate ( 9 ):

Let us expand (15) into a power series in the

problem of studying vibrations of distant objects, as a rule |2

| ≪ 1, therefore series ( 15 ) converge rapidly. This allows us to confine ourselves to the first two terms of the expansion. Otherwise, higher-order terms should be taken 1 ⃗⃗⃗⃗⃗⃗⃗⃗ = 1 1 − 2 √ 2 √ 2

− 1 √ 1 1 + ⋯.

Let’s substitute the first two terms (16) into (1):

ln ( )= − +

Let’s take a logarithm (10):

ln ( )= ln (0)+ ( )ln .

Then we substitute the initial conditions into the obtained equations: with = 0, ( )= 0; then: = ln (0). Uniting ( 11 ) and ( 12 ), we get: =

2 с ln( )−1 =

2(⃗⃗⃗1 1+⃗⃗⃗2 2) с(⃗⃗⃗1+⃗⃗⃗2)ln(⃗⃗⃗1+⃗⃗⃗2)−1 .

In expression (13) the values ⃗⃗⃗1, ⃗⃗⃗2 are summed

up. This means that the reflection coefficients:

, ⃗⃗⃗2 = √ 2 ( phases, and interfere when added.

The result of interference

: + ), have different ⃗⃗⃗⃗⃗⃗⃗ = 1(1 + 2 Accordingly, √ 2

√ 2 √ 1 1 − ).

−1 1 ⃗⃗⃗⃗⃗⃗⃗ 1 = (1 + 2 √ 2 2 1 1 ( 12 ) ( 13 ) ( 14 ) ( 15 ) ( 16 ) ( 17 ) ( 18 ) ( 19 )

We have obtained the photon balance equation for Р-model LFI.

Here is the number of pump electrons per

unit of time—a quantity proportional to the pump current.

In the resulting equation, the terms in the second term of the right-hand side represent the feedback radiation energy in the main resonator and the resonator of the object. Both radiations, when they enter the active medium of the laser diode, interfere: = ( ) ( )− ( ) (1 + 1 1 2 2 2 − 4 √ 2

√ 2 √ 1 1 − ), where resonator: 1 1 (1 + 2 2 2) = 1 1 1

—the average level of the inverse lifetime of photons in a composite ( )=

1 where: is interference term, which is also a variable component of the reciprocal lifetime of a

According to [13] and our scheme (Fig. 1), the radiation in the laser cavity interferes with the . Useful

radiation reflected by the object: ( )⁄2; = photon in a compound resonator— signal effect ( 8 ) is equal: ( )= Ω is the phase angle between the radiation reflected by the mirror of the main resonator— 1 and radiation reflected by the surface of the object—a mirror 2; Ω is frequency АОМ.

In the formula for the interference of two

quasi-coherent beams, the interference term is multiplied by the correlation coefficient. In our model (Fig. 1), this coefficient is close to one.

Not formally, this can be explained as follows: the radiation of both beams does not leave the composite resonator. It is known that a standing wave in a resonator is coherent throughout its length, which is a condition for generation. For a single-mode laser, this is due to the tendency to concentrate the energy in the highest quality mode. Thus, → . Therefore, the average value of the equivalent lifetime in the composite resonator will be maintained at the maximum possible level.

Although 2 ≪ 1, but because 2 ≫ 1, the corresponding products can be comparable in values: 2 2 ≅ 1 1, from which it follows that the quality factor of the target resonator may well exceed the quality factor of the main resonator, which means that the condition for the maximum quality factor of the composite resonator is the coincidence entropy production ( 6 ) to the distribution of energy between partial resonators if we represent it as a function of the reflection phase difference.

In the task of analyzing the vibrations of

distant objects, the greatest possible independence of the Signal-to-Noise Ratio (SNR) from an increase in the distance to the object is of decisive importance. Since the level of the device’s noise does not depend on the distance to the object, SNR is determined only by the level of the useful signal—S.

Let’s compare the dependence of the useful

signal S on the distance to the object - 2 between the parametric, on the one hand, conventional method, and LFI—method in the interpretation of Lang-Kobayashi [13], on the other. The useful signal, regardless of the schemes, is proportional to the interference term. where: = 2 1 , = (1 − 1)2√ 2

. 1

We accept that: (1 − 1)≈ 1, ≈ 1, = Ωt. 1 is laser resonator length, Ω is frequency АОМ. is frequency and с is the speed of light.

Taking into account the accepted assumptions: = √ln 1−1 1

1 √ 2 cos Ω . 1 1

In the conventional scheme, the -th part of the radiation reflected branched from

the resonator and, by the object (1 − )-th part: ( )(√(1 − ) ( ) + √ 2 ( ) ( + )).

Interference term:

= 2√ − 2 1

√ 2 cos Ω .

Let us accept that = 0.5, because an interference term is at its maximum.

It is known [23] that the power of radiation reflected by an object falls in proportion to the square of the distance to the object. This is equivalent to a corresponding decrease in the reflection coefficient. Let’s replace the constant reflection coefficient 2 with its function from 2: 2( 2)=

0 ( 2⁄ 0) 2 = 0 2 2 equal to the length unit 2.

Here 0—object surface albedo, that is reflection coefficient of an object at a distance 0, ( 20 ) ( 21 ) ( 22 ) ( ( ) + ( ) ( + )).

Interference term according to [2]: = 2 √ 1 1

√ 2 , 1 Ilk(l2) Ic(l2)

l2 = 20 log (

4√ 2 √ 1 ln 1−1 )

The blue-colored upper curve shows the ratio of the useful signal level , calculated according to the P-model, to the level , calculated for the conventional model.

= 20 log

2√ 2 √ 1√( − 2) 1 (24)

From Fig. 2 it can be seen that the P-model demonstrates an excess of the useful signal level at a distance of 100 m from the object by more than 32 dB and by 36 dB at a distance of 300 m about the predicted

value according to the equation [13].

At the same (100 m) object distance, the excess of the

P-model signal over the

conventional one is already more than 40 dB and more than 45 dB at a distance of 300 m.

The conventional model as well as the LFI models, demonstrate equal sensitivity to vibration amplitude, the second sensitivity parameter. Thus, the advantage in sensitivity to reflected radiation is decisive in favor of the design of measuring instruments according to the LFI method. 3. Justification of the Estimation of the

Photon

Lifetime in a

Compound Resonator

Let the radiation energy be at the output of the laser diode (LD). Then the feedback energy is model to the model calculated according to the Lang-Kobayashi equation:

( 23 ) coefficient. coefficients:

, where P is the radiation power, t is the time of formation (filling the resonator) with feedback radiation, and r is the feedback

Considering that in the diagram in Fig. 1, the power P without loss reaches mirror 3, which reflects only 3% of the incoming power, and 97% reaches mirror 5, we can approximately assume that all the radiated power reaches both mirrors— 3 and 5. In practice, the radiated power LD is equal to the power reflected by the main mirror and the mirror of the object: = 1 1 + 2 2 (25)

Let’s take into account that

= 1 + 2, radiation fill time of the resonator = , where —the length of the resonator, —speed of light. We represent (25) in the form: 1 1 + 2

2. Reducing by 2)= 1 1 + 2 2, hence: , we get: = 1 1+ 2 2.

1+ 2

= ( 1 + (26)

− ) 1 √ 1

We have obtained the equivalent lifetime of

photons in a compound resonator in the form ( 18 ).

We are interested in the dependence of the is equivalent photon lifetime in a composite resonator on the distance between the radiation source and the measurement object. Let us replace the reflection coefficient 2 with its dependence on 2 according to ( 22 ): follows that ̃ is the variable component of the equivalent lifetime of photons in a composite resonator, depending on the distance to the object like a square root.

4. Practical Testing of the Proposed

Model

To solve the problem of restoring the form of oscillations of the surface of distant objects, a prototype of a laser Doppler vibrometer (hereinafter referred to as the device) based on feedback interferometry—LFI was created (Fig. 3). The vibrometer has the following characteristics: SNR > 26 db at a distance of at least 300 m from the majority of unprepared surfaces.

Sensitivity to vibration amplitude—not worse

than 1 nm at a vibration frequency of 1 kHz and a bandwidth of at least 10 kHz. Radiation power— no more than 5 mW. Laser diode type—single mode, AR-coated, external cavity—320 mm.

Feedback coefficient—not more than 3%.

The results of the device prototype demonstrated its extremely high sensitivity. So, when the measurement object was removed from the device from 10 to 300 meters, the useful signal dropped from 58 dB to 40 dB. The same high sensitivity was confirmed by the results published in [12], where interference was observed after the probing beam had passed a distance of 40km (in an optical fiber).

Such a high sensitivity cannot be explained

within the framework of generally accepted models [13, 14]. Moreover, our data and the data published in [12] indicate that the achieved measurement distances exceed the calculated coherence lengths of the lasers used.

The discrepancy between the results of experiments and the previously existing theoretical estimates prompts a critical review of the generally accepted LFI model. As a result, a model is proposed that otherwise explains the interaction of individual components of the feedback energy in the formation of a useful signal. This reveals the causes and allows you to determine the quantitative difference in the estimates of the useful signal between the models: generally accepted and proposed.

5. Conclusions and

Further Research

Direction for

As part of the study, the following results were obtained: • The energy equivalence of the useful signal extracted from the resonator radiation and the useful signal extracted from the pump current is shown. Quantitative estimates have been obtained. • The concept of the equivalent length of a composite resonator, which depends on the reflection coefficient, is introduced. The optimality of the introduced estimate in the root-mean-square sense is proved, which gives grounds to consider the lifetime of photons in a composite resonator as an information parameter.

The qualitative advantage of the LFI method

over the conventional one is revealed. As the distance increases, its SNR advantage increases in proportion to the square root of the distance to the object. The LFI model, represented by the LangKobayashi equation, has no qualitative superiority over the conventional method. According to the Lang-Kobayashi model, as the distance to the object increases, the advantage over the conventional method in terms of SNR does not increase. The calculation of the long-range action, performed according to the P-model, shows that this is not the case. The growth of SNR about the LK model, as well as about the conventional model, is proportional to the square root of the distance to the object. This fact makes it possible to radically reconsider the limits of the range of laser measurements.

The representation of the LFI model as a parametric one allows us to consider it as a system that allows parametric resonance, which opens up new possibilities for improving the LFI efficiency.

6. References

References

[1] C. Chen , et al., A New Method of Safety Detection in High-Rise Building Curtain Walls Based on Natural Vibration Frequency , IOP Conf. Ser. Earth Environ. Sci . 687 ( 2021 ) 1 - 11 . doi: 10 .1088/ 1755 - 1315/687/1/012092

[2] Z. Huang , et al., Tension Force Detection of Short Suspender by Using Laser Doppler Vibrometer , Appl. Sci. 12 ( 13 ) ( 2022 ). doi: 10 .3390/app12136536

[3] L. Li et al., Analysis of Backscattering Characteristics of Objects for Remote Laser Voice Acquisition, Applied Optics , 53 ( 5 ) ( 2014 ) 971 . doi: 10 .1364/ao.53.000971

[4] A. Glinsky , Theremin: Ether Music and Espionage, Urbana, Illinois: University of Illinois Press, 2000 , 486 .

[5] S. Lenkov , D. Peregudov , V. Khoroshko , Methods and Means of Protecting Information, Unauthorized Receipt of Information, Kyiv, Arius, 1 , 2008 , 484 .

[6] A. Horev , Acoustic Intelligence Means: Directed Microphones and Laser Acoustic Intelligence Systems, Special Equipment and Communication , 3 ( 2008 ) 34 - 43 .

[7] M. Martseniuk , P. Skladannyi , V. Astapenya , Experimental Studies of the Stand of Simulation Modeling of Laser Microphone Operation for Removal of Acoustic Information, Cybersecur . Educ. Sci. Tech . 2 ( 14 ) ( 2021 ) 131 - 147 . doi: 10 .28925/ 2663 - 4023 . 2021 . 14 .131147

[8] V. Dudykevych , et al., ( 2022 ). Research of Optimization of Lames of the Laser Sensor of Vibration to Counter Laser Intelligence Systems , Cybersecur. Educ. Sci. Tech . 3 ( 15 ) ( 2022 ) 110 - 123 . doi: 10 .28925/ 2663 - 4023 . 2022 . 15 .110123

[9] H. Hulak , G. Otto, Methods and Models of Construction of Information Technologies of Remote Measurement of Nanobration, Cybersecur. Educ. Sci. Tech . 4 ( 8 ) ( 2020 ), 22 - 33 . doi: 10 .28925/ 2663 - 4023 . 2020 . 8 . 2233

[10] A. Horev , A. Savin , Efficiency Research of Sun Protection Window Films for Speech Information Protection from Leakage by Optoelectronic Channel, V 2021 IEEE Conference of Young Researchers in Electrical and Electronic Engineering . IEEE, 2021 . doi: 10 .1109/elconrus51938. 2021 .9396253

[11] Y. Zeng , et al., Test and Analysis of Window Vibration for Anti-Laser- Eavesdropping , Applied Acoustics, 176 ( 2021 ), 107871 . doi: 10 .1016/j.apacoust. 2020 .107871

[12] Y. Tu , et al., Optical Feedback Interferometry with 40km Transmission Distance Based on Adistributed Feedback Fiber Laser , Optic letters, 46 ( 15 ) ( 2021 ) 3805 - 3808 . doi: 10 .1364/0L.433658

[13] R. Lang , K. Kobayashi , External Optical Feedback Effects on Semiconductor Injection Laser Properties , IEEE Journal of Quantum Electronics , 16 ( 3 ) ( 1980 ) 347 - 355 . doi: 10 .1109/JQE. 1980 .1070479

[14] P. Groot , G. Gallatin, S. Macomber , Ranging and Velocimetry Signal Generation in a Backscatter-Modulated Laser Diode , Appl. Opt. 27 ( 21 ) ( 1988 ) 4475 - 4480 .

[15] V. Astapenia , V. Sokolov , P. Skladannyi , Problematic Issues of the Accelerating Lens Deployment , in: IEEE 2nd Ukrainian Microwave Week ( 2022 ) 337 - 341 . doi: 10 .1109/UkrMW58013. 2022 .10037124

[16] V. Astapenya , et al., Last mile technique for wireless delivery system using an accelerating lens , in: IEEE Int. Conference on Problems of Infocommunications. Science and Technology ( 2020 ). doi: 10 .1109/picst51311. 2020 . 9467886 .

[17] V. Astapenya , et al., Analysis of Ways and Methods of Increasing the Availability of Information in Distributed Information Systems , in: IEEE 8th International Conference on Problems of Infocommunications, Science and Technology ( 2021 ). doi: 10 .1109/picst54195. 2021 .9772161

[18] I. Gonorovsky , Radio Circuits and Signals Victor Kamkin , 1981 , 639 .

[19] H. Takuma , Physics of Semiconductor Lasers, 1989 .

[20] K. Petermann , Laser Diode Modulation and Noise , Kluwer Academic, Dordrecht, The Netherlands, 1988 , 315 . doi: 10 .1007/ 978 - 94-009-2907-4

[21] D. Lavis , P. Milligan , The Work of E. T. Jaynes on Probability, Statistics and Statistical Physics, Br. J. for Philos. Sci . 36 ( 2 ) ( 1985 ) 193 - 210 . doi: 10 .1093/bjps/36.2. 193

[22] C. Shannon , Collected Papers Wiley-IEEE Press, 1st Edition, January 1993 , 968 .

[23] M. Skolnik , Radar handbook McGraw Hill, 3rd Edition , 2008 .