Jan K.Jabczyski,Przemys?aw Gontar

Military University of Technology,Gen.S.Kaliskiego 2,Warsaw,00-908,Poland

Keywords: Laser beams Beam combining Coherence Atmosphere turbulences

ABSTRACT The performance of a laser weapon system based on coherent beam combining(CBC)depends on its propagation properties in the atmosphere.In this study,an analytical model based on partial coherent beam combining(PCBC)for assumed coherence coef ficients between beams in a CBC lattice was developed.The Kolmogorov model of atmospheric turbulence and the Hufnagel-Valley model of Cn2 dependence on atmospheric parameters were implemented.Novel simpli fied metrics were proposed to assess the CBC performance.Several beam pro files(super-Gaussian,truncated Gaussian,etc.)and geometries were analyzed in terms of maximal intensity in the far field.An approximate formula for PCBC ef ficiency dependent on the Fried radius was proposed.The results of CBC modeling were compared to those of the Gaussian beam propagation model in a turbulent atmosphere.The dependence of CBC performance on the Cn2 parameter,range,and elevation angle was analyzed.It could be concluded that the application of CBC for medium and long range propagation is impractical without an effective adaptive optics system.

1.Introduction

At present,depending on the application,laser weapon systems(LaWS)require the development of laser sources with a near-single spatial mode and output continuous wave(CW)powers of dozens of kW nearing MW[1-3].However,fundamental physical effects that limit the available output power from a single aperture laser source to a few kWhave been identi fied(see e.g.Refs.[3-7]).The use of beam combination techniques appears to be the most promising approach to increase the CWlaser output power beyond 100 kW[8-12].In principle,these techniques can be divided into two categories:1)those that use serial devices with a singleaperture output and 2)those that use parallel devices with a “tiled” aperture,composed of a two-dimensional(2D)array of laser beams.In 2016,power values above 30 kW were obtained by applying spectral beam combination to a case of single-aperture output[13];however,certain fundamental physical barriers were still identi fied.In the case of parallel(or “tile” )addition,coherent beam combining(CBC)and incoherent beam combining(ICBC)techniques have been proposed and extensively explored both theoretically and experimentally.In the ICBC technique,the farfield power density is proportional toN(the number of emitters).Such systems have been experimentally demonstrated and found to be feasible with respect to long propagation distances in the atmosphere[14-18].

According to a simple physical model of the CBC system[8,9],the maximal intensity is proportional toN2.However,owing to the lattice architecture,beam pro file,technical imperfections,and atmospheric turbulence,realistic estimates of the averaged power density in the far field can be much lower.Thus,the usefulness of CBC and its ability to outperform ICBC come into question.This has been considered in several works[9-12,19,20],but no de finite conclusion has been reached thus far.The experimental demonstration of effective CBC for horizontal paths on 7 km range with Target in the Loop setting[21-23]was demonstrated in last decade.

This study aims to analyze the feasibility of CBC for LaWS while taking into account the limitations posed by atmospheric turbulence.The semi-analytical model of partial CBC(PCBC)[23,24]has been implemented here.This model was developed to calculate the intensity distribution of the CBC of a laser beam array in the far field based on known coherence coef ficients for partially combined beams.Kolomogorov’s model of atmospheric turbulence[27],which enables the calculation of coherence radius based on thestructure parameter,and the Hufnagel-Valley(H-V)model[29]of theparameter dependent on the height over ground were assumed.

In Section 2,the proposed model is described,and simpli fied metrics are suggested to assess CBC performance based on the laser transmitter and atmosphere parameters.The analysis of several beam pro files(super-Gaussian,truncated Gaussian)and geometries is then presented.In the next part of the paper,the analysis of the impact of the atmospheric turbulence level and elevation angles on CBC performance is discussed.

2.Model

2.1.PCBC model

2.1.1.Definition of 2D laser beam array and beam profile in farfield

The proposed here analytical model can be applied to any 2D architecture of lattice.The architectures of 2D array could be different.The most compact consisted of central beam and two outer crowns of 6 and 11 beams(Fig.1),was considered with the parametersb(half period of lattice)and(cm,x,cm,y)b(coordinates ofm-beam center).The alternative architecture of 3 hexagonal prototype sublattices consisted of 7 beams each was applied in experimental works[22,23].

The main assumptions are discussed below,the details and mathematical rigorous description was given in Ref.[24].Each beam of the given amplitude functionA1(r1)defined in the exit pupil of the laser transmitter is propagated at a distanceLaccording to the Fourier-Bessel transform

Fig.1.Architecture of CBC lattice:19 beams ordered in hexagonal lattice:b-lattice half period,D=10 b-diameter array aperture,raper-radius of aperture of individual beam let,w0-gaussian beam radius of beamlet,(cm,x,cm,y)b-cooridnates of mbeamlet center.

where:λ-wavelength,r1-radius in exit pupil of collimator in transmitter,(x2,y2)-point coordinates defined in target plane distant onr2-radius to the center of target plane,raper-radius of collimator aperture.

2.1.2.Partial coherent summation of N beams

During propagation in turbulent atmosphere laser beam experiences random wavefront distortions.Usually to analyze such process the numerical model based on paraxial wave equation with a set of phase random screens is applied(see e.g.Refs.[11,20,27]).Instead of it,we proposed the analytical approach consisted in partially coherent summation of coherent beamlets defined by formula(1).We have limited the scope of analysis to the case of focusing i.e.the distance to the targetLis equal to the wavefront radiusFof exiting laser beam(F=L).The amplitude functionA2,mof eachm-beam in target plane is represented by the fieldA2multiplied by a phase function corresponding to the tilted wavefront at the(αm,x,αm,y)tilt angle as follows:

where

The different as a rule,tilts of each beamlet enable to focus the beam in the center point of target plane.In terms of geometric optics each beamlet chief ray crosses the target plane in the center point of target plane(0,0).The practical realization of such precise aiming for each beamlet is made by micromovement of fiber tip[23]or outer segmented mirror.

We assumed that the main physical effect of turbulent atmosphere on propagation of beamlets is the decrease in partial coherence.To calculate average,partial coherent superposition of beamlets in target plane the analytical expression for partial coherence summation was applied with a priori defined partial coherence coef ficients(see details in Refs.[24]).The average over propagation distance for the so-called‘long exposure case’partial coherence coef ficientμm,lbetween themth andlth beamlets is defined by Kolmogorov’s model of atmosphere turbulences(see e.g.Refs.[27])as follows:

where:ρcohis the coherence radius of atmosphere,andΔcm,lis the relative distance between centers of themth andlth beam as follows:

The final formula for the PCBC intensityIPCBC[24]is:

where:

2.1.3.CBC metrics

The Power in the Bucket(PIB)metric has to be known in order to assess the beam quality[23-27].To determine PIB,the intensity distribution has to be calculated in an area of at least 3-5 Airy radiirAiry=1.22λL/Din the far field.The bucket radius at targetbTis taken to be equal to Airy radiusbT=rAiryusually.This is a laborious procedure that does not lead to simple,general conclusions,because the effects of the lattice geometry,beam pro file,and partial coherence are largely indistinguishable.

Therefore,a simpli fied approach is proposed here based on the principle of the Strehl ratio(see e.g.Ref.[28])which corresponds to a case of small bucket radiusbT?rAiry.We intend to find the maximal intensity in the center point of far field(x2=0,y2=0).Let us note that

whereff=raper/b-filling factor.

Thus maximal intensity in far field is given by:

where:ηPCBC,0-PCBC ef ficiency dependent on lattice structure and coherence coef ficients as follows:

IBL,max-maximal intensity in far field for individual beamlet defined in such a way that forff=1 and top hat distributionIBL,max=1:

where

Such an approach enables separation of the analysis of the optimal shape/pro file of a laser beam and the atmospheric turbulence effects using formulae(11)and(12),respectively.

Speci fically,the lattice structure information is included in formula(11).Moreover,the results depend on the applied model of coherence radius based on atmospheric parameters(see p.2.2).We assume here‘long exposure time case’and fully coherent individual beamlet at the target plane,which may be the most controversial aspect of the proposed approach.However,as will be shown in p.2.2-2.3,this method led us to fairly general conclusions.

2.2. models of turbulent atmosphere

The level of turbulences in atmosphere is characterized by-refractive index structure constant depending on local meteorological conditions as:temperature,air pressure,wind speed,insolation,season,time of day,height over ground,topography,etc.[27].

In the proposed model,the value of thefor slant rays at elevation angleα(see Fig.2)was calculated based on the approximated formula of the Hufnagel-Valley model(H-V)ofdependence on height over ground[29]:

whereR0,HV=0.1 km is the height for which theCn,HV2drops to level 1/e as per the H-V model,-refractive index structure constant on the ground.

To compare the different turbulence conditions for given propagation length it is convenient to use the unidimensional parameter called Rytov variancedefined for slant path and H-V model as follows[27]:

The level of signi ficant turbulences corresponds to>1.

The H-V model enables the calculation of the effective Fried radius of slant rays valid for Gaussian beam case[27],assuming‘focusing case’i.e.the distance to target is equal to wavefront radius of Gaussian beam at exit pupil of emitter(L=F),as follows:

Fig.2.Scheme of CBC-LaWS focused at a distance L for slant path with elevation angleα.

where:k=2π/λ-wavenumber,β=sin（α）L/RHVandNF=Lλ/πw02-Fresnel number.Let us note,that the coherence radius defined by(16)dependent on beam radiusw0is close to the‘spherical wave’case and signi ficantly different from‘plane wave’case for which the Fried radius was defined[27].

In formulae(7)and(11),it is assumed that the intensity of individual beamlet in far field is not dependent on,which is an excessive approximation for a high level of.To correct this approximation,a formula for the relative Gaussian beam intensity ηGAwas applied[27,30]with respect to ideal “non turbulent” case=0)as follows:

whereWf-beam radius at target plane in vacuum=0),WLTbeam radius at target plane for long exposure time.Following the approach given in Ref.[27]for a focusing case(L=F)we obtained the following formula enabling calculationηGAfor slant paths:

Finally,a correction to the PCBC ef ficiency in formula(11)is proposed as follows:

2.3.Approximate formula on PCBC ef ficiency

PCBC ef ficiency(formula(11))depends on lattice architecture(aperture,distance between beamlets,geometry)atmospheric structure parameter,elevation angleαand rangeL.In some cases these parameters are not available and exact calculation of formula(11)is not possible.Thus,we have tried to find an approximate formula on PCBC ef ficiency(see Fig.3).By analyzing several cases of PCBC,it has been found that the PCBC ef ficiency can be approximated by the following formula:

Fig.3.PCBC ef ficiency vs.ratio of aperture diameter D to Fried radius r0-rhombsexact formula(11),continuous blue curve approximated formula(21),crosses-errors of approximation.

whereDis the aperture diameter of the CBC lattice(for 19-beamsD=10b).

This approximation simpli fies the analysis of the impact of atmospheric turbulence in the case of CBC systems.The primary factor here isD,which represents the whole aperture of the lattice.Thus,a CBC system can be considered as a type of partially coherent single aperture laser system with an effective aperture depending on the ratioD/r0.In a limiting case in which the‘hypothetical’adaptive optics system can increase the effective Fried radius to be signi ficantly greater than the apertureD,the PCBC ef ficiency tends to 1.However,it must be noted that this is only the first order approximation,and it does not take into account the particular lattice structure or the speci fics of the given CBC lattice 2Dintensity distributions in the far field.

In general,formulae(12)and(21)both enable the performance estimation of any CBC system.The main advantage is that by applying such a simpli fied approach,the limiting operational parameters of an existing or designed CBC system can be assessed for atmospheric turbulence without extensive details such as the compensation of phase errors and residual aberrations.

3.Analysis

Following the method explained in Section 2,the impacts of two typical laser beam pro files have been analyzed.

3.1.Properties of truncated Gaussian beams in CBC

Fig.4.Intensity in far field(red curve)IBL,max for truncated Gaussian beam and truncation losses(blue curve)vs.relative Gaussian beam radius(w0/raper).

The simplest method to manipulate the laser beam pro file is to truncate it by applying a‘hard’diaphragm.In this case,the Gaussian beam radius at aperturew0related to the single emitter aperture radiusraperis a control parameter.Such a case has been analyzed(see Fig.4),the truncation losses of the Gaussian beam defined as γtrunc=exp[-2(w0/raper)2]were calculated as well.

The obtained here optimal relative radiusw0/raper=0.847 corresponding to truncation losses of 8.1%was found as a maximum of intensity in far field(formula(11))is consistent with results of[10]and with our previous results[25].In such a case,73.5%of the maximal intensity is obtained.It should be noted that this approach is permissible only for low power CBC systems.

3.2.Properties of super-Gaussian beams in CBC

In the case of a high-power laser system,the admissible level of truncation losses has to be defined,and it is generally much lower than 1%(see Fig.4).Considering the Gaussian beam pro file,this leads to very low CBC intensity.Therefore,several designs of beam shapers have been considered to make the Gaussian pro file more convenient(see e.g.Refs.[31,32]).To analyze the optimal laser beam pro file for CBC with a negligible level of truncation losses,the family of super-Gaussian(SG)beams[26,27]has been used here,defined as follows:

As shown in Fig.5a,the pro file of the SG beam depends on the exponentpand radiuswp.Assuming the truncation losses are below 0.05%,the speci fic radiuswphas been found for eachp.Under such an assumption,the maximal intensity is solely dependent onp(see Figs.5b and 6).

Fig.5.(a)Intensity pro files of super-Gaussian beams in the near field.(b)IBL,max vs.p for super-Gaussian beam.

The dependence of the maximal PCBC intensity of SG beams on the ratio between the aperture diameter and Fried radius is shown in Fig.6.Forp=4,the same value of CBC intensity as in a case of Gaussian beam with optimal truncation level is achieved.

Fig.6.Maximal PCBC intensity of super-Gaussian beams(each having truncation losses below 0.0005)vs.ratio of aperture diameter D to Fried radius r0.

The further increase in thepexponent results in an increase in the PCBC intensity up to the limit of 90%defined for 95%of the filling factor and top hat pro file.However,the increase in thepexponent results in more challenging design and manufacturing requirements for an appropriate beam shaper.Thus,a reasonable compromise is needed.We consider that ap≈8 would enable a compromise between the requirements of maximal CBC ef ficiency and manufacturability of the beam shaper.

3.3.Impact of elevation angle and Cn2 on far-field intensity for a single aperture Gaussian beam

In general,a LaWS deployed on ground-based or maritime platforms operates over slant paths with elevation angles of a few to a few dozen degrees.Owing to the exponential decrease of thewith the height over ground,the performance of the LaWS significantly depends on the elevation angle.

To more clearly show the change in the LaWS ef ficiency with respect to the elevation angle anddefined at a ground,such dependencies have been presented for LaWS based on a singleaperture Gaussian beam(see Figs.7 and 8).The presented at Fig.7 Rytow varianceσ2Rdata were calculated for horizontal pathL=1 km.It was found that averaged over slant path Rytov variance deceases about 40 times between horizontal and vertical directions for 1 km range.

Let us note that for slant pathsCn,HV2decreases according to HV model,the relative intensity of Gaussian beam is calculated according to formula(18).

The same Gaussian beam radius as that in the case of an individual beamlet of a CBC system(analyzed in p.3.4 and p.3.5)was assumed in calculation.It is evident that the horizontal path is the worst case for LaWS operation,while between the horizontal path and 30°elevation,the admissible level ofcan differ(see Fig.8).

Fig.7.Maximum intensity for Gaussian beam vs.elevation angle for several =0.5 10-14 m-2/3,=0.5-red curve,=1 10-14 m-2/3=1-blue curve,=2 10-14 m-2/3=2-red curve;L=1 km,w0=16.1 mm.

Fig.8.Maximum intensity for Gaussian beam vs.Cn2 for several elevation angles;L=1 km,w0=16.1 mm.

3.4.Impact of elevation angle and on CBC performance

The typical dependence of LaWS ef ficiency based on a CBC systemvs.and elevation angles does not qualitatively differ from the LaWS based on a single aperture beam.

To quantify these differences,the maximal intensities taking into account diffraction and turbulence effects were calculated for equivalent cases:the single aperture beam with Gaussian beam radius equivalent to the CBC single emitter beam(Fig.9),the single aperture beam with Gaussian beam radius equivalent to the whole apertureDof 19-beam lattice(Fig.10)and the final case of CBC of 19-beams(Fig.11).The comparison of three cases for horizontal and vertical paths was presented in Fig.12.For slant pathsCn2decreases according to H-V model(formula(14)),the relative intensity of Gaussian beam is calculated according to formula(18),Fried radius is determined by formula(16).Fig.9.2D map of intensity in logarithmic scale of Gaussian beam,w0=16.1 mm,Cn2=10-14m-2/3.;X-horizontal(0.125 km,1 km),Yvertical axis(0.125 km,1 km).

Fig.10.2D map of intensity in logarithmic scale:Gaussian beam,w0=80 mm,Cn2=10-14 m-2/3.X-horizontal(0.125 km,1 km),Y vertical axis(0.125 km,1 km).

The Gaussian beam with the lowest width is evidently the least sensitive to atmospheric turbulence(Figs.7-9);however,it also offers the lowest laser power.The choice of appropriate aperture for an individual beamlet depends on the operation range of the LaWS,as well as on the technical limits imposed by the detrimental effects of the thermal-optics in the optical system of the laser effector.

There is no signi ficant difference between the case of CBC(Fig.11)and a single aperture beam with equivalent Gaussian beam diameter(Fig.10)for horizontal path(Fig.12).However,the advantage of CBC manifests for slant paths(see Figs.10-12).

Fig.11.2D map of CBC intensity in logarithmic scale:CBC-19-beams,D=200 mm,w0=16.1 mm,Cn2=10-14 m-2/3.X-horizontal(0.125 km,1 km),Y vertical axis(0.125 km,1 km).

Fig.12.Intensity vs range L for horizontal(solid lines)and vertical(dashed lines)and three cases of Gaussian beam of individual beamlet,w0=16.1 mm(red color),Gaussian beam of full aperture w0=80 mm(blue color)and CBC case D=200 mm,w0=16.1 mm,(black color).

As was shown in Fig.12 for vertical path the intensity of CBC beam is close to single wavelet beam(dashed black and red curves)whereas the intensity of full aperture Gaussian beam falls much faster.

It is caused by decreased impact of turbulence on coherence radius of focused Gaussian beam for slant paths(formula(16)).

It should be noted that a wider aperture(Fig.10)does not indicate higher output power in a single fundamental mode,whereas CBC(Fig.11)offers N times higher power with comparable performance to single aperture(Figs.9 and 10)for elevation angles>20°.

4.Conclusions

The dependence of CBC performance on theCn2parameter,elevation angle,and range was analyzed.Simpli fied CBC performance metrics based on maximal intensity in the far field were proposed.Applying this approach the analysis of the optimal beam pro file and lattice geometry can be separated from the impact of atmospheric turbulence.The proposed formulae on maximal intensity of beamlet(11)and approximated PCBC ef ficiency(21)based on the Fried radius can generally enable the estimation of performance of any CBC system without requiring technical details.

Optimal laser beam shapes were determined for cases of truncated Gaussian beam pro files with de finite truncation losses and super-Gaussian pro files with negligible truncation losses.The advanced optical technologies of optical coatings and glass production has pushed the limit of admissible CWpower density to the level of dozens kW/cm2[33].Thus,the barrier of admissible power density in laser optics elements has improved signi ficantly.Considering truncation loss limitations,the main challenge is related to the beam shaper technology.

The analysis of CBC performance was carried out for the socalled‘long exposure time’model of atmospheric turbulence;thus,any effects of beam wandering and wavefront compensation have not been taken into account.The results of CBC modeling were compared to those of single aperture Gaussian beam propagation in a turbulent atmosphere.Compared to the Gaussian beam of diameter equivalent to the CBC lattice diameter,the CBC system presents the comparable performance for horizontal paths.However,for slant paths(elevation angles>30°)the advantage of CBCLaWS is evident.

Analyzing formula(21)is it evident that the signi ficant decrease in CBC performance for a LaWS without adaptive optics(AO)appears for the Fried radiusr0?D.Fried radius for horizontal path of a few kms is of a few cm for typical atmospheric conditions,thus it limits practical application of CBC without AO to very low turbulence level(σ2R<0.2).On the other hand,it was shown in theoretical analysis[10,11]and con firmed experimentally[21-23],that the CBC system with advanced AO based on Target in the Loop concept can operate successfully for 7-km horizontal path up to the Fried radius comparable to individual beamlet collimator aperture(of 3-4 cm).The similar effect shown in our analysis for vertical path(see Fig.12 dashed lines)can be explained by the fact that turbulences occurs here in near field at a propagation distance ofRHV=0.1 km and do not disturb the coherent summation in target plane.

Overall,it was concluded that the application of CBC-based LaWS for medium and long range propagation is impractical without an effective adaptive optics system.

Declaration of competing interest

We declare there is not any con flict of interest connected with this publication.Both authors have seen manuscript and approved to submit to this journal.The work has not been published or submitted for publication elsewhere,either completely or in part.No materials are reproduced from another source.

Acknowledgments

This work was financed in the framework of the strategic program DOB-1-6/1/PS/2014 funded by the National Center for Research and Development of Poland.I would like to thanks Dr D.Sabourdy from CILAS for inspiration and critical remarks.