KBlueLeaf/taica-nlp-hw1 · Datasets at Hugging Face

text string
the two terms in the right - hand side correspond to the two regions of four - wave correlations .
the quantity @xmath43 entering the right side of eq . [ five ] is the operator of photon density in phase space ( the photon distribution function in @xmath44 space ) .
it was used in refs . @xcite,@xcite and
@xcite for the description of photon propagation in turbulent atmospheres . by analogy
, we can define the two - time distribution function @xmath45 then eq . [ five ] can be rewritten in terms of the distribution functions as @xmath46 let us represent @xmath47 in the form @xmath48 .
we assume that @xmath49 , as explained in the text after eq.[one ] . in this case
the hamiltonian of photons in a turbulent atmosphere can be considered to be independent of time . as a result , both functions defined by eqs .
[ six ] and [ seven ] satisfy the same kinetic equation , i.e. @xmath50 @xmath51 where @xmath52 is the photon velocity , @xmath53 is a random force , caused by the turbulence .
this force is equal to @xmath54 , where @xmath55 is the frequency of laser radiation .
@xmath56 is the refractive index of the atmosphere .
the general solution of the equation for @xmath48 can be written in the form @xmath57 where @xmath58 @xmath59 the functions @xmath60 and @xmath61 obey the equations of motion @xmath62 with the boundary conditions @xmath63 .
the instant @xmath64 is equal to @xmath65 , where @xmath66 is the speed of light .
@xmath64 is the time of the exit of photons from the source .
this choice of @xmath64 makes it possible to neglect the influence of the turbulence on the initial values of operators @xmath67 ( their dependence on time is as in vacuum ) .
the term for @xmath68 can be obtained from eq . [ twelve ] by putting @xmath69 . substituting both distribution functions into eq .
[ eight ] , we obtain @xmath70 @xmath71 @xmath72:\big>,\ ] ] where @xmath73 and @xmath74 are solutions of eqs .
[ twelve ] with the initial conditions @xmath63 and @xmath75 , respectively .
the operators on the right side of eq . [ thirteen ] are related through matching conditions with the amplitudes of the exiting laser radiation ( see ref .
@xcite ) by the relation @xmath76 where @xmath77 is the operator of the laser field which is assumed to be a single - mode field and the subscript ( @xmath78 ) means perpendicular to the @xmath28-axis component .
the function @xmath79 describes the profile of the laser mode , which is assumed to be gaussian - type function [ @xmath80 .
@xmath1 desribes the initial radius of the beam . to account for the effect of the phase diffuser
, a factor @xmath81 or @xmath82 should be inserted into the integrand of eq . [ fourteen ] .
the quantity @xmath83 is the random phase introduced by the phase diffuser .
a similar consideration is applicable to each of four photon operators entering both terms in square brackets of eq .
[ thirteen ] .
it can be easily seen that the factor @xmath84},\ ] ] describing the effect of phase screen on the beam , enters implicitly the integrand of eq .
[ thirteen ] ( the indices @xmath78 are omitted here for the sake of brevity ) .
there are integrations over variables @xmath85 as shown in eq .
[ fourteen ] .
furthermore , the brackets @xmath16 , which indicate averaging over different realizations of the atmosperic inhomogeneities , also indicate averaging over different states of the phase diffuser .
as long as both types of averaging do not correlate , the factor ( [ fifteen ] ) entering eq .
[ thirteen ] must be averaged over different instants , @xmath64 . to begin with ,
let us consider the simplest case of two phase correlations @xmath86}\big > .\ ] ] it is evident that in the case @xmath87 , as shown schematically in fig .
1 , the factor ( [ sixteen ] ) is sizable if only points @xmath19 and @xmath88 are close to one another .
two curves correspond to different instants @xmath18 and @xmath89 . ] therefore , the term given by eq .
[ sixteen ] can be replaced by @xmath90 where @xmath91 is considered to be a gaussian random variable with the mean - square values given by @xmath92 ^ 2\rangle = \langle [ \frac { \partial \varphi ( { \bf r},t_0)}{\partial y}]^2\rangle = 2\lambda _ c^{-2}$ ] , where @xmath93 is the correlation length of phase fluctuations . ( see fig.1 ) . as we see , in this case the effect of phase fluctuations can be described by the schell model @xcite-@xcite,@xcite-@xcite . a somewhat more complex situation is for the average value of @xmath94 given by eq . [ fifteen ] .
there is an effective phase correlation not only in the case of coincident times , but also for differing times . for @xmath95 , two different sets of coordinates contribute considerably to phase correlations .
this can be described mathematically as @xmath96}\big > \approx \big < e^{i[\varphi ( { \bf r},t_0)-\varphi ( { \bf r^\prime},t_0)]}\big > \times\ ] ] @xmath97}\big > + \big < e^{i[\varphi ( { \bf r},t_0)-\varphi ( { \bf r^\prime _
1},t_0+\tau ) ] } \big > \big < e^{i[\varphi ( { \bf r_1},t_0+\tau ) -\varphi ( { \bf r^\prime } , t_0)]}\big > .\ ] ] repeating the arguments leading to eq .
[ seventeen ] , we represent the difference in the last term @xmath98 as @xmath99 then , considering the random functions @xmath100 and @xmath101 as independent gaussian variables , we obtain a simple expression for @xmath102 . it is given by @xmath103}+ e^{-\lambda _ c^{-2}[({\bf r - r^\prime}_1)^2+({\bf r^\prime -r_1})^2]-2\nu^2\tau ^2},\ ] ] where @xmath104 ^ 2\rangle = 2\nu^2 $ ] . as we see , the effect of the phase screen can be described by two parameters , @xmath93 and @xmath105 , which characterize the spatial and temporal coherence of the laser beam . in the limiting case , @xmath106 , the second term in eq .
[ twenty ] vanishes and the problem is reduced to the case considered in refs .
@xcite,@xcite . in the opposite case , @xmath107 , both terms in eq .
[ twenty ] are important .
this is shown in ref .
@xcite . in
what follows , we will see that these two limiting cases have physical interpretations where where @xmath108 ( slow detector ) and @xmath109 ( fast detector ) , respectively .
there is a specific realization of the diffuser in which a random phase distribution moves across the beam .
( this situation can be modeled by a rotating transparent disk with large diameter and varying thickness . )
the phase depends here on the only variable @xmath110 , i.e. @xmath111 where @xmath112 is the velocity of the drift .
then we have @xmath113}+e^{-\lambda _ c^{-2}[({\bf r - r^\prime_1+v}\tau)^2+({\bf r^\prime -r_1+v}\tau)^2]}.\ ] ] comparing eqs .
[ twenty ] and [ twtw ] , we see that the quantity , @xmath114 , stands for the characteristic parameter describing the efficiency of the phase diffuser .
the criterion of slow " detector requires @xmath115 .
qualitatively , the two scenarios of phase variations , given by eqs . [ twenty ] and [ twtw ] , affect in a similar way the intensity fluctuations . in what follows
, we consider the first of them as the simplest one .
( this is because the spatial and temporal variables in @xmath102 , given by eq . [
twenty ] , are separable . )
_ vs _ propagation distance @xmath28 in the case of `` slow '' detector : @xmath116 .
the parameter @xmath117 indicates different initial coherence length . in the absence of phase diffuser @xmath118 ( solid line ) .
@xmath119 is the conventional parameter describing a strength of the atmospheric turbulence . ]
substituting the expressions for operators given by eq .
[ fourteen ] with account for the phase factors @xmath120 and averaging over time as shown in eq .
[ one ] , we obtain @xmath121 @xmath122\bigg > , \ ] ] where the notation @xmath16 after sums indicates averaging over different realizations of the atmospheric refractive index .
the parameter @xmath123 describes the initial coherence length modified by the phase diffuser .
other notations are defined by following relations @xmath124
@xmath125 @xmath126 further calculations follow the scheme described in ref @xcite .
2 illustrates the effect of the phase diffuser on scintillations in the limit of a slow " detector ( @xmath127 ) .
we can see a considerable decrease in @xmath128 caused by the phase diffuser . at the same time , the effect of the phase screen on @xmath128 becomes weaker for finite values of @xmath129 .
moreover , comparing the two upper curves in fig .
3 , we see the opposite effect : slow phase variations ( @xmath130 ) result in increased scintillations .
there is a simple explanation for this phenomenon : the noise generated by the turbulence is complemented by the noise arising from the random phase screen .
the integration time of the detector , @xmath3 , is not sufficiently large for averaging phase variations generated by the diffuser .
the function , @xmath131 , has a very simple form in the two limits : ( i)@xmath132 , when @xmath133 ; and ( ii ) @xmath134 , when @xmath109 .
then , in case ( i ) and for small values of the initial coherence [ @xmath135 , the asymptotic term for the scintillation index ( @xmath136 ) is given by @xmath137 the right - hand side of eq .
[ twfo ] differs from analogous one in ref .
@xcite by the value @xmath138 that is much less than unity but , nevertheless , can be comparable or even greater than @xmath139 . in case ( ii )
, the asymptotic value of @xmath26 is close to unity , coinciding with the results of refs . @xcite and @xcite .
this agrees with well known behavior of the scintillation index to approach unity for any source distribution , provided the response time of the recording instrument is short compared with the source coherence time .
( see , for example , survey @xcite ) .
a similar tendency can be seen in both figs . 3 and 4 : the curves with the smallest @xmath129 , used for numerical calculations ( @xmath130 ) , are close to the curves without diffuser " in spite of the small initial coherence length [ @xmath140 .
it can also be seen that all curves approach their asymptotic values very slowly .
describing diffuser dynamics .
the solid curve is calculated for @xmath118 ( without diffuser ) .
other curves are for @xmath141 . ] . ]
it follows from our analysis that the scintillation index is very sensitive to the diffuser parameters , @xmath0 and @xmath142 , for long propagation paths .
on the other hand , the characteristics of the irradience such as beam radius , @xmath143 , and angle - of - arrival spread , @xmath144 , do not depend on the presence of the phase diffuser for large values of @xmath28 . to see this , the following analysis is useful .
the beam radius expressed in terms of the distribution function is given by @xmath145 straightforward calculations using eq .
[ ten ] with @xmath69 ( see ref .
@xcite ) result in the following explicit form : @xmath146 where @xmath147 and @xmath148 is the inner radius of turbulent eddies , which in our previous calculations was assumed to be equal @xmath149 m . as we see , the third term does not depend on the diffuser parameters and it dominates when @xmath150 .
a similar situation holds for the angle - of - arrival spread , @xmath144 .
( this physical quantity is of great importance for the performance of communication systems based on frequency encoded information @xcite . )
it is defined by the distribution function as @xmath151 simple calculations , which are very similar to those while obtaining @xmath152 , result in @xmath153 ^ 2=\frac 2{r_1 ^ 2q_0 ^ 2}+12tz-\frac { 4z^2}{q_0 ^ 4r^2}(r_1^{-2}+3tq_0 ^ 2z)^2.\ ] ] for long propagation paths ,
. [ twei ] reduces to @xmath154 , which like @xmath152 does not depend on the diffuser parameters .
as we see , for large distances @xmath28 , the quantities @xmath152 and @xmath144 do not depend on @xmath93 and @xmath105 .
this contrasts with the case of the scintillation index .
so pronounced differences can be explained by differences in the physical nature of these characteristics .
it follows from eq .
[ two ] that the functional , @xmath26 , is quadratic in the distribution function , @xmath155 .
hence , four - wave correlations determine the value of scintillation index .
the main effect of a phase diffuser on @xmath26 is to destroy correlations between waves exited at different times .
( see more explanations in ref .

the two terms in the right - hand side correspond to the two regions of four - wave correlations .

the quantity @xmath43 entering the right side of eq . [ five ] is the operator of photon density in phase space ( the photon distribution function in @xmath44 space ) .

it was used in refs . @xcite,@xcite and

@xcite for the description of photon propagation in turbulent atmospheres . by analogy

, we can define the two - time distribution function @xmath45 then eq . [ five ] can be rewritten in terms of the distribution functions as @xmath46 let us represent @xmath47 in the form @xmath48 .

we assume that @xmath49 , as explained in the text after eq.[one ] . in this case

the hamiltonian of photons in a turbulent atmosphere can be considered to be independent of time . as a result , both functions defined by eqs .

[ six ] and [ seven ] satisfy the same kinetic equation , i.e. @xmath50 @xmath51 where @xmath52 is the photon velocity , @xmath53 is a random force , caused by the turbulence .

this force is equal to @xmath54 , where @xmath55 is the frequency of laser radiation .

@xmath56 is the refractive index of the atmosphere .

the general solution of the equation for @xmath48 can be written in the form @xmath57 where @xmath58 @xmath59 the functions @xmath60 and @xmath61 obey the equations of motion @xmath62 with the boundary conditions @xmath63 .

the instant @xmath64 is equal to @xmath65 , where @xmath66 is the speed of light .

@xmath64 is the time of the exit of photons from the source .

this choice of @xmath64 makes it possible to neglect the influence of the turbulence on the initial values of operators @xmath67 ( their dependence on time is as in vacuum ) .

the term for @xmath68 can be obtained from eq . [ twelve ] by putting @xmath69 . substituting both distribution functions into eq .

[ eight ] , we obtain @xmath70 @xmath71 @xmath72:\big>,\ ] ] where @xmath73 and @xmath74 are solutions of eqs .

[ twelve ] with the initial conditions @xmath63 and @xmath75 , respectively .

the operators on the right side of eq . [ thirteen ] are related through matching conditions with the amplitudes of the exiting laser radiation ( see ref .

@xcite ) by the relation @xmath76 where @xmath77 is the operator of the laser field which is assumed to be a single - mode field and the subscript ( @xmath78 ) means perpendicular to the @xmath28-axis component .

the function @xmath79 describes the profile of the laser mode , which is assumed to be gaussian - type function [ @xmath80 .

@xmath1 desribes the initial radius of the beam . to account for the effect of the phase diffuser

, a factor @xmath81 or @xmath82 should be inserted into the integrand of eq . [ fourteen ] .

the quantity @xmath83 is the random phase introduced by the phase diffuser .

a similar consideration is applicable to each of four photon operators entering both terms in square brackets of eq .

[ thirteen ] .

it can be easily seen that the factor @xmath84},\ ] ] describing the effect of phase screen on the beam , enters implicitly the integrand of eq .

[ thirteen ] ( the indices @xmath78 are omitted here for the sake of brevity ) .

there are integrations over variables @xmath85 as shown in eq .

[ fourteen ] .

furthermore , the brackets @xmath16 , which indicate averaging over different realizations of the atmosperic inhomogeneities , also indicate averaging over different states of the phase diffuser .

as long as both types of averaging do not correlate , the factor ( [ fifteen ] ) entering eq .

[ thirteen ] must be averaged over different instants , @xmath64 . to begin with ,

let us consider the simplest case of two phase correlations @xmath86}\big > .\ ] ] it is evident that in the case @xmath87 , as shown schematically in fig .

1 , the factor ( [ sixteen ] ) is sizable if only points @xmath19 and @xmath88 are close to one another .

two curves correspond to different instants @xmath18 and @xmath89 . ] therefore , the term given by eq .

[ sixteen ] can be replaced by @xmath90 where @xmath91 is considered to be a gaussian random variable with the mean - square values given by @xmath92 ^ 2\rangle = \langle [ \frac { \partial \varphi ( { \bf r},t_0)}{\partial y}]^2\rangle = 2\lambda _ c^{-2}$ ] , where @xmath93 is the correlation length of phase fluctuations . ( see fig.1 ) . as we see , in this case the effect of phase fluctuations can be described by the schell model @xcite-@xcite,@xcite-@xcite . a somewhat more complex situation is for the average value of @xmath94 given by eq . [ fifteen ] .

there is an effective phase correlation not only in the case of coincident times , but also for differing times . for @xmath95 , two different sets of coordinates contribute considerably to phase correlations .

this can be described mathematically as @xmath96}\big > \approx \big < e^{i[\varphi ( { \bf r},t_0)-\varphi ( { \bf r^\prime},t_0)]}\big > \times\ ] ] @xmath97}\big > + \big < e^{i[\varphi ( { \bf r},t_0)-\varphi ( { \bf r^\prime _

1},t_0+\tau ) ] } \big > \big < e^{i[\varphi ( { \bf r_1},t_0+\tau ) -\varphi ( { \bf r^\prime } , t_0)]}\big > .\ ] ] repeating the arguments leading to eq .

[ seventeen ] , we represent the difference in the last term @xmath98 as @xmath99 then , considering the random functions @xmath100 and @xmath101 as independent gaussian variables , we obtain a simple expression for @xmath102 . it is given by @xmath103}+ e^{-\lambda _ c^{-2}[({\bf r - r^\prime}_1)^2+({\bf r^\prime -r_1})^2]-2\nu^2\tau ^2},\ ] ] where @xmath104 ^ 2\rangle = 2\nu^2 $ ] . as we see , the effect of the phase screen can be described by two parameters , @xmath93 and @xmath105 , which characterize the spatial and temporal coherence of the laser beam . in the limiting case , @xmath106 , the second term in eq .

[ twenty ] vanishes and the problem is reduced to the case considered in refs .

@xcite,@xcite . in the opposite case , @xmath107 , both terms in eq .

[ twenty ] are important .

this is shown in ref .

@xcite . in

what follows , we will see that these two limiting cases have physical interpretations where where @xmath108 ( slow detector ) and @xmath109 ( fast detector ) , respectively .

there is a specific realization of the diffuser in which a random phase distribution moves across the beam .

( this situation can be modeled by a rotating transparent disk with large diameter and varying thickness . )

the phase depends here on the only variable @xmath110 , i.e. @xmath111 where @xmath112 is the velocity of the drift .

then we have @xmath113}+e^{-\lambda _ c^{-2}[({\bf r - r^\prime_1+v}\tau)^2+({\bf r^\prime -r_1+v}\tau)^2]}.\ ] ] comparing eqs .

[ twenty ] and [ twtw ] , we see that the quantity , @xmath114 , stands for the characteristic parameter describing the efficiency of the phase diffuser .

the criterion of slow " detector requires @xmath115 .

qualitatively , the two scenarios of phase variations , given by eqs . [ twenty ] and [ twtw ] , affect in a similar way the intensity fluctuations . in what follows

, we consider the first of them as the simplest one .

( this is because the spatial and temporal variables in @xmath102 , given by eq . [

twenty ] , are separable . )

_ vs _ propagation distance @xmath28 in the case of `` slow '' detector : @xmath116 .

the parameter @xmath117 indicates different initial coherence length . in the absence of phase diffuser @xmath118 ( solid line ) .

@xmath119 is the conventional parameter describing a strength of the atmospheric turbulence . ]

substituting the expressions for operators given by eq .

[ fourteen ] with account for the phase factors @xmath120 and averaging over time as shown in eq .

[ one ] , we obtain @xmath121 @xmath122\bigg > , \ ] ] where the notation @xmath16 after sums indicates averaging over different realizations of the atmospheric refractive index .

the parameter @xmath123 describes the initial coherence length modified by the phase diffuser .

other notations are defined by following relations @xmath124

@xmath125 @xmath126 further calculations follow the scheme described in ref @xcite .

2 illustrates the effect of the phase diffuser on scintillations in the limit of a slow " detector ( @xmath127 ) .

we can see a considerable decrease in @xmath128 caused by the phase diffuser . at the same time , the effect of the phase screen on @xmath128 becomes weaker for finite values of @xmath129 .

moreover , comparing the two upper curves in fig .

3 , we see the opposite effect : slow phase variations ( @xmath130 ) result in increased scintillations .

there is a simple explanation for this phenomenon : the noise generated by the turbulence is complemented by the noise arising from the random phase screen .

the integration time of the detector , @xmath3 , is not sufficiently large for averaging phase variations generated by the diffuser .

the function , @xmath131 , has a very simple form in the two limits : ( i)@xmath132 , when @xmath133 ; and ( ii ) @xmath134 , when @xmath109 .

then , in case ( i ) and for small values of the initial coherence [ @xmath135 , the asymptotic term for the scintillation index ( @xmath136 ) is given by @xmath137 the right - hand side of eq .

[ twfo ] differs from analogous one in ref .

@xcite by the value @xmath138 that is much less than unity but , nevertheless , can be comparable or even greater than @xmath139 . in case ( ii )

, the asymptotic value of @xmath26 is close to unity , coinciding with the results of refs . @xcite and @xcite .

this agrees with well known behavior of the scintillation index to approach unity for any source distribution , provided the response time of the recording instrument is short compared with the source coherence time .

( see , for example , survey @xcite ) .

a similar tendency can be seen in both figs . 3 and 4 : the curves with the smallest @xmath129 , used for numerical calculations ( @xmath130 ) , are close to the curves without diffuser " in spite of the small initial coherence length [ @xmath140 .

it can also be seen that all curves approach their asymptotic values very slowly .

describing diffuser dynamics .

the solid curve is calculated for @xmath118 ( without diffuser ) .

other curves are for @xmath141 . ] . ]

it follows from our analysis that the scintillation index is very sensitive to the diffuser parameters , @xmath0 and @xmath142 , for long propagation paths .

on the other hand , the characteristics of the irradience such as beam radius , @xmath143 , and angle - of - arrival spread , @xmath144 , do not depend on the presence of the phase diffuser for large values of @xmath28 . to see this , the following analysis is useful .

the beam radius expressed in terms of the distribution function is given by @xmath145 straightforward calculations using eq .

[ ten ] with @xmath69 ( see ref .

@xcite ) result in the following explicit form : @xmath146 where @xmath147 and @xmath148 is the inner radius of turbulent eddies , which in our previous calculations was assumed to be equal @xmath149 m . as we see , the third term does not depend on the diffuser parameters and it dominates when @xmath150 .

a similar situation holds for the angle - of - arrival spread , @xmath144 .

( this physical quantity is of great importance for the performance of communication systems based on frequency encoded information @xcite . )

it is defined by the distribution function as @xmath151 simple calculations , which are very similar to those while obtaining @xmath152 , result in @xmath153 ^ 2=\frac 2{r_1 ^ 2q_0 ^ 2}+12tz-\frac { 4z^2}{q_0 ^ 4r^2}(r_1^{-2}+3tq_0 ^ 2z)^2.\ ] ] for long propagation paths ,

. [ twei ] reduces to @xmath154 , which like @xmath152 does not depend on the diffuser parameters .

as we see , for large distances @xmath28 , the quantities @xmath152 and @xmath144 do not depend on @xmath93 and @xmath105 .

this contrasts with the case of the scintillation index .

so pronounced differences can be explained by differences in the physical nature of these characteristics .

it follows from eq .

[ two ] that the functional , @xmath26 , is quadratic in the distribution function , @xmath155 .

hence , four - wave correlations determine the value of scintillation index .

the main effect of a phase diffuser on @xmath26 is to destroy correlations between waves exited at different times .

( see more explanations in ref .