bjoernp/1-sentence-level-gutenberg-en_arxiv_pubmed_soda

sentences stringlengths 1 243k
on the other hand , there is a technical restriction on phase diffusers : up to now their characteristic times , @xmath2 , are not smaller than @xmath7 .
besides that , in some specific cases ( see , for example , ref .
@xcite ) , the spectral broadening of laser radiation due to the phase diffuser ( @xmath8 ) may become unacceptably high .
the factors mentioned above impose serious restrictions on the physical characteristics of phase diffusers which could be potentially useful for suppressing the intensity fluctuations .
an adequate choice of diffusers may be facilitated if we know in detail the effect of finite - time phase variation , introduced by them , on the photon statistics .
in this case , it is possible to control the performance of communication systems .
in what follows , we will obtain theoretically the dependence of scintillation index on @xmath9 without any restrictions on the value of this ratio this is the main purpose of our paper .
further analysis is based on the formalism developed in ref .
@xcite and modified here to understand the case of finite - time dynamics of the phase diffuser .
the detectors of the absorbed type do not sense the instantaneous intensity of electromagnetic waves @xmath10 .
they sense the intensity averaged over some finite interval @xmath3 i.e.
@xmath11 usually , the averaging time @xmath3 ( the integration time of the detector ) is much smaller than the characteristic time of the turbulence variation , @xmath12 , ( @xmath13 ) .
therefore , the average value of the intensity can be obtained by further averaging of eq .
[ one ] over many measurements corresponding various realizations of the refractive - index configurations .
the scintillation index determining the mean - square fluctuations of the intensity is defined by @xmath14\bigg /\big < \bar{i}\big > ^2= \frac{\big < : \bar i(t ) ^2:\big>}{\big<\bar i \big>^2}-1,\ ] ] where the symbol @xmath15 indicates the normal ordering of the creation and annihilation operators which determine the intensity , @xmath10 .
( see more details in refs .
@xcite,@xcite ) .
the brackets @xmath16 indicate quantum - mechanical and atmospheric averagings .
the intensity @xmath17 depends not only on @xmath18 , but also on the spatial variable @xmath19 .
therefore , the detected intensity is the intensity @xmath20 averaged not only over @xmath18 as in eq .
[ one ] , but also over the detector aperture .
for simplicity , we will restrict ourselves to calculations of the intensity correlations for coinciding spatial points that correspond to `` small '' detector aperture .
this simplification is quite reasonable for a long - distance propagation path of the beam .
in the case of quasimonochromatic light , we can choose @xmath20 in the form @xmath21 where @xmath22 and @xmath23 are the creation and annihilation operators of photons with momentum @xmath24 .
they are given in the heisenberg representation .
@xmath25 is the volume of the system .
it follows from eqs .
[ two],[three ] that @xmath26 can be obtained if one knows the average @xmath27 it is a complex problem to obtain this value for arbitrary turbulence strengths and propagation distances .
nevertheless , the following qualitative reasoning can help to do this in the case of strong turbulence .
we have mentioned that the laser light acquires the properties of gaussian statistics in the course of its propagation through the turbulent atmosphere .
as a result , in the limit of infinitely long propagation path , @xmath28 , only diagonal " terms , i.e.
terms with ( i ) @xmath29 or ( ii ) @xmath30 , @xmath31 contribute to the right part of eq .
[ four ] .
for large but still finite @xmath28 , there exist small ranges of @xmath32 in case ( i ) and @xmath33 , @xmath34 in case ( ii ) contributing into the sum in eq .
the presence of the mentioned regions is due to the two possible ways of correlating of four different waves ( see ref .
@xcite ) which enter the right hand side of eq .
[ four ] .
as explained in ref .
@xcite , the characteristic sizes of regions ( i ) and ( ii ) depend on the atmospheric broadening of beam radii as @xmath35 , thus decreasing with increasing @xmath28 .
in the case of long - distance propagation , @xmath36 is much smaller than the component of photon wave - vectors perpendicular to the @xmath28 axis .
the last quantity grows with @xmath28 as @xmath37 .
( see ref .
@xcite ) .
for this reason , the overlapping of regions ( i ) and ( ii ) can be neglected .
in this case eq .
[ four ] can be rewritten in the convenient form : @xmath38 @xmath39 where the value @xmath40 , confining summation over @xmath41 , is chosen to be greater than @xmath42 but much smaller than the characteristic transverse wave vector of the photons ; this is consistent with the above explanations .
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 .

on the other hand , there is a technical restriction on phase diffusers : up to now their characteristic times , @xmath2 , are not smaller than @xmath7 .

besides that , in some specific cases ( see , for example , ref .

@xcite ) , the spectral broadening of laser radiation due to the phase diffuser ( @xmath8 ) may become unacceptably high .

the factors mentioned above impose serious restrictions on the physical characteristics of phase diffusers which could be potentially useful for suppressing the intensity fluctuations .

an adequate choice of diffusers may be facilitated if we know in detail the effect of finite - time phase variation , introduced by them , on the photon statistics .

in this case , it is possible to control the performance of communication systems .

in what follows , we will obtain theoretically the dependence of scintillation index on @xmath9 without any restrictions on the value of this ratio this is the main purpose of our paper .

further analysis is based on the formalism developed in ref .

@xcite and modified here to understand the case of finite - time dynamics of the phase diffuser .

the detectors of the absorbed type do not sense the instantaneous intensity of electromagnetic waves @xmath10 .

they sense the intensity averaged over some finite interval @xmath3 i.e.

@xmath11 usually , the averaging time @xmath3 ( the integration time of the detector ) is much smaller than the characteristic time of the turbulence variation , @xmath12 , ( @xmath13 ) .

therefore , the average value of the intensity can be obtained by further averaging of eq .

[ one ] over many measurements corresponding various realizations of the refractive - index configurations .

the scintillation index determining the mean - square fluctuations of the intensity is defined by @xmath14\bigg /\big < \bar{i}\big > ^2= \frac{\big < : \bar i(t ) ^2:\big>}{\big<\bar i \big>^2}-1,\ ] ] where the symbol @xmath15 indicates the normal ordering of the creation and annihilation operators which determine the intensity , @xmath10 .

( see more details in refs .

@xcite,@xcite ) .

the brackets @xmath16 indicate quantum - mechanical and atmospheric averagings .

the intensity @xmath17 depends not only on @xmath18 , but also on the spatial variable @xmath19 .

therefore , the detected intensity is the intensity @xmath20 averaged not only over @xmath18 as in eq .

[ one ] , but also over the detector aperture .

for simplicity , we will restrict ourselves to calculations of the intensity correlations for coinciding spatial points that correspond to `` small '' detector aperture .

this simplification is quite reasonable for a long - distance propagation path of the beam .

in the case of quasimonochromatic light , we can choose @xmath20 in the form @xmath21 where @xmath22 and @xmath23 are the creation and annihilation operators of photons with momentum @xmath24 .

they are given in the heisenberg representation .

@xmath25 is the volume of the system .

it follows from eqs .

[ two],[three ] that @xmath26 can be obtained if one knows the average @xmath27 it is a complex problem to obtain this value for arbitrary turbulence strengths and propagation distances .

nevertheless , the following qualitative reasoning can help to do this in the case of strong turbulence .

we have mentioned that the laser light acquires the properties of gaussian statistics in the course of its propagation through the turbulent atmosphere .

as a result , in the limit of infinitely long propagation path , @xmath28 , only diagonal " terms , i.e.

terms with ( i ) @xmath29 or ( ii ) @xmath30 , @xmath31 contribute to the right part of eq .

[ four ] .

for large but still finite @xmath28 , there exist small ranges of @xmath32 in case ( i ) and @xmath33 , @xmath34 in case ( ii ) contributing into the sum in eq .

the presence of the mentioned regions is due to the two possible ways of correlating of four different waves ( see ref .

@xcite ) which enter the right hand side of eq .

[ four ] .

as explained in ref .

@xcite , the characteristic sizes of regions ( i ) and ( ii ) depend on the atmospheric broadening of beam radii as @xmath35 , thus decreasing with increasing @xmath28 .

in the case of long - distance propagation , @xmath36 is much smaller than the component of photon wave - vectors perpendicular to the @xmath28 axis .

the last quantity grows with @xmath28 as @xmath37 .

( see ref .

@xcite ) .

for this reason , the overlapping of regions ( i ) and ( ii ) can be neglected .

in this case eq .

[ four ] can be rewritten in the convenient form : @xmath38 @xmath39 where the value @xmath40 , confining summation over @xmath41 , is chosen to be greater than @xmath42 but much smaller than the characteristic transverse wave vector of the photons ; this is consistent with the above explanations .

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 .