ecommons University of Dayton Monish Ranjan Chatterjee University of Dayton, Fathi H.A. Mohamed University of Dayton

Transcription

1 University f Daytn ecmmns Electrical and Cmputer Engineering Faculty Publicatins Department f Electrical and Cmputer Engineering Investigatin f Prfiled Beam Prpagatin thrugh a Turbulent Layer and Tempral Statistics f Diffracted Output fr a Mdified vn Karman phase Screen Mnish Ranjan Chatterjee University f Daytn, mchatterjee1@udaytn.edu Fathi H.A. Mhamed University f Daytn Fllw this and additinal wrks at: Part f the Cmputer Engineering Cmmns, Electrical and Electrnics Cmmns, Electrmagnetics and Phtnics Cmmns, Optics Cmmns, Other Electrical and Cmputer Engineering Cmmns, and the Systems and Cmmunicatins Cmmns ecmmns Citatin Chatterjee, Mnish Ranjan and Mhamed, Fathi H.A., "Investigatin f Prfiled Beam Prpagatin thrugh a Turbulent Layer and Tempral Statistics f Diffracted Output fr a Mdified vn Karman phase Screen" (214). Electrical and Cmputer Engineering Faculty Publicatins. Paper 3. This Cnference Paper is brught t yu fr free and pen access by the Department f Electrical and Cmputer Engineering at ecmmns. It has been accepted fr inclusin in Electrical and Cmputer Engineering Faculty Publicatins by an authrized administratr f ecmmns. Fr mre infrmatin, please cntact frice1@udaytn.edu, mschlangen1@udaytn.edu.

2 Investigatin f Prfiled Beam Prpagatin thrugh a Turbulent Layer and Tempral Statistics f Diffracted Output fr a Mdified vn Karman Phase Screen Mnish R. Chatterjee 1,* and Fathi H. A. Mhamed 1 1 Department f Electrical & Cmputer Engineering University f Daytn, Daytn, Ohi * Crrespnding authr ABSTRACT Gaussian beam prpagatin thrugh a turbulent layer has been studied using a split-step methdlgy. A mdified vn Karman spectrum (MVKS) mdel is used t describe the randm behavir f the turbulent media. Accrdingly, the beam is alternately prpagated (i) thrugh a thin Fresnel layer, and hence subjected t diffractin; and (ii) acrss a thin mdified vn Karman phase screen which is generated using the pwer spectral density (PSD) f the randm phase btained via the crrespnding PSD f the medium refractive index fr MVKS turbulence. The randm phase screen in the transverse plane is generated frm the phase PSD by incrprating (Gaussian) randm numbers representing phase nise. In this paper, numerical simulatin results are presented using a single phase screen whereby the phase screen is lcated at an arbitrary psitin alng the prpagatin path. Specifically, we examine the prpagated Gaussian beam in terms f several parameters: turbulence strength, beam waist, prpagatin distance, and the incremental distance fr Fresnel diffractin fr the case f extended turbulence. Finally, n-axis tempral statistics (such as the mean and variance) f the amplitude and phase f the prpagated field are als derived. Keywrds: Atmspheric turbulence, Gaussian beam, Mdified vn Karman spectrum, split-step beam prpagatin methd, randm phase screen. 1. INTRODUCTION Atmspheric turbulence effects may have a strng influence n several laser applicatins whereby the turbulence causes fluctuatins in bth the intensity and the phase f the received light signal. Experimental studies f atmspheric turbulence effects n laser beam prpagatin is f imprtance in relating envirnmental parameters t beam prpagatin effects and estimate magnitudes f perturbatins which degrade the perfrmance f laser systems [1]. Theretical descriptins in the intermediate and strng turbulence regimes are less well develped than fr weak turbulence. Knwledge f atmspheric turbulence effects is helpful in the develpment f a wide class f atmsphericptics systems including laser cmmunicatin, energy transfer, remte sensing, and active and passive imaging systems. Inhmgeneities in the temperature and pressure f the atmsphere lead t variatins f the refractive index alng the transmissin path. Index inhmgeneities can deterirate the quality f the received signal and cause fluctuatins in bth the intensity and the phase f the received signal. Several atmspheric turbulence spectral mdels use randm phase screens t mdel the turbulence [2-5]. Cmmn amng these are the Klmgrv, Tatarski, vn Karman and mdified vn Karman spectra (MVKS). The phase fluctuatins f the phase screen used t mdel the randm phase distributin within the aperture are parameterized by the Fried parameter, which describes the transverse cherence length, and the inner and uter scales that determine the amunt f aberratin seen by the prpagating beam. The splitstep prpagatin methd invlving the Fresnel-Kirchhff diffractin integral is used t mdel the prpagatin f the Free-Space Laser Cmmunicatin and Atmspheric Prpagatin XXVI, edited by Hamid Hemmati, Dn M. Brsn, Prc. f SPIE Vl. 8971, SPIE CCC cde: X/14/$18 di: / Prc. f SPIE Vl

3 electrmagnetic wave thrugh turbulence media wherefrm the scintillatin index (SI) and fringe visibility (FV) may be calculated [3]. Mdeling laser beam prpagatin thrugh turbulence using successive phase screens prvides an efficient tl fr tracking the effect f atmspheric turbulence laser beam prpagatin. In this wrk, a split-step beam prpagatin methd (SSBPM) is used whereby either a single randm phase screen r extended (multiple) randm phase screens are placed at arbitrary lcatin(s) alng the prpagatin path. In additin, the PSD f the MVKS spectrum is used in this wrk due t its simplicity while still including bth the inner and uter scales defining the inertial range fr turbulent eddies. In this paper, we have studied the effect f atmspheric turbulence n the prpagatin f a Gaussian beam prfile. This study was prmpted by the eventual desire t prpagate a mdulated chatic wave generated frm an acust-ptic cell with feedback thrugh the turbulence. In the initial stages f the wrk, tw scenaris were fllwed: a single randm phase screen lcated at a specific distance frm the aperture plane fr which the field intensity at the bservatin (image) plane is numerically calculated, and extended (multiple) randm phase screens that are placed an infinitesimal distance (Δz) apart and nce again the field intensity at different psitins and at the image plane is calculated using the split-step algrithm. Three parameters were chsen fr bth cases: strength f turbulence (weak r strng), width f the prfiled Gaussian beam (with beam waist w either wide r narrw), and (c) prpagatin distance L. Mre details abut theses parameters will be discussed with the simulatin results and interpretatin. We als nte that in this paper, we nly reprt the results fr a single planar phase screen placed alng the prpagatin path. Als, sme wrk regarding n-axis time statistics f diffracted amplitude and phase in the image plane during prpagatin in atmspheric turbulence is discussed fr the case f a thin turbulent layer represented by a planar randm phase screen. The utline f this paper is as fllws. In sectin 2, the split-step prpagatin methd incrprating the Fresnel-Kirchhff diffractin integral is discussed briefly. A general verview f the turbulence, its mdels and randm phase screen generatin is presented in sectin 3. The simulatin results and interpretatins are reprted in sme detail in sectin 4. Sectin 5 prvides cncluding remarks and an utline f nging and future wrk. 2. THE SPLIT-STEP BEAM PROPAGATION METHOD Mst atmspheric turbulence simulatins use the split-step algrithm t mimic the turbulence and it is a cmmn methd fr the analysis f laser beam prpagatin thrugh inhmgeneus media. In this paper, we have applied the SSBPM t all numerical simulatins fr single and multiple phase screens and fr time-statistical analyses f atmspheric turbulence. We intrduce here a brief descriptin f SSBPM. It is knwn that the beam prpagatin methd is used widely fr the numerical simulatin f prpagatin thrugh inhmgeneus, anistrpic, and nnlinear materials including waveguiding structures with weak variatins alng the prpagatin directin [9]. The methd relies n cmputing the slutins in infinitesimally small steps taking int accunt the linear and nnlinear (r nndeterministic) steps separately. The key methdlgy behind the SSBPM are tw-fld. First, the expected prpagatin distance is sectined int infinitesimal segments. Secndly, the linear prcesses (such as diffractive integrals) are perfrmed between the spatial frequency dmain r k-space and the spatial crdinates, which are interrelated. Crrespndingly, the randm phase functin defined by the MVKS mdel ( ()) is in the spatial frequency dmain as such; it is then prcessed via a series f transfrmatins s that we finally btain a spatial phase distributin, where the subscripts (i,j) imply spatial crdinates f pints n the chsen grid within which the phase distributin is applied. In the case f extended turbulence (with multiple randm phase screens), nt reprted here, the prpagatin distance, L, is divided int increments Δz =L/n, where is the number f randm phase screens [1]. In the SSBPM, we begin with the prfiled electrmagnetic (EM) beam at z =. The prfiled beam (Gaussian in ur case) travels the first lngitudinal increment (Δz) (frm z t z + Δz). In this distance (Δz), the field is subjected t the familiar Fresnel-Kirchhff diffractin integral with z replaced by Δz as fllws: Prc. f SPIE Vl

4 (, ) = (, ) ( ) ( ), (1) where (, ) is the input prfiled beam, (, ) is the field after distance Δz, k is the unbunded wave number and λ is the wavelength in the medium. When the prfiled EM wave reaches the first randm phase screen, the secnd peratr in the algrithm describes the effect f prpagatin in the absence f diffractin and in the presence f the medium inhmgeneities (randm phase screen in ur case); this is incrprated in the spatial dmain. Hence, after a distance f Δz, the phase perturbatins caused by refractive index fluctuatin arising frm turbulence effects are represented by multiplying the field by a phase functin e (,) as: (, ) = (, ) e (, ), (2) where (, ) is the field amplitude immediately after randm phase screen, and (, ) is the field befre randm phase screen. The abve prcess is repeated until the field has traveled the desired distance. All the simulatin results presented in this paper were carried ut using SSBPM. In the case f a single phase screen, the prpagatin path is divided int tw segments: frm the input prfiled Gaussian beam t the phase screen, and frm the phase screen t the bservatin plane at z = L. The single phase screen may be cnsidered as a special case f the extended phase screen. Fr the extended phase screen (representing a wide range f turbulence), it is fund that a tradeff is needed between the accuracy and cmplexity. Thus, fr a given prpagatin distance, if the number f phase screens is increased, the crrespnding incremental diffractive distance (Δz) will be lwer. This leads t an increase in the prcessing time. Ideally, fr better representatin f the atmspheric turbulence ver an extended medium, ne must use the cnditins Δz and number f phase screens. 3. ATMOSHERIC TURBULENCE In this sectin, we briefly intrduce a brief verview f atmspheric turbulence and in particular the mdified vn Karman (MVKS) phase turbulence mdel. We als discuss numerical generatin f the randm phase screen which describes the randm behavir f the turbulence ver a thin transverse layer in the prpagatin directin. The characteristics f the ptical wave transmitted thrugh atmspheric turbulence can underg dramatic changes resulting in ptential system perfrmance degradatin. In standard turbulence mdeling, three parameters characterizing atmspheric turbulence play an imprtant rle in describing medium behavir: the refractive index structure parameter (C ) which describes the strength f the atmspheric turbulence, and the inner ( ), and uter (L ) scales f turbulence eddies. The wrk reprted here is a part f nging research invlving prpagatin f chatic (and encrypted) EM waves thrugh atmspheric turbulence. 3.1 vn Karman Spectrum The pwer spectrum density f the vn Karman Spectrum (als called the mdified vn Karman spectrum (MVKS) in the frm shwn) is given by [11]: () =.33, (3) where C is the medium structure parameter, = 5.92 is an equivalent wavenumber related t the inner scale, = 2 is a wavenumber related t the uter scale, and k is the unbunded nn-turbulent wavenumber in the medium. In the abve equatin, () represents the s-called pwer spectral density (PSD) f the refractive index f the medium. Prc. f SPIE Vl

5 3.2 Thin phase screen generatin In this sectin we discuss the generatin f a phase screen t mimic the statistical behavir f the phase fluctuatins due t a turbulent atmsphere using a discrete grid and generating the phase screen frm the given spectrum based n fast Furier transfrm (FFT) techniques. The purpse f a phase screen is t simulate the randm phase perturbatins resulting frm randm index fluctuatins in extended atmspheric turbulence [12]. The generated randm phase screen (either planar r extended) in this paper is characterized by several different parameters: C (r Fried parameter r ), inner and uter scales, and L respectively, and the incremental spatial frequencies,. The prcedure f phase screen generatin is as fllws: beginning with the MVKS mdel with given parameters as mentined, and by using a standard scheme based n Furier transfrm generatin, a set f randm cmplex numbers (fllwing a Gaussian distributin) is generated n the chsen grid. Fllwing eq. (4), the randm numbers are multiplied by the square rt f the phase pwer spectrum (PPS) wherefrm an inverse Furier transfrm prduces the phase screen. The real part f the result is taken t be the randm phase functin φ(x, y) due t atmspheric fluctuatins based n the MVKS mdel. The discrete phase distributin in 2-D is given as: = ( + ), (4) where IFFT represents the inverse fast Furier transfrm peratin, is the pwer spectral density (given belw) evaluated in the transverse plane, and a and b are randm numbers generated in rder t apprpriately mimic the randm nise-like characteristics f the vn Karman phase. The MVKS pwer spectral density may be expressed as: () =3. (5) 4. NUMERICAL SIMULATIONS, RESULTS, AND INTERPRETATIONS In this sectin, we present numerical simulatin results f the prfiled Gaussian beam passing thrugh turbulence medium. In this paper, we present results fr a planar randm phase screen placed at a pre-determined lcatin alng the prpagatin. We als reprt sme time statistics crrespnding t the diffracted field amplitude and phase n axis at the image plane. Sme f the parameters are kept cnstant during the simulatin prcess. These parameters cnsist f: number f sample pints (grid reslutin) = 512x512, wavelength λ=1µm, physical size f grid = mmxmm, inner scale l = 1mm, and uter scale L = 3mm respectively. As we mentined earlier, the ther parameters (C r r, w,and L) are varied depending n weak r strng turbulence, wide r narrw Gaussian beam, and different prpagatin path lengths. Fr an extended phase screen we als have the lngitudinal increment (Δz), which is the physical distance between tw successive phase screens. 4.1 Single phase screen In the fllwing, we place the randm phase screen at the lcatins,.5l, and L. A Gaussian beam is then prpagated first frm the bject plane t the phase screen under near-field cnditin, and thereafter, fllwing passage thrugh the screen, nce again via the diffractin integral t the image plane. The resulting beam prfile, phase distributin and the scalar field amplitudes are then determined at the image plane. Case I. Turbulence strength In standard atmspheric turbulence literature, it is knwn that the numerical ranges f the structure parameter C defining strng, intermediate and weak turbulence are defined as: C >1 m-2/3 strng 1 < < 1 m-2/3 intermediate C <1 m -2/3 weak Prc. f SPIE Vl

6 While the abve represents the necessary values f the structure parameter defining a certain turbulence regime, we als need t specify the s-called Fried parameter r crrespnding t the chsen C. This is btained via the relatin: r =.185, (6) where k is the unbunded wave number, and L is the prpagatin distance. Weak turbulence Using the abve infrmatin, ne may arrive at a weak turbulence regime defined by: r =1 mm r C =1.67x1-18 m -2/3. O.4 r O - - Y [m] x [rn] -s - 5 [ml X [m] 1 Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km r= cm ( a=1.671e-18 j3) w=mm Parameters after prpagatin: Maximun Intensity =.7188 (w/ m2 ) Beam waist = mm Fig.1. Gaussian beam prpagatin t distance z=l (phase screen at the bject plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane..--; ti C cu... (ml" - x - L - x h.] 67g' a a) _z - - P-; - -1 [ml 1 y ht2 - X [M] Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km r= cm ( 6=1.671e-18 ni2/3 ) w=mm Parameters after prpagatin: Maximun Intensity = (w / n-? ) Beam waist = mm Fig.2. Gaussian beam prpagatin t distance z=l (phase screen at.5l). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Prc. f SPIE Vl

7 - 5 c E X (MI um- - )(fin") Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km r= cm ( a=1.671e-18 rri2/3 ) w=mm Parameters after prpagatin: Maximun Intensity =.7116 (w / m2 ) Beam waist = mm Fig.3. Gaussian beam prpagatin t distance z=l (phase screen at L r image plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Strng turbulence Similarly, fr this case, we chse r =.1mm r C =1.67x1-13 m -2/3. F) C a) A..... [rnr2 - VI] E - - [m].8.8 a) - - E - -1 X Im! 1 Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km r=2743cm ( a=1.671e-13 rri22 ) w=mm Parameters after prpagatin: Maximun Intensity = (w / m2 ) Beam waist = mm Fig.4. Gaussian beam prpagatin t distance z=l (phase screen at the bject plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Prc. f SPIE Vl

8 I I - - x trrq 1.8 c1.5 c.5 1 a) - n x Ern] 1 Y DT X Im] Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km r=2743cm ( a=1.671e-13 ni213 w=mm Parameters after prpagatin: Maximun Intensity = (w / m2 ) Beam waist =.97852mm Fig.5. Gaussian beam prpagatin t distance z=l (phase screen at.5l). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. "E Si Im ii R ID A CI) c O) - - 7g: O - -1 X [al] 1 O) ca 5 as a -5? r=2743cm ( a=1.671e-13 rrin3 w=mm in.d- - x (m) Parameters after prpagatin: Maximun Intensity =.7116 (w / n-12 ) Beam waist = mm Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km Fig.6. Gaussian beam prpagatin t distance z=l (phase screen at L r image plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Frm Figs. 1-6, we bserve that the Gaussian beam suffers mre amplitude distrtin and likely mre phase fluctuatins fr strng turbulence cmpared t weak turbulence. Regarding the randm phase screen placement, we find that the phase fluctuatins increase when the phase screen is placed at the beginning f the prpagatin distance (at ) r the middle (.5L) cmpared t the end (L) relative t the prpagatin path. An intuitive interpretatin might be as fllws. Thus, when the phase screen is placed at, the prpagated beam passes thrugh the randm phase screen (turbulence) befre any kind f self-diffractin alng the prpagatin path, while when the phase screen at the image plane (at z=l), the prpagated beam is subjected t self-diffractin befre it reaches the randm phase screen at the end. In ther wrds, when an EM beam acquires a randm phase prfile, the resulting phase fluctuatins are mre Prc. f SPIE Vl

9 prnunced as the beam prpagates ver an arbitrary distance under self-diffractin. On the ther hand, when the beam initially self-diffracts with a deterministic phase prfile, and encunters a randm phase at the end, the exiting beam des nt experience a cmparable rate f phase fluctuatins. We may als nte that under weak turbulence, while the phase fluctuatins may still be prnunced, the amplitude r intensity prfile remains relatively unaffected. Als, under weak turbulence, the incident Gaussian underges peak amplitude (r intensity) decay during prpagatin. Since the medium is cnsidered lssless, this decay simply implies a lwering f the Gaussian peak as the prfile bradens due t diffractin. Under strng turbulence, n the ther hand, fr earlier screen placement, the diffracted beam nt nly splits and cnsequently distrts, it als experiences apparent peak intensity increase. This might seem t be cntradictry fr a lssless prpagatin; hwever, we nte that the increased intensity peak is prbably misleading, and mre likely there ccurs split utput amplitudes f ppsite plarities that wuld still cnserve net energy and pwer. Since the plts shw intensity (which is amplitude-squared), this feature is nt visible in the plts. Further quantitative analyses t affirm these findings are pending. Case II. Waist f the Gaussian beam In this case, we cnsider mderate turbulence ( r =.5 mm, r C = x 1-16 m -2/3 ). We cnsider als the physical size f the Gaussian beam, which we define in the fllwing as narrw r wide. In this series, bth narrw and wide will imply Gaussian beams whse spt size (2w ) fits well within the size f the diffractin grid. Thus, narrw and wide beams are nly defined in terms f the relative spt sizes f the beams, and nt necessarily in terms f cmparisn with the aperture. Narrw Gaussian beam (w=1 mm) :.1 cm c- - -tini -1 1 [rn] `dl ftl r= cm (C?i= e -16 n1213 ) w =lmm - Parameters after prpagatin: Grid size =mm x mm Grid Reslutin = 512x512 Pixels Ttal distance L =5 km Maximun Intensity = 42 (w / m2 ) Beam waist = mm Fig.7. Gaussian beam prpagatin t distance z=l (phase screen at the bject plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Prc. f SPIE Vl

10 [m].3.1 Ñ c m.1 - Ê X [m] s 1 Grid size =mm x mm Grid Reslutin = 512x512 Pixels Ttal distance L =5 km r= cm (C F1.5725e -16 rr t3 ) w =lmm Parameters after prpagatin: Maximun Intensity = 4236 (w / m2 ) Beam waist = mm Fig.8. Gaussian beam prpagatin t distance z=l (phase screen at.5l). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane ) L Ern] 1 5 -,,.. 5 I' lifffriril - Irr,T2 x lm1 Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km r= cm ( a=1.5725e-16 rri2/3 ) w=1mm Parameters after prpagatin: Maximun Intensity = 423 (w / n-12 ) Beam waist =2227mm Fig.9. Gaussian beam prpagatin t distance z=l (phase screen at L r image plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Prc. f SPIE Vl

11 Wide Gaussian beam (w=1 mm ) Y E c _ 1 y [m] - - [M] Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km r= cm ( CR=1.5725e-16 m2/3 ) w=1mm Parameters after prpagatin: Maximun Intensity = 1.19 (w / m2 ) Beam waist =99.886mm Fig.1. Gaussian beam prpagatin t distance z=l (phase screen at the bject plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane c X [m] 1 m 5 c a _s,, yj1...::...,.. - [imi- - X [m] Grid size =mm x mm Grid Reslutin = 512x512 Pixels Ttal distance L =5 km r= cm (a= e -16 ri12'3 ) w =1mm Parameters after prpagatin: Maximun Intensity = (w 1 n12 ) Beam waist = mm Fig.11. Gaussian beam prpagatin t distance z=l (phase screen at.5l). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Prc. f SPIE Vl

12 -1 g a) - - :E X [1 - - y Im] Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=5 km r= cm ( Cg=1.5725e-16 ni23 ) w=1mm Parameters after prpagatin: Maximun Intensity =.9752 (w / m2 ) Beam waist =1.7871mm Fig.12. Gaussian beam prpagatin t distance z=l (phase screen at L r image plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Figs.7-12 shw that fr narrw Gaussian beams under mderate turbulence, placement f the phase screen early in the prpagatin likely nce again creates greater verall phase fluctuatin prfiles in the utput beam. Additinally, there is pssibly sme beam amplitude splitting that happens at the utput fr early phase screen placements cmpared with placements further alng the diffractin path. Fr wider Gaussian beams, similar amplitude-splitting behavir is nce again evident; mrever, we als bserve a phase clustering effect arund the edges f the grid when the phase screen is placed well befre the end f the prpagatin path. As befre, there is als peak amplitude r intensity decay in the utput beams. As mentined befre, further quantitative analyses f these phenmena are currently pending. Case III. Prpagatin distance In this case, we again cnsider mderate turbulence ( r =.5 mm r C = x 1-16 m -2/3 ). Fr the cases studies presented here, the Fraunhfer r far-field distance (D 2 /λ, where D is the grid size) happens t be abut 2 km. Thus, all prpagatin cases reprted herein apply nly t near-field diffractin. In the reprt presented here, we simply discuss the effect f prpagatin distance upn the diffracted EM beam within the Fresnel regime, except that we cmpare relatively shrt versus lnger lngitudinal prpagatin distances. Prc. f SPIE Vl

13 L=1km - 1 E. - x [ml.5 c 'L= - O [rr] 1 5 as a. -5 imf... i r= cm ( a=7.8625e-16 r'?... w=mm - Parameters after prpagatin: x Maximun Intensity = (w / ) Beam waist =49.943mm Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=1 km Fig.13. Gaussian beam prpagatin t distance z=l (phase screen at bject plane). 3-D Gaussian beam, Gaussian beam crsssectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane..8 m c O - Ê -.8 C - - Ê x[m] 1 Grid size =mm x mm Grid Reslutin = 512x512 Pixels Ttal distance L =1 km r= cm (C2rF7.8625e -16 g13 ) w =mm Parameters after prpagatin: Maximun Intensity =.9873 (w / m2 ) Beam waist =.8828mm Fig.14. Gaussian beam prpagatin t distance z=l (phase screen at.5 L). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Prc. f SPIE Vl

14 E X [MI 1 trr.f2 Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=1 km r= cm ( a=7.8625e-16 n1213 ) w=mm - Parameters after prpagatin: Maximun Intensity =.9845 (w / m2 ) Beam waist =.8828mm Fig.15. Gaussian beam prpagatin t distance z=l (phase screen at L r image plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. L=1 km.3 77;.1 a) "6.....;.....; [m] x E. - - [m) E X [m] 1 and 3 (.13 _c a Y Inq., le, - x Ern] Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=1 km r= cm ( a=7.8625e-17 ni23 ) w=mm Parameters after prpagatin: Maximun Intensity = (w / m2 ) Beam waist = mm Fig.16. Gaussian beam prpagatin t distance z=l (phase screen at bject plane). 3-D Gaussian beam, Gaussian beam crsssectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Prc. f SPIE Vl

15 .3 > 'y.1 c m Y ] : - - Ê naiti - -1 X [m] 1 5 a) as a -5 Y [m1- fll, : - x[m] Grid size =mm x mm Grid Reslutin = 512x512 Pixels Ttal distance L =1 km r= cm (C?i= e -17 ni213 ) w =mm Parameters after prpagatin: Maximun Intensity =.381 (w 1 m2 ) Beam waist = mm Fig.17. Gaussian beam prpagatin t distance z=l (phase screen at.5l r image plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. rt 3.3 'c7.1 C a) : O [m] - - [m] - 1: - 6? c "e" - O -.."11111! Itill3111"a 1 I" r - -1 X [M] 5. I y [m]- - x[m] Grid size=mm x mm Grid Reslutin=512x512 Pixels Ttal distance L=1 km r= cm ( Ca=7.8625e-17 n[2/3) w=mm Parameters after prpagatin: Maximun Intensity = (w / m 2) Beam waist = mm Fig.18. Gaussian beam prpagatin t distance z=l (phase screen at L r image plane). 3-D Gaussian beam, Gaussian beam crss-sectin, (c) Gaussian beam prfile, Randm phase screen prfile, and Gaussian beam phase distributin in transverse utput plane. Frm Figs.13-18, we find that fr prpagatin t shrter distances, the utput EM beam suffers very little amplitude distrtin r decay, and likely a small amunt f diffractive bradening. On the ther hand, the phase fluctuatins are nce again greater fr earlier screen placements cmpared with placements later alng the prpagatin path. Fr prpagatin t lnger distances, we bserve greater diffractive bradening, as expected. Additinally, there is als greater amplitude decay and greater phase fluctuatin fr earlier screen placement, as befre. There is likely als a small amunt f diffractive beam splitting in the utput fr early phase screen placement. Further quantitative analyses are als pending fr this case. Prc. f SPIE Vl

16 4.2 Tempral statistics We nte at this stage that investigating the prpagatin f EM waves thrugh atmspheric turbulence in the current wrk has been mtivated by the prblem f chatic wave prpagatin thrugh a turbulent layer. While the majrity f turbulence mdels are in the spatial r inverse dmains, the chatic acust-ptic first-rder light is characterized by time-dependent chas. As a result, it becmes necessary t develp the means t track the turbulence as a functin f time. One way t accmplish this is t realize that the randm phase fluctuatins assciated with MVKS is als inherently a time-dependent phenmenn. Thus, the phase f an EM wave will nt nly fluctuate randmly in space (via inner and uter scales), but als with time. In the simulatins presented here, this time fluctuatin enter directly int the prcess because the randm numbers used as part f the generatin f the MVKS phase functin will als autmatically change randmly with time. Thus, ne may evaluate tempral statistical measures f the utput field in the system under study fr bth amplitude and phase in rder t develp an apprpriate tempral mdel. In this sectin, the tempral mean and variance f the amplitude and phase f the diffracted field are cmputed by placing the single randm phase screen at arbitrary psitins alng the prpagatin path (say in steps f.1l frm the bject t the image planes) and evaluating the diffracted Gaussian beam at image plane (z=l). Fr each phase screen psitin, the n-axis mean and variance f the amplitude and phase f the field are calculated. A turbulence strength f r =.1mm r C = 2.7 x 1-13 m -2/3, Gaussian beam waist f w =mm, prpagatin distance L=5km, and number f sample pints (grid reslutin) f 513x513 are used fr this cmputatin. The number f samples (iteratins) used t calculate the mean and variance f the amplitude and phase f the field is times (N=), i.e., the randm phase screens acrss which the EM wave prpagates are re-created times. Fig.19 shws the mean and variance f the amplitude and phase f the diffracted Gaussian beam at z=l fr different randm phase screen psitins Mean and variance Amplitude(mean) Amplitude(variance) Phase(mean) Phase(variance) Nrmalized phase screen lcatin Fig.19. On-axis mean and variance f the amplitude and phase f diffracted Gaussian beam at Z=L fr different randm phase screen psitins. We nte here that tempral statistics fr pints in the paraxial regin as well as sufficiently ff-axis have als been cmputed. Clearly, the verall scales f the time statistics in terms f crrespnding inner and uter scales in time need t be examined and derived further. These and ther relevant results will be analyzed and presented in fllw-up wrk. Prc. f SPIE Vl

17 5. CONCLUDING REMARKS In this wrk, we have studied the influence f atmspheric turbulence n prfiled Gaussian beams prpagating ver distances in the 1 t 1 km range. A PSD f the MVKS was used t generate the randm phase screen representing the randm behavir f the turbulent atmsphere. Furthermre, a split-step prpagatin methd invlving Fresnel- Kirchhff diffractin cmbined with transmissin thrugh a thin randm phase screen was applied. Numerical simulatins are presented fr a narrw regin f turbulence. Diffracted utput fields are derived fr different system parameters cnsisting turbulence strength, Gaussian beam waist and the prpagatin distance. The simulatin results shw that the diffracted Gaussian beam underges greater distrtin fr extended phase turbulence (results fr this will be discussed elsewhere). Als, the Gaussian beam waist and the prpagatin distance are fund t have a direct impact n the diffracted field under narrw atmspheric turbulence. Finally, n-axis tempral statistics (mean and variance) f the amplitude and phase f the diffracted Gaussian beam have been calculated in anticipatin f the use f this infrmatin in tracking subsequent prpagatin f (mdulated) chatic waves thrugh the turbulence. The gal wuld be t examine if a chatic wave exhibits any degree f immunity relative t turbulence cmpared with deterministic, nn-chatic waves. REFERENCES [1] L. Sjöqvist, M. Henrikssn and O. Steinvall, Simulatin f laser beam prpagatin ver land and sea using phase screens a cmparisn with experimental data, Prc. SPIE 5989, (5). [2] V.S. Ra Gudimetlaa, R.B. Hlmesb, T.C. Farrella, and J. Lucas, Phase screen simulatins f laser prpagatin thrugh nn-klmgrv atmspheric turbulence, Prc. SPIE 838, (May 211). [3] E.M. Whitfield, P.P. Banerjee and J.W. Haus, Prpagatin f Gaussian beams thrugh a mdified vn Karman phase screen, Prc. SPIE 8517, (Oct. 212). [4] L.C. Andrews, R.L. Phillips and A.R. Weeks, "Prpagatin f a Gaussian-beam wave thrugh a randm phase screen," Waves in Randm Media 7, (1997). [5] X. Zhu and J.M. Kahn, Free-space ptical cmmunicatin thrugh atmspheric turbulence channels, IEEE Trans. Cmm., n. 8 (Aug. 2). [6] R.R. Kumar, A. Sampath and P. Indumathi, Secure ptical cmmunicatin using chas, Opt. Cmm., Indian J. Science and Tech. 4, n. 7, , (July 211). [7] L. Larger and J. Gedgebuer, Encryptin using chatic dynamics fr ptical telecmmunicatins, C. R. Physique 5, (4). [8] M. R. Chatterjee and M. Alsaedi, Examinatin f chatic signal encryptin and recvery fr secure cmmunicatin using hybrid acust-ptic feedback Opt. Eng., n. 5, (May 211). [9] J. D. Schmidt, Numerical Simulatin f Optical Wave Prpagatin with Examples in Matlab, SPIE Press: Bellingham, WA (21). [1] T. C. Pn and T.Kim, Engineering Optics with Matlab, Wrld Scientific: Singapre (6). [11] L.C. Andrews and R.L. Phillips, Laser Beam Prpagatin thrugh Randm Medium, 2nd Ed., SPIE Press: Bellingham, WA (1998). [12] M.C. Rggemann and B.M Welsh, Imaging Thrugh Turbulence, CRC Press: Bca Ratn, FL (1996). Prc. f SPIE Vl