Scece Cha Seres F: Iformato Sceces 008 SCIENCE IN CHINA PRESS Sprger www.sccha.com fo.sccha.com www.sprgerlk.com Statstcal characterstcs of the ormalzed Stokes parameters LIU Tao 1 WANG XueSog 1 & XIAO ShuPg 1 1 School of Electroc Scece ad Egeerg Natoal Uversty of Defese Techology Chagsha 410073 Cha; School of Electroc Egeerg Naval Uversty of Egeerg Wuha 430033 Cha The statstcal propertes of the ormalzed Stokes parameters a Gaussa stochastc plae wave feld are descrbed detal. Va the expresso of the three ormalzed Stokes parameters the mea varace ad hgh-order momets are calculated whch smplfy C. Brosseau s results. The ew dsperso ormalzed cotrast fucto skewess ad kurtoss are defed to descrbe the o-gaussa dstrbuto characterstcs whch ca be appled to Gaussa wave felds relatg to depolarzato of lght by a spatally radom medum. stataeous polarzato Stokes vector statstcal aalyss the ormalzed Stokes parameters 1 Itroducto Polarzato s oe of the propertes of electromagetc waves whch ca be descrbed usg a determate ellpse whe the beam s moochromatc [1]. However for tme-varyg electromagetc waves the waves vector tp wll ot stll be a ellpse track ay loger. The above waves are commoly amed partally polarzed waves [1]. Uder such codtos the polarzato should be expressed from statstcal aspects. Targets movemet ad shver ca cause the wave s polarzato to vary by tme ad felds composed or backgroud speckle wll also cause such results. Barakat [1] studed the statstcs of partally polarzed lght waves ad preseted ts aalytcal expressos of the jot probablty desty fucto (PDF) or margal PDF s of may parameters (such as ts ampltude phase phase descrptors Stokes vector etc.). Elyahu [34] gave the aalytcal expresso of the Stokes parameters Gaussa stochastc felds ad hgh-order momets were derved. Theoretcal calculatos of the ormalzed stokes parameters PDF were reported by Kurashov et al. [56] the cotext of polarzato propertes of speckle felds. Brosseau [7] derved a expresso the geeral case for a partally polarzed wave. However Receved March 14 007; accepted November 6 007; publshed ole August 8 008 do: 10.1007/s1143-008-0115-0 Correspodg author (emal: lutao1018@sa.com) Supported by Natoal Excellet PhD Paper Fuds Supportg (Grat No. 08100101) ad the New Cetury Excellet Talet Pla Cha (Grat No. NCET-04-0997) Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606
there s far from a systemc theory thus far because these studes are qute decetralzed. Moreover most of them are based o a covetoal hypothess that the electromagetc waves are harmoc arrow-bad or partally polarzed. Ufortuately these theores caot reflect the statstcs of the ucovetoal electromagetc waves. Especally the covetoal descrpto methods caot adapt to descrbe the reflected waves or the targets of wdebad polarmetrc radars whch wll be wdely used moder battlefelds. Wag [8 9] poted out that usg stataeous polarzato theory ca perfectly descrbe the polarzato formato whether electromagetc waves are covetoal or ucovetoal. As the developmet of the ormalzed Stokes parameters statstcs the hgh-order momets of the ormalzed Stokes parameters are derved ths paper; ad the ew cocepts of dsperso ormalzed cotrast skewess ad kurtoss are defed to descrbe the dstrbuto characterstcs. Secto gves a revew of the probablty desty fucto of the ormalzed Stokes parameters Gaussa felds assumpto; secto 3 calculates the mea varace ad hgh-order momets whch smplfes Brosseau s [7] results. Secto 4 ad secto 5 defe the ew cocepts of the frst ad secod kd of dsperso ormalzed cotrast skewess ad kurtoss to descrbe the dstrbuto characterstcs. I fact the frst kd of propertes comes from the Stokes parameters ad the secod kd of the defto s gve for the ormalzed Stokes parameters. Secto 6 s the cocluso whch shows that the statstcs of the ormalzed Stokes parameters ca be also appled to K-dstrbuto felds hypothess f the multplcatve model the depedet varables wth Square Root of Gamma dstrbuto ca hold the same the two orthogoal polarzed chaels whle the Gaussa depedet varables determe the polarzato states. PDF for the ormalzed Stokes parameters I geeral the electromagetc waves ca be expressed as the Stokes vector s defed as HV * HV E wth the bass of ( hv) ˆ ˆ thus HV R E E where deotes the esemble average (tme averagg for temporal radomess or spatal averagg scatterg problems). deotes the Kroecker product. Smlar to Stokes vector 1 0 0 1 1 0 0 1 R 0 1 1 0 0 j j 0 the ormalzed Stokes parameters are defed as * T J RE ( E ) [ g g g g ] [ g g T ] T HV HV HV HV [0] HV [1] HV [] HV [3] HV [0] HV T g ghv [1] ghv [] ghv [3] HV g HV [ g HV [1] g HV [] g HV [3]]. (1) ghv ghv [0] ghv [0] ghv [0] For coveece of the descrpto of the other compoets the degree of polarzato P s used where deotes the esemble average. T LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606 1595
1 + + 3 g g g P () g where 0 P 1. Besdes the above defto the Stokes vector compoets weght (SW) s gve whch wll vary wth the polarzato bass. As show there s ω P ad HV 0 0 ghv [] ω 1 3. (3) g 1 3. P ω + ω + ω The PDF of the frst ormalzed Stokes parameters ca be expressed by [7] 1 P 1 ω1 g HV [1] f 1 ( g HV1) 3/ HV1 [ 11 G g ]. (4) (1 + ω1 P ω1 g HV [1] + P g 1 HV [1] ) By the method of polarzato bass trasform [7] we have the PDF expresso of the ormalzed Stokes parameters respectvely 1 P 1 ωg HV[] f ( g [] HV [] ) 3/ HV [] [ 11] 1 G g 3. (5) (1 + ω P ωg HV[] + P g 1 HV[] ) 1 Whe P 0 t wll be f 1 ( g HV [] ) g G HV [] [ 11] 1 3. 3 Momets of the ormalzed Stokes parameters The statstcal characterstcs of the ormalzed Stokes parameters startg from momets wll be aalyzed as follows. Obvously the mea of the ormalzed Stokes parameters s gve by [7] The secod momet s 1 P 1 x(1 ω x) 1 P HV [] 1 3/ 1 ω Ω g dx ( I I ). 1 P HV [] ω 3 g ( I I ). The th order momet s wrtte as 1 P g HV [] ( I ω I ( + 1) ) where 1 x I dx Ω 1 1 3/ + ω P ω g HV [] + P g 1 HV []. Ω Evaluato of I gves ω (1 P ω ) Δ I1 I + 3 (1 P )(1 ω ) P (1 P )(1 ω ) P 1596 LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606
I I I 4 ω(3ω ω P + P + P 3) 3ω 3 + 4 5 P (1 P )(1 ω ) P 6 4 4 4 P P ω ω ω + P + ω ω P + ω P + ω 4 + 6 7 P (1 P )(1 ω ) P Δ 5 (1 ) 15 (1 ) (3 10 13 ) 3( 5 (1 )) Δ 8 6 4 4 4 ω P + P ω ω ω + P + ω ω + P + ω + ω 5 8 3 P (1 P )(1 ω ) [6 4 (1 ) 105 (1 ) (45 70 115 ) ( 75 59 16 )] 4 P + P + 9 5 ω (3 7ω 3 (1 ω )) + Δ P 1+ P where Δ l whch are ot accordg to the forms of eq. (B4) ad eq. (B6) 1 P Brosseau s paper [7]. I Brosseau s paper f we sert eq. (B4) to eqs. (B1) ad (B) we caot obta the results expressed by eqs. (B7) ad (B8). The mor mstake les the error eq. (B4). After my reevaluato of I the exact expresso s as follows: (1 P ω1 ) Δ I + 3 P (1 P )(1 ω1 ) P The orgal expresso ref. [7] msses a egatve symbol the frst tem whch may be just a prt mstake. That s I ref. [7] Δ s defed as (1 P ω1 ) Δ I 3 P (1 P )(1 ω1 ) P. (6) +. (7) 1 P + ω1 1 P ω1 Δ sh sh. 1/ + 1/ (8) ( 1 P )( P ω1 ) ( 1 P )( P ω1 ) Ths expresso s wdely used ths area [10] whle the expresso s complcated. It s of terest to obta a smpler form. I fact eq. (8) ca be smplfed as follows: 1+ P Δ L. 1 P (9) The the frst-to-forth momets ad the varace ca be easly derved g g ω ω (1 P ) Δ P P HV [] 3 1 ω P P ω P HV [] 4 5 (3 ) ( 3 )(1 ) 1 + + Δ P P P (1 P ) 4 HV [] ω 6 Δ ω Δ + ω D[ g ] [ P( P ) (1 P ) 8 P 4P ]. 4P g 5 3 4 3 ω P + Pω P + ω ω P P + ω P + ω HV [] 7 7 [18 30 (9 13 )] 3 (1 )( 5 (3 )) Δ 4P 4P LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606 1597
g 8 4 6 4 4 P P ω P ω P ω ω P ω HV [] 9 [6 + 105 3 (5+ 8 ) 5 (18+ 3 ) + (9+ 114 + 16 ω )] 1P 6 4 4 ω ω ω ω 9 3(1 P )(3P 35 + 15 P ( + ) 3 P (1 + 6 )) + Δ. 1P If the covarat Hermta matrx s smplfed dagoal [3] 1 0 HV σ 0 λ momet s 1 P HV [1] 1 ω1 1( + 1) HH g ( I I ) λ 1 1 λ ( λ 1) ( 1) F13 + + 3 + F13 + + 3 λ λ 1 λ 1 + + 3 ( λ + 1) ( + ) 1 λ ( λ + 1) ( 1) F1+ F13 + 1 + λ 1 + +. ( + 1) Usg the same method we ca obta the other compoets hgh-order momet g HV [] 31+ 3+ P (1 + ( 1) ) F1 P 1 (1 + ) 1 P 31+ 3 + (1 λ) (1 + ( 1) )(1 + λ) F1 4λ 3. 4(1 + ) λ As s see whe s odd the equato above always remas to be zero. Whe result s g HV [1] λ λ + 1 l λ λ 1 ( λ 1) λ 1 0 λ 1 g HV [] ghv [3] 0 Dg HV [1] 4 1 the th 4 λ λ(l λ) 1 λ 1 ( ) ( λ 1) ( λ 1) 1 λ 1 3 4λ λ( λ + 1)l λ + 3 λ 1 ( λ 1) ( λ 1) Dg ( HV [3]) Dg ( HV []) 1 λ 1. 3 For the coveece of uderstadg the dstrbuto of the ormalzed Stokes parameters ad gvg the dstrbuto curves we descrbe the mea ad varace as a fucto of the degree of polarzato P ad the polarzato weght of the Stokes parameters (SW) ω the followg fgures. the 1598 LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606
Fgure 1 Mea of the ormalzed Stokes parameters of dfferet P ω. Fgure Varace of the ormalzed Stokes parameters of dfferet P ω. I Fgure 1 ad Fgure t ca be see that the mea ad varace of the ormalzed Stokes parameters equals zero f P < ω. If the degree of polarzato P s varat the absolute value of the mea of the ormalzed Stokes parameters s proportoal to the varable ω. From Fgure we ca see that the polarzato weght ω s less effectve tha the degree of polarzato P for the varaces. The ormalzed Stokes parameters satsfy the followg costra va eq. (3) ad t s oly determed by the degree of polarzato P: 3 3 1 (1 P ) ghv [] ghv [] 1 P P 1 1 Δ 1. It s kow that P s the varat varable for the ormalzed Stokes parameters. The curves of the mea ad varace are gve as follows wth a kow P. It ca be see that the mea of the ormalzed Stokes parameters s lear wth the parameter ω. 4 Dsperso ad ormalzed cotrast of the ormalzed Stokes parameters To descrbe the dsperso of the ormalzed Stokes parameters the dsperso s gve as the secod defto P (1 P ) Δ (1 P ) Δ 1 Dv [][ g] 1 g 1 g HV [1] + g HV [] + g HV [3] 1 1 +. (10) P P P The frst dsperso s defed by the Stokes vector whch s gve as Dv [ g ] 1 P. (11) [1] Here we compare the two kds of dspersos P (1 P ) Δ Dv [][ g] Dv [1][ g ] P 00 P 1. P It s show that the secod dsperso s always greater tha the frst oe. The followg s the compared curves of the two dspersos. I Fgure 3 the upper curve s the secod dsperse ad the oe below s the frst kd of dsperso. They all become smaller whe the degree of polarzato P becomes bgger. That s to LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606 1599
say whe the degree of P s zero the dspersos become 1 whle whe the degree of P s 1 the dspersos become 0. These represet the focusg degree of the polarzato state of the partally polarzed lght. Fgure 4 shows the mus of the two kds of dsperso. Fgure 3 Dspersos of dfferet P. Fgure 4 Dfferece betwee the dspersos varyg of dfferet P. Obvously the dspersos are oly determed by the frst momet whch s the degree of polarzato P. To obta more formato from the dstrbuto we defe a ew varable C the ormalzed cotrast fucto (NCF) as the secod CF whch s deoted as HV [] HV [] HV [] g g g C 1 13. (1) g HV [] ghv [] Isertg g HV [] HV[] g to eq. (1) we ca obta 1 C 3 P ( P Δ) P (PΔ 8 P + (1 P ) Δ ) ω 3 3 P ( P Δ)(1 P ) P(6P 4P 3(1 P ) Δ) ω Just by the same method we defe the frst cotrast of fluctuato (CF) as HV [] HV [] HV [] HV [] ghv []. (13) g g g C 1 13. (14) g It s easy to obta the expresso of the frst CF detal [38] ghv [] HV [0] (1 ) + 4 P HV [ ] C 1 1 13. (15) g P g (1 ) / ω + 4 As see the CF s ot correlated wth the eergy of the waves. Fgures 5 ad 6 show the comparso betwee the frst kd ad the secod kd of CFs. Fgure 7 shows the rule that the frst CF vares wth the degree of polarzato P ad the weght of polarzato w. Fgure 8 shows the rule of the secod CF varyg wth P ad w. Fgure 9 s the dfferece betwee the two CFs varyg wth P ad w. It ca be see that the dfferece s ot bg but almost stable. The CF ca be used to target detecto ad recogto [10]. We ormalzed the cotrast va the ormalzed cotrast fucto defto. The ormalzed CF les betwee the tervals [0 1] ad [0.5 1] whle the orgal terval [10] s [0 ]. From Fgures 5 ad 6 we ca see that f P0 the frst ormalzed CF ad the secod ormalzed CF are all equal to 1 whle f P1 the frst ormalzed CF equals 0.5 ad the secod ormalzed CF equals 0 whch s a useful characterstc of a stochastc wave or a target. 1600 LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606
Fgure 5 The frst NCF of dfferet w whe P0 0.5 0.8 1. Fgure 6 The secod NCF of dfferet w whe P0.1 0.5 0.9 1. Fgure 7 The frst NCF of P ad w. Fgure 8 The secod NCF of P ad w. Fgure 9 The dffereces betwee the two NCFs. 5 Polarzato skewess ad kurtoss To study the characterstcs of the ormalzed Stokes parameters that are o-gaussa dstrbuto we preset two varables to descrbe ths property whch are called polarzato skewess ad polarzato kurtoss. Skewess s a parameter that descrbes asymmetry a radom var- LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606 1601
able s probablty dstrbuto. Kurtoss s a parameter that descrbes the shape of a radom varable s probablty desty fucto (PDF). The dstrbuto ows to a symmetry dstrbuto f skewess equals zero. By the PDF of the ormalzed Stokes parameters the polarzato skewess ad kurtoss are gve by [11] ( g ) 3 HV [] g HV [] ( g ) 4 HV [] g HV [] S HV [ ] 3/ Dg [ HV [] ] K HV [ ] 3 Dg [ HV [] ] By the same method the frst polarzato skewess ad kurtoss are defed as S ( g ) 3 HV [] ghv [] 1 HV [ ] 3/ Dg [ HV [] ] K ( g ) 4 HV [] ghv [] 1 HV [ ] 3 Dg [ HV [] ] 1 3. (16) 013. (17) From the hgh-order momets of Stokes parameters t s easy to obta the followg equatos detal [38]. ( ) 3/ ω(3 3P + 4 ω ) 1 SHV [0] (1+ 3 P )/ 1 + P 1 SHV [ ] 3/ 13. (18) 1 P + ω S HV [ ] K 4 ( P + 4P + 1) ( P + 1) 1 HV [0] K ( ) 4 (1 P + P ) 3 13. (19) 1 HV [ ] (1 P + ω ) ( ( )) ( ( )) 5 3 3 ( P ) P P ( P ) P( P ) ( P ) P ( P ω ) 3/ ( 1+ P ) ( 1+ P ) ω Δ + P P( P ω ) + ( P ω ) Δ ω 1 6 ω 1 3 ω Δ 1 ω Δ + 3 3 Δ 1 3. ( ) 6 4 4 6P + 15ω + 3P ( 3+ 8ω ) P ω ( 9+ 8ω ) 3( 1+ P ) 4 K 4 HV [ ] 8P P( P ω ) Δ+ 3 4 4 4P ( 3P 6P ω ω ) 3( 1 P ) ω Δ + + Δ ω 8P ( 1 P ) (( 1 P ) ( 1 P ) P P( P ) ( P ) ) 13. ( + + Δ ) ( ω ( ω ω )) + + Δ + + Δ (1) The 3D fgures of the polarzato skewess ad kurtoss are gve the followg. Fgure 10 gves the frst polarzato skewess varyg wth the degree of polarzato P Gaussa felds for. Fgure 11 shows the frst polarzato skewess varyg wth P ad w (the polarzato weght SW) for Stokes parameters. Fgure 1 presets the frst polarzato kurtoss of varyg wth P. Fgure 13 gves the Stokes parameters frst polarzato kurtoss varyg g HV [0] g HV [0] wth P ad w. Fgure 14 dcates the polarzato kurtoss of the Stokes parameters varyg wth P ad w. (0) 160 LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606
From Fgure 10 t ca be see that the skewess of g HV [0] s more tha 3 ad whe the degree of polarzato P becomes zero the skewess wll be 6. The deflecto of the symmetry becomes larger whe the P turs from 0 to 1. From Fgure 11 we ca see the skewess s maly decded by ω. The least value of the frst skewess of the last three Stokes parameters s m HV 1 S [ ] 13. From Fgure 14 t ca be see that the ormalzed skewess s smlar to the frst kd of skewess of the Stokes parameters whle the values are smaller tha the latter. Therefore t s more symmetrc tha the frst kd of skewess. From Fgure 13 ad Fgure 15 we ca see the kurtoss s almost holdg the same value though P ω vary a large area whch shows that there s lttle fluctuato. Moreover we ca see as the degree of polarzato crease the kurtoss shall become larger. That s to say the PDF becomes flatter wth the degree of polarzato decreased. I Fgure 11 ad Fgure 14 we ca see that the skewess s maly determed by the polarzato weght. Fgures 16 18 show that the skewess s almost determed by the weght of polarzato. Moreover f the degree of polarzato s fxed the skew s proportoal to the weght of polarzato whch may be useful for later formato processg ad ca be used to characterze the Gaussa radom fled. Fgure 10 g HV[0] the frst skewess. Fgure 11 Stokes parameters the frst skewess. Fgure 1 g HV[0] the frst kurtoss. Fgure 13 Stokes parameters the frst kurtoss. LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606 1603
Fgure 14 The ormalzed skewess. Fgure 15 The ormalzed kurtoss. Fgure 16 The slce of the skewess wth P0.5. Fgure 17 The slce of the skewess wth w0.5. 6 Applcato Fgure 18 The slope of skewess ad w. As a applcato of the above formalsm we shall cosder the same stuato for the descrpto of partally polarzed lght terms of ormalzed Stokes parameters [7]. It s kow that optcal waves propagatg through a radom multple-scatterg medum composed of a collecto of ucorrelated optcally actve sphercal partcles whose sze s very small compared wth the wavelegth of the scattered radato (.e. Raylegh regme) wll be depolarzed. The trasmsso of partally lght through such a medum the weak scatterg lmt s characterzed by 1604 LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606
where ω j ( ) ( ) ( ) ( ) ( ) + ( ) ω o 0.7 o 0.7 o 0.5 1 ω 13 3 3 33 0.7 ω ω 0.7 ω + + ω 0.7 j 13 are the compoets of the cdet average Stokes vector are the compoets of the trasmtted average Stokes vector ad scatters. The degree of lear polarzato s defed as [1] PL ω j. j 1 o ω j () j 13 + 1 deotes the umber of Now the characterstc parameters are gve wth a comparso betwee a cdet learly polarzed state ( ω 1 1 ω ω 3 0) ad a crcularly polarzed state ( ω3 1 ω1 ω 0). Here there s PL 1 o ω for the scattered wave. The slope k s rotato varat determed by the degree of polarzato P. The curves of P wth dfferet umber of scattergs are draw as follows (Fgure 19). Fgure 19 The degree of polarzato P (a) ad the slope k (b) as a fucto of the umber of scatterg evets. The cotuous le s for the learly polarzed state ad the dot-le s for the crcularly polarzed state. It s easy to see that depolarzato of learly polarzed requres more scatterg evets [10]. The dsperso (c) of the scattered wave also has the same propertes. The dfferece s show curves. The secod cotrast of the scattered wave s curved as follows wth the umber of the scatterg evets (Fgure 0). It ca be see that the effect s smlar to the dsperso. Whe the dsperso s larger the cotrast turs to agree wth t. Fgure 0 The cotrast of the scattered wave (a). If the cdet wave s learly polarzed the cotrast of the frst ormalzed parameter s the real le ad the thrd parameter s cotrast equals 1. If the cdet wave s crcularly polarzed the cotrast of the frst ormalzed parameter equals 1 ad the thrd parameter of the ormalzed Stokes parameters s dotted the fgure whch s larger tha the wave learly polarzed. The geeral skewess of the ormalzed Stokes parameters (b). The real le s for the learly polarzed wave ad the dotted le s for the crcularly polarzed wave. Also the dstrbuto s rght skew. The kurtoss (c) becomes smaller as the umber of the scatterg evets s creased the learly polarzed cdet wave s super tha the crcular oes. Moreover f the weght of the polarzato becomes larger so s ts kurtoss. I the fgures of kurtoss from top to bottom the frst ad thrd are learly polarzed cdet waves whch are g HV[] ad g HV[] respectvely ad so are the forth ad the secod represet crcularly polarzed waves. LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606 1605
j I fact the skewess s oly defed as the thrd-order momet wthout ormalzato. That s from eq. (0). The real le represets that the learly polarzed wave s asymptotcally symmetrc. Moreover the symmetry of the crcularly polarzed wave s more dstct tha that of the learly polarzed wave. Whe ω 0 the skewess becomes 0. The symmetry of the dstrbuto becomes more dstct wth the larger umber of scatterg evets ad the dstrbuto becomes flatter as the kurtoss s earer to 0. Ths also cofrms the cocluso of Brosseau [7]. 7 Coclusos The statstcs characterstcs of the ormalzed Stokes parameters are studed ths paper. We have obtaed some mportat coclusos whch drectly buld up statstcal foudatos for further polarzato formato processg such as polarzato detecto polarzato recogto. The theores above are mostly based o the Gaussa hypothess. Our aalyss smplfes the results gve by Brosseau [7] the orgal paper. The hgh-order momets are calculated to descrbe the stochastc wave felds. The applcato relatg to depolarzato of lght by a spatally radom medum should be doe the future. 1 Barakat R. The statstcal propertes of partally polarzed lght. Opt Acta 1985 3(3): 95 31 Barakat R. Statstcs of stokes parameters. J Opt Soc 1987 4(T): 156 163 3 Elyahu D. Vector statstcs of correlated Gaussa felds. Phy Rev E 1993 74(4): 881 89 4 Elyahu D. Statstcs of Stokes varables for correlated Gaussa felds. Phy Rev E 1994 50(3): 381 384 [DOI] 5 Kurashov V N Mareko V V Molebaya T V. Statstcs of ormalzed Stokes parameters of speckle felds. Opt Spectrosc (USSR) 1991 70: 36 38 6 Kurashov V N Mareko V V Molebaya T V. Codtoal probablty dstrbuto of ormalzed Stokes parameters. Opt Spectrosc (USSR) 1991 71: 573 575 7 Brosseau C. Statstcs of the ormalzed Stokes parameters for a Gaussa stochastc plae wave feld. Appl Opt 1995 34: 4788 8 Wad X S. Study o wde-bad polarzato formato processg ( Chese). Ph. D. Thess. Chagsha: Natoal Uversty of Defese Techology. 1999. 6 9 Wag X S L Y Z Xao S P et al. Istataeous polarzato statstcs of electromagetc waves. Sc Cha Ser F-If Sc 004 47(5): 63 634 10 Korotkova O. Chages statstcs of the stataeous Stokes parameters of a quas-moochromatc electromagetc beam o propagato. Opt Com 006 61(): 18 4 [DOI] 11 Evas M Hastgs N Peacock B. Statstcal Dstrbutos. 3rd ed. New York: Joh Wley & Sos 000 1 Cheault D B Pezzat J L Chpma R A. Meuller matrx algorthms. SPIE Polarzato ad Measuremet 199 1746: 31 46 1606 LIU Tao et al. Sc Cha Ser F-If Sc Oct. 008 vol. 51 o. 10 1594-1606