The development of polarization speckle based on random polarization phasor sum CHUNMEI ZHANG,1 SEAN HORDER,1 TIM K. LEE,2,3 WEI WANG1,* 1 SUPA, Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, Eh14 4AS, UK; 2 Cancer Control Research Program, BC Cancer Agency, Vancouver, BC V5Z 1L3, Canada 3 Department of Dermatology and Skin Science, University of British Columbia, Vancouver, BC V5Z4E8, Canada *Corresponding author: W.Wang@hw.ac.uk

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX The random walk approach has been extended and applied to study the development of polarization speckle by taking the vector nature into account for stochastic electric fields. Based on the random polarization phasor sum, the first and second moments of the Stokes parameters of the resultant polarization speckle have been examined. Under certain assumptions about the statistics of the component polarization phasors that make up the sum, we present some of the details of the spatial derivation that leads to the expressions for the degree of polarization and the newly proposed Stokes contrast which are suitable for describing the polarization speckle development. This vectorial extension of the random walk will provide an intuitive explanation for the development of the polarization speckle.

© 2018 Optical Society of America

OCIS codes: (000.5490) Probability theory, stochastic processes, and statistics; (030.6140) Speckle; (030.6600) Statistical optics; (110.5405) Polarimetric imaging; (260.5430) Polarization

http://dx.doi.org/10.1364/AO.99.099999

1. INTRODUCTION As a stochastic process, a random walk describes a path consisting of a succession of random steps on some mathematical space; explains the observed behaviors of many processes; and serves as a fundamental model for the recorded stochastic activity. Due to its theoretical importance and practical interest, random walk has been applied to many scientific fields including physics, chemistry, biology, psychology, ecology, computer sciences, as well as economics [1,2]. In optics, speckle phenomena are such examples, which is composed of a multitude of independently phased additive complex components with both random lengths (amplitudes) and directions (phases) in the complex plane. When these complex components are added together, they constitute what is known as a “random walk.” The resultant of the sum may be large or small, depending on whether constructive or destructive interference dominates the sum, and the squared length of the resultant is what we usually observed intensity with a typical granularity distribution [3-5]. On the other hand, we often deal with vectorial signals with two orthogonal components being the complex-valued signals in many areas of physics, and particularly in polarization optics [6]. Up until now, most of the investigation of the random walk problem in speckle has treated the optical fields as scalar fields without considering their polarization properties. In this paper, we will extend the random walk theory by taking the vector nature into account for stochastic electromagnetic fields referred to as polarization speckle [7-25]. This vectorial extension of the random walk will provide an intuitive

explanation for the development of the polarization speckle. Based on the random polarization sum, we will calculate the mean and the second moment of the Stokes parameters where the average is replaced with the ensemble average. for the polarization speckle. Following these results, we examine the relations between the degree of polarization and the newly proposed Stokes contrast with the phase standard deviations and the random walk number. 2. FIRST AND SECOND MOMENTS OF THE STOKES PARAMETERS OF THE RESULTANT POLARIZATION PHASOR We begin with explicit expression for the random polarization phasor sum of interests. Polarization speckle, known as stochastic electric fields, is a vectorial signal which could be understood as a sum of many contributions from polarization phasors with random vector directions (random polarization states) and random complex amplitudes (random amplitudes and random phases) representing a monochromatic or nearly monochromatic vectorial electric field disturbance. For each component of polarization speckle a complex addition of many small independent contributions from the scalar components of the polarization phasors constitutes a random walk in the complex plane and the resultant phasor of the sum has its total complex amplitude. Figure 1 illustrates an example of polarization phasor sum based on the vector random walks for xE

~ and yE~ , respectively.

ex

whelealoarpopothvecointi.eappopopa

whar

Fig. 1 SchematiGiven that thexpressible as

1

~ˆ

~

1N

ExEE

N

n

x

=

+=

=

here E represeectric field vectorong x and y dire the lengths anolarization compolarization phasoe nth componenector), kna and omponents of troduced here ane. 2| |E< > even wpproach infinity. Because of theolarization opticsolarization phasarameters can be

*0

1 1

1

x x

N N

n m

S E E E

N = =

= +

=

*1

1 1

1

x x

N N

n m

S E E E

N = =

= −

=

*2

1

1

x y

N N

n m

S E E E

N = =

= +

=

*3

1

( x y

N N

n m

S i E E

iN = =

= −

=

here * indicatesrbitrary statistics

ic diagram of rane complex electr

ˆ(

ˆˆ~

1

axea

xeayE

xn

x

yi

xn

ixy

ϕ

ϕ

+

+=

ents the resultanr), xE~ and yE

~ arirections, with thnd xϕ and ϕonents, respectivor components innt of polarizationknϕ are the lennE respectivelynd in what followwhen the numbere great importans, we present oursor sums. Fromgiven by

*

(

1

[ xn xm

y y

ixn xm

E E

a a e φ φ−

*

( )[ xn xm

y y

ixn xm

E E

a a e φ φ−

*

(

1

[ yn xm

y x

iyn xm

E E

a a e φ φ−

*

(

1

)

[ yn xm

y x

iyn xm

E E

a a e φ φ−

−

s a complex cons for both the a

ndom polarizationric field of the

,)ˆ

1ˆ

ye

Nyea

yn

y

iyn

iy

ϕ

ϕ =+

nt polarization pre two Cartesian hem being unit vey are the phasevely. N represen the random wn phasor in thength and phase y. The scaling ws in order to prer of component ponce of the Stokr derivations basm their definit

() ynm iyn yma a e φ −+

() yniyn yma a e φ φ−−

) (m xnixn yma a e φ+

) (m xnixn yma a e φ−njugate. Allowingamplitudes xna

n phasor sum. random walk

1

EN

N

nn

=

(phasor (a complcomponents of ectors. xa and aes of the resultaents the number walk, nE

represene sum (a complfor xk = or factor N1serve finite energolarization phasokes parameters sed on the randotions, the Stok) ] ,ymφ−

() ] ,ymφ (

) ] ,ymφ− (

) ] ,ymφ− (

g for the momeand yna and t

is 1) lex E

yant of nts lex y is gy, ors in om kes

(2) (3) (4) (5)

nt, the

phases thn elemeach othpolariza

0S<

=

1S<

=

2S<

= 3S<

=

Whenkna ( ksecond identicafunction We nthe seco

20S< >

21S< >

22S< >

23S< >

xnφ and ynφ , bmentary polarizther and of the aation phasor. We 0

1 1

2 2

1[

( 1)

N N

n m

x y

y

Na a

N a

= =

>=

=< > + <

+ − <

11 1

2 2

1[

( 1)

N N

n m

x y

Na a

N a

= =

>= <

=< > − <

− − <

21 1

1[

N N

n m

x y

NN a a

= =

>=

= < ><

31 1

[N N

n m

x

iN

iN a= =

>= <

= − < ><

n Eqs. (6)~(9) hx= or y ) aremoment 2

ka< >ally distributed n )(ωϕkM given bnow turn attentiond moment of th

21 1

(

1

[ xp

N N N

n m p

ixp xq

N

a a e φ

= = =

>=

×

21 1

(

1

[ xp

N N N

n m p

ixp xq

N

a a e φ

= = =

>=

×

21 1

(

1

[ yp

N N N

n m p

iyp xq

N

a a e φ

= = =

>=

×

21 1

(

1

[ yp

N N N

n m p

iyp xq

N

a a e φ

= = =

>=

×

but assuming thazation phasor aramplitudes and pcan express the m(

2

( 1)

( 1)

xn xixn xm

y y

a a e

N a

M M

φ φ

φ

−<

> + − <

> −

(

2

( 1)

( 1)

xn xixn xm

y y

a a e

N a

a M M

φ φ

φ

−<

> + − <

> −

(

[ (1) (

xn yixn ym

y x y

a a e

M M

φ φ

φ φ

−<

>

(

[ (1)

yn xmiyn xm

y x y

a a e

a M M

φ φ

φ φ

−<

>

have been derivee identically distr> . We have alsand therefore by ( ) exkM

φ ω =<ion to the more dhe stokes parame1 1

) (

[

p xq

N Ni

xn xmq

iyp yq

a a e

a a eφ φ

= =

−

<

+

1 1

) (

[

xq

N Ni

xn xmq

iyp yq

a a e

a a eφ φ

= =

−

<

−

1 1

) (

[

xq

N Ni

yn xmq

ixp yq

a a e

a a eφ φ

= =

−

<

+

1 1

) (

[

xq

N Ni

yn xmq

ixp yq

a a e

a a eφ φ

= =

−

<

−

at amplitude andre statistically inphases of all othemean stokes para)

2 ( 1)

(1) ,

xmyn ym

x x

y

a a

a M M

M

φ

φ

> + <

> −

)

2 ( 1)

(1) ,

xmyn ym

x x

y

a a

a M M

M

φ

φ

> − <

> −

)

( 1) ( 1)

ymyn xm

x

a a

M

φ

φ

> + <

− + −

)

( 1) ( 1

mxn ym

y x

a a e

Mφ φ

> − <

− − − ed, we have assributed, with meso assumed thahave common

xp( )kjωφ > . difficult problem eters. ( )

) ] ,

xn xm

yp yq

yn yma aφ φ

φ φ

−

−

+

>

( )

) ] ,

xn xm

yp yq

yn yma aφ φ

φ φ

−

−

−

>

( )

) ] ,

yn xm

xp yq

xn yma aφ φ

φ φ

−

−

+

>

( )

) ] ,

yn xm

xp yq

xn yma aφ φ

φ φ

−

−

−

>

d phase of thendependent of er elementary ameters as ( ) ]

(1)

yn ymim

x

e

M

φ φ

φ

− >

(6) ( ) ]

(1)

yn ymim

x

e

M

φ φ

φ

− >

(7) ( ) ]

(1)] ,

yn xmim

y

e

M

φ φ

φ

− >

(8) ( ) ]

1) (1)] .

xn ymi

y

e

M

φ φ

φ

− > (9) umed that all ean ka< >and at all knϕ are characteristic

of calculating ( ) ]yn ymi

meφ φ−

(10) ( ) ]yn ymi

meφ φ−

(11)

( ) ]xn ymime

φ φ− (12)

( ) ]xn ymime

φ φ−

(13) where we have assumed again that the amplitudes and phases of the component polarization phasors are statistically independent. For these summations corresponding to each Stokes parameter, there are fifteen different cases that must be considered, as follows [4,5]. (1) qpmn === N terms (2) pnqpmn ≠== ,, )2)(1( −− NNN terms (3) nqpmn ≠≠= , )2)(1( −− NNN terms (4) mnqmpn ≠== ,, )1( −NN terms (5) nqmpn ≠≠= , )2)(1( −− NNN terms (6) mnpmqn ≠== ,, )1( −NN terms (7) npmqn ≠≠= , )2)(1( −− NNN terms (8) qnpmn ≠== , )1( −NN terms (9) pnqmn ≠== , )1( −NN terms (10) mnqpn ≠== , )1( −NN terms (11) mnmqp ≠== , )1( −NN terms (12) qpmn ≠≠≠ )3)(2)(1( −−− NNNN terms (13) pmnqp ≠≠= , )2)(1( −− NNN terms (14) pmnqm ≠≠= , )2)(1( −− NNN terms (15) qmnpm ≠≠= , )2)(1( −− NNN terms After summation of all fifteen terms, we have our desired second moments for each individual Stokes parameter. These results are 2 4 4 2 2 2 20

2 2 3 2 4

4 4 4 2

3 2 2 2

2 2

2

1{ 2( 1)( )

2 ( 6 11 6)[

( (1)) ( (1)) ] ( (1)) [4( 1)

2( 3 2)

( (2) 2) 2 ( 1) ]

( (1)) [4( 1)

x y x y

x y x

x y y x

x x x x

x x y

y y

S a a N a aNN a a N N N a

M a M M N

a a N N a aM N N a a

M N a a

φ φ φ

φ

φ

< >= < > + < > + − < > + < >

+ < >< > + − + − < >

+ < > + −

< >< > + − + < > < >

+ + − < > < >

+ − < >< 3 2

2 2 2 2

2 2 2 2 2 2

2 2 2 2 2

2( 3 2)

( (2) 2) 2 ( 1) ]

( 1)[ ( (2)) ( (2)) ]

2 ( 1) ( (1)) ( (1)) } ,

y

y y y y x

x x y y

x y x y

N N

a a M N N a a

N a M a M

N N a a M M

φ

φ φ

φ φ

> + − +

< > < > + + − < > < >

+ − < > + < >

+ − < > < > (14)

2 4 4 2 2 2 2

1

2 2 3 2 4

4 4 4 2

3 2 2 2

2 2 2

1{ 2( 1)( )

2 ( 6 11 6)[

( (1)) ( (1)) ] ( (1)) [4( 1)

2( 3 2)

( (2) 2) 2 ( 1) ] ( (1))

[4( 1)

x y x y

x y x

x y y x

x x x x

x x y y

y

S a a N a aNN a a N N N a

M a M M N

a a N N a aM N N a a M

N a a

φ φ φ

φ φ

< >= < > + < > + − < > + < >

− < >< > + − + − < >

+ < > + −

< >< > + − + < > < >

+ − − < > < > +

− < >< 3 2 2 2

2 2

2 2 2 2 2 2 2

2 2 2 2

2( 3 2)

( (2) 2) 2 ( 1) ] ( 1)

[ ( (2)) ( (2)) ] 2 ( 1)

( (1)) ( (1)) } ,

y y y

y y x

x x y y

x y x y

N N a a

M N N a a N

a M a M N N

a a M M

φ

φ φ

φ φ

> + − + < > < >

+ − − < > < > + −

< > + < > − −

< > < >

(15) 2 2 2 2 22

2 2 2

2 2

2( 1) ( (1)) [2( 1) ( (1))

(1 (2))] [2( 1)

(1 (2))( (1)) 2 (1 (2) (2))] ,

x x y y

y y x y

x y y x y

S N a M N a M

a M a N a

M M a M M

φ φ

φ

φ φ φ φ

< >= − < > − < >

+ < > + + < > − < >

+ + < > + (16) 2 2 2 2 23

2 2 2

2( 1) ( (1)) ( (2) 1)

[2( 1) ( (2) 1)( (1)) 2

( (2) (2) 1)] .

x y x y x

y x y y

x y

S N a a M M a

N a M M a

M M

φ φ

φ φ

φ φ

< >= − − < > < > − − < >

− < > − + < >

− (17) When these results have been derived, we make use of the Hermitian symmetry, *)]([)( ωω ϕϕkk MM =− due to the fact that the probability density functions of kϕ is real-valued. All these resulting expressions in Eqs. (14)-(17) are sufficiently complex, we prefer to make some simplifying assumptions at this point. Let the lengths of all the elementary polarization phasors be unity, i.e. 1=kna , for all nand xk = or y , and hence

4 4 3 3 2 2

1.

x y x y x y x

y

a a a a a a aa

< >=< >=< >=< >=< >=< >=< >

=< >=

(18) In addition, we assume that the phases kϕ are zero mean Gaussian random variables with standard deviations ϕσk [4] )2(}])(2[exp{)( 22 ϕϕϕ σπσϕϕ kkkkP −= ,

(19) in which case the characteristic functions are given by ]2)(exp[)( 22 ϕϕ σωω kkM −= . (20) To describe the statistical properties of the polarization speckle, two quantities are of great importance: the spatial degree of polarization and the Stokes contrast. The formula for the spatial degree of polarization is 2)](Tr[)](Det4[1 JJPS −= , (21) with

whthspby0SthSto

(2whonsp(1 sp

corewhsp3.D of staNwhindimfro

1

2J

=

here Tr and Dee coherency matNote the fact thapeckle has been dy the average inte0 is equivalent toe stochastic electokes contrast can [

SC<

=

3) hich is a measuren a Poincare sphpherical radius forAfter considerab8) and (20) we h)1{(

)[(

2

2−×

−=ϕσ xe

NPs

The resulting expeckle is

(

cosh((

((sinh

11(8{

)(

2

2−×

+

×

−=

ϕσ

σ

xe

NCs

It’s interesting orresponding standuces to [{

)1(8{

−×

−=

NN

NC

hich is the same apeckle, as of cours. STATISTICEVELOPEDFigure 2 shows f partially develandard deviation

2,10N = and100hen the numbedicating that thmportant role wheom the figure tha

0 1

2 3

S SS i S

< > + << > − <

et signify the tracrix J , respectiveat the well-knowndefined as the stanensity, i.e. 1C σ=o the intensity Itric field discussen be introduced b2 2

1 2S SS

> + < ><e of how strong there surface as cor 0S (intensity). ble simplificationhave the spatial de

2{}

(

21])(

)(22

22

2

+

−

×

+ϕ

ϕ

σ

σ

y

x Ne

xpression for the

)]

((sinh)])

1(8)2)(

[)

1)(

22

2

)(2

22

2

−−

−

+

+

−

ϕ

ϕ

σ

ϕ

ϕ

σ

σσ

σ

y

x

e

NeN

y

x

to note that fondard deviation e}]1

cosh(1)[

12)( 2 −+−

+−

ϕσe

N

as the conventionse it should be [3]CAL PROPED PLARIZATthe relationship loped polarization of ϕσ x and σ . There is no ser of random whe number of ren the value is chat the spatial deg

2

0

S iS

> < > + <> < > − <ce operation and ely. n contrast for thendard deviation o

1 I [4]. Since theI and 2 2

0 1S S= +ed in this paper isby 23

20

]S SS+ < > − <

>he fluctuations ofompared with thn, by using these aegree of polarizat

)[1(2

)1

(

)(222

−

−

−+

+−ϕ

σ

σ y

eN

eN

e Stokes contrast

.

)}2)

)1

)cosh((1

212

)(2 2−

×

−

+−ϕ

ϕ

σ

ϕ

σ

σyeN

y

x

or 0~

=yE (or ~Eequal to zero, the,

((sinh)])((

21

22ϕσ

nal contrast used]. ERTIES OFTION SPECKbetween the degon speckles, SP

ϕσ y for various significant changwalks increased random walks dhosen to be largergree of polarizati

3

1

SS

< >< >

, (2the determinant e conventional lasof intensity divide Stokes paramet2 22 3S S+ + becaus fully coherent, t

20 ,S > f Stokes vectors ahe average value assumptions in Eqtion as

]}

2(2

)()

2

22 −+

−++ϕϕ σσ yx e

NN

(2of the polarizati

)1(2[

1[

)])2

−+

−

N

N

(20

~ =xE ) with te expression abo)}2)( 212

ϕσ

(2d widely in the las PARTIALLKLE gree of polarizati, and the phavalues of N , ge in these grapfrom 10 to 10does not play r than 10. It is cleon starts at one f

22) t of ser ded ter use the are of qs.

.}

)1

1−

−

4) on

5) the ove

26) ser LY

on ase i.e. phs 00, an ear for

smalleralso be polarizarandomthat theof ϕσ x arapidly.

r values of ϕσ x annoted that the pation falls to zerom walks. For incre spatial degree oand ϕσ y , howev.

nd ϕσ y , and thepoint and rate ato is dependent oreasing values of of polarization rever, the rate at wen begins to fall tt which the spaton the value of thf random walks, iemains at one for which it falls to z

to zero. It can tial degree of he number of it can be seen r higher values zero increases

povaporathtreHocotow. Iσpo

poσ

poσinctoweqwa

Fig 2. Spatial olarization specklarious values of N Figure 3 ploolarization of paandom walk numat for smaller valend of the decreowever, as N ionverge towardswards one can beIt can also be se

πϕ 2x , must beolarization closer

Fig 3. Spatial dolarization specklϕσϕxy / .

The relationshipolarization speckϕσ y for various valcrease, the Stokwards a constanqual to zero for bave plates, the Sto

degree of polarkles vs. phase stanN . ts the relationshartially developember N for varioulues of N , increaeased values for increases, all ths one. The rate e seen to decreaseen that for large kept low, as thto one for all valu

degree of polariles vs. random wap between the Stokle SC and the plues of N is illustrkes contrast Cnt 22 [7]. Nobirefringent matokes contrast van

rization SP of pndard deviation hip between theed polarization us values of ϕ σσyasing πσ ϕ 2x withe spatial degrese values for at which thesese with increasingger spatial degrehis will keep theues of N .

ization SP of palk number N fookes contrast of pphase standard drated in Fig. 4. AsSC becomes satute that when botterial with a flat nishes as expecte

partially developof ϕσ x and ϕσ y fe spatial degree speckles SP an

ϕσx . It can be sell cause the generree of polarizatioSP will eventuae values converg values of ϕ σσyees of polarizatioe spatial degree

partially developor various valuespartially developdeviations ϕσ x ans both ϕσ x and σurated and tenth ϕσ x and ϕσ y asurface, such asd.

ped for of nd een ral on. ally rge ϕσxon,of

ped s of ped nd

ϕσ ynds are s a Fig 4speckle

N . Figur

ϕσ y . Foshows falling to

4. Stokes contrae vs. phase standare 5 plots the Stor a fixed ratio σya pronouncedtowards zero. Sim

ast SC of parard deviation ϕσ x

tokes contrast Cϕϕ σσ xy , the behad maximum, firmilar to the contra

rtially developedϕx and ϕσ y for varSC vs N for valuavior of the Storst rising withast C for the lase

d polarization rious values of es of ϕσ x and okes contrast h N and then er speckle, the

suevbuhaondethintnuvedelarpsecrecoenN

sp4.papoasrethprpoposprainvco

urprising fact, not ventually fall towut for large standappen is extremen the different deviations of the Ste Stokes contratensity on N is umerator of SCector, is proportioescribes a path corge applicationssychology, compuconomics, randomcorded stochastiThus, for any σontrast will beginnough N , althou

N required to see t

Fig 5. Stokes peckles vs. random. CONCLUSDue to its greataper an extensionolarization propesumptions for thsults of the first e polarization sprovides an intuolarization specolarization speckpatial degree of pandom walk anvestigated. Furthomplex vector an

t visible in the figuwards zero, regarddard deviations, thely large. An expldependencies of tokes vectors on ast in equation 0S< > for large

S ,indicating the onal to N . Asonsisting of a sucs to many scuter science, phym walks serve c activity. ϕσ x and ϕσ y , evenn to determine anugh for large phathis effect may be

contrast SC of m walk number NSION t importance in sn of the random erties for stochahe random polarizand second mompeckle. The vectouitive explanatiokle. To charackle, the newly inpolarization overnd the phase her a systematic nd the associated

ure, is the that all dless of the valuhe value of N wlanation for this the average intN . As shown in(25), the depene N , while the expected fluctuas a stochastic proccession of randoientific fields iysics, chemistry, as a fundamen ntually the numend will cause SCase standard deve extremely large

f partially develN for various valstatistical optics, walk approach stic electric fieldzation phasors, wments of the Stokorial extension ofon for the devcterize the devntroduced Stokesr various values standard deviaanalysis of the rad random polariz

the curves for Ce of ϕσ x and ϕσ ywhere this begins phenomenon restensity 0S< > ann the expression fndence of averadependence of tation of the Stokocess, random waom steps. Due to including ecologbiology as well ntal model for terator of the StokS to fall for a larviation, the value e.

loped polarizatilues of ϕϕ σσ xy / .we present in thby considering tds. Based on somwe have derived tkes parameters ff the random wavelopment of tvelopment of ts contrast and tof the number ations have beandom walk of tzation phasor su

SCϕσ y , to sts nd for age the kes alk its gy, as the

kes rge of

on his the me the for alk the the the of een the um

will facpolarizawill opein statisFundingSUPAScience04932) Refere

1. G.

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