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Scattering of partially coherent electromagnetic beams

by water droplets and ice crystals

Jianping Liu a,n, Lei Bi b, Ping Yang b, George W. Kattawar a

a Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USAb Department of Atmospheric Sciences, Texas A&M University, College Station, TX 77843, USA

a r t i c l e i n f o

Article history:

Received 31 August 2013

Received in revised form

6 November 2013

Accepted 8 November 2013Available online 16 November 2013

Keywords:

Partially coherent light

LorenzMie

Mueller matrix

Water droplet

Ice crystal

a b s t r a c t

The conventional LorenzMie theory is generalized for a case when the light source is

partially spatially coherent. The influence of the degree of coherence of the incident field

on the generalized Mueller matrix and the spectral degree of coherence of the scattered

light is analytically studied by using the vector field instead of the scalar field to extend

previous results on the angular intensity distribution. The results are compared with the

Mueller matrix obtained from the Discrete Dipole Approximation (DDA) method, which is

an average over an ensemble of stochastic incident beams. Special attention is paid to the

Mueller matrix elements in the backward direction, and the results show some Mueller

matrix elements, such as P22, depend monotonically on the coherence length of the

incident beam. Therefore, detecting back scattering Mueller matrix elements may be a

promising method to measure the degree of coherence. The new formalism is applied to

cases of large spherical droplets in water clouds and hexagonal ice crystals in cirrus

clouds. The corona and glory phenomena due to spheres and halos associated with

hexagonal ice crystals are found to disappear if the incident light tends to be highly

incoherent.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In the conventional theories of light scattering, such as

LorenzMie theory[1], Discrete Dipole Approximation (DDA)

[2,3], and Finite-Difference Time-Domain (FDTD) method[4],

the incident light is generally assumed to be fully coherent in

both space and time. However, the assumption is not always

justified. In reality, light acquires some degree of incoherence

due to light source fluctuations or to interactions with randommedia such as a turbulent atmosphere. Although the general

framework of optical coherence theory has long been well

established[57], scarcely any attention has been paid to the

effect of coherence on light scattering by deterministic media.

The influence of spatial coherence on scattering by a particle

was partially investigated by Cabaret et al. [8]and Greffet[9].

The extinction cross section of rotationally invariant scatterers

was found not to depend on the transverse (or spatial)

coherence length, but the extinction cross section of aniso-

tropic scatterers did depend on the state of coherence of the

illuminating field. Further research on the topic was con-

ducted by van Dijk [10] and Fischer [11] who studied the

effects of spatial coherence on the angular distribution of

radiant intensity scattered by a sphere. By using the angular

spectrum representation of a random field[6]and half wave

expansion of the scattering amplitude, the intensity of ascattered field was analytically obtained. The angular distribu-

tion of radiant intensity depends strongly on the degree of

coherence, but the extinguished power does not. Sukhov [12]

numerically studied the effect of spatial coherence by a

random medium on the properties of scattered fields, and

the DDA method was used to demonstrate that the statistical

properties of the scattered light from an inhomogeneous

medium were altered due to coherence effects.

Previous theories have been limited to the study of the

scattered field intensity, which does not fully describe

a scatterer with respect to light scattering. A common

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n Corresponding author. Tel.: 1 979 587 9918.

E-mail address: [email protected] (J. Liu).

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generalization is to study the Mueller matrix, which relates

the Stokes vector of the incident beam to the Stokes vector of

the scattered beam. For a large class of light beams such as

the so-called Gaussian Schell-model beam, the Mueller

matrix for a sphere is derived analytically by extending the

LorenzMie theory. We developed a DDA code [3] by

modifying the source field to incorporate incoherence, and

the Mueller matrix was obtained by averaging over anensemble of stochastic incident fields. In addition, we

investigated the spectral degree of coherence of the scat-

tered light. As applications of the formalism developed, we

performed the computation of light scattering by water

droplets and hexagonal ice crystals, which are the major

scatterers inside atmospheric clouds. The coherence effects

of spherical water droplets were studied by modifying the

LorenzMie theory. With the Invariant Imbedding T-matrix

method (see[13]and references cited therein), we were able

to study the coherence effects on halos of large hexagonal ice

crystals.

2. Scattering of partially coherent light by a sphere

We consider the scattering of electromagnetic beams of

an arbitrary state of spatial coherence by a homogenous

sphere on the basis of a generalization of previous results

in[10,11]. Note that the procedure is quite similar to that

used by Lahiri and Wolf [14], where the refraction and

reflection of partially coherent electromagnetic beams

were considered.

2.1. Theory of coherence

We use some of the important results from the coher-

ence theory of electromagnetic beams [7]. The stochasticnature of incoherent monochromatic light is represented

by an ensemble of random fields fEr;g, where is the

frequency, and for each realization, the field component is

transverse to the direction of propagation. By choosing the

same plane of reference as in the previous section, each

random field can be expressed as

Er; Elr;

Err;

!: 1

Following the same formalism used by Wolf [7], the

second-order correlation properties of the stochastic field

are fully characterized by the 22 cross-spectral density

matrix (CSDM) defined by

Wr1; r2; Enr1;UE

Tr2;

Enl r1;Elr2; E

n

l r1;Err2;

Enrr1;Elr2; En

rr1;Err2;

!;

2

where denotes the ensemble average. From the CSDM,

the spectral density can be derived at point r and at

frequency

Sr; TrWr; r;; 3

which can be interpreted as a contribution to the intensity

at point r from the field component of frequency .The spatial degree of coherence of the random field is

defined by[7]

r1; r2; TrWr1; r2;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sr1;p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Sr2;p ; 4

which unifies both polarization and coherence. 0r j

r1; r2;jr1 with 0 representing complete incoherence

and 1 representing complete coherence.

2.2. Incident light

Incoherent beams can be represented by an ensemble

of random fields and are realized using the angular

spectrum representation [6]. A partially coherent beam

propagating in the zdirection can be expressed as

Eir;

Zju

?jo1

eiu? ;expikuUr d2

u? ; 5

where the coefficient eiu? ; is a two-component ran-

dom vector defined by

eiu? ; eil u

? ;

eir u

? ;

0@ 1A; 6k =cis the wavenumber, u is the direction of propaga-

tion of each plane wave component, u? ux; uy is the

projection ofuonto thez0 plane, and the unit vector u

points into the z40 half space, i.e. uzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1ju? j2

q . The

scattering plane is chosen as the reference plane.

The incident field is uniquely defined by the source at

thez0 plane, which is characterized by the CSDM

Wi1;2; E

in1;UEiT2;; 7

where 1 and 2 are 2D vectors in the z0 plane. UsingEq.(5), we have

Wi1;2;

Z d

2u? d

2u?

~W

iu? ; u

? ;expiku

? U1u

? U2;

8

where the angular correlation matrix ~W

i

u? ; u

? ;

einu? UeiTu? is defined as a four-dimensional

Fourier transformation of the CSDM at thez0 plane, i.e.

~W

i

u?

; u?

;

k

2

4 Z d

21d

22W

i

1;2;expiku

? U1u

? U2:

9

We consider a widely used class of partially coherent

beams, the so-called Gaussian Schell-model beams, which

have the following CSDM elements

Wilm1;2; alamblmexp

2122

4s2S

!exp

122

2s2

!:

10

The independently chosen parameter sS can be inter-

preted as the width of the beam, and s as the coherencelength. The remaining parameters in Eq. (10) have the

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following constraints[7]

bij 1 when i j; 11

bij r1 when iaj; 12bij b

n

ji; 13

ai Z0: 14

By substituting Eq.(10)into Eq.(9)and performing the

four dimensional integration, one can obtain the angular

correlation matrix[10]

~Wi

lmu

? ; u

? ; alamblmk

2ssef f

2

!2

exp k

2

2 u? u

? 2s

2Su

? u

? 2s

2ef f

4

" #( ); 15

where

1

s2ef f

1

s2

1

4s2S: 16

In order to obtain the Mueller matrix, knowledge of the

Stokes vector of the incident beam is required. According

to the definition, the Stokes vector is closely related to the

22 polarization matrix[7]

Wir; r; Einr;UEiTr;

E

in

l r;Eil r; E

in

l r;Eir r;

Ein

r r;Eil r; E

in

r r;Eir r;

0

@

1

A:

17

From Eq.(17),we get the Stokes vector I ir; at point r

Iir;

Will r;Wirrr;

Will r;Wirrr;

Wirl r;Wirl r;

iWirlr;W

ilrr;

0BBBBBB@

1CCCCCCA

: 18

However, such a Stokes vector is spatially dependent

and would make the Mueller matrix ill defined. One

solution is to define the Stokes vector of the incident beam

at the z0 plane and let the width of the beam go toinfinity. From Eq.(10), we immediately have the Gaussian

Schell-model beam

Wilm;; alamblm; 19

which is a constant 22 matrix. Analogously, the angular

correlation matrix in the same limit can written as

~Wi

lmu

? ; u

? ; alamblmk

2s

2

2

exp 1

2k

2s

2 u

? j2

2u? u

? :

20

Thus, under this condition, we can generalize theMueller matrix to the case of incoherent beam scattering.

2.3. Scattered light

In the previous section, the incident field amplitude is

defined with reference to the main scattering plane, which is

defined with respect to the whole incoherent beam. How-

ever, for each plane wave component of the incident beam,

the scattering plane is different from the main scattering

plane because the wave vector is different from that of theincident beam (the comparison with Mie scattering is

illustrated in Figs. 1 and 2). Therefore, each plane wave

component must experience a series of coordinate rotations

in order to be placed in the fixed scattering plane.

Due to the rotational symmetry of homogenous

spheres and the incident beam about the z axis, the

scattering direction can be chosen to be

u sin ; 0; cos; 21

which implies the main scattering plane coincides with

the meridian plane that contains vector u (see Fig. 2).

Now consider the plane wave component ei

u? ;exp

ikuUr, where the amplitude is defined with reference to

the meridian plane ofu. First, we transform the reference

plane to the meridian plane of u, which can be done

X Y

Z

u u'

r'

l'

r

l

Fig. 1. The geometry of Mie scattering, where uis the incident direction,

u is the scattering direction, and is the scattering angle.

XY

Z

u u'

i2 i1

l'r'

lr

Fig. 2. Scattering of one plane wave component of the partially coherent

beam, where Z is the direction of whole incident beam,uis the incident

direction,u is the scattering direction, and is the scattering angle. i1 is

the angle between the meridian plane containing u and the scattering

plane, and i2 is the angle between the meridian plane containing u andthe scattering plane.

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through a rotation about the zaxis, i.e.

eil

eir

0@

1A-

cos sin

sin cos

! e

il

eir

0@

1A; 22

or in a more compact form ei-Rei, where the fact is

used that the unit vector u has a spherical coordinate

;

. Second, we consider the scattering of a plane wave

from direction uinto direction u. Since in the LorenzMie

scattering the change of amplitude is defined with respect

to the scattering plane, the incident field amplitude has to

be transformed from the meridian plane to scattering

plane by a counterclockwise rotation about vector uwith

an angle i1 (see Fig. 2), which is the angle between the

meridian plane of uand the scattering plane spanned by

vectors uand u. Thus, the amplitude of the incident field

can be written as

Ri1Rei: 23

By applying the results of the LorenzMie theory, we

obtain the scattered field in the scattering plane ofuand u

SRi1Rei; 24

where is the angle between u and u, and S is the

scattering amplitude matrix of LorenzMie scattering,

which is given by[15]

S S2 0

0 S1

!: 25

Third, the scattered field has to be expressed with

respect to the main scattering plane, and is obtained by a

clockwise rotation about vector u with an angle i2,

where i2 is the angle between the scattering plane

spanned by vector u and u. The resulting scattered field

reads

R1

i2SRi1Rei; 26

which is equivalent to

Ri2SRi1Rei; 27

when using the relation R1

i2 Ri2.

Note that the above argument only works when o

or u uU ^z40. When u uU ^zo0, the scattered field

in the main scattering plane is given by

R1

i2SRi1Rei R i2SRi1Re

i;

28

which is essentially the same as Eq. (27) except that the

signs of sin i1 and sin i2 are reversed. Therefore, after

obtaining cos i1 and cos i2 from spherical trigonometry,

namely

cos i1 cos cos cos

sin sin ; 29

cos i2 cos cos cos

sin sin ; 30

sin i1 and sin i2 are given by

sin i1 ^

u ^

u

U ^

zju uU ^zjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi

1 cos 2i1p

; 31

sin i2 u uU ^z

ju uU ^zj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi1 cos 2i2

p ; 32

where cos uUu.

We obtain the total scattered field referenced to the

main scattering plane

Esru; expikr

ikrZ Au; uUeiu? ; d2u? ; 33

where the 22 matrix Au; u is defined by

Au; u Ri2SRi1R: 34

2.4. Mueller matrix

Once the scattered field is known, we can calculate the

CSDM

Wsr1 u1; r2 u2;

exp ikr1r2

k2

r1r2

Z d

2u? d

2u?

An

u1; uU ~W

i

u? ; u

? ;UATu2; u; 35

with the definition of the angular correlation matrix of the

incident field given by

~W

i

u? ; u

? ; einu? ;Ue

iTu? ;: 36

Taking the limit that the width of the Gaussian Schell-

model beam goes to infinity, which implies that Eq. (20)

holds, we obtain

Wsr1 u1; r2 u2;

exp ikr1r2

k2

r1r2

k2s

2

2

Z d

2u?A

n

u1; uUWiz 0UA

Tu2 u

exp 12

k2s

2 u

? j2

; 37where W

iz 0 is the CSDM of the incident beam

defined at the z0 plane. From Eq. (37), we have the

polarization matrix at point ru

Wsru; ru;

1

k2

r2

k2s

2

2

Z d

2u?A

n

u; uU

Wiz 0UA

Tu; u exp

1

2k

2s

2ju

? j2

: 38

Using the relation between the Stokes vector and the

polarization matrix(18), and the definition of the Mueller

matrix

Isu 1

k2

r2PuUIiz 0; 39

we get the 44 Mueller matrix Pu, given by

Pu k

2s

2

2

Z d

2u? exp

1

2k

2s

2ju

? j2

Mu; u: 40

Here, the 44 matrix Mu; u is the Mueller matrix

defined from the effective amplitude matrix Au; u. Some

selected matrix elements ofMu; uare listed below

M11 jA1j2jA2j

2jA3j2jA4j

2=2;

M12 jA2j2jA2j

2jA4j2jA3j

2=2;


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M22 jA1j2jA2j

2jA3j2jA4j

2=2;

M33 ReA1An

2A3An

4;

M43 ImA1An

2A3An

4;

M44 ReA1An

2A3An

4; 41

whereA1 A22, A2 A11,A3 A12, andA4 A21. Substitut-

ing Eq.(34)into Eq.(41), we find the relation between the

matrixMu; u and the Mueller matrix of a sphere in the

fully coherent case P0

M11 P011 ;

M12 P011 cos 2i1;

M22 P011 cos 2i2 cos 2i1P

033 sin 2i2 sin 2i1;

M33 P011 sin 2i2 sin 2i1 P

033 cos 2i2 cos 2i1;

M43 P043 cos 2i1;

M44 P044 : 42

Together with Eq. (40), we can calculate the Mueller

matrix of a sphere when the incident light has an arbitrary

degree of coherence.

2.5. Spectral degree of coherence

Once the CSDM is obtained, we can study the spectral

degree of coherence of the scattered field. According to

definition (4), the degree of coherence sr1 u1; r2 u2;

between two directions in the far field zone can be written as

sr1 u1; r2 u2; TrW

sr1 u1; r2 u2;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiSsr1 u1;q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS

sr2 u2;q

: 43

We consider a special case where the incident beam is

completely unpolarized, i.e. the CSDM at z 0 has the

value

Wi

lmz 0 Ilm: 44

Therefore, the CSDM of the scattered field is given by

Wsr1 u1; r2 u2; I

expikr1r2

k2

r1r2

k2s

2

2

Z d

2u?A

n

u1; uUATu2; u exp

1

2k

2s

2 ju

? j2

;

45

from which one can obtain the spectral density at point

ru by

Ssru; I 2

k2r2

k2s

2

2

Z d

2u? P

011 uU uexp

1

2k

2s

2 ju

? j2

:

46

P011 has the same meaning as in the previous section, and

the numerical results of the spectral degree of coherence

will be presented inSection 4.

2.6. Water droplets and ice crystals

We apply the developed formalism to randomlyoriented ice crystals. In general, the scattering amplitude

matrix in Eq. (25)is no longer diagonal but is given by

SS2 S3

S4 S1

!: 47

Following the same procedure as inSection 2.4,one can

derive similar results in Eqs. (40) and(42), with Eq. (42)

modified to be

M11 P011 ;

M12 P011 cos 2i1P

013 sin 2i1;

M22 P022 cos 2i2P

032 sin 2i2 cos 2i1

P033 sin 2i2P

023 cos 2i2 sin 2i1;

M33 P022 sin 2i2P

032 cos 2i2 sin 2i1

P033 cos 2i2P023 sin 2i2 cos 2i1;

M43 P043 cos 2i1P

042 sin 2i1;

M44 P044 : 48

After orientation averaging,P0

becomes block-diagonal,

which means the Mueller matrix elements P013 , P

023 , P

032 ,

andP042 in the previous equation vanish. Therefore, we get

the same formalism as for the sphere, except that P0

is the

orientation averaged Mueller matrix in the fully coherent

case.

As an application, we studied the coherence effects on

light scattering by spherical water droplets and hexagonal

ice crystals. Here, the coherent Mueller matrix for an ice

crystal was obtained by using the invariant imbedding

T-matrix method[13]. The numerical results are presented

inSection 4.

3. DDA simulation

We present a general numerical method to calculate

the Mueller matrix of a particle of arbitrary shape and

refractive index when the incident light is partially coher-

ent. To link the numerical simulations to the analytical

results in the previous section, we limit the incident beam

to the Gaussian Schell-model beam with infinite width.

Thus, the only free parameter of the incident beam is the

coherence length. Moreover, the incident field can have an

arbitrary polarization state because the Mueller matrix isindependent of the CSDM at the source plane (z0). In the

simulation, we choose to linearly polarize the incident

field. After simplification, the electric field is essentially

reduced to a scalar random field, and the second-order

correlation matrix is reduced to a scalar function. The first

step of the problem is to generate a 2D source that satisfies

the required correlation function, and once the source is

known, the field at any point in space can be obtained by

propagating the source using the Green's function [7].

Several numerical methods are available, such as the

random pulse method[16], random phase screens method

[17], and shift-invariant filter method [18], but we use a

more straightforward general method. As an example, letus consider the second-order correlation function on the


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z0 plane

1;2; Un2; tU1; t A1A221;

49

where A is the spatial profile, 21 is the spatial

degree of coherence, and is the temporal degree of

coherence. The goal is to express the random field U; t as

a superposition of some random variables that havepredefined correlation functions, i.e.

U; t Ak

hkheikUe itIkxIkyI; 50

where Ikx, Iky, and I are complex random numbers

with correlation functions

InkxInkyI

nIkxIkyI kxkxkyky: 51

From Eqs.(49) to(51), one can obtain

1;2; A1A2k

jhkj2eikU 2 1

jhj2e i;

52

which implies that

k

jhkj2eikU 2 1 21; 53

jhj2e : 54

Thus, the coefficients hk and h in the random field

U; t can be obtained by solving the two equations.

For the monochromatic case, the correlation function is

given by

1;2 exp 12

2

2s2

!12; 55

and the random field is written as

U k

hkeikUIkxIky: 56

The coefficient hk in Eq. (50)can be easily solved from

Eq.(53). In the actual DDA simulation, the incident field

Ei;z; can be obtained directly from the above results,

i.e.

Ei;z; jk? jrkhk? IkxIky

exp i k? U

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik

2jk? j

2

q z

ei; 57

whereei is a unit vector denoting the polarization of the

incident field. For a specific realization of the random field,

one can get the scattering amplitude matrix Su and

corresponding Mueller matrix Mu in the direction u.

After averaging over the ensemble of the incident fields,

one can get the Mueller matrix for a sphere. Numeric

results are discussed inSection 4.

4. Results

We applied the analytical formalism to calculate the

Mueller matrix of a sphere when the incident light is

partially coherent. The results are shown in Fig. 3. The

sphere has a refractive index of n 1:5 and a radius ofa , where is the wavelength of the incident light. The

coherence length shas values fromto 10. Note thatP11is essentially the intensity reported in [10,11]based on a

scalar incident field and similar conclusions were reached.

One of the most important conclusions is that the phase

function tends to be more isotropic as the incident field

becomes less coherent and gradually approaches the

coherent case as the coherence length increases. Other

Mueller matrix elements also become more isotropic ascoherence deteriorates. Moreover, the reduced Mueller

matrix elements P22=P11, P33=P11, and P44=P11 in the

backward direction are much more sensitive to the coher-

ence length than those in the forward direction, and can

be explained by the fact that interference in the forward

direction is more robust to random phases than that in the

backward direction.

To validate the formalism we derived, we compared the

analytical results with DDA simulations. The comparisons

of Mueller matrix elements are shown inFig. 4. The sphere

has the same parameters as before and two cases, s

and 4, are computed. In the DDA simulations, 160 plane

waves components are used to construct the incident field,a total of 1000 ensemble averages are taken, and the inter-

dipole spacing used isd =20. The results show that DDA

simulations agree very well with the analytical formalism.

We investigated the Mueller matrix elements in

the backward direction (1801 degree scattering). The

sphere has the same parameters as before, and the

coherence length is chosen from 0 to 10. As can be seen

from Fig. 5, for Mueller matrix elements P12=P11 and

P43=P11, no significant dependence on coherence length

is observed. In comparison, other Mueller matrix elements

all change monotonically and gradually plateau as the

coherence length increases. Evidently, all approach coher-

ent values when the coherence length reaches a largeenough value.

In addition, we computed the spectral degree of coher-

ence of the scattered field. The results are shown in Fig. 6,

where the sphere has the same parameters as previously

and the coherence length is chosen fromto 10. Here, the

spectral degree of coherence is defined as the correlation

between two directions, the zand the polar angle . We

find that at any angle, the norm of the degree of coherence

always decreases as the incident light becomes less coher-

ent. Note that jj is not always unity even when the

coherence length approaches infinity due to the comple-

tely unpolarized special form of the incident beam. A

simple calculation gives the degree of coherence whenthe coherence length is infinitely large

1; 2 jSn21S22Sn11S12jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jS11j2jS21j

2q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jS12j2jS22j

2q ;

58

whereS1 and S2 are defined in Eq. (25). It can be shown

that Eq.(58) is not always equal to unity.

As applications of the formalism developed, we com-

puted the Mueller matrix elements for water droplets and

hexagonal ice crystals that are the constituents for atmo-

spheric clouds. The results for a water droplet are shown in

Fig. 7. The water droplet has a size parameter x 80 and arefractive index n 1:33. The coherence length parameter


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kshas values from 5 to 100. Apart from similar conclusions

to the case of a small sphere, we found that both the coronaand glory phenomena will be eliminated as the incident

field becomes incoherent. Since both are consequences of

interference, which can be extinguished by the introductionof incoherence, the elimination is reasonable. The results for

Fig. 3. Mueller matrix elements for a sphere of radius a and refractive index n1.5. The coherence length s has values of, 4, and 10.

Fig. 4. Comparison of Mueller matrix elements computed from DDA and Mie methods. Two cases, s and 4are simulated. The remaining parameters

are the same asFig. 3.


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the hexagonal ice crystal are shown inFig. 8. The ice crystal

has a size parameter kL 2ka 150, whereL is the height

andais the semi-width, and a refractive indexn 1:31. The

coherence length parameter ks has the values 5, 10, and

100. The results show that both the 221 and 461 halos

gradually disappear as the coherence length decreases,

which can be explained by the same reasoning as used for

the corona and glory for water droplets. The Mueller matrix

elements for the water droplet and ice crystal in the

backward direction are shown in Figs. 9 and 10, respec-tively. We found similar results to those of small spheres.

5. Conclusions and discussions

The conventional LorenzMie formalism is generalized

to the case when the incident light is partially coherent.For the Gaussian Schell-model beams, the Mueller matrix

and spectral degree of coherence for the scattered field is

derived analytically and the formalism is validated by DDA

simulation. The results suggest that the reduced Mueller

matrix elements P22=P11, P33=P11, and P44=P11 in the

backward direction strongly depend on the degree of

coherence of the incident light. The same formalism is

applied to randomly orientated water droplets and hex-

agonal ice crystals. We find that the corona and glory

phenomena associated with water droplets and halos with

hexagonal ice crystals disappear if the incident light is

highly incoherent.

The relevance of these results to atmospheric radiationis three-fold. First, despite the fact that the solar source is

completely incoherent, which is presumably thermal

emission, the sunlight reaching the surface of the earth

acquires a certain degree of correlation through the

process of propagation. This is a straightforward result of

the van CittertZernike theorem, and a rough estimation

reveals that the sunlight has a spatial coherence length of

about 60mm [7]. Given the size of large atmospheric

particles, the finite coherence length of sunlight will have

observable effects on the pattern of the scattered field.

Second, when using a laser beam instead of natural solar

light in remote sensing, the light beam becomes partially

coherent when passing through atmospheric turbulence.Therefore, it is of practical interest to investigate the effect

Fig. 5. Mueller matrix elements for a sphere in the backward direction (1801). The coherence length s is from 0 to 10. The remaining parameters are the

same asFig. 3.

0 45 90 135 180

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 6. The spectral degree coherence of the scattered field from a sphere.

is the angle between scattering direction and the z direction. Five

cases, including the coherent case are simulated. The parameters are the

same asFig. 3.


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of spatial coherence in light scattering. Third, even though

there is not a variant of the radiative transfer equation inwhich the scattering matrix derived in this paper can be

inserted, the results obtained here shed light on what

results can be expected in a multiple scattering situation.As it is shown here, all Muller matrix elements become

Fig. 7. Mueller matrix elements for a water droplet of size parameter x80 and refractive indexn 1:33. The coherence length parameter ks has values of

5, 10 and 100.

Fig. 8. Mueller matrix elements for a hexagonal ice crystal with size parameterkL2ka150, whereLis the height andais the semi-width, and refractive

indexn 1:31. The coherence length parameter ks has values of 5, 10 and 100.


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Thus our results provide a potential method to deter-

mine coherence length and for possible applications in

atmospheric remote sensing.

Acknowledgments

George Kattawar acknowledges support by the National

Science Foundation (OCE-1130906). Ping Yang acknowl-

edges support by NASA (NNX11AK37G).

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