polarization and atmospheric backscatter coefficient measurements

Polarization and atmospheric backscatter coefficientmeasurements

Richard Anderson

Recently, it was pointed out that polarization effects must be considered in hard target calibration of lidars.A vector theory of radiometry is developed, and it is demonstrated for a real nonideal target that thereflectance is a matrix quantity and not a scalar quantity, and all its components must be measured. Theseconcepts can be extended to actual field measurements of the volume backscatter coefficients. The volumebackscatter coefficient at range R is an averaged (4 X 4) matrix, which is averaged over the sampling depth dR

= cr2. The transmitted beam is polarized in a definite sense, the received beam is still polarized, and bothare represented as (4 X 1) Stokes vectors so that the interaction must be represented by a (4 X 4) matrix calledthe volume backscatter coefficient 3. Present experiments are in error for data are considered a scalarquantity with only one value not a matrix with sixteen components. Some of these components may be zerobut many are not.

1. Introduction

The 1987 paper by Kavaya1 presented the idea thatpolarization effects must be included in the calibrationof hard targets. Previous to this theory all calibrationmethods and atmospheric measurements were per-formed as if light were depolarized and a scalar theorywas used. Laser radiation used in lidar is polarized,and the radiation field must be treated as a vectorquantity. This means that optical components, scat-tering medium, etc. must be treated as Mueller matri-ces and the fields are Stoke's vectors. Any correctcalibration technique or field measurement techniquemust be treated in this context. The reflectance p,BRDF, and backscattering parameter p* of a target are(4 X 4) matrices, and a correct calibration involves thedetermination of all sixteen components of these ma-trices. Similarly, in a volume backscatter experimentthe measured beta is a (4 X 4) matrix, and it can becorrectly represented when all sixteen matrix compo-nents are measured.

Nicodemus et al.2 discussed the scalar theory ofvarious radiometric quantities as the BSSRDF,BRDF, reflectance, and reflectance parameter. The

The author is with University of Missouri-Rolla, Physics Depart-ment, Rolla, Missouri 65401.

Received 1 February 1988.0003-6935/89/050865-10$02.00/0.© 1989 Optical Society of America.

BSSRDF is actually a (4 X 4) scattering matrix relatingthe incident STOKES vector flux to the emergentStokes vector radiance. The target scattering matrixis defined by the vector equation

dLi(Oi,0i;xi,yi;Orr;xr,yr) = S(0ieti;Xii;orPr;xryr)dfi. (1)

The symbols and represent Stokes vectors andMueller matrices, respectively, and • is the bidirec-tional surface scattering reflectance distribution func-tion matrix BSSRDF. The BRDF matrix is related tothe BSSRDF matrix by an integration over the irradi-ated area. Each component of the BSSRDF is a func-tion of the 'incident angles and coordinates and theemergent angles and coordinates, and the BSSRDFmatrix is component by component integrated overthe irradiated area (xi,yi) to give the BRDF (4 X 4)matrix, which is no longer a function of xi,yi. TheBRDF matrix is represented by fr(0iji;0r,0r;Xryr) or

hr(0iki;0r,'r;xrYr) = i 9(i,,ii;xi,yi;Or,r;xr,yr)dAi. (2)

The reflectance of a target must also be represented bya (4 X 4) matrix when polarized radiation is used. Nononideal real target totally depolarizes an incidentbeam, so the polarized incident flux is represented by adefinite incident, (4 X 1) Stokes vector, and the emergentflux is represented by another (4 X 1) Stokes vector:

dfr = X(0i,0Si;0rr)d4i, (3)

where p is the reflectance matrix and dqi and dobr arethe incident and emergent fluxes on an area dAi of thetarget. The backscatter parameter p* is a (4 X 4)matrix for by definition it is given by the relationship

1 March 1989 / Vol. 28, No. 5 / APPLIED OPTICS 865

Figure 1 shows the irradiation and observation ge-ometry. The incident flux is related to the incidentradiance by the equation

dfi(i,oi) = di(0i,0i)dAi

(5)= {I Li(oj,oi) cosoidwi} dAi,

where (i,oi) are the irradiation angles, dAi is the irra-diated area, and the projected solid angle is d~i =cosidcoi. The irradiance dEi is within the solid angledwi at (0i,oi) and may not be uniform over the area dAi.The emergent irradiance from the irradiated area Ai is

Fig. 1. Schematic of irradiation and observation pattern.dLr(iki;0rcr;XrYr) = JAi dLr(Oi,0i;XiYi;Or,4 r;XrsYr)

and from Eqs. (1) and (2)

dLr(0is(i;XiYi;0r,0r;XrYr)

= S(oiCi;Xiyi;orr;xryr)dAiLi(OiOi) cosOidwi,

Wr Wr

= *(0i,0i;-0i,0i~d¢i, (4)

where 'Wr is the emergent solid angle and b* is thebackscatter parameter. For any real target whichmight exhibit retroreflection, specular reflection, anddiffuse scattering, the reflectance and backscatter pa-rameter matrices are different for each set of angles ofirradiation and each set of angles of emergence.

Bickel and Bailey3 presented an excellent discussionon the measurement of the scattering matrix. Thesescattering matrix components are products of (4 X 4)matrices of the optical components and the scatteringmatrix taken in the proper order. The matrices of theoptical components should be known so that the scat-tering matrix can be evaluated. As an example, forvertically polarized incident light and horizontal po-larized detected light the first Stoke's vector, which isonly sensed by the detector, is S11 - S12 + S21 - S2 2. Itrequires three additional polarization measurementsto determine S11, S12, S21, and S22, and these are verti-cal-vertical, horizontal-vertical, and horizontal-hori-zontal measurements. For hard target measurementsin lidar calibration the scatter matrices become thecomponents of p, fr, and p*. For atmospheric volumebackscatter measurements it would be f. An alternatemethod of determining the scattering matrix is givenby Shumaker,4 and it is discussed later.

II. Theory

A. Backscattering Parameter and Reflectance Matrix

In this discussion it is assumed that the reflectancematrix components have been determined in sometype of backscattering experiment used to calibratefield targets. It is necessary to determine the relation-ship between the reflectance matrix components andthe backscatter parameter matrix components. Thereflectance is defined as the ratio of the reflected fluxby the incident flux and is given by Eq. (3). Thereflectance and backscatter parameter matrix can berelated to each other and the BRDF matrix.

so

dL,(09q5i;0r 0r;1r Yr) = 1r(0ii; 0rr;Xr r)Li(O0,¢'i) cosOidwi. (7)

Generally, it is assumed the incident flux is uniformand isotropic, and the surface is a diffuse scatterer so itscatters uniformly and isotropically with respect to theposition of irradiation and observation. This meansthat the BSSRDF and BRDF matrices depend only onthe separation distance between the points of observa-tion and irradiation. In these conditions Eq. (2) be-comes

ir(0it0i;0rtr) = IA 0(Oi,'Oi;OrOr;r)dAi, (8)

where r = [(Xr - X,) 2 + (Yr - yi) 2]1/2. Since Ei(Oi,oi) isuniform and isotropic this is designated by the symbolEi in Eq. (7) or

dLr(OiOki;Orcr;Ei) = Ir(0i0i;0rskr)Li(0i,0ki) cosOidwi.

Now

Lr(0r,4r) = i dLr(0icki;0rfr;Ei)

- J Ir(Oiski;0rcr)Li(Oi,¢i) cosOidwi

and

d~r = [| Lr(Or,'Or) COSOrdwr]dAr

[f r

= [Jf | ?r(Oi,(Pi;0r,(r)Li(Oiki) COSOi CoSOrdwidwr] dAr-

Now from Eqs. (3) and (5)

dqkr = J co(siki;Orr)Li(Oiisfi) cOaidwi} dAi,

(9)

(10)

(11)

so from Eqs. (10) and (11) the relation between thereflectance and BRDF matrices is

866 APPLIED OPTICS / Vol. 28, No. 5 / 1 March 1989

(6)

[L A(0j,,0;0,,,0,)Lj(0j,0j) cosOidwi dAi

= [| f r(Oiki;Orkr)Li(°iri) cosOi cosOdwidw ] dAr. (12)

The backscatter parameter becomes

[A *(6j,,0j;0j,0j)L(0j,0,) coOidwi] dAi

= 1 r(0,4j,0j,0)Lj(0j,0j) cos2Oidwidwr] dAr, (13)

where Wr is the total emergent solid angle subtended atthe detector by the target. In hard target calibrationcertain conditions are assumed to hold. It is assumedthat the incident radiance is uniform and isotropic andnot a function of Oi and Xi. The surface scatters in auniform and isotropic manner so it is the same in the Orand 'r directions. The solid angles of observation andincidence are small, and the detector observes an areagreater than the irradiated area Ai but less than theentire target At > Ar > Ai. Then in the integrals Ar =Ai. Then Eqs. (12) and (13) become

PWtisi r) = (r(Oiki;r,r0) C °rwr (14)

and

b*(0j,0;180--0j,-0) = (fr(0j,0j;180 - oi,oi)) cosoi, (15)

where (fr(0iki;0rr)) is the averaged BRDF matrixover the small solid angles wi and Wr-

The main point of the discussion above is that for areal target (field target) used to calibrate the lidar thereflectance cannot be represented by a scalar numberp. For a nonideal hard target it will be a differentmatrix for each angle of incidence and emergence. Atbackscatter it and the backscatter parameter are alsomatrices. It will be assumed that b or p* have beenproperly evaluated at backscatter.

Determination of the matrix components in Eq. (3)will be discussed. In this experiment a laser beam is-expanded and is polarized in turn in each of the statesof polarization: horizontal, vertical, +45°, -45°, andright- and left-hand circular. This radiation strikes ascattering medium which can change the intensity andthe state of polarization of the incident radiation. Fi-nally, the emergent radiation is detected by using aquarterwave plate and linear polarizer preceding thedetector. The light is focused on the detector. Thedetector can be swung in an arc about the scatterer inthe usual goniometer-reflectometer arrangement (Fig.2). The angle of observation can be varied from thenormal direction to the scatterer to angles approaching490'. At backscatter angles Or = -i + 60 and -0i - 60are measured, where 60 is the smallest angle the detec-tor can be placed to backscatter.

The backscatter reflectance for each angle of inci-dence Oi must be interpolated from data taken from 0r= +90 to -90° to the target normal and at the closestpossible angle to backscatter. In hard target calibra-tion targets far from ideal are used (sandpaper, sand-

DETECTOR CIRCLE

POLARIZER R ,

LASER X - - -A - -…-BEAM 7 \ \ _' ,/ SAMPLE ONBEAM 9,,/ ROTATABLEEXPANDER LENS Ac \ / / TABLE

X' ZLENS7 >e _ANALYZER '

/ ROTATABLESPECULAR DETECTOR

REFLECTEDBEAM

Fig. 2. Schematic of the reflectometer.

blasted aluminum, etc.), and depending on the targetthey may exhibit retroreflection, off-specular reflec-tion, specular reflection, diffuse reflection, etc. In eachregion the reflectance is different. The fact that thereflectance matrix may be different for each fixed an-gle of incidence at many different emergent angles isdemonstrated from the scattering matrix measure-ments made by Iefelice and Bickel.5 The reflectancematrix is directly related to the scattering matrix, ex-cept that reflectance requires absolute measurements.

The laser will be assumed to be temperature andfrequency stabilized. Before each measurement foreach state of polarization the incident flux /i is mea-sured by measuring the power of the incident beamwith the target removed, allowing the light to fall di-rectly on the analyzer and detector (straight through).The analyzer is set in the various positions it is used foreach polarization of the light, and this signal is theincident signal with the analyzer attenuation normal-ized out of the measurements. The laser is continu-ously monitored to correct for intensity variations.

The sample is replaced, and measurements are madeon the sample for each incident polarized state, and foreach state the analyzer is set at its four orientationsbetween the quarterwave plate and the linear polariz-er. For each incident polarized beam and for eachsetting of the analyzer-detector the measured flux is

(16)

in which ir is the emergent beam Stoke's vector, M isthe analyzer Mueller matrix, b is the reflectance matrixof the medium, and hi is the incident polarized flux. Itis assumed that the detector is polarization-indepen-dent.

The quarterwave plate is set at an angle a to thehorizontal and the polarizer at fi. In measurements, atypical set of a and : is (0,0°),(0,90°),(45,45°),(0,45°).The quarterwave plate and linear polarizer Muellermatrices are seen in Table I where

SI(a,o) = MX4(a)Mp(O- (17)

Measurement of the components of the p matrix ateach angle of observation requires sixteen measure-ments. The detector senses only the first component


f = 90A,

of the Stoke's vector. One typical calculation will bemade for incident horizontally polarized light scat-tered from the target. The analyzer is set at angles aand a of (0,0°). One has from Eq. (16)

1 1 0 0\P11 P12 P13 PI¢ = 0i/4 1 1 0 O)l P21 P22 P23 P24 111

0 0 00 P31 P32 P33 P34 0

0 0 0 P41 P42 P43 P44/0

P11 + P12

= /4 P11 + P12+ P21 + P22

+ P21 + P22o

0

In Table II are all the first components of Eq. (16)which is sensed by the detector for various states ofincident polarization and the four settings of the ana-lyzer for each incident polarization. From these six-teen equations all the matrix components of p can beevaluated. For each angle of incidence the reflectancemust be measured at many emergent angles and fromplots of Pij vs emergent angles allow the scatterer to becharacterized.

B. Volume Backscatter Coefficient

Details of the measurement at large R of the volumebackscatter coefficient of the atmosphere neglecting

the effects of turbulence and speckle have been givenby Kavaya et al.7- 9 To measure the volume backscat-ter coefficient of the atmosphere at large R it is neces-sary to calibrate the lidar system from the hard targetfor which the backscatter matrix p* is known at theangle of incidence on the hard target in the field cali-bration. Consider a lidar system with separate trans-mitters and receiver so a choice can be made on thepolarization of the incident and received signals. Thelaser is pulsed to simplify the analysis, and the pulse isassumed to be rectangular, of width r. The powerreceived by the receiver from the hard target at rangeR is

Prt(t) = *P(t - 2R8/c)A/R2 70(R)

[ JR ] (19)

in which Pit(t - 2R8/c) is the incident polarized powerand is a Stoke's vector, Prt(t) is the Stoke's vector of thereceived field, p* is the (4 X 4) backscatter parametermatrix of the target, changing the incident beam inten-sity and polarization. A/RS is the solid angle at whichthe receiver subtends at the target, where A is theeffective receiver area. (Rs) is the telescope overlapfunction at range R,8 is the optical system efficiency,and at(r) is the atmosphere extinction coefficient along

Table 1. Components of the M(a,#) Matrix

(1 02M (a) = 0 Cos 2a

X/ 4 a 0 Sin2a os2aSin2a

0Sin2a os2a

Sin 2a-Cos2a

IM () 1 /2 00.23p 13) / |sin2p3

0

Cos2aCos26Cos2Cos2aCos2SSin2ACos2aCos2S

0

Sin2aCos26Cos2PSin2aCos26Sin2pSin2aCos2&

0

M(0°.00) = 1/2

M(45 ,45°) - 1/2

1 1 0 01 1 0 00 0 0 00 0 0 0

1 0 1 0,I 0 0 0

1 0 1 0O O O

M(00 ,900) = 1/2

M(00 ,450) 1/2

1 -1 0 0-1 1 0 00 0 0 1 O 00 0

10 0 1)0 0 0

10 0 1 .O O0


o-Sin2aCos2a

o

COSPCos 2P

Sin2p Cos2p0

t Cos2pSin2p

O

Sin2pSin2p Cs2p

Sin 2P0

o0o0

6 - p - a.

Sin26Cos2PSin26Sin2PSin26

0

Wap)=M WM (a)-IP X/4 2

the target path. The atmospheric backscatter overthe short target path is negligible compared to theeffect of the scattering of the target. Since the inci-dent pulse has a width -, the pulse only interacts withthe target over this time. The entire detected signalfrom the target is obtained by integrating the pulsebetween 2R,/c and (2R,/c) - r, so Eq. (18) becomes

J(2R /) - RIt = J' Pdt = A/Rs i0(R,)

2R,/c

X exp- [2 J at(r)dr c/2 .i- P*Pitdt,

but

J b*Pt(t)dt = P* Pit(t)dt = p* Wit, (21)

in which Wit is the total energy in the incident pulse,and, since the radiation was polarized, it is a Stoke'svector, which indicates the beam's polarization prefer-ence. Equation (19) becomes

It = AIR',qO(Rs) exp - [2 J at(r)dr p* Wic/2. (22)

It should be mentioned that It is also a Stoke's vector(20) for the scattered light is still polarized, and its prefer-

ence of polarization is indicated by the values of theStoke's vector components Q, U, or V.7

Table II. First Matrix Components of the Equation 0, = Mpo from which the Matrix Components of p can be Determined

(P1 + 1 2 + 21 + 22] =

101 + 12 - 21 - 22 = (4 r/0iJh(0O.900)

(P11 + P12 + P31 + P32 (4r/0ijh(450,45)

(P11 12 + 41 + P42) = (40r/0igh(0O,450)

(P11 - 1 3 + 2 1 - 2 3 3

-P1 P1 3 - 2 1 + 23]

( 00r/i) -(0 .0-

(0r/1_ij -( o90

(P11 - P1 3 + P3 1 P33) = (40r/0i)-(450,450)

(Pil - P1 3 P41 - P43] = (40r/0 i-(0O 450)

(2 Or/0sV(O°.o0)

(P11 - P12 - P21 + 22] = (2 r/oi.v(^0 0 .900)

(P11 + P1 4 + P21 + P2 4 = (4Or/Oigr(0o,0o)

(Pl + 1 4 P21 - P2 4 3 = 40r/0jir(o0 ,90)

(P11 - 1 2 + 3 1 - 3 2 ) = (2r/Oiv(45045°) (P11+ P1 4 + P3 1 + P3 4 ] = (40r/0i1 r(450,450)

(P11 - 1 2 + 4 1 - 4 2 3 = (20r/0i]V(0°.450). (P1 1 + 1 4 + P41 + 44] = (40r/0ijr(0',450)

IP1 + 1 3 + 2 1 + P 2 3 ]

(P+l P1 3 - P2 1 - 233

= (40r/0i)+(0°,00)

= (40r/01i]+(00,900)

(P11 - 1 4 + 2 1 P24

(P11 - P14 -

= (40r/0i 1 1(00, 00)

P21 + P2 4 = (40r/0i 1 1(00,900)

[Pll + 13 + P31 + P33 ] = (40r/0 i)+(45',450) (p11 - P14 + P31 n34] (4 r/0i]1(45',450)

(P11 + P13 + P4 1 + P4 3 3 = (40r/oi]+(00,450) (p11 - P14 + P41 P44) = (40r/0i)1(0°,450)

The subscript designation is p(a,p) where p = h,v,+,-,r or 1.


(P11 - 1 2 + 2 1 - 22 =

(4 Or/0i]h(0',0')

The lidar is now directed into the atmosphere, andthe signal corresponding to the range Rb is examined todetermine the volume backscatter coefficient (Rb).Again, the transmitted pulse emitted at time t - 2Rb/cgives the received signal at time t. This pulse is polar-ized and is represented by the Stoke's vector Pit. Thispulse will sample a slab of atmosphere of thickness crT2 at Rb = c/2(t-r). The backscattered signal receivedat time t, in general, is not depolarized and must berepresented by a Stoke's vector or

ct/2Prb(t) = I (Rb)Pb(t - 2R/c)AR2O(Rb)n

c/2(t-,)

X exp -[2 fRb b(r)dr] dR. (23)

in which A/Rb, q, and O(Rb) were defined previouslyand ab(r) is the extinction coefficient along the atmo-spheric path. The volume backscatter coefficient atrange R is a (4 X 4) matrix, which changes both theintensity and polarization of the incident field. As-sume an aerosol density for only single backscatter,and this backscatter is on the average represented bythe volume backscatter coefficient, and the backscat-ter process changes the intensity and polarization ofthe incident beam. The $ measured is a spatial aver-age over a distance of width dR = c/2r.

If one assumes PAb is nearly constant over the rangeof integration, a number of terms can be removed fromthe integral, and it may be written in the form

Prb(t) = 1A/RIO(Rb) exp - [2 J ab(r)dr (Rb)

ct/2X Pi(t - 2R/c)dR

b /2(t-r)

= qA/R2 O(Rb) exp - [2 J ab(r)d] (Rb)cI2 J Pfb(t)dt

(24)

= 7A/RbO (Rb) exp -[2 Jb ab(r)dr] 4(Rb) Wibc/2, (25)

in which Wib is the total energy in the incident polar-

ized pulse, and it must be represented by a Stoke'svector.

Equations (22) and (25) give the received signal ifthe detector were not preceded by an analyzer de-scribed in the previous section. If one desires to ob-tain the components of the 3(Rb) matrix both in cali-bration and in atmospheric sampling, the incidentlight must be polarized horizontally, vertically, +45°to the horizontal, -45° to the horizontal, and right-and left-hand circularly, and for each type of incidentlight signal it is detected at the four orientations of theanalyzer. The analyzer was described previously.Then Eqs. (22) and (25) become

It(t) = Kt(a,3)j*Wit(t) (26)

and

Prb(t) = Kb2f(a,f3)/(Rb) Wib(t),

in which

Kt = A/R 210(R) exp - 2 J at(r)dl c/2

and

Kb = A/RMnO(Rb) exp - 2 ab(r)dr c/2.

The detector measures only the first component of theStoke's vector, and only these components will be list-ed. Consider the example in which Wit and Wib aremeasured for horizontal polarized light and for theanalyzer angles a = 00 and = 00. Then intensitiesmeasured by the detector are the first components ofthe Stokes vectors in Eq. (26):

IMt) = KtWi 1 (t)/4(P11 + P12 + P21 + P22)

Prb(t) = KbWib(t)/4(01ll + '312 + I21 + '322). (27)

Table III shows the ratio of the measured backscattersignals for a horizontal incident polarized beam andsimilar sets of equation result for the other incidentpolarized beams. Table IV is the volume backscattercoefficient matrix components evaluated from them.

Table . Ratio of the Backscatter Signal from the Atmosphere to the Hard Target forHorizontally Polarized Incident Light

(P11+P12+P21+P22 = Kb[It (t) Wb(t) Jh(O o 0)(P 11+P 12+P 21 P*22)

(311~/312/321-/32 = Pb(t) Wit(t) o

+A11/12/3 /32 = 't [ rb (t) Wt(t) Jh(5,5 1 1 2(11 12 P31032] Kb btt) W ](t) ]h ° °[P*l1+P*12+P*31+P*32)

[p1+p1+p4+p43 =K It (t) Wb(t-) h(0° 45o)[P ll+P*12 P*41 P*42)


Table IV. Matrix Components of the Spatial Averaged Volume Backscatter Coefficient of the Atmosphere at Range Rb

4b ([Itb] h 0° 0°) 1P 11 + +12 P +21 22 + f Wit 12 - -P11 " Kt 11, t lb h~~~ 1 Wb h00,00 1 +P2Ij

+ t(Pr b't]V (o 0,) IP 1 1 - P12 + P21 - P 2 2]

2 [(It Wib h(0 0QO) (P 11 + P'1 2 + 21 + *22)

+ (rb Wit ](*0,00

t b vrb w, 0 (P 11

+ frb it h, ,0°) (P 11

- P12 21+ P22])

+ P 1 2 - P 21 - P22)

(t b t] ] 00) - P'12 + P 21 - 22)

- b t b (00 00 ) lli si 2

021 4Kb f h( 00o (P'll + P'1 2 + P*21 + P*22

[P lb h( P( 11 0 12 P 21 P22)

02 Wt +[h 0 , ° ( 1 P*12 P*21 + P22)

[rb Wit]v(oo 90 )PI 11 - P 12 - 21 + P22]IV

-t Ibib t]h(,90 ( 11 + P12 P*21 P 22]

-(Pb Wibt (v(o ) [ 11 P*12 - P 21 + P*22]

- hWib ((,90°)Ps P 12 - P 2 1 - P*2 2 ]]

- [(Pb Wit] (Psl- 1 P12 + *21 - 22 (Ie Wit ]V(00,90) -P 1 2-P 2 1 +

3 2Kb ([r Wit vh(4545) P1 + P1 2 + 31 - P*32 ]

=2 2Kb ([b Wit 450) IP + P12 + +

(b WitJ v(45 0 450)1 11 - 12 + P*31 - P* 3 2 ]A -

P4 1 2Kb rb Wit] P11 + 12 + 41 +P*2

b ([tP lb h(450,450) -

I e Wit l (v(p09 s0) [ 11 - *12 + *4 1 - P 42 ]] A ll

P* 223]1

cont inued


Table IV, continued

Kt ([rIrb Witl 4 -'042 - 2 III It;b .0 L ("l '2 '1' 42)2

b LL.t Wib, h(0 .5 ) ~ +p 1 4

frb W1 t

.bj( ,45 0(

Kt (P rb Wit

13 2 b ([ t ib + .(0,O)

Ii -P'12 + P41 - Ps42)3 - 12

(Pil + +:3 P; 1 4.)(3 +, r o 9ot (X +1 ~;3- 21 P -3 11+ (1 r Wi J+0 0,900) JJ

Kt K b[(I Witi l + *P1 P23 P23 P2Al

2Kt lb I 00 41/ P3

(P P;j - 42

93' Kt I[(' ib, +(0 .0 + '

_ (rb 't (4. + pl- 41- 43~ -21

I.t Wib .1+(0O .900) tAi 1 3 P 1 P2JJ 2

33 2Kb ([(tWb J +(45 45 [ + 13 + 31 + P3

fprb Witj -- * - - 4~3, -1It Wb (450 450) [ 11 ; 3 3 1 3) 1

22 [Ilt +(0°,450) [#;l1 P; + ': + ';3)(t43 } t ( ;1 + P;1 + ;r W 3 13

b it) -

(I't Wib J .100.450) (P 43JjPJ P4 -P1] 13

014 2Kb ttt1t Wib Jr0° 0 ) Pll + P1 4 + 21 + 4

+ [Prb Witi ~f It Wb r)00,900) ( 11 P4 P2 1 P

2 4J

14 A

- rt rb it +t

2Kb; ([(It lb J r)0 0

.00) (P1 + # A21+ '24j

[t Wb r00

,go) (Pll P1 4

P 2 1P

2 4]} - 21

[Rb L t1r44,45) ;1 ;4P3 P34] +PI4

K34 ([[PtW ( l 4 1

2Kb rb ir(t0 4s + P 4 + p 1 + p44

[44 t ([yb Wib ('1504 4 ; 3

'4

[~ it1

134 rbfb 110r4(00,45

0) * 1 + *4 P4 4

[ I-rWb nt 11 -

In all the measurements the ratios of beam powers andthe total energies of the incident beams are required,and this means that relative values and not absolutevalues may be measured.

111. Discussion

In beta measurements there are two different lidarconfigurations in use. One consists of a separate

transmitter-receiver optics. In this system it is possi-ble to transmit a beam of any state of polarization, and,since the laser is vertically polarized, this natural stateof polarization of the laser is generally transmitted.The detected radiation may be of the same polariza-tion or at an angle of 900 to the transmitted beam.These are the usual co- and cross-polarization mea-surements. The other lidar system uses a single tele-


scope to transmit and receive the radiation. This sys-tem transmits right-hand circularly polarized lightand detects left-hand circularly polarized light.

In this discussion, only matrix components presentin the measurements outlined above are considered forthe case of a general scattering medium. Let i be theincident flux on the target and fr the detected flux.These quantities are polarized and must be represent-ed by Stoke's vectors. The scattering medium (atmo-spheric medium and optical components) is represent-ed by the matrix S. For the linear polarized 3 lidar thetransmitted signal is vertically polarized and exam-ined at both vertical and horizontal polarization or inco- and cross-polarization. For the co-polarizationmeasurements one has

S12 S13 S14\ 1S22 S23 S24 -1

S32 S33 S34 0 )S42 S43 S44 0 /

Sll - S1 2 - S21 + S22

= i 8S11 + S12 + 21 - S22

o )

so the first component of the Stoke's vector is

Or = (Sll - S12 - S21 + S22)0i- (29)

In the case of a cross-polarization measurement thefirst component of the Stoke's vector is

Or = (ll - S12 + S21 - S22)i. (30)

In the case of circular polarization a quarterwave plateand a linear polarizer give a pseudoanalyzer with thematrix

1 0 0

A = 110 0(0 0 0

OO

/1

0 j and notA= 0/0

0 00 00 00 0

This difference is discussed by Bickel and Bailey3 andSchumaker, 4 but if the polarizer also consists of alinear polarizer and a quarterwave plate it is represent-ed by the pseudomatrix

1 1 0 0\P-=OO 0 0

00 O11 0 0

1 0 0 1\

and not P =0 00 1 0 0 0

Theory demonstrated that the same final matrixcomponents occur in the flux using the pseudopolar-izer and analyzer as would be obtained with the idealdevices. The correct polarizer and analyzer, whichyield the second set of matrices above, consist of acombination of a quarterwave plate-linear polarizer-quarterwave plate, and the polarizer is its own analyzerwhen light traverses it in the opposite direction. Forthe pseudoanalyzer one has

(28)

1 0 0 -1\ Sil1 0 0 -1 S21

= O OO O S31

o o o S41

S12 S13 814 1

S22 23 S24

S32 S33 S34 0

S42 S43 S44 1

11 + S1 4 - S41 - S44

= i Sll + S14 - S41 - S44 I

so the measured power is

0 = qS1(S1 + S 14 - S41 - 44).

Then the /3 backscatter signal for incident verticalpolarized light in both co- and cross-polarization de-tection is given by the following equations:

co-polarized detection (vertical)

Kb (PIJt)Wib(t)

X (Pll - P12 - P21 + P22);

cross-polarized detection (horizontal)

( - 12 + 21 - 22) = (t) )h,

X (Pll- P12 + P21- P22).

In the case in which the incident light is right-handcircularly polarized, and the detected light is left-handcircularly polarized, the backscatter signal gives

('3l + 14- 341 - 344)Kb ( IJ(t)Wb(t) )l,r

X (Pll + P;4 - P41 - P44)'

Equation (33) contains two of the previous sixteenequations used to determine /, and they are shown inTable III. Equation (34) is not identical to previousequations, because the angles between the quarter-wave'plate and linear polarizer do not produce a matrixequivalent to the analyzer matrix for left-hand circu-larly polarized light.

From this discussion it is apparent that a 3 lidarusing separate optics for the transmitter and receiver ismost adaptable and could be modified to measure thesixteen matrix components of /. In comparing Eqs.(33) and (34), the measured expression for ,B termsinvolves totally different sums and differences ofterms. This means that 3 measurements could verywell give ,B values, which are different for the two typesof instrument when observing the same portion of thesky.

In all previous theory similar ratios of signals weremeasured, but both / and p* were taken as scalarquantities, which can only be true if the scattered lightis totally depolarized. This is not generally observed,and the / and p* scalar values are really the bracketand terms in Eqs. (33) and (34). The scalar relation-ship used by most investigators is

Kb ( Iz(t) Wib(t) )P


(31)

(32)

(33)

(34)

1 -1 0 0 Sli

�, = 0i -1 I 0 0 S21

0 0 0 0 S31

0 0 0 0 S42

The two sets of measurements taken by differentinstruments would be expected to yield different re-sults unless the scattered light is completely depolar-ized. In that case the /3 matrix is

/11 0 0

( 0 0 0 (35)0 0 0/

Even in a scattering from highly symmetric particles(spherical particles) where the matrix is symmetric,the two different / instruments would not measure thesame , values. Only in cases where the matrix isdiagonal and one diagonal component is much largerare the values nearly the same. There may be cases ofdiagonal matrices in which /3ll >> 22, 33, and 44, butthis means the scattering medium has nearly depolar-ized the light and the diagonal matrix is virtually thesame as in Eq. (35).

IV. Conclusions

This paper was written to call the attention to prob-lems of polarized light in volume backscatter measure-ments. The problem must be formulated in terms ofMueller matrices and Stoke's vectors. The presenttheory of backscattering measurements considers thescattered light to be totally depolarized, and a scalartheory is appropriate. This condition may be met insome limited circumstances, but in general the correcttheory must treat the volume backscatter coefficientas a matrix quantity and the incident and detectedfields as Stoke's vectors. If in the scattering processthe detected light is polarized, the matrix theory mustbe used and the backscatter measurements taken bythe different apparatus should not agree.

Development of a backscattering instrument tomeasure all the /3 components will be much more com-plex than the present apparatus, and measurementswill have to be completely automated and analyzed ona real-time computer system. If the present measure-

ment technique is not changed, the measurement mustbe made with the same type of apparatus to comparemeasurements. The instrument involving separatetransmitters and receivers is most adaptable. If onlyco- and cross-polarization measurements are made,these will not be the true but will be representative of/ and is given in Eq. (33).

This research was performed under a NASA/ASEESummer Faculty Fellowship during the summer of1987.

References1. M. J. Kavaya, "Polarization Effects on Hard Target Calibration

of Lidar Systems," Appl. Opt. 26, 796 (1987).2.. F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, and

T. Limperis, "Geometrical Consideration and Nomenclature forReflectance," in Self-Study Manual on Optical Radiation Mea-surements: Part I-Concepts, Natl. Bur. Stand. U.S. Monogr.160 (Oct. 1977), Chap. 6.

3. W. S. Bickel and W. M. Bailey, "Stokes Vectors and MuellerMatrices and Polarized Scattered Light," Am. J. Phys. 53, 463(1985).

4. J. B. Shumaker, "Distribution of Optical Radiation with Respectto Polarization," in Self-Study Manual on Optical RadiationMeasurements, F. E. Nicodemus, Ed., Natl. Bur. Stand. U.S.Tech. Note 910-3 (1977).

5. V. J. Iefelice and W. S. Bickel, "Polarized Light Scattering MatrixElements for Selected Perfect and Perturbed Optical Surfaces,"Appl. Opt. 26, 2410 (1987).

6. W. A. Shurcliff, Polarized Light (Harvard U. P., Cambridge,1962).

7. M. J. Kavaya, R. T. Menzies, D. A. Haner, U. P. Oppenheim, andP. H. Flamont, "Target Reflectance Measurements for Calibra-tion of Lidar Atmospheric Backscatter Data," Appl. Opt. 22,2619(1983).

8. M. J. Kavaya and R. T. Menzies, "Lidar Aerosol BackscatterMeasurements: Systematic, Modeling, and Calibration ErrorConsiderations," Appl. Opt. 24, 3444 (1985).

9. M. J. Kavaya and R. T. Menzies, "Aerosol Backscatter LidarCalibration and Data Interpretation," JPL Publ. 84-6 (1 Mar.1984).

ICO Prize Committee Seeks Nominations for 1989 Award

The International Commission for Optics awards an annual prize of $1000 for outstandingachievement in optics. The recipient of the ICO Prize must not have reached the age of 40 be-fore December 31 of the year for which the Prize is awarded. The Prize Committee welcomesnominations for this prize. The Nominator should briefly sketch the name and presentaffiliation of the nominee, his or her academic and employment history, major publications, anda short description of the scientific achievement for which the candidate is nominated for theaward. The nomination should be sent, no later than 31 March 1989, to the Chairman, ICOPrize Committee, Henri H. Arsenault, LROL, Universite Laval, Quebec, P.Q., GlK 7P4, Canada.


polarization and atmospheric backscatter coefficient measurements

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