Recent Developments in the Study of the Polarization of Sky Light*

Recent Developments in the Study of the Polarization of Sky Light* ZDENEK SEKERA University of California at Los Angeles, California Page 1. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 46 2. Theory of the Polarization of Sky Light.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Stokes Polarization Parameters. . . . . . . . . . . 2.2. Scattering of Light; Rayleigh and Mie 2.3. Equation of Radiative Transfer and It 2.4. Polarization of Sky Light in a Molecul 2.5. Polarization of Sky Light in a Turbid Atmosphere.. . . . . . . . . . . . . . . . . . 72 3. Measurements of the Sky-Light Polarization.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.1. New Developments in tjhe Technique of Polarization hleasurements.. . . . 76 3.2. Results of the Recent Photoelectric Measurements of the Sky-Light Polarization.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2.1. Measurements of the Positions of the Neutral Points. . . . . . . . . . . . 82 88 3.2.2. Position of the Maximum Polarization.. . . . . . . . . . . . . . . . . . . . . 3.2.3. Maximum Degree of Polarization in the S 3.3. Dispersion of the Sky-Light, Polarization in Relation t,o the Turbidit Atmospheric Optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 99 103

1. INTRODUCTION The investigation of the polarization of sky light is one of the studies in the geophysical field, in which rather significant advances were achieved during the last few years. Much progress has been made both in the theoretical studies as well as in the measurement of the actual polarization of the diffuse light from a clear sky. In the theoretical studies the correct formulation and the solution of the basic problems of radiative transfer, as presented by Chandrasekhar in a series of papers in the Astrophysical Journal [ 11and then summarized in his treatise “Radiative Transfer” [2] gave rise to a new line of attack on the theoretical problems of sky-light polarization. First, by means of the polarization parameters, introduced for the first time by Stokes in 1852 and then again by Chandrasekhar, not only the state of polarization

* Much of the work reported on in this paper has been made possible by sponsorship extended by the Geophysics Research Directorate, Air Force Cambridge Research Center. Their support is gratefully acknowledged. 43

44

ZDENEK SEKERA

can be correctly formulated, but also a great simplification is introduced in the mathematical treatment. Furthermore, Chandrasekhar's derivation of the expressions for the polarization parameters of the diffuse sky radiation from the equation of radiative transfer offers the great advantage over the previously used direct computations in that the effects of multiple (higher order) scattering is incorporated in the most rational way. In this method the cumulative effect of each successive order of scattering is obtained by successive iterations from certain systems of integral equations, while in the previous methods each higher order scattering could be included by performing two additional integrations of progressively more complicated expressions. Finally, by expressing the solution of the equation of radiative transfer in terms of the so-called scattering and transmission matrices, with the use of Chandrasekhar's principles of invariance, i t is possible t o reduce the problem to the solution of a few systems of integral equations. Once the solutions of these equations are knownthe so-called X - and Y-functions-the derived quantities in almost all problems concerning the illumination and polarization of the diffuse sky radiation can be expressed in terms of these functions and thus computed without great difficulties. For the case of Rayleigh scattering, it is possible to determine uniquely all necessary constants and thus to arrive a t the exact solution of the transfer problem (with all orders of scattering included). Although the scattering in the real atmosphere is governed by a more complicated law than that of Rayleigh scattering, the knowledge of the exact solution in terms of Rayleigh scattering is of the utmost importance for the theoretical discussion of the sky-light polarization. Because of the fundamental assumption that the scattering particles are of a size negligible with respect to the wavelength, the Rayleigh scattering is applicable only to a pure molecular atmosphere. However, the ratio of the number of molecules in a unit volume to the number of aerosol particles in a turbid atmosphere is so large that the molecular scattering is responsible for the major part of the observed polarization. Therefore, the effect of larger aerosol particles (i.e., of a size comparable to or larger than the wavelength) can be computed as a first-order correction to the values corresponding to the molecular atmosphere, in the computation considered as the first approximation. When the theoretical deviations of the polarization of sky light in a turbid atmosphere from those of a molecular atmosphere are computed for different size distributions of the aerosol particles, and then compared with the measured deviations, the actual size distribution of the aerosol particles can be indirectly determined. Thanks to the recent rapid development of electronics, especially in the construction of better photoelectric tubes, it has been possible t o improve the photoelectric methods of polarization measurements t o such

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

45

a high degree t hat they supersede in many respects all previous visual measurements. The great sensitivity of modern photomultiplier tubes makes measurements in narrow spectral regions possible. They can be extended t o wavelengths beyond the sensitivity of the human eye (ultraviolet and infrared regions). If a rotating retardation plate in front of a fixed analyzer is used as the measuring element, the values of the total intensity, of the degree of polarization, and of the position of the plane of polarization can be obtained instantaneously. With the previously used Martens polarimeter, the measurement of just the degree of polarization required from 3 to 5 min. With such a new instrument, yielding instantaneous values, i t is further possible to measure the degree of polarization along any vertical plane by scanning from one horizon through the zenith to the other horizon, measuring simultaneously in several wavelengths, the period of one scan being restricted only by the time of response of the recording instruments. If the scanning is performed along the sun's vertical, the positions of the neutral points can be determined as well. With a proper amplification even a very low degree of polarization (of the order of 0.001) can be detected. In such a case the position of a neutral point is indicated by a sharp minimum in the record of the degree of polarization even in the close vicinity of the sun and of the horizon. I n this way all the neutral points can be measured with the same facility continuously from sunrise to sunset, while by previous methods the intense background of the field in the Savart polariscope decreased the visibility of the fringes and thus prevented the measurements of the Babinet point for high solar elevations. The Brewster point, for the same reason, was observed only on very rare occasions and with a special arrangement t o eliminate the glare of the environment of the point. The measurements made quite recently with such a photoelectric polarimeter have confirmed the results of the relatively few existing visual measurements which were sporadically made in the past in narrow spectral regions. Systematic and continuous measurements throughout t.he day have revealed very interesting diurnal variations. I n order to eliminate the changes in the sky-light polarization due to the changes in the sun's elevation, the theoretical values for a molecular atmosphere of the composition and density distribution of the actual atmosphere were subtracted from the measured values. The deviations obtained in such a way have shown a rather general characteristic, namely, the increase in magnitude for the longer wavelengths. The intensity of skylight caused by the molecular scattering and by the scattering from large aerosol particles varies with the wavelength according to different laws, namely, and as ( b I l), respectively. Hence, the effect of large particles on the sky-light polarization is more pronounced for the longer wavelengths. The increase of the magnitude of the deviations with the wave-

46

ZDENEK SEKERA

length therefore indicates that the major part of the deviations is an effect of the large particle scattering. Moreover, the deviations show an asymmetric daily variation, with the maximum in the afternoon, closely following the diurnal variation of dust and haze particle content-in itself a result of the diurnal variation in the atmospheric turbulence and in the production of air contaminants in industrial areas. In addition, distinct short periodical fluctuations can be noticed in the deviations, increasing with the wavelength and superimposed upon the diurnal variations just described. This is caused by horizontal and vertical inhomogeneities in the aerosol particle content. I n contrast t o the above, some quantities, such as the position of the Brewster neutral point, show deviations from the theoretical values which are larger in the shorter wavelengths and exhibit variations in magnitude from day to day. They can be attributed to the reflection from the ground or a very low haze layer, rather than to the effects of large aerosol particles. All these points and observations will be discussed in detail in the following text. *In the first part the method of solving the problem of radiative transfer as developed by Chandrasekhar [2] is outlined after a few introductory chapters. The theoretical values and distribution of skylight polarization in the molecular atmosphere are discussed, and the extension of the theory to a turbid atmosphere is indicated. In the second part the new developments in the measurements of sky-light polarization are presented, and the results of recent measurements, performed by a newly constructed photoelectric polarimeter, are summarized. In the last part the contribution of the study of sky-light polarization to the solution of other problems of atmospheric optics is mentioned. 2. THEORY OF

THE

POLARIZATION OF SKY LIGHT

2.1. Stokes Polarization Parameters

As is well known, radiation fields such as those represented by the diffuse sky light, may be assumed to consist of trains of electromagnetic waves, in which the electric and the magnetic field intensities (electric and magnetic vectors E and H)oscillate in a plane normal to the direction of propagation of the waves (given by the unit vector n). If any two orthogonal components of the electric (or magnetic) vector oscillate in such a regular way that the phase difference 6 between these two oscillations and the ratio of their amplitudes remain constant, the corresponding electromagnetic waves or the radiation is said to be polarized. Under such circumstances the end point of the electric vector describes an ellipse (elliptical polarization), if the phase difference is different from a multiple

DEVELOPMENTS I N THE STUDY OF POLARIZATION OF SKY LIGHT

47

of T. If 6 = 0 or nT (n being an integer), then the ellipse degenerates into a straight line (linear polarization). If 6 = (n ,l.i)~ and, in addition, the amplitudes of both oscillations are equal, the ellipse becomes a circle (circular polarization). On the other hand, if the two components oscillate in such a random fashion that the phase difference is not a constant, the mean position of the end point of the electric vector cannot be determined, and the radiation is unpolarized or neutral. The human eye and other photosensitive elements which are used for measuring the light intensity, are unable to distinguish the different characteristics of the oscillation of the electric vector directly. They respond only to the energy of the corresponding electromagnetic wave. I n the case of diverging radiation from a point source, the radiant energy is measured by the intensity,' and in the case of parallel radiation, by the net flux2 The relationship between these quantities and the electric or magnetic vector can be established by means of the Poynting vector. If, in the first case, the electric and magnetic vectors in a sufficiently large distance R from the source can be written in the form

+

(2.la) and similarly for the parallel radiation, E = Aei(kz--wt), H = Bei(kz-ot) (2.lb) the intensity and the net flux are proportional to (2.2)

Re { A X B * . n ]

or

%Re (A-A*)

where the star denotes the complex conjugate. If 1, r are two perpendicular unit vectors, forming with the direction of the wave propagation n an orthogonal triple (1 X r = n), and in the expressions (2.la) and (2.lb) the amplitude vector A is written as (2.3)

A

=

At1

+ AJ

then-in complete analogy to the expression (2.2)-it introduce the following quantities [3]:

is convenient to

where the constant C depends on the electromagnetic units used, and the 1 The amount of radiant energy passing through a cone of unit solid angle per unit time and per unit frequency interval. * The amount of radiant energy passing through a unit area normal to the direction of propagation per unit time and per unit frequency interval.

48

ZDENEK SEKERA

bar represents the mean over eventual short periodic fluctuations. After (2.3) is substituted in (2.2),it is easy to see that the sum I = Il I , is equal to the intensity (or the net flux) of the radiation. Then, 11and I , represent the intensities (or the net fluxes) of the wave when only one of the components is present. Furthermore, if the phase difference 6 is explicitly expressed in At and A,, for example by putting Al = a, A , = be-", then

+

-

I,

(2.5) I l = Ca2,

=

-

Cb2,

U

=

2Cab cos 6,

V

=

2Cab sin 6

Moreover, it can be shown [2] that if x is the angle between the direction of the major axis of the ellipse and of the unit vector 1, then U

(2.6)

= (Il

- I,) tan 2x

And finally, if /3 = arctan @ / a ) , where a, b are proportional to the length of the major and minor axes of the ellipse, respectively, then

V (2.7) It can also be shown, that (2.8) 11- I , so that (2.9)

=

=

(It

+ Ir)sin 20

+

(11 I,) cos 2p cos Zx,

+

(11 1,)' = (11-

+

U

= (Il

+ I,) cos 2p sin 2x

+ U2+ V2

If the four quantities I = 11 I,, Q = I I - I,, U , V are known, the shape and the orientation of the axis of the ellipse can be determined, and thus the state of polarization completely defined. These quantities are therefore called the Stokes polarization parameters, after Sir G. G. Stokes, who introduced them for the first time in 1852. From (2.7) it follows that linear polarization is characterized by V = 0, and circular polarization by Q = U = 0, but V # 0. Since in neutral radiation the ellipse as well as the angle x become undetermined, Q and U must vanish, and since the mean of the phase difference also vanishes, it follows from (2.5) that V = 0. The neutral radiation is thus characterized by Q = U = V = 0. The Stokes polarization parameters can be directly measured if the analyzed radiation is transmitted through a retardation plate of a known retardation e followed by an analyzer, and the intensity of the radiation leaving this analyzer is measured. If the retardation plate is oriented with the fast axis along the unit vector 1, and the pipe of transmission of the analyzer is deviated from the direction of 1 by the angle $, then the measured intensity varies with $ according to the relation (2.10)

I($) = W 1 I

+ Q cos 2$ + ( U cos

B

- V sin e) sin 21))

DEVELOPMENTS I N THE STUDY OF POLARIZITION O F SKT LIGHT

49

This relation reveals an important characteristic of the Stokes parameters, namely, their additivity for incoherent streams of radiations. As a consequence of the additivity of the Stokes parameters, it follows that for a mixture of polarized and neutral radiations (so-called partial polarization) I > (Q2 U 2 V 2 ) ) $In . such a case it is convenient to introduce the degree of polarization, defined by

+ +

(2.11)

P

=

+

(Qz U 2+ V 2 ) ) + / 1

which compares the intensity of all polarized components in the mixture with the total intensity of all components. Apparently, if there is no polarized component in the mixture, P = 0; if there is no neutral component present (case of total polarization), P = 1. The use of the Stokes parameters for the complete definition of the state of polarization has-in addition to their additivity-the great advantage that all these parameters have the same dimension (that of the intensity or of the net flux). They are therefore more suited for an analytical discussion than the quantities used previously, namely, the total intensity (I),the degree of polarization ( P ) , the position of the plane of polarization (angle x ) , and the ellipticity /3 [ = arctan @ / a ) ] . 2.2. Scattering of Light; Rayleigh and Mie Theory of Scattering

The polarization of the diffuse light from a clear sky is one of the several manifestations of light scattering by air molecules and other atmospheric particles. Hence, the understanding or interpretation of the observed properties of sky-light polarization has to be based on the knowledge of the law of light scattering. It is possible to formulate the problem of light scattering referred to a scatterer having an arbitrary shape [4]. But only the law of scattering by spherical particles is so far known in all the details necessary for practical use in a theoretical computation. In deriving the law of scattering by spherical particles, it is assumed that parallel radiation of the wavelength X is incident on a single dielectric sphere of the radius a and of the dielectric constant e. The incident radiation excites electromagnetic waves both in the interior and exterior of the sphere. At a large distance from the sphere, the solution of Maxwell’s equations, which satisfies the boundary conditions on the surface of the sphere, has the form of a diverging spherical wave from the center of the sphere. This spherical wave is identified with the scattered radiation. From the electromagnetic wave theory, the expressions for the components of the electric vector of such a wave in the direction normal and parallel to the plane of scattering (i.e., the plane containing the directions of the incident and scattered waves) can be derived and, when

50

ZDENEK SEKERA

substituted in (2.4), the Stokes parameters of the scattered radiation can be easily obtained. They are related to the corresponding parameters of the incident wave by a linear relation, expressible in the form

I(*)= (p,j]* F(0)

(2.12)

(i,j = 1, 2, 3, 4)

where Va>denotes a one-column matrix with the elements (Ill(*),II(a), V a )V*)) , and F(O) the matrix corresponding to the incident flux. From a quite general consideration [5], as well as from the direct computation based on the electromagnetic theory, it follows that the matrix { P,) contains only the following elements which are different from zero, (2.13)

Pi1

Pzz = Fz,

= F1,

P33

=

= F3,

P44

= -P43

P34

= F4

If the solution of the Maxwell equations is written in the form of an infinite series, then (2.14)

1

Fa = Re {SLSR*) Fq = Im {SLSR*)

F I = SLSL*, Fz = SRSR*,

where SLand S R are certain series appearing in the Mie t h e ~ r y If . ~ the particle is of a size negligible with respect t o the wavelength, a -+ 0, then only the first term in each of the series SL and S R should be considered, yielding the expressions (2.16)

+ 2),

S R = (X/27r)a3(t - l > / ( e

SL

=

SR cos e

e

o

The relationship (2.12) assumes the form

(

(%+>”

(2.17) I(u’= ( 2 ~ / X ) ~ u ~

e o

o

1

0

COS~

0

o o

o o

cos

o

COS

)

. FCo)

e

The intensity of the scattered radiation in this case has the form which represents the radiation from an oscillating dipole, having a polarizability

!g

(2.18)

= U3(€

1

- 1)/(€ + 2)

m

8

sda,~;e)

(X/2r)

+ bn(a,c)tn(8))

{an(a,e)pn(e)

n= 1

(2.15) s R ( a , s ; e ) = (x/2r)

2

n=l

+ bn(a,e)pn(e)

Ian(a,e)tn(e)

where anj bn are complex coefficients, a = 2ra/X, and p n ( e ) = dP,(z)/dz, t,(e) = l)Pn(z)- x dP,(z)/dz, z = COB 0, 0 being the scattering angle between the incident and scattered ray, P&) being the Legendre polynomial of the order n. n(n

+

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

51

Lord Rayleigh arrived at the same result by considering the scattering by a volume very small compared to the wavelength with a dielectric constant approaching unity, without any specification of its shape. This independence of scattering from the size and shape of a very small scattering particle is inherent in the expression (2.18) for the polarizability which can be related to the macroscopically measured refractive index m by the Lorentz-Lorenz law

v,

(2.19)

r0

=

(%N)(m2 - l>/(mz

+ 2)

If (2.19) is used instead of (2.18), the intensity and the polarization of the scattered radiation becomes a function of the number N of scattering particles in a unit volume, independent of their size and shape. For larger scattering centers of a size comparable to the wavelength, the law of scattering can be derived from the infinite series (2.15). For large values of a, however, these series converge very slowly, and they should preferably be replaced by asymptotic expansions in (llcr). Several approximations for the solution of the problem for large a have recently been suggested (see [4], pp. 45-77), but the explicit form of the terms in the matrix ( P i j }have not yet been derived. In the theoretical solution of the problem of scattering by a dielectric sphere, the radiation field at a large distance from the sphere is assumed to be composed of the incident and scattered radiation, so that in computing the intensity (or the Stokes parameters) the sum of these two fields should be used. When this is done, the result contains separate terms corresponding to the intensity of the incident and of the scattered radiation and, in addition, terms resulting from the interference of these two fields. It can be shown (see Section 111 in [4]), and it is easily understood that the interference occurs only in the forward direction of the incident radiation. From the principle of the conservation of energy it follows that in the case of perfect scattering, when no loss of the radiant energy occurs during the interaction of the incident wave with the scattering particle, these interference terms are equal to the total amount of energy scattered by the particle in all directions with a negative sign (the so-called cross-section theorem). As a result of this interference the net flux of the incident radiation in the forward direction is decreased by the amount J-(IL@) I p ) dw

+

where the integration is to be performed over all solid angles. For a neutral incident radiation (F,(O) = F II(O) = X F o ) this expression reduces to (2.20)

XFoJ(F1

+ F2) dw

=

K(cY,a)na2Fo

52

ZDENEK SEKERA

where the quantity K(a,e), called the scattering cross section, can be computed from (2.14) and (2.15).* In the real atmosphere, every unit volume contains a very large number of molecules as well as of larger aerosol particles, all of them acting as scattering centers. Each of these centers is thus illuminated also by the scattered radiation from all other centers. Although the problem of scattering in such a case (multiple scattering) can be formulated without great difficulty, an attempt to solve this problem has been made only for the case when the scattering centers are so far apart that the scattered radiation can be considered as a pure radiation field (purely transverse waves). The expressions for the Stokes parameters in such a case have not yet been derived. However, as all the scattering centers in the atmosphere are randomly distributed, and moreover are subjected to a quite irregular motion, it can be assumed that the various components of the scattered radiation are incoherent. In such a case the Stokes parameters are additive, and the parameters of the radiation scattered by a unit volume can be obtained by summation of the Stokes parameters referred to the radiation from all the particles. Similarly, the total amount of energy scattered by the unit volume illuminated by neutral radiation is given by the sum of the corresponding expressions in (2.20). Hence, the volume scattering coefficient By, defined as the total amount of the radiant energy scattered by a unit volume per unit net flux of the incident radiation has the form (2.22) a

t

where N(a,e) denotes the number of particles of the radius a and of the dielectric constant c, and the summation is to be extended over all different values of a and of e. As a result of multiple scattering the radiation incident on an elementary volume dV will be decreased by the amount of the energy scattered by this volume in all directions, but it is increased by the part of the scattered radiation from the environment of dV, which is rescattered into the direction of the incident radiation. The law expressing losses or gains during this process, when expressed in a mathematical form, leads to the equation of radiative transfer in a scattering medium such as the earth's atmosphere. The solution of this fundamental equation can be

DEVELOPMENTS I N T H E STUDY O F POLARIZATION O F SK Y L I G H T

53

used to describe quantitatively the properties of the diffuse sky radiation, one of which is the polarization. 2.3. Equation of Radiative Transfer and I t s Solution

For simplicity, let us assume that the propagation of energy in a scattering medium can be measured by a scalar quantity, the specific intensity5 I p ( p , c p ) , depending in every point of the medium on the directional parameters (for example, the cosine of the zenith angle p and the azimuth cp) and the frequency v. The law of scattering can then be expressed by a single function P(p,p;p’,p’), called the phase function, defined in such a way, that the amount of energy received (per unit time and unit frequency interval) by the volume element d V through the solid angle dw’ from the direction (p’,cp’) and scattered into the solid angle dw in the direction (p,cp) is given by the expression (2.23)

(%)PV d V d u P(p,p;p’,cp’)~”(p’,cp’) dw‘

where ,By is the volume scattering coefficient defined above. From this definition it follows that the phase function satisfies the normalization condition

(2.24)

do’

(%)SP(P,co;P’,V’)

=

1

where the integration is performed over all solid angles. The energy of the radiation passing through the volume d V = da ds6 in the direction (p,cp) in the solid angle dw is decreased by the amount of radiation scattered by the volume d V in all directions of magnitude -8, d V Iy(p,cp) dw, and increased by the amount of radiation scattered , by the volume d V into the solid angle dw in the direction ( p , ~ )originally received by the volume d V from all directions (p’,cp’). The magnitude of this contribution is apparently obtained by the integration of the expression in (2.23) over all solid angles dw‘. When the net change in the radiant energy is expressed in terms of the specific intensity, the equation of radiative transfer can be easily derived in the form (2.25) d I v (P,(o)

=

- B Y

+

d s I”(P,P)ds P Y

s

P(p,p;p’,d)

dw ‘

l&’,d)41

The second term on the right side represents the virtual emission of the volume d V as a consequence of multiple scattering in the medium. If, however, the scattered radiation has a definite polarization, the radiational field no longer has scalar properties and must be defined by 5 6

I.e., the intensity as defked in per unit normal area of the emitting surface. Where ds is measured along the direction ( p , ~ ) .

54

ZDENEK SEKERA

a n intensity vector or intensity matrix Iy(p,cp)with the elements Il(p,cp), I,(p,cp), U(p,cp), V(p,cp),where the subscripts I and T refer to the intensity parallel and normal, respectively, to the vertical plane of azimuth ‘p, in which the zenith angle (arccos p ) is measured. Furthermore, if the phase function is replaced by the phase matrix

P(p,v;d,v’) = {Pi,}, ( i , j = 1, 2, 3, 4) obtained from the matrix ( Pij} in (2.12) by applying the normalization condition

(2.26) the equation of the radiative transfer has a form identical to (2.25) with the functions Iu(p,cp),P(p,(p;p’,q’) replaced by the corresponding matrices mentioned above. The solution of the integro-differential equation (2.25) may be difficult in a quite general case (see [2]); however, if the atmosphere is assumed to be plane parallel with a stratified density distribution, some simplifications can be achieved. In such a case ds = d z / p , By becomes a function of the vertical coordinate only, and the optical thickness (2.27)

r =

P y ( z ) dz

can be introduced as a parameter measuring vertical distances. Moreover, it is convenient to separate the parallel solar radiation illuminating the atmosphere, from the radiation scattered by the atmosphere, the diffuse sky radiation. If TF denotes the matrix of the Stokes parameters of the net flux of the solar radiation per unit area normal to its direction (-po,po), incident on the top of the atmosphere,’ then this flux is attenuated to the value rF exp (-7/pO) at the level of the optical thickness 7 . The volume dV is illuminated by this reduced flux, and the virtual emission in (2.25) has to be correspondingly increased. The equation of radiative transfer in a plane-parallel atmosphere then assumes the form

- (,l’4r 1

-1

/2u 0

P(P,cp;d,P’)

. Ib;p’,d) dcc’ d d

where the subscript Y is omitted since the frequency dependence appears implicitly in the optical thickness. 7 In the following, the upward direction is specified by +p, the downward direotion by - p .

DEVELOPMENTS I N THE STUDY OF POLARIZATION OF SKY LIQHT

55

The form of the equation (2.28) suggests two methods for an approximate solution; first, by successive iteration, and second, by approximative quadrature. In the first method the first approximation, obtained by setting I(r;p’,p’) E 0 in the integrand, has an obvious physical meaning, namely, i t gives the intensity and polarization for primary scattering only. Substituting this solution in the integrand, the second approximation gives the values of the sky radiation with the primary and secondary scattering included. I n both these approximations the equation reduces to a simple differential equation of the first order and can be expressed in a closed form in terms of known exponential integrals. When this method is applied to equation (2.25) for the phase function corresponding to Rayleigh scattering, the results obtained by Chapman and Hammad [6] can be easily derived. These results, however, cannot be very well used for the description of the polarization of sky light since the use of the phase function a priori excludes any polarization. In the second method the integration is replaced by a summation over finite intervals. This method is described in detail in [2] and was also successfully used by Robley [7]. The physical meaning of this approximation is quite obvious; the continuous distribution of the intensity in different directions is replaced by discrete streams of equal intensity. The integro-differential equations in (2.28) can be reduced to a few systems of simultaneous integral equations by an ingenious method developed by Chandrasekhar [2]. The intensities of the radiation emerging from the atmosphere at its upper and lower boundaries can be related to the extraterrestrial flux TF of the sun’s radiation by means of two matrices, the reflection matrix8 S and the transmission matrix T, such that a t the upper boundary: (2’29) at the lower boundary:

l(o;plq) = (j/P)s(rl;kJP;PO,PO)

1(rl;-p,v) =

‘

( ~ ~ ) T ( ~ ~ ; P , P ;* F ~o,(Po)

with r1 denoting the optical thickness of the entire atmosphere. Furthermore, if the intensities of radiation leaving any atmospheric layer at its upper or lower boundary are expressed in terms of these matrices, a series of fundamental relationships, the principles of invariance, can be derived. Helmholtz’s principle of reciprocity can be used €or the determination of the form of these matrices S and T. From the principles of invariance and their derivatives with respect to r , it is possible to derive a sufficient number of equations so that it is possible to separate the individual functions in the matrices and finally to derive the set of integral equations they have to satisfy. The fact that in the elements of the phase matrix the scattering angle

* The matrix S is called in’ 121 the scattering matrix, which name should be used preferably for the matrix in (2.12) defining the appropriate law of scattering.

56

ZDENEK SEKERA

enters only in cos 8, can be utilized in the reduction of the integro-differential equations in (2.28). Since cos e = Pp’

+ (I -

P2)$*(~

-p)5*

cos (cp’ -

cp)

the elements of the phase matrix P(p,cp;p’,cp’) can be developed into harmonic series in (cp’ - cp), Consequently, the intensities I and the other matrices can be written as series

1

M(~;P,P;PO,PO) = MJl(k)(7;~,v;~o,cpo); (M E b=O

P,S, T)

where I(O) and Mt0) are independent of azimuth, and and M(k)contain the elements with cos [k(cpo - cp)] or sin [k(cpo- cp)]. If the series in (2.30) are substituted in (2.28), and the corresponding terms in (cpe - cp) on both sides are compared, separate equations for I(k)(~;p,cp)are easily obtained. Moreover, if the matrices P(k)satisfy the relation (2.31)

(%I

lo2*P ( k ) ( ~ , ~ ; ~. P’ (, k~)‘()~ ’ , ~ ’ ; -

PO,PO)dv‘ =

p m ( p,cp; - Po,cpolf‘k’ (P’)

then the equations for Ick)admit the solutions of the form (2.32)

I c k ) ( ~ ; ~ , c=p )pck’(p,cp;- ~ o , c p o ) * F ~ ‘ ” ( ~ ; P , P O )

where q W ( ~ ; p , p are ~ ) only scalar functions. For the solution of the standard problem, in which it is assumed that the atmosphere is not illuminated from the bottom by reflected radiation, the following boundary conditions have to be satisfied: (2.33)

I(O;-p,cp) = 0,

I(Tl;P,cp) 3

0

In such a case the matrices S”) and Tck)assume the following form

The functions X ( k )and Y ( k )are solutions of the simultaneous integral

DEVELOPMENTS IN THE STUDY OF POLARIZATION O F SKY LIGHT

57

equations c,

where the form of the characteristic function q ( k ) ( p ‘ )is given by the form of the phase matrix, specifically by the form of the function f ( k ) ( p ’ ) in (2.31). The equations for the azimuth-independent matrices So)and T(O) can be obtained only by elimination based on the invariance principIes. The phase matrix for Rayleigh scattering can be obtained from the matrix in (2.17) by introducing the normalization factor ( W ) and by its transformation to obtain the intensities I t and I,, parallel and normal, respectively, to the vertical plane through the direction given by (p,cp). This transformation is described in detail in [2], pp. 40-42; as its result the phase matrix has the following properties: (a) the series in (2.30) each has only three terms k = 0, 1, 2 (as a consequence of the elements cos2O), and (b) the azimuth independent matrix PcO)has only two columns and two rows. The solution of the equation for the azimuth-independent terms then has the form

where

and the tilde denotes the transposed matrix. The eight functions in the matrices in (2.37) are linear functions of the two pairs of functions X t ( p ) , Yt(p), X & ) , Y,(p), which satisfy the equations (2.35) with the characteristic functions (2.38)

*lG)

=

(3/4>(1-

P2>,

*?(PI =

(%)(1 - P’)

In the linear relations between the functions in (2.37) and the Xi,Yi-functions (i = Z,r), all constants can be uniquely determined by the moments of the Xi, Yi-functions. From the explicit form of the matrices P(k)( k = 1,2) the characteristic functions for the X - and Y-functions appearing in (2.34) can be determined, (2.39)

*“’(P)

=

(3.8)(1 - p2)(1

+ 2~1’1,

*(‘)(P)

=

(%16)(1

+

P’)’

58

ZDENEK SEKERA

Finally, when the phase matrix is reduced with respect to the last column and row, the equation for the Stokes parameter V is obtained in the scalar form (2.28) with the phase function P(COS

e)

=

(34) cos e

The corresponding elements of the reflection and transmission matrix for the parameter V become linear functions of the functions X&), Y , ( p ) , X&), Y&); the characteristic function for the last pair of functions is of the form

q4.4 = 3P2/4

(2.40)

With the derivation of the last expression the reduction of the integrodifferential equation (2.28) is accomplished. Once the solutions of the five pairs of the integral equations of the form (2.35) with the characteristic functions given in (2.38), (2.39), and (2.40)-the X- and Y-functions-are known, the intensity and the polarization of the radiation emerging from the upper and lower boundaries of the atmosphere can be computed without great difficulty. Furthermore, with the use of the invariance principles, the intensity and the polarization of the downward or upward diffuse sky radiation a t any level within the atmosphere can be determined in terms of the reflection and transmission matrices corresponding to the layers above and below [8]. If the earth surface reflects the incident radiation at every point according to a fixed law, the additive correction to the intensity of the diffuse sky radiation can also be computed. If I, represents the intensity matrix of the reflected radiation, then the intensity of the diffuse sky radiation at the bottom of the atmosphere is increased by the amount

In the case of Lambert’s law of reflection, according to which the reflected radiation is unpolarized and isotropic in the outward direction, independent of the polarization and of the angular distribution of the incident radiation, and the outward normal flux of the reflected radiation is a fraction (albedo XO) of the inward normal flux of the incident radiation, the expression (2.41) reduces to a simple expression (3 = constant)

(2.42)

where the elements of the matrix A (2.43)

Ai, = [1

- ri0l>lrho>

(i1.i

= 2,

DEVELOPMENTS I N THE GTUDY OF POLARIZATION OF SKY LIGHT

59

contain the functions r t ( p ) , r r ( p ) linearly dependent on the functions X d p ) , Yt(1.4)~ X r ( p ) , Yr(p). These functions ~ ( p )introduced , by Chandrasekhar in [2], are proportional to the contributions corresponding to the two components of the global radiationlg received by a unit horizontal area of the earth’s surface [9]. For the case of specular reflection, the matrix I, can be computed by applying Fresnel’s law to the reduced flux of the sun’s radiation, to the diffuse sky radiation, and to the reflected radiation. When substituted in (2.41), the integration with respect to the azimuth angle can be performed, resulting in the following form for 1‘:

(2.44) *

+ +

(3p8o/32){D(plpo) 4po(1 - p2)’*(l - 1.4o~)” P cos (PO - p 2 COB 2(cpO 0 E(1.4,po) F(P,Po) cos 2(cp0 - sin(cp0 - cp) 21.4sin 2(cp0 -

I’(~i;-p,p)=

(

)

(

cp> cp>

cp)

if the solar radiation is assumed to be neutral (Ft = F , = %Po, F , = F , = 0). The one-column matrix D(p,po)and the functions E ( p , p o ) , F ( p , p o ) satisfy integral equations, containing only the functions appearing in (2.37) and the X ( i ) -and Y(i)-functions (i = 1, 2). Their solutions can be obtained by simple successive iteration [lo]. 2.4. Polarization of Sky Light in a Molecular Atmosphere

If the presence of haze or dust particles or other particulate matter in the atmosphere is disregarded, then the theory of Rayleigh scattering can be applied, and the intensity and polarization of the sky light in such a molecular atmosphere can be computed as indicated in the preceding section. From (2.17)’ (2.20) the expression for the volume scattering coefficient is easily obtained, (2.45)

32r3 ( m - 1)2

As is well known, the refractivity (m - 1) of a gas is proportional to the density, and thus the variation of By with height for each air constituent can be computed from the known composition and density distribution of the atmosphere. When the contributions to By from each air constituent are added, the optical thickness r for a given height and for a given wavelength is obtained by the integration according to (2.27). The computations based on the most recent atmospheric data have shown (see [ll]). that the mass scattering coefficient k, = ( & / p ) is almost constant up to 9

That is the total amount of radiant energy from the sun and from the sky.

60

ZDENEK SEKERA

80 km because there is little change in the composition of the atmosphere in this region. The results of the computation are summarized in Fig. 1, where the variation of the optical thickness with the height and with the wavelength is clearly demonstrated.

FIG. 1. Variation of the optical thickness of the molecular atmosphere with the height and with the whvelength.

The sun’s radiation may very well be considered to be unpolarized, in which case Ft = F, = XF,F, = F, = 0. Because of the reducibility of the matrix P(p,‘p;p’,cp’) in (2.28) with respect to the last row and colum6, the parameter V of the sky radiation in a molecular atmosphere is de; pendent only on F,, and hence it vanishes identically in a molecular a& mosphere. The diffuse sky radiation in a molecular atmosphere is therefore only linearly polarized.

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

61

For the computation of the intensity and polarization of the sky radiation in the Rayleigh atmosphere, it is thus necessary to know the values of only four pairs of the X - and Y-functions with the characteristic functions given in (2.38) and (2.39). These functions were computed by successive iterations of the corresponding equations (2.35) , performed independently a t the Watson Scientific Computing Laboratory, New York, and at the Institute for Numerical Analysis (National Bureau of Standards) , Los Angeles, with the use of high-speed computers. From the values of the X - and Y-functions, the eight functions in (2.37) and the functions yz, y, have been computed for different values of r and P. The intensities IZ and I , along the sun’s vertical have been computed for T = 0.15, 0.25, 1.00, for other azimuths only for T = 0.15. The results are published in [12] and [13]. Once the intensities Iz and I , along the solar vertical are known, the corresponding values for other azimuths can be obtained from the relationship (2.46)

where

I

+

Iz(P,P) = 11cos2%’(PO - cp) 11,sin2%’(PO -

+ p2Z(d sin2 P)

(PO - P> - Z(P)sin2 (PO - cp) U(P,P)= ( 2 ~ ) - ’ [ 1 t Ita]sin (PO - P) - PZ(P)sin 2 (PO- P)

Ir(P,P) = I r

+

IZ Iz(P,Po), I z ~ IZ(P,PO Ir Ir(Pcc,cpo)

are the intensities along the sun’s vertical and (2.47)

Z ( P ) = (%6P0)(1

- P2)(P - PO)-’{x(2’(PO)y(2)(P) - Y(2’(Po)x(2)(P) 1

From (2.46) it follows that U = 0 along the sun’s vertical so that the plane of polarization is according to (2.6) parallel or normal to the solar vertical, and the expression for the degree of polarization reduces to the form (2.48)

P

= (Ir

-

Il)/(Ir

+ 11)

or

(Ir

-

Ita)/(Ir

+ 11.1

for the solar and antisolar side of the sun’s vertical, respectively. In this form, the degree of polarization is positive, when the plane of polarization is normal to the sun’s vertical, and is negative, when the plane of polarization is parallel to the sun’s vertical, in agreement with the original definition of the positive and negative polarization, introduced by Arago. The distribution of the degree of polarization computed from (2.48) is represented in Fig. 2, for the sun close to the zenith ( p o = 0.98) and close to the horizon (PO = 0.20) for the optical thickness T = 0.15 (see [14]).The basic properties of the sky-light polarization in the sun’s verti-

62

ZCMllH INILL OC OBICMVITIOM

FIQ.2. Distribution of the degree of polariration along the sun’s vertical in a Rayleigh atmosphere for two different solar elevations and for the optical thickness of 0.15.

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIQHT

63

.90

.80

.m 60

30

A0

.30

.2 0

.I 0

0

-.I 0

FIG. 3. Distribution of the degree of polarization along the sun's vertical in a Rayleigh atmospherefor a different optical thickness (zenith distance of the sun 53.1').

cal in a molecular atmosphere can be easily seen from these two curves, namely, (a) the maximum polarization close to 90" from the sun, (b) the negative polarization in the vicinity of the sun and of the antisolar point, and (c) the neutral points (P = 0), the Babinet and Brewster points for the higher solar elevations, and the Babinet and Arago points for the sun close to the horizon.

64

ZDENEK SEKERA

.90

Z E N I T H ANGLE O F OISERVATION

FIG.4. Distribution of the degree of polarization along the sun's vertical in a Rayleigh atmosphere for different values of the albedo (zenith distance of the sun 53. lo).

The dependence of the degree of polarization on the optical thickness can be appreciated by examining Fig. 3, where the distribution of the degree of polarization is shown for r = 0.15, 0.25, 1.00 for the zenith distance of the sun 53.1'. The maximum degree of polarization decreases with increasing optical thickness, and its position is shifted slightly to-

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

65

wards the sun. The neutral points appear a t greater distances from the sun or from the antisolar point, the larger the optical thickness. If in (2.48) the intensities are increased by I { , I,’ in (2.42), the effect of the isotropic ground reflection (according to the Lambert law) can be studied and is illustrated in Fig. 4 for 7 = 0.25 for three different values of the albedo (0, 0.25, 0.80). The effect of such ground reflection on the maximum polarization is similar to that of the increased optical thickness, namely, a decrease in the degree of polarization with increasing z

90.

04.

0.

10-

zcr

I

I

3P

40.

I

5060. SUN’S ZENITH ANGLE

70.

00.

90.

FIG.5. Position of the maximum degree of polarization in the sun’s vertical in a Rayleigh atmosphere for different zenith distances of the sun and for different values of the optical thickness (A, = 0).

albedo. However, the positions of the neutral points, especially of the Babinet point, are unaffected by the isotropic, neutral reflection. The exact position of the maximum polarization in the sun’s vertical can be determined from the condition d P / d p = 0, leading to the equation d[ln (It.

+ It’)I/dp = dtln ( I , + I i ) l / d ~

A graphical solution of this equation gives the positions of the maximum,

plotted as a function of the zenith distance of the sun in Fig. 5 for different optical thicknesses and in Fig. 6 for different values of the albedo (see [15]). Similarly, the positions of the neutral points (see [IS]) in the sun’s vertical are given by the condition or

I, I,

+ I,’

+ I,’

-

I1 - II’

=

0

II, - It’

=

0

for the Babinet and Brewster points or for the Arago point, respectively. I n Fig. 7 the distances of the neutral points from the sun and from the antisolar point are shown for different sun elevations. The curves give distances for three optical thicknesses (7 = 0.15, 0.25, and 1.00), the full

66

EDENEK SEKERA

lines corresponding to zero albedo, the dotted line to an albedo of 0.80. The upper curve from the origin up to the cusp, corresponds t o the position of the Brewster point, from the cusp on t o that of the Arago point. A closer inspection of the diagram will reveal the fact that all the cusps of the different curves are situated on a straight line indicating the position of the neutral point on the horizon. In a molecular atmosphere, the Brewster point should rise at the same time as the Arago point sets, and vice. versa.1° The broken curve corresponds to the position of the

0.

10.

20.

30.

40.

SO'

Sun's Zenith Angle

60.

70.

80.

90'

FIG.0. Position of the maximum degree of the polarization in the sun's vertical in a Rayleigh atmosphere for different zenith distances of the sun and for different values of the albedo (optical thickness 7 = 0.60).

Babinet point. Its relative position with respect to the other curves indicates that the distance of the Babinet point from the sun is much smaller than that of the Brewster point or that of the Arago point from the antisolar point. For a large optical thickness, such as 7 = 1.00, the curve for the Arago point suggests the existence of two neutral points for sun elevations between 26.2" and 23.9". These two points appear first as one point 6.4"above the horizon; and for lower sun elevations they separate, one of them eventually disappearing below the horizon, while the upper one, the regular Arago point, rises slowly above the horizon. The dotted curves in Fig. 7 show the position of the neutral points when the ground is reflecting according to the Lambert law with the albedo of 0.80. The neutral points are shifted away from the sun with respect to their positions for zero albedo. This shift is appreciable only for large optical thicknesses. The shape of the curve corresponding to the position of the Arago point is changed so that the possibility of the appearance of a double Arago neutral point is eliminated. It should be 10 This property of the neutral points in a molecular atmosphere wae mentioned for the first time in [13].

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIQHT

67

mentioned, however, that the albedo of 0.80 is much higher than its average value, and it corresponds to the reflectivity of the ground when covered by fresh snow. While in the sun’s vertical the state of polarization is completely determined by one quantity, namely, the degree of polarization P in (2.48) with a positive or negative sign, defining the position of the plane of polarization (normal or parallel to the sun’s vertical), away from the

-bO,Brewsler

and Arago; - - - - L a o , Babinet;.~~.....A~.80,allothers.

Fro. 7. Distances of the neutral points from the sun or from the antisolar point in a Rayleigh atmosphere for different values of the sun’s elevation, of the optical thickness and of the albedo.

sun’s vertical the state of polarization of the M u s e sky light is determined by two quantities, the degree of polarization (2.49) P

=

{ ~ [ I I ( c c ,v )I r G , p ) 1 2

+ U ( P , V > * } / [ ~ I ( +C CI~(cc,v>I ,V)

always a positive quantity, and the deviation of the plane of polarization from the vertical plane, given according to (2.6) by (2.50)

tan 2~ = ~ ( c c , c P ) / [ I I ( cc , cI~(cc,cP>I P)

As an example, the distribution of the degree of polarization over the entire sky is illustrated for 7 = 0.15 in Fig. 8 for the higher sun elevations,

68

ZDENEK SEKERA

and in Fig. 9 for the lower sun elevations. Since the distribution is symmetrical with respect to the sun’s vertical, only half of the hemisphere is shown. The isolines of the degree of polarization are numbered in percentages (see [17]).

FIG.8.Distribution of the degree of polarization over the entire sky for a higher sunk elevation (in a Rayleigh atmosphere of the optical thickness 0.15). w-

P

0

FIG.9.Distribution of the degree of polarization over the entire sky for a low sun’s elevation (in a Rayleigh atmosphere of the optical thickness 0.15).

From the expression in (2.50) it follows that the plane of polarization is parallel or normal to the local vertical (i.e., the vertical plane through the direction toward the point of observation) wherever U(p,cp) = 0, and inclined by 45’ wherever Il(p,cp) - Ir(p,p) = 0, i.e., along the so-called neu-

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

69

tral lines. At the intersection of the neutral lines and the lines U ( p , ( p ) = 0, the position of the plane of polarization is undetermined, and according to (2.49) this point is a neutral point ( P = 0). From (2.46) it is evident =O)

FIG.10. Position of the lines U ( p , p ) = 0 for different zenith distances of the sun and for different values of the optical thickness in a Rayleigh atmosphere.

that U ( p , q ) = 0 for sin (PO(2.51)

cp) =

0, and at points where

11- 11, = 4 g 2 2 ( p ) cog (PO - cp)

The first condition is satisfied along the sun’s vertical. Beyond the sun’s vertical the points of the curve U(p,(p) = 0 are given by (2.51), and the position and the shape of this curve can be appreciated from Fig. 10, showing a set of these curves for different zenith distances of the sun and for different optical thicknesses. All these curves have a similar

70

ZDENEK SEKERA

character, they intersect the sun’s vertical at right angles at the zenith and at a point a few degrees above the sun. For a given position of the sun, the curves corresponding to different optical thicknesses are situated very close together as, for example, for the zenith distance of the sun of 43.9’’ they are not far from the curve corresponding to the primary scattering only. The positions of the point of intersection of these curves with the sun’s vertical are shown in Fig. 11 in their dependence on the position of the sun and on the optical thickness.

0.

Zenith Dirlontm of Sun

FIQ.11. Position of the intersection of the line U ( p , q ) = 0 with the sun’s vertical for diflerent values of the zenith distance of the sun and of the optical thickness in a Rayleigh atmosphere.

The neutral lines, as discussed in detail in [13] and [18],intersect the sun’s vertical at right angles at the neutral points; at the zenith they have a double point with the tangent inclined by 45” from the sun’s vertical, The neutral lines have the shape of a lemniscate, if the Arago point is above the horizon; otherwise, they consist of two branches, the inner branch passing through the Babinet point, the outer branch through the Brewster point. The inner branch through the Babinet point is situated inside of the curve U(c(,v)= 0, so that these two lines do not intersect each other except at the zenith. Consequently, there are no neutral points in a molecular atmosphere outside the sun’s vertical. At the zenith the angle x becomes undetermined because no particular azimuth can be attributed to the zenith, and thus the reference for the angle x is not determined. However, the position of the plane of polarization is well

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

71

defined, being normal to the sun’s vertical. From the behavior of the X - and Y-functions it can easily be seen that U(p,cp) is negative inside the curve U(p,cp) = 0 (for sin (cpo - cp) > 0), and positive outside. Similarly, II(p,cp) - II(p,cp) is positive inside the inner branch and outside

FIG.12. Lines of equal deviation of the plane of polarizationfrom the vertical for a high solar elevation in a Rayleigh atmosphere of the optical thickness of 0.15.

FIG.13. Lines of equal deviation of the plane of polarizationfrom the vertical for a low solar elevation in a Rayleigh atmosphere of the optical thickness of 0.15.

the outer branch of the neutral line, for high solar elevations or inside the lemniscate, when the Arago point is above the horizon. The curve U(p,cp) = 0 and the neutral lines divide the entire hemisphere into four regions, where the sign of tan 2 x can be easily determined, thus allowing

72

ZDENEK SEKERA

a determination of the proper values for x. If x is counted positive when the plane of polarization is rotated to the right from the vertical plane, then the orientation of the plane of polarization follows, which is represented in Fig. 12 and Fig. 13 by the lines of equal deviation of the plane of polarization from the local vertical (of the azimuth 9).

2.6. Polarization of Sky Light in a Turbid Atmosphere When the theoretical values and the distribution of the polarization of diffuse sky radiation are to be determined in a turbid atmosphere, two major difficulties are encountered. Because of the wide variation in the content, size, and nature of the aerosol particles, it is very difficult or almost impossible to formulate the law of scattering by an atmospheric aerosol exactly. The measurements of the size distribution of aerosol particles are quite difficult, and the less reliable the smaller the size of the particles. The determination of the physical nature (e.g., the refractive index, etc.) of aerosol particles is much more difficult and can be done for the larger particles only. However, recent measurements [19] give quite a definite indication of the type of size distribution under average conditions. Furthermore, from an extension of the Mie theorv t o cover the scattering by two concentric spheres of different dielectric constants [20], it can be concluded that a sphere with a thin cQating scatters the light as if the inner core were not present, provided that the outer shell has a moderate refractive index. Similarly, the effect of deviations from a spherical shape in the scattering particle becomes important only for the higher value of the refractive index [21]. Since, under normally prevailing humiilities, it can be expected that all aerosol particles are coated with a thin layer of water or a diluted salt solution, it seems quite reasonable to assume that the aerosol particles scatter the light like small spherical water droplets. Hence the law of scattering can be derived from the original Mie theory for a more or less realistic model with respect to the size distribution of aerosol particles. The second difficulty which appears is the problem of multiple scattering by large aerosol particles. It is not a priori obvious that the scattered components of the radiation from individual particles in a given volume are incoherent. The study of this problem is, of course, not an easy one, and it will require some time and effort before a definite answer can be given, If, however, the assumption of incoherent scattering can be extended to cover large particles also, then the matrix of scattering by a given volume is obtained by the addition of matrices for single scattering {PiJ\in (2.12) for all particles, as indicated in Section 2.2. Although for a given value of the scattering angle 0 it is not difficult to perform the addition of the functions Fi(i = 1, 2, 3, 4)in (2.14) over all particles in a

DEVELOPMENTS I N THE STUDY OF POLARIZATION O F SKY LIGHT

73

given volume, especially with the use of the extensive tables of these scattering functions, such an addition is not valid in analytical considerations. For such purposes the scattering angle should be separated from the other variables a and E . If the series SL and SB in the form (2.15) are used in (2.14), such a separation of variables can be accomplished most conveniently by expanding the functions Fi in (2.14) in series of Legendre polynomials. This can be done first by expanding the expressions for S L and S E in series of Legendre polynomials and then by applying the known formula for the product of two Legendre series, as is done in detail in [22]. In this way it is possible to arrive a t the expressions for the functions Fi in the form

2 p,,(~(cu,e)P,,(~~~ (D

(2.52)

Fi(a,e,B) =

0)

n=O

The matrix corresponding to the scattering by a unit volume containing N(a,e) particles of the radius a and dielectric constant e, has the same form as the matrix for single scattering with the functions Fi given by the expressions in (2.52), in which the coefficients P,,(~) are replaced by the sums over all values of a and E, similar to (2.22). If the size distribution is continuous, as it seems to be in the case of a turbid atmosphere, then the sums have the form

where now N(a,e) represents the number of particles of the radius within the interval ( u p da). the With the values in (2.53) substituted in (2.52) instead of matrix (Pi2}is obtained, defining the scattering by a, unit volume, in terms of the Stokes parameters, oriented parallel or normal to the scattering plane. I n the equation of radiative transfer however, the corresponding matrix refers to the Stokes parameters oriented parallel or normal to the vertical plane through the direction of the scattered or of the incident radiation. If the advantage of the simple integration with respect to the azimuth is to be used, as indicated in (2.31), the matrix P(cos 0) = ( Pij) with Pij given in (2.13) has to be transformed to the form P(p,p;p’,p’), which refers to the orientation of the Stokes parameters parallel and normal to the vertical plane and the directions are defined by the pairs ) (p’,cp’), respectively. This transformation can be perof values ( 1 , ~and formed either by the method used in the derivation of the corresponding form for Rayleigh scattering in [a], or by a method based on the transformation of the corresponding components of the electric vector [23].

+

74

’

ZDENEK SEKERA

Let L(y) denote the matrix by which the intensity matrix Bas to be multiplied in order to obtain the intensity matrix with the elements oriented along the axes rotated by the angle y in the positive direction. Then, in the first method, the transformation of the matrix consists in performing the product (2.54)

-

p(P,la;P!,laf) = L(q) ~ ( ~ 0)0- L(T s

- P>

where p,q denote the angles between the vertical plane and the plane of scattering in the spherical triangle in Fig. 14, formed by the direction

FIG.14. Relationship between the scattering angle, the zenith distances and the azimuth difference of the source and of the observed point.

of the incident radiation (k’),the zenith (Z), and the direction of the scattered radiation (k). The second method has the advantages that the two additional multiplications of Legendre series involved in the first method, can be avoided by performing the separation of the variables a,e, and 0 as the final step in this transformation. When the normalization of this transformed matrix is performed according to (2.26), then the equation of radiative transfer in a turbid atmosphere will acquire a form identical to the equation for a Rayleigh atmosphere, except that the term rBvP(p,p;~’,(p’) is to be replaced by (2.55)

+

Bv(R)P(R) (P, la;p’,(P’) B”(L’p(L’ (P,P ;P’, la’) where the superscripts R and L denote the quantities related to the molecular and large particle scattering, respectively. The last matrix, however, can be split into two terms

+

PCL)(P,P&’, 9’)= P@)( P , G P ’ , la’) P‘( P ,p;CL’,la’) where the matrix P’ evidently contains only terms responsible for the deviation of the large particle scattering from the Rayleigh scattering. (2.56)

DEVELOPMENTS I N THE STUDY OF POLARIZATION O F SKY LIGHT

75

If, furthermore, the optical thickness 7 is introduced by the expression (2.57)

7

=

/,“

[P”(R)

+ p”‘L’] dz

where the dependence of on z is owing to the variation of N(a,e) with height, then the equation of radiative transfer in a turbid atmosphere assumes the form 1 cc ~ ( 7 , c c , r ) l d=~ I(~,P,P) - 4 e-r’roP(R)(cc,(p;-cco,PO) F 3

+ W)

1 -1

J Z0T

P‘(cc,r;ccrlr’)* I(T,cc’lP’> dcc’ dr’)

This equation shows clearly the twofold effect of aerosol particles, consisting of an increase in the optical thickness, and of a change in the intensity and polarization owing to the difference in the large-particle scattering from Rayleigh scattering, expressed in the last two additional terms. This change, which leads, to the deviation of the polarization of sky light in the turbid atmosphere from that of a molecular atmosphere, described in the preceding section, is apparently larger the larger the , as seems to be the case for quotient /3Y(L)/(/3Y(R) If N(a,c) the most typical distribution of aerosol particles over the continent, and this is used in the expression for /3Y(L),which for a continuous size distribution has the form

-

+

pY‘L) =

c /om

7 f K ( c Y , € ) N ( U , € ) U ~da

f

it is easy to see that varies with A-l. In the coastal regions (see [24]), where the sea spray adds a quasi-Gaussian distribution of larger drops, is independent of the wavelength or may even increase with wavelength, in the region of the shorter wavelengths. Since, on the other hand, B Y ( R ) decreases with increasing wavelength as A-4, the quotient in front of the last bracket in (2.58) has a smaller value for the short wavelengths and increases with longer wavelengths. Hence, a stronger effect caused by aerosol Scattering can be expected in the longer wavelengths, with greater deviations in the sky-light polarization from the theoretical values corresponding to the molecular atmosphere, and with wider daily and local variations, in response to the strongly varying content of aerosol particles.

76

ZDENEK SEKERA

The solution of (2.58) is substantially more complicated than for f l y ( L ) ) is a Rayleigh scattering, especially if the quotient /3V(LJ/(/3V(R) complicated function of the optical thickness T . The form of the equation (2.58) suggests the possibility of an approximate solution by successive iterations, with the solution for a Rayleigh atmosphere as the first approximation, This is definitely possible for short wavelengths, where is much larger than p V ( L ) ,and the deviations from the Rayleigh solution are small.ll

+

3. MEASUREMENTS OF THE SKY-LIGHT POLARIZATION 3.1. New Developments in the Technique of Polarization Measurements

The first photoelectric measurements of sky-light polarization with continuous recording (see [25]) were limited by the low efficiency of the photovoltaic cells used. The output of the photocells was too low to give a measurable response for narrow band filters; furthermore, the photocell did not permit any amplification, because of the noise inherent in the first tube of the amplifier. The u8e of modern photomultiplier tubes can easily eliminate the difficulty mentioned above. The much higher output and a suitable amplification can provide a measurable response even for very narrow band filters. However, the original arrangement-a uniformly rotating Nicol prism in front of the photocell-cannot be used efficiently, as the output of the photomultiplier tube is dependent on the orientation of the plane of polarization of the light striking the photosensitive surface. This difficulty can be removed by placing a depolarizer in front of the photocathode (see [26]). The action of the depolarizer is limited to a very narrow spectral region, so that such a method can be conveniently used only if the measurement is restricted to a given wavelength. It becomes impractical if the polarization measurement is to be extended over several narrow spectral regions. In such a case it is preferable to use a system in which the analyzer (Nicol or Glan-Thompson prism) can be kept in a fixed position with respect to the photomultiplier tube. A simple computation will show that a combination of a rotating retardation plate and of a fixed analyzer in front of the phototube can be very well used for the measurement of the polarization. This has the advantage that it can provide the measurement of all the Stokes parameters, and thus can be used for a determination of the intensity, the degree of polarization, as well as of the ellipticity and the position of the polarization plane of the measured light (see [27]). The recorded photoelectric "The validity and the convenienceof such a procedure for other wavelengthsis being currently studied as a part of an investigation sponsored by the Air Force Cambridge Research Center (Contract No. AF 19(604)-1303).

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

57

current has then the character of a composite alternating current. Let 'p, $ be the position angles of the polarization plane of the measured light and of the transmission plane of the analyzer, respectively, and e the retardation of the plate rotated with the frequency w ; then the intensity of the photoelectric current is proportional to the luminous intensity of the light leaving the analyzer IA, which is given by the relationship (3.1)

IA= I[1

+ P cos2 ( 4 2 ) cos 2('p - #)] + V sin

sin (2wt - 2#)

+ I P sin2 (e/2) cos (4wt - 2$ - 2'p) E

where I , P denote the total intensity and the degree of linear polarization (i.e., the expression in (2.11) with V = 0 ) , respectively, and V denotes the Stokes parameter measuring the ellipticity of the measured light, defined in (2.7). From this relationship it follows that such a system of a rotating retardation plate with a fixed analyzer in front of the phototube can be used for the measurement of the parameters mentioned above, if the output of the multiplier tube is amplified by separate amplifiers, tuned to the second and fourth harmonic of the frequency w , and the dc component, the amplitude and the phase of the second and fourth harmonics are measured. The preliminary measurements, made with such a system including commercially available retardation plates, showed that the amplitude and the phase of the second harmonic varied with the intensity, the degree of polarization, and the position of the polarization plane of the measured light. By a detailed analysis, it was found that the observed dependence of the amplitude and of the phase of the second harmonic might be the consequence of the slightly different transmissions along the fast and slow axes of the retardation plate. Under such an assumption it is not difficult to compute the relationship corresponding to (3.1), (see [28J).The effect of the uneven transmissions along the fast and slow axes (tf, t8) can be noticed in a slight change in the factors cos2 (e/2), sin2 ( 4 2 ) and sin e (to be replaced by a = %(l e, cos e), b = % ( l - e, cos a), and c = er sin e) and in an addition of two more terms of the form

+

where

p,I[cos (2wt - 2$) pr = ( t i - L ) / ( t f

+ P cos (2wt - 2'p)]

+ L),

er

=

(1 - pr2)'5

The value of p , is, however, small (of the order 0.01) and the effect of uneven transmissions changes the factors mentioned above only slightly and, in the case of linear polarization, can be nullified by proper calibration. It affects more seriously the measurement of the ellipticity, especially if the ellipticity is small as is the case for the sky light. This undesirable effect and its elimination are currently being studied.

78

ZDENEK SEKERA

When the parameters of the sky-light polarization are to be measured from dawn twilight through noon to sunset twilight, or along the sun's vertical, another difficulty appears. During such measurements the total intensity I of the sky light varies through such a large range that, if the voltage applied to the photomultiplier is set for the measurement a t low intensities, the operating point for the highest intensities is shifted out of the region of linear characteristics, and in addition the effect of fatigue appears. This difficulty can be removed by introducing a voltage regulation which adjusts the voltage on the photomultiplier in such a way that the output of the photomultiplier remains constant for the entire range of the intensity variation and a t a sufficiently low value to insure operation along the linear part of the characteristics and to avoid the fatigue effect. I n such an arrangement, the voltage applied on the photomultiplier is related to the dc component in (3.1) and can be used for its measurement. If S ( E ) denotes the sensitivity of the photomultiplier, then the voltage E is adjusted by the control circuit in the voltage regulator to such a value that

S ( E ) I d . = f = constant

(3.2)

where f is the photomultiplier output, and from (3.1) Ido

=

CZ[1

+ UP cos 2((p - #)]

C being a constant. The amplitude of the fourth harmonic is then given by A4 = S(E)gCIP

(3.3)

where g is an instrumental constant containing the gain factor of the amplifier, etc. If the sensitivity S ( E ) and other factors in (3.3) are eliminated by the use of (3.2), (3.3) assumes the form A4 = gfbP/[l

(3.4)

+ UP cos 2((p - +)]

from which the degree of linear polarization P can be computed, (3.5)

P

= kAa/[l

- kA4u cos 2((p - #)I,

Ic

= l/gfb = constant

By a proper choice of the retardation of the plate, or by a proper setting of the analyzer, the second term in the denominator in (3.5)can be made sufficiently small to be neglected. This can be done, evidently, by setting the analyzer 45" to the plane of polarization of the measured light, or by using a half-wave plate (E = T ) . From (3.1) it follows that the constant a is then minimized, while the constant b reaches its maximum, and the sensitivity of the measurement of the degree of polarization is increased. In,both of these cases, the amplitude of the fourth harmonic is directly

DEVELOPMENTS IN THE STUDY O F POLARIZATION OF SKY LIGHT

79

proportional to the degree of polarization; the total intensity can be measured by the voltage applied to the photomultiplier, and the position of the plane of polarization can be derived from the measured phase of the fourth harmonic. Based on the theory just described, a photoelectric polarimeter has been built and used for the polarization measurement of sky light12a t the Department of Meteorology, University of California, Los Angeles. The

To Hysteresis /-Synchronous Drive

Power

Photo

Multiolisr

Photo Control

Brown Recorder Dynode Wltoge

-

Ilntmsifyl

FIQ.15. Schematic diagram of the photoelectric polarimeter for the measurement of sky-light polarization.

optical system of the polarimeter, as indicated in Fig. 15, consists of a coaxial baffle collimator, with an angle of view of about 3 O , a retardation plate rotated by a synchronous motor a t a speed of 10 rps, a Corning monochromatic filter, a Glan-Thompson prism as an analyzer, and finally, a photomultiplier. To the gear train of the synchronous motor a small alternator is attached, whose output is used to standardize the “Twin-T” tuned amplifier, and after passing through an adjustable phase shifting network, is used as a reference signal for the phase measurement of the fourth harmonic of the photomultiplier output. The phases of the reference signal and the amplified fourth harmonic are compared both on an oscilloscope and on a phase meter, the reading of which is recorded on a 18 Under the Contract AF 19(122)-239 with the Air Force Cambridge Research Center, Cambridge, Massachusetts.

80

ZDENEK SEKERA

Brown recording potentiometer. The amplitude of the fourth harmonic is recorded on another Brown potentiometer. The dynode voltage of the photomultiplier is regulated by a control circuit, holding the output on a low constant value. The value of the

FIG.16. Photoelectric polarimeter used for measurements of sky-light polarization a t the Department of Meteorology, University of California a t Los Angeles.

varying dynode voltage is recorded on a third Brown potentiometer, as the measure of the total intensity of the measured light. The complete instrument consists of two identical systems (see Fig. 16) mounted close together, with the retardation plates driven by the same synchronous motor. Each system has three Corning monochromatic filters (which together cover the spectrum from the ultraviolet to the far red), exchangeable by remote control. The complete system is mounted in a yoke, whose orientation with respect to the sun can be automatically

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

81

controlled by a special sun-following device. The basic element of this device is a pair of dual photocells, one for the sun's elevation and the other for the azimuth. When the sun's image moves out of the central position in the field of view of the telescope attached to the yoke, onehalf of the photocells are illuminated more than the other half, which activates the corresponding motor, driving the yoke back until the sun's image assumes the initial central position. The photoelectric polarimeter itself can then scan in the sun's vertical from horizon to horizon. The direction and the extent of the scan are remotely controlled. The angular distance of the scan is measured with a coded protractor from the sunfollowing telescope as zero reference, and recorded by auxiliary pens on the records of the Brown potentiometers. A shield is attached to the sunfollowing telescope, carrying neutral filters of calibrated density. These reduce the direct solar radiation and thus protect the photosensitive surface of the tubes during the scan through the sun's disk and at the same time provide a measure of the intensity of the direct solar radiation. Once the filter and the direction of the scan are remotely selected, the instrument automatically records the dynode voltage and the amplitude and the phase of the fourth harmonic of the photomultiplier output. After the instrumental constants are determined by proper calibration, the instantaneous values of the degree of polarization, of the position of the plane of polarization, and of the total intensity of the sky light along the sun's vertical can be determined. With a moderate scanning speed (about 26" per minute), it is possible to use an automatic switching device which alternates the two channels at seven-second intervals, in order to allow an electrical equilibrium of the electronic components to be established. I n this way the distribution of the degree of polarization along the sun's vertical at two different wavelengths can be easily determined, by means of a single scan from horizon to horizon. During the scan the position of the neutral points is recognized by a minimum in the amplitude curve of the fourth harmonic, with a simultaneous shift of its phase by 180". With an appropriate increase of the amplifier gain in the vicinity of the neutral points, the minimum can be made sufficiently sharp to allow the determination of the position of a neutral point with an accuracy of 50.2". A phase shift during the passage through a neutral point establishes its existence and distinguishes it from an incidental minimum in the degree of polarization. From the recording of the dynode voltage, it is further possible to determine the direct solar radiation when scanning over the sun's disk. This quantity can be used in determining the atmospheric transmission and the turbidity factors for different Wavelengths, which serve as very useful parameters for the interpretation and discussion of the observed

82

ZDaNEK SERERA

deviations of the sky-light polarization from the theoretical values computed for a molecular atmosphere. 3.2. Results of the Recent Photoelectric Measurements of the Sky-Light

Polarization 3.2.2. Measurements of the Positions of the Neutral Points. The measurements of the sky-light polarization with the photoelectric polarimeter described in the preceding section were initiated in 1952. During the first period of these measurements only the position of the neutral points was measured as this did not require the time-consuming calibration of the instrument accomplished about two years later. The first preliminary measurements, made a t the University of California campus a t Los Angeles (later denoted as the UCLA campus), brought several unexpected results. The Babinet and Brewster points could both be identified very easily for low solar elevations, but their position was about 5" to 10' closer to the sun than predicted by the theory. For higher solar elevations they were too close to the sun to be measured exactly. Their position behind the protective shield attached to the telescope of the sun-follower, was suggested by an observed minimum of the degree of polarization, without any phase shift. These measurements were repeated in October, 1952 at the Smithsonian Astrophysical Observatory a t Table Mountain, California, at an altitude of 7500 f t . Here, too, the same character of the deviations of the Babinet and Brewster points from their theoretical positions was found, but with smaller deviations, consistent with the lower turbidity at this level. The visual measurements of the position of the Arago point, made with a Savart polariscope, agreed with the results of the photoelectric measurements within the limits of observational error. The positions of the Babinet and Brewster points could not be found by the Savart polariscope, although their positions were easily determined by the photoelectric polarimeter . Systematic measurements of the neutral points were subsequently carried out at Table Mountain (November, 1952 and 1953)) a t Los Angeles (September, 1953 and 1954)) a t Cactus Peak (at the Observatory of the Naval Ordinance Test Station, China Lake, California, at an altitude of 5400 ft, July, 1954), and a t Pasadena, California (at the California Institute of Technology, a t an altitude of 850 ft, November, 1954). The measurements were made first in four narrow spectral bands, centered around 365 mp, 460 mp, 515 mp, and 625 mp, at the rate of one neutral point every minute. During the measurements at Cactus Peak, when the scanning speed was doubled, two additional bands around 405 mp and 580 mp were added. Furthermore, from this period on, meas-

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

83

urements of the degree of polarization along the sun’s vertical were introduced, and the measurements of the neutral points were incorporated as a part of this broader program. In order to eliminate the effect of the altitude of different locations, as well as the diurnal variations, the measured positions of the neutral

: i

I

I

D

PO

1

I

30

40

1

1

I

30

PO

I0

5 tor 365 m)

0

S65

5

460

0

450

5

515

,

I

0

; 5

6LS

6

6L5

-5

625

-10

625

-I5

6P5

50

50

40

Fro. 17. Deviations of the measured position of the Babinet neutral point for the theoretical values for the molecular atmosphere, as measured at the UCLA campus, Los Angeles, September, 1953.

points were compared with the theoretical values for a molecular atmosphere, and the deviations of the actual from the theoretical positions were studied. It was found that these deviations varied within quite large limits from day to day, or even during one single day, and that they did not correlate too well with the changes in the visibility. As an illustration

b

84

ZDENEK SEKERA

of such variations and deviations, those measured during clear days in September, 1953 at the UCLA campus are plotted for different solar elevations in Fig. 17 for the Babinet points, and in Fig. 18 for the Brewster and Arago points, for the four different wavelengths mentioned above. :

f

I

I

,I

S for 365 m p

0

315

5

460

0

4t30

5

515

0

515

5

625

0

625

-5

625

-10

125

-I5

625

-

L

E

2 Y

n

I

0

10

I

20

30 L.Y.

40

50

SO

1

40

SUN'S ELEVATION (Dagraas)

I

30

P.M.

I

20

I

10

FIQ.18. Deviations of the measured position of the Brewster and Arago neutral points from the theoretical values for the molecular atmosphere, as measured at the UCLA campus, Los Angeles, September, 1953.

The curves for the Babinet and Brewster points for longer wavelengths stop for a small solar elevation, indicating that these points were not found for higher solar elevations. Because of the protecting shield, mentioned above, neutral points cannot be identified by the instrument if their distance from the sun is less than 4 ' or 5". From the theoretical positions of the neutral points in Fig. 7, the deviations from the theo-

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

85

retical positions can be found corresponding to the distance of 4" from the sun. They are plotted in Figs. 17 and 18. It becomes then quite evident, that for the longer wavelengths the neutral points disappear a t much larger distances from the sun than a t those corresponding to the width of the shield. I n such a case their disappearance is most likely to be attributed to the changes in the sky-light polarization in the vicinity of the sun, caused by the presence of large particles in the turbid atmosphere. From Figs. 17 and 18 it is also evident that the daily variations increase in magnitude with the increasing wavelength. This fact is not a feature peculiar to the sky-light polarization a t Los Angeles alone, but it can be found in the measurements from all other locations. Despite these daily variations the curves show a few characteristic features which can be studied from the arithmetic means of the deviations, measured on days with less than one-tenth cloudiness and with relatively good visibility. The curves of these means for four different locations, mentioned above, are shown in Figs. 19 and 20. The deviations measured a t Pasadena and a t Cactus Peak show relatively small differences between morning and afternoon hours, and they are thus included in one mean. The corresponding differences for the UCLA campus and Table Mountain are much greater, and hence separate curves for morning and afternoon hours are drawn. The curves of the deviations of the Babinet point show for all locations a sudden change of the slope for the low solar elevations (5 5"), with the slope increasing slightly with increasing wavelength. This feature is probably a consequence of the simplifying assumption of a plane-parallel atmosphere introduced in the theoretical computations. The deviations of the Arago point, however, do not show any systematic changes in the curves for very small solar elevations. The curves of mean deviations of the Babinet point show distinctly the effects of turbidity variations depending on the location; the curves for Table Mountain and Cactus Peak are closer to the line of zero deviation than the corresponding curves for Los Angeles and Pasadena, for all wavelengths except the red, where the curve for Cactus Peak is close to the curves for locations with higher turbidity. With increasing wavelength the curves for morning and afternoon hours a t Table Mountain are more and more separated, with the afternoon values not too far from the curves for lower locations. This indicates that the smog and industrial pollution from the Los Angeles basin reaches as far as the Table Mountain area, as has been actually observed in terms of the change of the blue color of the sky over the site of observation during the afternoon. A similar separation of the morning and afternoon curves can be noticed for UCLA as indicative of a typical asymmetry in the curves with respect

86

ZDENEK SEKERA

SUN'S ELEVATION

I-

m -.-.--.

............. TABLE

U.C.L.A. (A.M.) (P.M.)

JvJ

TABLE MT. (A.M.)

n----.CAL

I------U.C.L.A.

MT. (P.M.)

p _..-..-.. CACTUS PEAK (A.Mond EM.) TECH (A.M.ond RM.)

FIG. 19. Mean deviations of the measured positions of the Babinet neutral point from the theoreticalvalues for the molecular atmosphere,as measured on four different locations.

to the meridian. All the deviations of the Babinet point, except in the shorter wavelengths at the mountain sites, are negative, with increasing magnitude for higher solar elevations and with increasing wavelength. The deviations of the h a g o point, on the other hand, are mostly positive, increasing in magnitude with increasing wavelength. While the devi-

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIQHT

87

a 10

ro 4

a (D 0

? I

d

2

'" 4

4

E

v, u) 4

20'

50°

40-

30-

60'

O70

SUN'S ELEVATION

1 -

U.C.L.A. (A.M.) U.C.L.A. (P.M.1 IU-.-.---. TABLE MT (A.M.)

x------

Ip

P-- -

PI--

TABLE MT ( P M I CACTUS PEAK ( A M and P M 1

-- CAL

TECH ( A M ond P M 1

FIG.20. Mean deviations of the measured positions of the Brewster and Arago neutral points from the theoretical values for the molecular atmosphere as measured on four different locations.

88

ZDENEK SEHERA

ations show the tendency to decrease with increasing solar elevation in the shortest wavelength] they do increase with increasing solar elevation for longer wavelengths, except for the deviations measured a t Pasadena. The latter show quite a different character from those measured at all other locations. I n the ultraviolet part the deviations increase from negative values to almost zero deviation with increasing solar elevation; in the red part the deviations are greater (by about 2") than those measured a t other locations. The largest local differences can be noticed in the curves of mean deviations of the positions of the Brewster point. The deviations measured a t Cactus Peak have the most regular character] being constant during the day, varying between 1' and - 1" in all wavelengths and dropping to the values around -2" in the red. The curves for Pasadena have a similar character, they drop to larger negative values for increasing solar elevations. The curves of the deviations for Los Angeles and Table Mountain are similar in that both show a very pronounced asymmetry with respect to noon. At Table Mountain this asymmetry increases with increasing wavelength, from a very small value in the ultraviolet t o the maximum in the red part; the curves for UCLA show the largest asymmetry in the ultraviolet part. The shape of the curve in the ultraviolet and in the red part indicates that these deviations are caused by two different effects. Besides the increase of the turbidity in the afternoon hours, existing at both of these locations, the varying type of the ground reflection in the vicinity of the UCLA campus may be the second effect responsible for these deviations. In the morning hours the solar radiation is reflected from the metropolitan area; in the afternoon the solar radiation is reflected from the ocean, or from the thick haze layer over the sea surface. The specular reflection from the sea surface is more likely to change the position of the neutral points, especially of the Brewster point, than the isotropic, neutral reflection considered in the theoretical computations and discussions. A definite answer to all these suggestions and evaluations of the effect of aerosol particles, of the sphericity of tho earth's surface, and of the specular reflection has to await the quantitative solution of these problems. 3.2.2. Position of the Maximum Polarization. Another point, which is useful for the estimate of the distribution of the polarization of the sky light along the sun's vertical, is the position of the maximum degree of polarization. I n previous investigations, the maximum degree of polarization was occasionally found to deviate from the point of 90' from the sun (its theoretical position assuming primary scattering only), No systematic measurements of the position of the maximum were performed in the past. If the measurement of the polarization along the sun's verti-

+

3AQM

--

DEVELOPMENTS I N THE STUDY O F POLARIZ.4TION O F SKY LIGHT HION31

I Y

I .4

89

90

ZDENER SEKERA

cal is done by the photoelectric polarimeter, the position of the maximum degree of polarization can be easily determined for each scan through the sun's vertical, with an accuracy of f l " , because the maximum is usually quite broad. The systematic measurement of the maximum degree of polarization was initiated in July, 1954, and the results of a few measurements are reproduced in Fig. 21. The measured positions are compared with the theoretical values computed for the albedo of 0.25. I n the shorter wavelengths the measured positions agree well with the theoretical for low solar elevations, but for higher solar elevations they are found closer to the point 90" from the sun, than the theoretical positions. This discrepancy may be attributed to a rather high value of the albedo used in the computations for these wavelengths. For longer wavelengths the position of the maximum is shifted further from the sun and has been found between 91" for high solar elevations, with values increasing to 94" for low solar elevations. All the curves indicate the tendency of slightly larger shifts in the afternoon hours. 3.2.3.Maximum Degree of Polarization in the Sun's Vertical. As is evident from the theoretical discussion in Section 2.4, and especially from Fig. 3, the maximum degree of polarization in the sun's vertical is one of the most variable parameters of the sky-light polarization. From its large variation with the amount of scattering particles, it can be concluded that the maximum degree of polarization will vary with the turbidity in a similar way as in a molecular atmosphere where it is proportional t o the optical thickness; i.e., it will decrease with increasing turbidity of the actual atmosphere. The measurements of the maximum degree of polarization, performed on the locations mentioned in 3.2.1 above, have proved this dependence on the turbidity, as expected. As an example, the results of the measurements from Cactus Peak, shown in Fig. 22, can be compared with those from Pasadena, in Fig. 23. The conditions at Cactus Peak can be considered quite uniform during the day; nevertheless the curves for all wavelengths show a definite asymmetry, with the minimum shifted from the noon to the afternoon hours. The similar effect at Pasadena is much more pronounced, with a deep minimum during the mid-afternoon. In Fig. 24 the change in the maximum degree of polarization is shown for the UCLA campus on September 17, 1954. On this day the haziness in the morning hours, resulting in a very low visibility (about 2 disappeared during the afternoon and was followed by an increase of the visibility to 12 miles; at the same time the maximum degree of polarization in all wavelengths increased by about 150%. The measured values of the maximum degree of polarization cannot be compared immediately with the theoretical values of the molecdar

mJes),

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

91

92

ZDENEK SEKERA

AM

SUN’S ELEVATION

PM

Fro. 23. Maximum degree of polarization in the sun’s vertical as measured at Pasadena, California on November 5, 1954.

AM

SUN’S ELEVATION

PY

FIG.24. Maximum degree of polarization in the sunk vertical as measured at the UCLA campus at Los Angeles, California on September 17, 1964.

DEVELOPMENTS IN TEE STUDY OF POLARIZATION OF SKY LIGHT

93

atmosphere as in the case of the positions of the neutral points. The maximum degree of polarization, as apparent from Fig. 4, depends upon the albedo of the ground reflection a t a rate comparable to that of the optical thickness. For the comparison of the measured and theoretical

LY

SUN'S ELEVATION

PY

FIG.25. Relative deviations of the maximum degree of polarization in the sun's vertical as measured a t Pasadena, California, from the theoretical value for the molecular atmosphere with the albedo of 0.10.

values, it is quite essential to know the albedo of the ground reflection. The determination of this is quite difficult and thus no reliable data are available. In order to get a t least the order of the magnitude of the deviations of the measured values from the theoretical, the albedo of 0.10 was used for the computation, a value which corresponds to the albedo

94

ZDENEK SEKERA

over an urban area. I n Fig. 25 the relative deviations, i.e., the difference of the measured values minus the theoretical, divided by the theoretical value, are presented, corresponding to the measurements a t Pasadena, shown in Fig. 23. The curves show relative deviations from 0% up to -SO%, increasing in magnitude for longer wavelengths. A systematic study of the maximum polarization and of the distribution of the degree of polarization along the sun's vertical is being carried on a t the Department of Meteorology, University of California a t Los Angeles, and the results will be available in the near future. 3.3. Dispersion of the Sky-Light Polarization in Relation to the Turbidity of the Atmosphere

The better understanding of the dependence of the sky-light polarization upon the wavelength-in the classical papers of Chr. Jensen, F. Linke and others (see [29]) denoted as the dispersion of the sky-light polarization-represents an achievement in this field of the same importance as the new development in the theory and the great improvement of the measuring technique. The contradictory results of the earlier measurements of the sky-light polarization, namely, the difference of the degree of polarization in the red and in the blue, by some authors found positive, by others negative, can be easily explained as the consequence of different degrees of the turbidity conditions during the measurements. Since the turbidity was neither actually measured (except in the most recent studies) nor estimated objectively from other phenomena (such as visibility, color of the sky), the comparison of the results of different authors has to be done with respect to the general character of the turbidity of the place of observation, or the secular variation of the turbidity (effect of volcanic eruptions). When this is done, the dispersion of the sky-light polarization follows a very simple law and the measurements of the different authors become quite consistent (see [29]). When it is realized that the degree of polarization itself can be considered as a measure of turbidity, then it is possible to compare the difference of the degree of polarization in the red and in the blue (P, - Pa) with the degree of polarization in the red (P,), as is shown in Fig. 26, and the law of the dispersion of the skylight polarization becomes quite evident. I n Fig. 26 all available visual measurements of the degree of polarization 90" from the sun measured with a red and with a blue filter are plotted. Although the points are considerably scattered, the correlation of the difference P, - Pa on P , is quite obvious (correlation coefficient T = 0.80). For very low turbidity (Tichanowski's values [30] from Crimean Peninsula, some of which are from an elevation of 1180 m), the difference is positive; while for high tur-

DEVELOPMENTS I N THE STUDY OF POLARIZATION OF SKY LIQHT

95

bidity (Dietze’s measurements [31] from the period of the blue sun and moon in Europe in 1950), the difference is negative. A comparison with the theoretical values for the molecular atmosphere suggests that the change of the sign of P, - Pb cannot be explained by Rayleigh scattering and Lambert’s type of ground reflection. For larger values of the albedo the curve of the theoretical values is shifted towards lower values of P , but the difference P, - Pb remains positive. .I2

Ir‘

.08

-

.04

-

-‘b

0.0-

-.04

-

-.08

D

-.I 2 -

.* -.I6

P

-

Oietre AKolitin

b

v Pikhikoff

cTichonmki

-.20I

I

I

I

I

I

FIG.26. Difference of the maximum degree of polarization in the red and in the blue (P, - Pb) as function of the maximum degree of polarization in the red (P?) as measured by different authors.

The results of the measurements referred t o in the previous section confirm, in general, the results of the few visual measurements of the degree of polarization made in the past in different wavelengths. The measurements, as reproduced in Figs. 22, 23, and 24, show the typical distribution of the degree of polarization with the wavelength, increasing from the ultraviolet to the maximum in blue or green. For longer wavelengths, the distribution is quite dependent on the turbidity and on local conditions. In the red part of the spectrum the degree of polarization is equal to or slightly higher than the maximum in blue or green for low

06

ZDENER SEKERA

solar elevation, and drops below the maximum in the early afternoon hours a t Cactus Peak.13 At Pasadena, however, the degree of polarization in the red is much smaller at low solar elevations and approaches the maximum in the green during the afternoon hours, when the turbidity reaches its maximum. When the curves for X = 460 mp and for X = 625 mp for Cactus Peak are compared, then the difference P, - Pb in relation to P , follows the mean character of the correlation in Fig. 26, being positive for higher P, and decreasing to larger negative values for small P,. If the corresponding curves for Pasadena are compared, then the difference P, - Pb is negative for larger P, and decreases in the magnitude for increasing P,.When plotted in a diagram similar to Fig. 2G, the points for Pasadena, and also for the UCLA campus, are grouped in more horizontal loops, for small P , having even the opposite slope than the corresponding curves for Cactus Peak. This suggests a different character of the turbidity increase a t Cactus Peak from that at the other two locations. I n fact, the increase of the turbidity a t Cactus Peak probably occurred at higher levels, as could be concluded from the development of occasional fair weather cumuli a t the top of an 8000 to 10,000 ft inversion: whereas a t Pasadena and a t the UCLA campus the increase of the turbidity most likely occurred at lower levels, as indicated by the decrease of horizontal visibility . The observed variations in the degree of polarization for different wavelengths, as well as in the position of the neutral points and in the maximum degree of polarization, corroborate very well the theoretical discussion in Section 2.5 on the effect of aerosol particles. The largest effect, as predicted, can be noticed for longer wavelengths in all elements mentioned above. The increase of the optical thickness is most apparent in the positive deviations of the Arago points, a t locations where the aerosol particles can be assumed to be coated with a sufficiently thick water layer to gain the character of transparent particles with a predominant forward scattering. In the opposite case, when the air is dry and opaque aerosol particles appear, the change of the polarization can be expected to occur also in the backward direction (the antisolar side), which may cause an opposite shift of the Arago point, i.e., the change of the sign of the deviations (from positive to negative). The increase of the optical thickness by the aerosol particles for longer wavelengths can also be noticed in the over-all decrease of the maximum degree of polarization, or in the increase of the relative deviations. The maximum degree of polarization in a Rayleigh atmosphere increases with the increasing wavelengths, and the decrease of the maximum polarization by large laRecent measurements of the sky-light polarization at the zenith [32]show very similar results.

DEVELOPMENT'S IN THE STUDY OF POLARIZlTION OF SET LlGH'P

97

particles itself increases with increasing wavelength. Hence, the maximum degree of polarization in the sun's vertical in a turbid atmosphere as a function of wavelength will increase from the ultraviolet to a maximum in the blue or green, and then decrease to the red, provided the turbidity factor of the aerosol particles in these wavelengths is sufficiently large. The effect of the change of polarization owing to the non-Rayl&gb scattering will be pronounced for the transparent particles mostly in the forward direction. It is quite pronounced in the negative deviations of the Babinet and Brewster points and also noticeable in the shift of the maximum degree of polarization from 90" from the sun toward the antisolar point. For locations with low humidity or farther from the coast, these effects may be compensated for a little by the presence of opaque particles, responsible for the same effects, but in the opposite direction, If the aerosol particles are located in a distinct layer in an upper level, their effect will be much less pronounced in the short wavelengths, compared with the corresponding effect of particles in lower levels. The molecular scattering in the layers below the turbid layer may produce so much polarization that the effect of the turbidity at the higher level could be largely compensated. In the long wavelengths, however, the optical thickness of the layers below the turbid layer is so small, that the molecular polarization below the turbid layer has only a negligible effect. This then results in :t much greater negative difference P, - Pa than if the turbid layer were at the ground, in agreement with the large negative differences P, - Pb measured by Dietze in 1950 (see [31]). This can also explain the different behavior of the differences P, - Pa in the measurements at Cactus Peak and in Pasadena or a t UCLA. The solution of the problem of radiative transfer in a turbid atmosphere, indicated in Section 2.5, will give more quantitative and definitive proof of the conclusions and observations made above, and it will allow a more detailed analysis of the size distribution and of the total amount of the aerosol particles in different levels of the actual atmosphere. 4. STUDYOF THE POLARIZATION OF SKYLIGHTIN RELATIONTO THE

RELATEDPROBLEMS OF ATMOSPHERIC OPTICS

The study of the sky-light polarization is useful for other problems of atmospheric optics for the simple reason that the polarization of scattered light is a very sensitive indicator of the scattering processes, and thus the polarization of sky light gives the best information about the scattering in the atmosphere, compared with other phenomena, dependent also on the light scattering in the atmosphere. This point can be illustrated by the following example. From the measurements of the sky-light polari-

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zation, especially of the position of the neutral points, it became quite evident, that the original Rayleigh theory of scattering in the atmosphere, with the consideration of the primary scattering only, is not sufficient to explain some of the essential properties of the sky-light polarization. Even Soret’s (see [29]) quite primitive consideration of the secondary scattering showed in 1889 the importance and the necessity of including higher order scattering in the theoretical discussions of the sky-light polarization. On the other hand, the measurements of the total intensity of the sky light in different wavelengths, or the measurements of the attenuation in the atmosphere, agreed quite well with the Rayleigh theory, and thus only recently the effect of the secondary and higher order scattering has been considered for the first time in the problems of the illumination by the sky light. Although Soret’s original discussion of the secondary scattering shows that the effect of the polarization of the primary scattered light on the total intensity is definitely not negligible, several attempts t o compute the total intensity of the sky light included the secondary scattering but disregarded the polarization of the scattered light; that is, the computation based on the use of the phase function only (see Section 2.3). A similar remark can be made with respect to the dust and haze particles in the atmosphere. Once the theory of light scattering in the atmosphere is worked out t o such a degree that it gives the values for the polarization parameters in agreement with the measurements, then the other parameters or quantities, depending on the light scattering, can be derived with much greater assurame of their correctness. The total intensity of the sky light, for different wavelengths and from different directions, appears in the polarization problems as one of the parameters which is used for computing the degree of polarization. In this computation, however, the relative intensity is used, corresponding t o the unit net flux of the extraterrestrial radiation. When the relative intensity is multiplied by the correct value of this extraterrestrial flux, the absolute total intensity is obtained, a quantity, which can be actually measured, and which enters in all problems of the illumination by the sky light, of the color of the sky, etc. With the knowledge of this quantity it is possible to compute also several basic quantities in the theory of visibility. By integration over all solid angles it is also possible to obtain the net flux of the sky radiation, through a unit area of a quite arbitrary orientation, a quantity, needed for several practical applications. A correct knowledge of the scattering processes in the atmosphere is furthermore a necessary condition for the study of the real absorption in the atmosphere, and the indirect optical methods of the ozone determination should be, for example, revised in this respect. The aerosol particles

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act as quite efficient scatterers also in the near and far infrared radiation, and thus they may play a very important role in the problems of the radiative equilibrium and of the heat balance of the atmosphere. As in every other field, there are still a few problems in the study of the sky-light polarization which remain to be solved. One of these is the problem of the radiative transfer in a spherical atmosphere. As mentioned above, the assumption of a plane-parallel atmosphere is quite legitimate for the higher solar elevations. For low solar elevations it is, however, definitely questionable and the measurements show deviations from the theory, which cannot be explained by any other means. Several quite interesting measurements of the sky-light polarization during the twilight brought results which, compared with a correct theory, may lead to valuable information about the molecular density and dust particle content in higher levels. LIST OF SYMBOLS Numbers in parentheses () and in brackets [J indicate the number of the equation or of the section, respectively, where the symbol is introduced or used for the first time. a scalar (real) amplitude along the direction of 1 [2.1], (2.5) major axis of the ellipse described by the electric vector 12.11

radius of the scattering dielectric sphere [2.2], (2.15) instrumental constant of the photoelectric polarimeter ~3.11

complex coefficient in SL and S R (2.15) scalar (real) amplitude along the direction of r [2.1], (2.5) minor axis of the ellipse described by the electric vector P.11

instrumental constant of the photoelectric polarimeter [3.11

bn C

complex coefficient in SL and SR (2.15) instrumental constant of the photoelectric polarimeter [3.11

instrumental constant of a retardation plate with uneven transmissions [3.1] photomultipler output (3.2) scalar function (2.31) instrumental constant containing the gain factor of the amplifier (3.3) k propagation constant (2. la), (2. lb), instrumental constant (3.5), integer (2.30) mass scattering coefficient [2.4] subscript denoting quantities parallel to the plane of reference (vertical) [2.3] 1 unit vector parallel to the plane of reference (vertical)

e,

(2.3) [2.3]

m

macroscopically measured refractive index (2.19)

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ZDENEK SEEERA

integer n unit vector in the direction of propagation of a electromagnetic wave (212) P angle between the plane of scattering and the vertical plane through the direction of the incident radiation

Ib

Pr

r

r ds tl,

4

Ac

B C

(2.54) function of the scattering angle in SL and Sn (2.15) coefficient in the Legendrc series of scattering functions for single scattering (2.52) coefficient in the Legendrc series of the scattering functions for atmospheric aerosol (2.53) instrumental constant of a retardation plate with uneven transmissions [3.1] angle between the plane of scattering and the vcrtical plane through the direction of scattering (2.54) subscript denoting quantitics normal to the plane of reference (vertical) [2.3] unit vector (2.3), normal to the vertical plane [2.2] elementary length along the direction given by (p,p) [2.3] transmissions along the fast or slow axes of a retardation plate [3.1) functions of the scattering angle in SL and S R (2.15) cosine of the scattering angle (2.15) elementary length in the vertical direction [2.3] amplitude vcctor (2.la) Rcalar amplitudes of the electric vector along 1 and r, respectively (2.3) measured amplitudc of the fourth harmonic (3.3) [3.1] amplitude vector (2.la) instrumental constant including the transmission through the optical part of the photoelectric polarimeter [3.1] one-column matrix of two elements relating to Fresnel reflection (2.44) functions relating to Fresnel reflection (2.44) clectric vcctor (2.111) scattering functions for sin& scattering (2.14) matrix of the net flux of the extraterrestrial radiation [2.3] matrix of the net flux of thc incident radiation (2.12) Stokes parameters, elements of the matrix F [2.3] net flux of a neutral extraterrestrial radiation [2.3] magnetic vector [2.1], (2.1a) Stokes polarization parametcrs, total intensity, intensities parallel and normal to the plane of reference, respectively 12.11, (2.4), [3.1] matrix for the intensity of the scattered radiation and its elements, the intensities parallel and normal, respectively, to the plane of Scattering (2.12) specific intensity of the diffuse sky radiation in the direction (p,p) of the frequency Y when the polarization is neglected 12.31

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intensity matrix of the sky radiation in the direction (~,I,o)of the frequency Y and its elements, the intensities parallel and normal to the vertical plane through the direction ( p , ~ [2.3] ) azimuth independent matrix and matrices containing terms sin [k(po - a)] and cos [ ~ ( P O - a)] (2.30) intensity matrix of the radiation reflected by the ground (2.41) intensity matrix of the additional sky radiation due to the ground reflection (2.41) 11, I I Z , I , intensities of the diffuse sky radiation along the sun’s vertical, parallel to the vertical on the solar and antisolar side, and normal to the sun’s vertical, respectively (2.46) luminous intensity of the light leaving the analyzer in the photoelectric polarimeter (3.1) luminous intensity of the dc component in the photoelectric polarimeter (3.2) scattering cross section (2.20), (2.21) two-column matrix of fourth elements, scattering functions (2.37) two-column matrix of fourth elements, scattering functions (2.37) matrix of the rotation of reference axes for Stokes parameters by the angle p in positive direction (2.54) azimuth independant matrix and matrices containing terms with sin [ k ( p o- a)] and cos [k((po - a ) ] (2.30) number of particles in a unit volume (2.10) number of sphcrical particles of the radius a and of the dielectric constant e (2.22) degreo of polarization (2.11), degree of linear polarization (3.1) elements of the ssattering matrix in (2.12) phase function for scattering without polarization (2.23) phasa matrix [2.3] s2attering matrix for Rayleigh scattering and for large aerosd particles, respectively (2.55) P‘ snattering matrix containing only terms responsible for the difference betwcen the Rayleigh and aerosol particles szattering (2.56) degree of polarizatio! measured in the red and blue part of the spectrum [3.3] Stokes parameter [2.1] distance from a point source (2.la), distance series in the Mie theory (2.15), (2.16) matrix of thc diffuse reflection by the atmosphere of thc optical thickness 7 illuminated from the direction (-PI,&, into the direction (p,p) (2.29), Sk) matrix containing the terms with sin or cosine of k ( p ~ a) sensitivity of the multiplier tube (3.2)

-

102

ZDENEK SEKERA

matrix of the diffuse transmission by the atmosphere of the optical thickness 7 , illuminated from the direction ( - p ’ , q ’ ) , in the direction ( - p , q ) , (2.29) U Stokes parameter, defining the position of the plane of polarization (2.4) Stokes parameter] element of the matrix 10) (2.12) Stokes parameter, element of the matrix I(p,(p) or L(P,(P) “2.31 V Stokes parameter] defining the ellipticity of the polariaation (2.4) dV elementary volume [2.2] Stokes parameter] element of the matrix I(*)(2.12) Stokes parameter, element of the matrix I ( p , q ) or I&,(P) solution of the integral equation (2.35) solution of the integral equation (2.35) for the characteristic functions %, *?, in (2.38) and (2.40) solution of the integral equation (2.35) solution of the integral equation (2.35) for the characteristic functions *g, qr,qvin (2.38) and (2.40) function of X(2)and Y(2)in (2.47) polariaability (2.18) nondimensional parameter in Mie theory (2.15) ellipticity of polarization (2.7) volume scattering coefficient (2.22) volume scattering coefficient for aerosol particles and for molecules, respectively, (2.55) functions related to the global radiation (2.43) phase difference [2.1] retardation of a retardation plate [2.1], [3.1], dielectric constant [2.2] scattering function in (2.37) scattering function in (2.37) scattering angle (2.15) scattering function in (2.37) wavelength [2.1] albedo of the reflection according to Lambert’s law (2.42) matrix related to the reflection according to Lambert’s law, A i j its elements (2.42), (2.43) cosine of the zenith angle cosine of the zenith angle of the sun frequency [2.3] scattering function in (2.37) density [2.4] scattering function in (2.37) elementary area [2.3] optical thickness (2.27) azimuth angle [2.3], position of the plane of polarization of the measured light [3.1], (3.1) scattering function in (2.37) scalar function in (2.32) . . x angle defining the position of the plane of polarization (2.6)

*,

DEVELOPMENTS IN THE STUDY OF POLARIZATION OF SKY LIGHT

103

scattering function in (2.37) angle defining the position of the analyzer (2.10), in the photoelectric polarimeter (3.1) $ ( p ) scattering function in (2.37) i&), *$(/A) characteristic functions in the integral equation (2.35) defined in (2.38), (2.39) and (2.40) w circular frequency (2.la), rotational speed of the retardation plate (3.1) dw, dw’ elementary solid angles x(p)

$

@)(/A),S’i(p),

REFERENCES 1. Chandrasekhar, S. (1946). On the radiative equilibrium of a stellar atmosphere. I-XXIV. AStTophys. J. 103-108. 2. Chandrasekhar, S. (1950). “Radiative Transfer.” Oxford, U. P., New York. 3. Fakoff, D. L., and MacDonald, J. E. (1951). On the Stokes parameters for polarized radiation. J. Opt. SOC.Amer. 41, 861-862. 4. Saxon, D. S. (1955). Lectures on the scattering of light. Sci. Rept. No. 9, pp. 1-991, Contract AF 19(122)-239, Dept. of Meteorology, Univ. of California. 5. Perrin, F., and Abragam, A. (1951). Polarisation de la lumihre diffusbe par des particules spheriques. J. phys. radium 12, 69-73. 6. Hammad, A., and Chapman, S. (1939, 1945, 1947, 1948). The primary and secondary scattering of sunlight in a plane-stratified atmosphere of uniform composition. T . Phil. Mag. [7] 28, 99-110; 11. [7] 36,434-440; 111.[7] 38,515-529; IV. [7] 39, 956-966. 7. Robley, R. (1952). La diffusion multiple dans l’atmosph8re deduite des observations crdpusculaire. 11. Ann. Ghphys. 8, 1-20. 8. Sekera, Z., Diffuse sky radiation in upper levels of the atmosphere. In preparation. 9. Deirmendjian, D., and Sekera, Z. (1954). Global radiation resulting from multiple scattering in a Rayleigh atmosphere. Tellus 6, 382-398. 10. Fraser, R. S., Jr., and Sekera, Z. (1955). The effect of specular reflection in a Rayleigh atmosphere. Final Rept. Appendix E, pp. 1-12, Contract AF 19(122)239, Dept. of Meteorology, Univ. of California. 11. Deirmendjian, D. (1955). The optical thickness of the molecular atmosphere. Arch. Meteorol. Geophys. Biokl. B6, 452-461. 12. Sekera, Z., and Blanch, G. (1952). Tables relating to Rayleigh scattering of light in the atmosphere. Sci. Rept. No. 3, pp. 1-85, Contract A F 19(122)-239, Dept. of Meteorology, Univ. of California. 13. Chandrasekhar, S., and Elbert, D. D. (1954). Illumination and polarization of the sunlit sky on Rayleigh scattering. Trans. Am. Phil. SOC.44(6) 643-728. 14. Coulson, K. L. (1952). Polarization of light in the sun’s vertical. Sci. Rept. No. 4, pp. 1-40, Contract AF 19(122)-239, Dept. of Meteorology. Univ. of California. 15. Fraser, R. S., Jr. (1955). Theoretical positions of maximum degree of polarization. Final Rept. Appendix C, pp. 1-8, Contract A F 19(122)-239, Dept. of Meteorology, Univ. of California. 16. Coulson, K. L. (1951). Neutral points of skylight polarization in a Rayleigh atmosphere. Sci. Rept. No. 7 , pp. 1-29, Contract AF 19(122)-239, Dept. of Meteorology, Univ. of California. 17. Coulson, K. L., and Sekera, 2. (1955). Distribution of polarization and the orientation of the plane of polarization of sky radiation over the entire sky in a Ray-

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leigh atmosphere. Final Rept. Appendix B, pp. 1-10, Contract 19(122)-239, Dept. of Meteorology, Univ. of California. 18. Chandrasekhar, S., and Elbert, D. D. (1951). Polarization of the sunlit sky. Nature 167, 51-54. 19. (1954). The determination of polydisperse aerosol size distribution from the analysis of light scattering data. Sci. Rept. No. 12, pp. 1-36. Contract AF 19(122)472. Armour Research Foundation, Illinois Institute of Technology. 20. Aden, A. L., and Kerker, M. (1951). Scattering of electromagnetic waves from two concentric spheres. J . Appl. Phys. 22, 1242-1246; Giittler, A. (1952). Die Miesche Theorie der Beugung durch dielektrische Kugeln mit absorbierendem Kern und die Bedeutung fur Probleme der interstellaren Materie und des atmospharischen Aerosols. Ann. phys. 11, 65-98. 21. Atlas, D., Kerker, M., and Hitschfeld, W. (1953). Scattering and attenuation by non-spherical particles. J . Atm. and Terrest. Phys. 3, 108-1 19. 22. Sekera, Z. (1952). Legendre series of the scattering functions for spherical particles. Sci. Rept. No. 6, pp. 1-25, Contract AF 19(122)-239, Dept. of Meteorology, Univ. of California. 23. Sekera, Z. (1955). Scattering matrix for spherical particles and its transformation. Final Rept. Appendix D, pp. 1-15, Contract AF 19(122)-239, Dept. of Meteorology, Univ. of California. 24. Deirmendjian, D., and Sekera, Z. (1956). Atmospheric turbidity and the transmiesion of ultraviolet sunlight. J . Opt. Soc. Amer 46, 565-571. 25. Sekera, Z. (1935). Lichtelektrische Registrierung der Himmelpolarisation. Gerl. Beitr. Geophys. 44, 157-175. 26. Hall, J. S., and Mikesell, A. H. (1950). Polarization of light in the galaxy as determined from observation of 551 early-type stars. Publs. U.S. Nav. Obs. 17, Part I, 1-61. 27. Sekera, Z. (1951). Theory of polarization measurement suitable for investigation of skylight polarization. PTogr. Rept. No. 2, Appendix A, pp. 1-7, Contract AF 19(122)-239, Dept. of Meteorology, Univ. of California. 28. Seaman, C. H., and Sekera, Z. (1955). Photoelectric polarimeter for measurement of skylight polarization. Final Rept. pp. 1-18, Contract AF 19(122)-239, Dept. of Meteorology, Univ. of California. 29. Sekera, Z. (1951). Polarization of skylight. in Compendium of Meteorology, pp. 79-90. American Meteorological Society, Boston. 30. Tichanowski, J. J. (1926). Resultate dcr Messungen der Himmelspolarisation in verschiedenen Spektrumabschnitten. Meteorol. Z . 43, 288-292. 31. Dietze, G. (1951). Die abnormale Triibung der Atmosphiire September/Oktober 1950. 2. Meteorol. 6, 86-87. 32. Dumont, R. (1954). Mesures de la dispersion du degre de polarisation au zenith entre 4000 et 8500. Compt. rend. 238 (26), 2512-2514.