Polarized light

PRAMANA © Printed in India Vol. 45, No. 4,

__journal of October 1995

physics pp. 319-326

Polarized light

G R A M A C H A N D R A N , A R USHA DEVI and N S S A A N D E E P K A N N A D A V A R D H A N A

Department of Studies in Physics, University of Mysore, Manasagangotri, Mysore 570006, India MS received 27 April 1995

Abstract. Following the recent work of Chandler et al on quasi probability distributions for spinol/2 particles, we show that polarized light can be interpreted in terms of trivariate probability distributions in two different ways by choosing the variates to correspond to (i) the co-ordinates on the Poincare sphere, (ii) the components of the spin operator of the photon. In either case, it is shown that the Margenau-Hi!l procedure leads to probability mass functions while the Wigner-Weyl approach leads to probability density functions and the well-known Stokes parameters are also realised as appropriate averages with respect to these distribution functions.

Keywords. Density matrix; polarization; Stokes parameters; quasi-probability distributions.

PACS Nos 14"80; 03"65; 02"50; 07"60

1. Introduction

The subject of polarization of light [1] is an ancient one going back, according to Born and Wolf [2], to Huygens [3] of the seventeenth century. The state of polarization was characterized by Stokes [4] in terms of a set of four parameters So, s I , s 2, s3, named after him. Poincare [5] developed a geometrical representation for all states of polarization in terms of points on a sphere known by his name. It is well-known [6] that photons, the quanta associated with light, are spin-1 particles which are massless and hence deprived of the ]10) or longitudinal state. The review articles by Fano [7] and McMaster [8] show how a partially polarized state of light can be represented in terms of a 2 x 2 hermitian and positive semidefinite matrix p employing the concept of the density matrix [9]. What is the composition in a partially polarized beam of light? If p has eigen values 21 and 22 with corresponding eigen states le 1 ) and

182 )

respectively, one can say that the beam consists of N21 photons with polarization le~) and N22 photons with polarization le2 ), where N denotes the total number of photons which is proportional to the Stokes parameter s o ~>(s 2 + s22 + s 2 ) ~/2 representing the total intensity I of the beam. The equality corresponds to completely polarized or pure states characterized by one of the eigenvalues being zero. The recent work of Chandler et al

[10] on spin-1/2 particles shows that one can interpret the contents of a 2 x 2 density matrix for spin-1/2 particles in probabilistic ways based on the correspondence rules of Margenau and Hill [11] on one hand and Weyl [12] and Wigner [i3] on the other.

More recently Usha Devi et al [14] have adduced arguments following Margenau and Hill to show that an aligned spin-1 system can be viewed in terms of bivariate probability distribution by identifying the associated characteristic function. It is interesting to note that a beam of photons is akin to an aligned spin-1 system when the 319

G R a m a c h a n d r a n e t al

Stokes parameter s 3 is zero. It is interesting to investigate whether a joint probability distribution interpretation can be given to an arbitrarily polarized beam of light. In this paper, we extend the Margenau-Hill procedure to provide a probabilistic interpreta- tion for partially polarized light, in terms of a trivariate probability distribution even in the general case when s3 # 0. We find, in fact, that there are two convenient ways in which this extension can be carried out viz., using the operators representing

(i) the co-ordinates on the Poincare sphere (ii) the components of the spin of the photon

as the variates. This leads to a probability mass function where the variates assume discrete values. On the other hand a probability density function can be obtained using the Wigner-Weyl correspondence rule where once again the variates may be chosen in two alternative ways (i) and (ii).

2. States of polarization and the density matrix for photons

Our notations are as follows: Using a right handed co-ordinate system where the z-axis is chosen parallel to the momentum vector k of the photon, all possible states of polarization can be represented by [15]

[ s ~ # ) = ( c o s f l c o s ~ - i s i n f l s i n ~ t ) l e x ) + ( c o s f l s i n c t + i s i n f l c o s ~ ) l ~ y ) (1) where [e x ) , [gy) denote linear states of polarization along x, y axes respectively. Here 0 ~< ct < ~ and - ~/4 <~ fl ~< n/4 are angles such that 0 = ~/2 - 2fl and th = 2~ designate points on the Poincare sphere where the north pole corresponds to right circular polarization and the south pole corresponds to left circular polarization. Clearly, [e~# ) and ]%.~/2-#) are orthogonal states in a two dimensional complex vector space and correspond geometrically to diametrically opposite points on the Poincare sphere. We choose the circular polarisation states

[~R) = I%./,) = (lex) + iler ) ) / x / ~

as our basis with respect to which the density matrix p assumes the form Trp ~,

p =--~-- t l + ~ ' V ] (3)

where e l , a2, a3 denote the standard Pauli matrices, I = Tr p = s o denotes the intensity of the light beam. Moreover,

s o P 1 = s I = Tr(pal) = I x - I r (4)

s o P 2 = s 2 = Tr(pa2) = I x, - It, s o P 3 = s3 = Tr(po3) = 1L -- le

where x', y' denote linearly polarized states with ~/= 0 and ~ = rr/4 and 37r/4 respectively.

Note that s 1 , s 2, s 3 are expectation values of the operators a 1 , a z, a 3 respectively and hence we could identify a l , a2, a3 as operators representing the co-ordinates of a point on the Poincare sphere. On the other hand, considering photon as a spin-1 particle, the 320 Pramana - J. Phys., Vol. 45, No. 4, October 1995

Polarized light

density matrix can be expressed in terms of the Fano statistical tensors [7] tkq through

" - - - r

Trp

(5)

k=O q=

where zkq(J).are spherical tensor operators constructed out of the spin operators J of the photon and are normalized such that

(lm'lz~q(J)l l m ) = c(lkl; mqm')x/~ + 1 (6) where c(lkl; mqm') are Clebsch-Gordan coefficients. An explicit 3 x 3 matrix form for p is given by (1) of [16]. Observing that the basis states I lm) are such that I l l ) = - l e R) and i l - 1) = leL) leads to the identification,

t ° = 1 (7)

/~) 1/2 S3

t l ⁼ - S o

1

Re(t~) = x//3 sl 2 s o Im(t 2) = x//3 s2

2 s o

t~+t =P+, =0.

Clearly the system is aligned [17] if s 3 = 0 and an unitary transformation 'e-i°~2 0 0

u = o 1 0 (8)

0 0 e +I°/2

with tan 0 = s 2/s t takes the photon density matrix into its principal axes of alignment frame (PAAF) [17],

it So

₀

O(S~+o~)~:~

₀

t

P""~=2

( ~ : + s l ) 1/~ o So /

(9) The conclusions drawn by Usha Devi et al [14] apply in toto here, together with the identification,

II x = ~l-s o - ( s ~ + s2) t/2 ] 1 (10)

1 s~)1/2 ]

ri, = ~ [so + (s~ +

Hz = 0

Pramana - J . Phys., Vol. 45, No. 4, October 1995 321

G Ramachandran et al

and we can explain the polarization in terms of the bivariate joint probabilities [14]

with any two of the spin components of the photon as variates. To describe the more general situation i.e., when s 3 ~ 0 we need a trivariate probability distribution where we have to treat all the three components of photon spin as variates.

3. Margenau-Hiil probabilities for photons

The Margenau-Hill trivariate characteristic function for three non-commuting oper- ators X1, X2, X3 is defined by,

. A . ^

t~Ml-l(l t, I2, I3) = ~ Tr[p {exp0X 1 I1 )exp(lX2Iz)exp(iX313) + exp(i:f 1 I1 )exp(i)(313) exp(i)f212 ) + exp (i,~ 212) exp(i)( 313)exp(i)~ 111 ) + exp(iX 212 )exp(iX x I1 )exp(i)f3 I3) + exp(i.~ 313)exp(i)( 111 )exp(L~z I2 )

+ exp(iXaI3)exp(iX212)exp(iXll 1)}]. (11) Choosing X 1, X2, X3 to be 0-1,0-:, 03 and p as given in (3), we can simplify (11) to obtain

~bMH(I1,12,13 ) = S o COS 11 COS 12 COS 13 q- Ce) is 1 sin 11 cos

12

^{cos 13}

+ is2cosllsinl2cosl 3 + is3cosllcosl2sinl 3. (12) Simplifying (12) further, ~hl*) 1I I2 , 13 ) could be cast into the form W M H ~ * I

CPMH(II'I2'I3)-

(e) E f~)n(ml,m2,m3)exp(i(mlll + m212 + rn313))

? n x , m ~ , n l 3 = - - 1 , + I

(13) where f~)n(mx, m2, m3) could be identified as the Margenau-Hill trivariate probability mass function for photons and is given by

f(~) t,, MHI.,,,1 , m 2 , m 3 ) = ~ [ s 0 + 1 mls 1 + m2s 2 d- m3s3]. (14) ml, m2, m 3 are the classical random variables representing the operators tr 1,0-2,0-3 and the Stokes parameters could be identified as first moments of the distribution f~)n(ml,m2,ma) i.e.,

Sk = Tr(p0-k) = ~ mkfMn(ml,m2,m3), t*) k = 1,2,3. (15)

m x , l l q ~ , n l 3 = - - I , + 1

On the other hand X 1, X 2 , X 3 in (11) could be chosen as J t , J 2 , J 3 , the photon spin components and the density matrix p is then given by (5). To simplify the characteristic function ~b~)n(ll, 12,13) we follow the approach outlined in [14] where we identify that the unitary matrices

11:) 1(+/:

⁽¹⁶⁾

322 Pramana - J. Phys., VoL 45, No. 4, October 1995

Polarized li#ht

where

diagonalise J1 and J2 respectively. We then replace

exp(iJxI~) = U~exp(iJ~I1)U~,exp(iJ212)= U~exp(iJ~I2)U 2 (17) in (11) where J~ and J~ are diagonal matrices with diagonal elements 1, O, - 1. We can now simplify the characteristic function further to yield

C ~ M H ( m l , m 2 , m 3 ) (J) = 2 f M H ( m l , m 2 , m 3) (J)

rat,ra2,m3 = - 1,0, + 1

x exp(i(mlI 1 + m2I 2 + m313) ) (18)

~j) 1 , ,

fMa(ml, m2, m3) = ~ Re[(pU1 )m.,., (U1 U2) . . . . (U2)m~,.~

+ ( U 2 P ) s , r , , ( U ~ ) r a , r a , ( V 1 V t 2 ) m t m 2 + ( U 2 p U t l ) m ~ r n , ( U t 2 ) r a , r a ~ ( U 1 ) r , ~ r a , ] . (19) Explicitly it may be seen that f ~ n ( m l , m z, m3) assumes the simple form

(j) 1

fMn(ml,m2,m3)

⁼ {2m 1 m2s 2 - m3s3} (20)

if m l , m 2 , m 3 = + 1;

~j) _ 1 +3mZ[so ( m Z - m 2 ) s l ] 2m3s3} (21)

fMH (mz, m2, m3) -- ~-~ {ml m2 s2 -- --

if one of the variables ma, m2, m 3 assumes the value 0 and 1

f(S) ( m M r l V " l , m 2 , m 3 ) = - - ~-~m3 s3 (22)

if more than one of m 1, m2, m 3 take the value O. Consequently the Stokes parameters could now be realised as the classical averages,

!

sl ₌ TrEp(j2 _-- j 2 ) ] ₌ ~ j M n , m x , m z , m 3 , , m 2 f i t ' s ) . . ¢ 2 _ m 2 )

fflx,m2,M3 = -- 1 1

s2 = T r [ p ( J 1 J 2 + J 2 J 1 ) ] = ~ JMH/'"I(IJ) t.., ,m2,m3)(2mxm2)

mx,mz,ra3 = - 1 1

s3 = _ T r [ P J 3 ] = _ ~ cu) JMH~ 1 t m , m2,m3)m3 (23)

ma,m2,m3 = - 1

4. W i g n e r - W e y l probabilities for photons

The Wigner-Weyl characteristic function is defined by

~ww(I1,12,13) = Tr [pexp(i()(111 + )(212 + )(f13))].

Substituting -~1 = al, X2 = a2, X3 = a3 and p given by (3) in (24), we get gP~v)w(lx,I2,13)= So [COs l + P'ISii----fl ]

12 -I- /'2~1/2

w h e r e I = ( l ~ + . 2 _ . 3 , •

(24)

(25)

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G Ramachandran et. al

Fourier transform of the characteristic function ~b~w(l~, 12,13) yields the Wigner- Weyl trivariate probability density function f~w(X t , x2, x3):

fff

f ww(Xl,X2,X3)=(2rr) -3 d3I~)~v)w(ll,12,13)exp(-i(xlll + x212 + x313))

= , 2 n ) - a S o f o f : f ~ ~ 12dlsinO1dO1d¢t

[COS

I +

i ( P ' I ) ~ ]

x e x p ( -

i(xlI 1 + x2I 2

⁺ x3/3) ). (26) Carrying out the integration over the solid angle d o t = sin 0td0tdCt, we obtain

f~)w(Xx,X2,X3)=(2n2x)-l [ f o d l lcoslsinlx-P'V~, f o dlsinlsinlx ]

(27)

= , , x = (x~ + x~ + x2) 1/23 . We can simplify (27) further to get where V x \c?xl c?x2 3

f~)w(Xl, X 2 , X3) = S O [Fo(x) 4- F 1 (x)P'x] (28) where

1 d

Fo(x) =

4nxdx[6(1

+ x) +6(1 - x ) ] 1 d [ 6 ( 1 + x) - 6 ( 1 - x ) ]

Fl(x) = ~n-~x [_ x _"

⁽²⁹⁾

Now the quantum mechanical expectation values could be obtained in terms of

fww(Xl,X2,X3)

as,

=fffdx~ww,Xx,X2,X3)X~,

k = 1,2,3. (30,

Sk

⁼ Tr(po. k) 3 (.)

Alternatively we can replace X~,,~2, X3 in (20) by J~, J 2 , J 3 in which case

~b(J) H 12 13) = Tr(pexp(iJ'l)). _{WW~*l, ,} We now express

exp(iJ-I) = R*(~b l, 0 t, 0)exp(iJ 3

l)R(dp I, 0 I, O)

(31)

(32) where R(~, fl, 7) denote rotations through Euler angles ~, fl, 7. Making use of(5) and (6) and the transformation properties of the spherical tensor operators zkq(J) under rotations viz.,

k

R(a, fl,7)z~(j)R+(cc, fl,7)= y" k

r¢(J)Dcq(~, fl, 7)

k

(33)

q'= - k

where D k denotes the standard (2k + 1) dimensional irreducible representation of 324 Pramana - J. Phys., Vol. 45, No. 4, October 1995

Polarized light

rotations [18], we obtain

# J ~ / r w w t ' l , I 2 , l , ) = ~s ° ^{- - - -} k

3 s 313

2 I ~ mexp(iml)

m = - 1,0,1

+ 4T{So(312 -

12) +

3s~(I 2 - lZt) - 682I 112}

X m = -EI,O,I (3m 2 -

2)exp(iml)]. (34)

Fourier transform of ~J)

C~ww(Ii,I2,13)

could now be carried out by generalising the techniques used in (26) to (29) to yield the Wigner-Weyl probability density function,

fo~ t.. w w ~ , x2, x3) = I

So,~o(X) - ,~l(x)s3x3 + ,~:(x)

where

2 m)

6nx ,,, ax

o ~ , ( x ) = Z ~ m d [ 6 ( X x m ) ]

1 _i)~I d ~I d (,~Im-x)l~ l ⁽³⁶⁾

" ~ 2 ( x ) = ~ - ~ (3m2 Lx~xx ~x~xx \ x JJJ

Here e(m - x ) denotes the step function

2 ~ f ® e ia'

{~ if a > 0

t(a)

= - - d t = (37)

-oo t if a < 0 Making use of the Weyl correspondence rule

xt xz m . ~ S {jtx j,~ j.3 }

(38)

where S stands for the symmetrizer, the Stokes parameters could be realised as the classical averages,

ww(X,,

x3)(x - x )d3x

r,,',

ww(Xl, Xz, x3)(2xt x2)d3x

P r a m n n a - J. Phys., Yol. 45, No. 4, October 1995 325

G Ramachandran et al

Acknowledgement

One of us (ARU) thanks the CSIR for financial assistance through the award of a Research Fellowship.

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