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Coherence properties of Stokes beams
Optics Communications 285 (2012) 4715–4718
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Coherence properties of Stokes beams Jing-Shyang Horng a, Yajun Li b, Emil Wolf c,n a b c
Department of Electro-Optical Engineering, National United University, No.1, Lien-Da, Kung-Ching Li, Miao-Li, Taiwan 360, R.O.C P.O. Box 975, Great River, NY 11739, USA Department of Physics and Astronomy and the Institute of Optics, University of Rochester, Rochester, NY 14627, USA
a r t i c l e i n f o
abstract
Article history: Received 14 November 2011 Accepted 22 May 2012 Available online 9 June 2012
A Stokes beam is a light beam which is expressible as a superposition of a completely polarized and a completely unpolarized beam. In this note we study coherence properties of a wide class of beams of this kind. The behaviors of the degree of coherence of a Stokes beam produced by a circular planar source are expressed in terms of Lommel functions and theoretical results are illustrated by computed examples. & 2012 Elsevier B.V. All rights reserved.
Keywords: Polarization Beams Coherence theory
1. Introduction The concept of a Stokes beam is a refinement of a notion introduced by G.C. Stokes in a classic paper [1], which may be regarded as the starting point of polarization optics. According to this theory, any light beam may be considered to be the sum of a completely polarized beam and a completely unpolarized one ([1], p. 413). A few years ago it was shown that such decomposition is not possible, in general [2]. In the present note we study coherence properties of a class of Stokes beams. However, it is appropriate to begin with a few words of caution relating to the fact that the concept of completely polarized and completely unpolarized beams is ambiguous, as will become clear later on. This note is organized as follows. In Section 2, a few results from the elementary theory of polarization of light beams are recalled. In Section 3, an example regarding the coherence properties of a Stokes beam from a planar and circular source is given and we found that the degree of coherence of the Stokes beam can be expressed in terms of the Lommel functions originally developed for the investigations of the three-dimensional light distribution near the focus of a converging spherical wave [3].
2. Cross-spectral density matrices of polarized and unpolarized beams Consider a statistically stationary light beam, which propagates into the half-space z4 0, with its axis along the z-direction, n
Corresponding author. Tel.: þ1 585 275 4397; fax: þ 1 585 473 0687. E-mail address: [email protected] (E. Wolf).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.05.039
as illustrated in Fig. 1. The spectral degree of polarization of the beam at any point r may be expressed in terms of two rotational invariants of the correlation matrix [4] " # /Enx ðr, oÞEx ðr, oÞS /Enx ðr, oÞEy ðr, oÞS 2 ð2:1Þ W ðr, oÞ ¼ /En ðr, oÞE ðr, oÞS /En ðr, oÞE ðr, oÞS : x y y y Here Ex and Ey denote the Cartesian components of the members of the statistical ensemble (assumed to be stationary and ergodic) of the electric field at frequency o, in the sense of coherence theory in the space–frequency domain ([5], Chapter. 4), and the asterisk denotes the complex conjugate. The spectral degree of polarization Pðr, oÞ of the beam at the point r, at frequency o, is given by the expression ([5], Eq. (14), p. 179) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 4Det W ðr, oÞ Pðr, oÞ ¼ u ð2:2Þ t1 l 2 m2 Tr W ðr, oÞ where Det denotes the determinant and Tr the trace. The degree of polarization is bounded by zero and unity ð0 r P r 1Þ. When P ¼ 0 the light beam is said to be unpolarized at the point P(r) at frequency o and in the other extreme case, when P ¼ 1, it is said to be fully polarized at that point and at that frequency. Until fairly recently it was generally assumed that unpolarized light has the same correlation properties as natural light ([3], p. 624; [6], p. 350) and that polarized light has the same behavior as monochromatic light ([3], p. 624; [6], p. 351). If that were so, the correlation matrices of light of this kind would necessarily have the forms 2ðuÞ 1 0 ð2:3Þ W ðr, oÞ ¼ A 0 1
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J.-S. Horng et al. / Optics Communications 285 (2012) 4715–4718
Further, as in [2], we will model a light beam that is completely polarized as a beam of monochromatic light. Its cross-spectral density (CDS) matrix in any transverse cross-section of the beam has, therefore, the ‘‘factorized’’ form ([6], Sec. 6.3.2): " n # ex ðr1 , oÞex ðr2 , oÞ enx ðr1 , oÞey ðr2 , oÞ 2ðpÞ ð2:6Þ W ðr1 ,r2 , oÞ ¼ en ðr , oÞe ðr , oÞ en ðr , oÞe ðr , oÞ : x 2 y 2 y 1 y 1 This expression for completely polarized beam may be obtained from Eq. (2.4) by the use of the relationship Enk El ¼ enk el , ðk, l ¼ x, yÞ, where enk and el are components of the complex vectors Enk and El . By the use of Eq. (2.2), one can readily verify that the matrices (2.3) and (2.4) represent beams, at each point, are, indeed, completely unpolarized and completely polarized, respectively. We will refer to the beams represented by the matrices (2.3) and (2.4) as canonical models of unpolarized and polarized light beams, respectively.
3. Coherence properties of Stokes beams
Fig. 1. Notation relating to the arguments of the correlation matrix at points in the far field of Stokes beam.
and " /Enx ðr, oÞEx ðr, oÞS 2ðpÞ W ðr, oÞ ¼ /En ðr, oÞE ðr, oÞS x y
/Enx ðr, oÞEy ðr, oÞS /Ey ðr, oÞEy ðr, oÞS n
# ,
ð2:4Þ
where the superscripts (u) and (p) refer to unpolarized and to polarized light, respectively. In Eqs. (2.3) and (2.4), A is a constant, Ex(r,o) and Ey(r,o) are the Cartesian components of the electric field at the point P(r) and the angular brackets represent the average of the product of two components of the electrical field taken over the statistical ensemble. 2 One can readily confirm that when W matrix has the form of Eq. (2.3), the degree of polarization of the beam at the point P(r) has zero value and that when it has the form given by Eq. (2.4) it has the value unity. It is important to appreciate that Eqs. (2.3) and (2.4) contain information about correlations between transverse, mutually orthogonal components of the electrical field vector at a single point. However, to elucidate the propagation of the correlation matrices, it is necessary to know the values of correlations between the fluctuating field components at pairs of points ([5], Sec. 9.4). The problem of specifying two-point correlation matrices of polarized and unpolarized light has attracted some attention in recent years (see, for example, [7,8,9]) and it was found that there is no unique form of two-point cross-spectral density matrices of such beams. In view of this fact, we will proceed as follows. We will model an unpolarized beam as a beam of natural radiation (see, for example, [6], Sec. 6.3.1). The cross-spectral density of such radiation in any transverse cross-section zQconst of the beam perpendicular to the axis (the z-axis) has the form [10] 2ðuÞ 1 W ðr1 ,r2 , oÞ ¼ Aðq1 , q2 , oÞ 0
0 1
:
ð2:5Þ
where r1 ¼(q1,z), r2 ¼(q2,z), q1 and q2 being the transverse vectorcomponents of r1 and r2, respectively, perpendicular to the beam axis (the z-axis) and z labels the longitudinal components of these vectors (i.e. components along that axis).
We will now examine the spatial coherence properties of some Stokes beams. To do so we need an expression for the CSD matrix of a beam which is a superposition of two beams. Such an expression follows at once from a general formula derived in [11] and one finds that if the two beams propagate along the same direction, which we take to be the z-direction, and if they are independent of each other then the spectral degree of coherence of the beams resulting from their superposition is 2ab 2ba given by Eq. (9) of Ref. [11], with W ¼ W ¼ 0, viz., 2ðaÞ
2ðbÞ
Tr W ðr1 ,r2 , oÞ þ Tr W ðr1 ,r2 , oÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Zðr1 ,r2 , oÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaÞ S ðr1 , oÞ þ SðbÞ ðr1 , oÞ SðaÞ ðr2 , oÞ þ SðbÞ ðr2 , oÞ
ð3:1Þ
where 2ðpÞ SðpÞ ðrq , oÞ ¼ Tr W ðrq ,rq , oÞ,
ðp ¼ a, b and q ¼ 1,2Þ:
ð3:2Þ
As an example, consider a Stokes beam produced by a planar, circular, secondary source of radius a in the x,y-plane as shown 0 0 in Fig. 1. In this figure, Q1(q1) and Q2(q2) is a pair of source points and P 1 ðr1 ÞRP1 ðq1 ,zÞ and P2 ðr2 ÞRP2 ðq2 ,zÞ is a pair of field points. Let us choose the CSD matrix of unpolarized light in the source plane z ¼0 to be proportional, in suitable choice of units, to the unit matrix 2ðuÞ 1 0 0 0 , ð3:3Þ W ðq1 , q2 ; z ¼ 0; oÞ ¼ Aðq1 , q2 Þ 0 1 0
0
where A(q1,q2) describes the source in the xy-plane bounded by a circle of radius a, that can be expressed as 0 0 r r ð3:4Þ Aðq01 , q02 Þ ¼ circ 1 circ 2 a a where circ(x) is the circle function and r0 is the distance from the origin O to a variable point in the source plane. The propagation law of CDS matrix from the source plane into the half-space z40 has been investigated by Mandel and Wolf (see, e.g. Eqs. (5.6–14) and (5.16–17) in [6]) and they found that the CSD matrix at a pair of points located in any transverse plane zQcont40 can be expressed in terms of the 2 2 correlation matrix of electric field vector at pairs of points in the source plane
J.-S. Horng et al. / Optics Communications 285 (2012) 4715–4718
zQ0 in the form 2ðuÞ W ðq1 , q2 ,z; oÞ ¼ ZZ 2ðuÞ 0 0 2 0 2 0 0 0 W ðq1 , q2 ; z ¼ 0; oÞKðq1 q1 , q2 q2 ,z; oÞd q1 d q2 , r0 r a
ð3:5Þ where the propagation kernel can be expressed as (see Eq. (3) on p. 182 of [5]) Kðq1 q01 , q2 q02 ,z; oÞ ¼ Gn ðq1 q01 ,z; oÞGðq2 q02 ,z; oÞ,
ð3:6Þ
and G is the Green’s function for paraxial propagation from the 0 source point Q(q ,z ¼0) to the field point P(q,z): ! 2 9qq0 9 ik 0 exp ik ; ð3:7Þ Gðqq ,z; oÞ ¼ 2pz 2z (k¼ o/c, c being the speed of light in vacuum). In the cylindrical coordinates system shown in Fig. 1, the Green function in Eq. (3.7) can be expressed as ik r2 þ r02 2rr0 cosðycÞ : ð3:8Þ exp ik Gðqq0 ,z; oÞ ¼ 2pz 2z On substituting Eq. (3.8) into Eq. (3.7) and then into Eq. (3.6), for the propagation kernel takes the form Kðq1 q01 , q2 q02 ,z; oÞ 2 0 2 02 0 ½r2 þ r02 k 1 2r1 r1 cosðy1 c1 Þ½r2 þ r2 2r2 r2 cosðy2 c2 Þ ¼ exp ik 1 2pz 2z
ð3:9Þ 2
2
The elements d r and d r on the right-hand side of Eq. (3.5) are given by the expression 0 1
2
d r0q ¼ r0q dr0q dyq
0 2
ðq ¼ 1,2Þ:
ð3:10Þ
On substitution of Eqs. (3.9) and (3.10) into Eq. (3.5), one finds, after a straightforward but long calculation, expression for the CSD matrix of the un-polarized beam in any transverse crosssection zQconst in the form 2 2ðuÞ 1 1 0 Y g , ð3:11Þ W ðq1 , q2 ,z; oÞ ¼ 4 0 1 q¼1 q where ( 2
) Z Z 0 0 r þ r02 2 r1 ¼ a y1 ¼ 2p 1 2r1 r1 cosðy1 c1 Þ g1 ¼ exp ik 1 r01 dr01 dy1 2z lz r01 ¼ 0 y1 ¼ 0 " Z 0 # r1 ¼ a r r0 r2 r02 k ¼ expðik 1 Þ 2 J 0 k 1 1 exp ik 1 r01 dr01 ð3:12Þ z 2z z 2z r0 ¼ 0 1
and J0(x) is the zero-th order Bessel function of the first kind. It is useful to introduce the dimensionless parameters u¼
ka2 , z
s0q ¼
r
0 q
a
,
sq ¼
rq a
and
vq ¼ usq ,
ðq ¼ 1,2Þ: ð3:13Þ
# " Z s0 ¼ 1 1 i i g 1 ¼ g 1 ðu,v1 Þ ¼ u exp us21 2 J0 ðv1 s01 Þexp uðs01 Þ2 s01 ds01 : 2 2 s01 ¼ 0
ð3:14Þ Similarly, we may express g2 on the right-hand side of Eq. (3.11) as i g 2 ¼ g 2 ðu,v2 Þ ¼ u exp us22 2
" Z 0 s1 ¼ 1 s02 ¼ 0
# i uðs02 Þ2 s02 ds02 ¼ g n1 ðu,v2 Þ: J 0 ðv2 s02 Þexp 2
The integrals on the right-hand sides of Eqs. (3.14) and (3.15) can be evaluated in terms of the Lommel U- or V-functions [3,12], i.e. Z s¼1 i expðiu=2Þ 2 ½U 1 ðu,vÞ þ iU 2 ðu,vÞ J 0 ðvsÞ exp us2 s ds ¼ 2 u=2 s¼0 expðiu=2Þ i v2 i exp uþ þ iV 0 ðu,vÞ þV 1 ðu,vÞ : ¼ ð3:16Þ u=2 2 u where U1, U2, V0 and V1 are the Lommel U- and V-functions defined by the following series of Bessel functions [4]: 1 u n þ 2s X ð1Þs J n þ 2s ðvÞ, U n ðu,vÞ ¼ ð3:17aÞ v s¼0 and V n ðu,vÞ ¼
1 X
ð1Þs
s¼0
u n2s v
J n2s ðvÞ:
ð3:17bÞ
in which n is an integer. The Lommel U- and V-functions are convenient for computations when 9u=v9 o 1 and 9u/v941, respectively. Next we investigate the fully polarized beam, for which we 2ðpÞ choose the one whose CSD matrix W in the source plane z¼0 has the form 2ðpÞ 1 1 0 0 : ð3:18Þ W ðq1 , q2 ; z ¼ 0; oÞ ¼ Aðq1 , q2 Þ 1 1 Following the same mathematical procedure developed above we 2ðpÞ can express W in the half-space z 40 in the form 2 2ðpÞ 1 1 1 Y g , ð3:19Þ W ðq1 , q2 ,z; oÞ ¼ 4 1 1 q¼1 q Comparing Eqs. (3.11) and (3.19), one readily finds that 2ðuÞ 2ðpÞ Tr W ðq1 , q2 ,z; oÞ ¼ Tr W ðq1 , q2 ,z; oÞ :
ð3:20Þ
On substituting of Eqs. (3.14) and (3.15) into Eqs.(3.11) and (3.19) and then into Eq. (3.20), we find that 2 ðuÞ 2ðpÞ g ðu,v1 Þg n1 ðu,v2 Þ Tr W ðq1 , q2 ,z; oÞ ¼ Tr W ðq1 , q2 ,z; oÞ ¼ 1 : 2 ð3:21Þ The spectral density at the point P(r) can be obtained from the trace of the autocorrelation of the CSD matrix: 2 Sðr, oÞ ¼ Tr W ðr,r, oÞ: It follows that 2
SðpÞ ðqq , oÞ ¼ SðuÞ ðqq , oÞ ¼
9g 1 ðu,vq Þ9 ðq ¼ 1,2Þ : 2
ð3:22Þ
On substituting of Eqs. (3.21) and (3.22) into Eq. (3.1), we are able to write an expression of the spectral degree of coherence for the Stokes beam in the form 2ðuÞ
On substitution of Eq. (3.13) into Eq. (3.12), we obtain
2
4717
ð3:15Þ
Tr W ðq1 , q2 , oÞ g ðu,v1 Þ g n1 ðu,v2 Þ
: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 Zðq1 , q2 , oÞR qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g 1 ðu,v1 Þ g 1 ðu,v2 Þ SðuÞ ðq1 , oÞ SðuÞ ðq2 , oÞ ð3:23Þ It is interesting to note that the modulus of the spectral degree of coherence 9Z(q1,q2,o)9¼1 (see Fig. 2(a)), which implies that the Stokes beam is a spatially fully coherent electromagnetic beam if the CSD matrices of the two sources are expressed in the forms (3.3) and (3.19). It is also interesting to consider the phase distribution of Z(q1,q2,o) over the transverse plane z¼ z0. Assuming that the
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J.-S. Horng et al. / Optics Communications 285 (2012) 4715–4718
Fig. 2. Notation relating to the calculations of the modulus and phase distributions of the spectral degree of coherence of the Stokes beam from a circular source in the xOy-plane.
field point P1 is an off-axis point, the field point P2 is on the axis and the two points are separated by a distance d as shown in Fig. 2. The dimensionless parameters defined in Eq. (3.13) now take the following values: u0 ¼
ka2 , z0
v1 ¼ u
d a
and
v2 ¼ 0
ð3:24Þ
On substituting of Eq. (3.24) into Eq. (3.23), we find that the phase variation can be expressed in the form
d Y ¼ arg Zðq1 , q2 , oÞ ¼ arg g 1 ðu0 ,u0 Þg n1 ðu0 ,0Þ a n
d ð3:25Þ ¼ arg½g 1 ðu0 ,u0 Þ þ arg g 1 ðu0 ,0Þ , a where n h i 8 o 2 > arg exp i u0 1 þ d2 ½iV 0 þ V 1 u0 ,u0 d i when d o 1, < 2 a a a d h i n arg g 1 u0 ,u0 ¼ o 2 > a : arg exp 2i u0 1 þ da2 ½U 1 þ iU 2 u0 ,u0 da otherwise:
ð3:26Þ
Fig. 3. Modulus and phase of spectral degree of coherence of Stokes beam from a planar circular source of radius a in the xOy-plane. (a) Modulus of the beam
9Z(q1,q2,o)9, and (b) the phase variationY ¼ arg Zðq1 , q2 , oÞ .
Acknowledgements This research was supported by the US Air Force Office of Scientific Research, under grant No. FA 9550-08-1-0417.
References
and
1J 0 ðu daÞcos u2 : arg g n1 ðu,0Þ ¼ arctan J 0 ðu daÞsin 2u
ð3:27Þ
On substituting of Eqs. (3.26) and (3.27) into (3.25), we are able to plot the three curves in Fig. 3(b) to show the phase of degree of coherence of the Stokes beam as a function of d/a for the three different values of the parameter u0 ¼50, 100 and 200. The curves in Fig. 3(b) show the rapid variations of the phase with the increasing values of u0. In summary, we have derived expressions for the spectral degree of coherence of Stokes beam generated by superposition of a completely unpolarized beam and a completely polarized beam. Results are illustrated by case example regarding the Stokes beam generated by a circular planar source. We found that the coherence properties of such a Stokes beam are rotationally symmetric with respect to the axis of beam and can be expressed analytically in terms of the Lommel functions.
[1] G.C. Stokes, Transactions of the Cambridge Philosophical Society 9 (1852) 399, Reprinted in W. Swindell, Polarized Light, (P. Ross, Dowden, Hutchinson, 1975), pp. 124–141. [2] E. Wolf, Optics Letters 33 (2008) 642. [3] M. Born, E. Wolf, Principles of Optics, 7th ed., Cambridge University Press, 1999, Section 8.8. [4] This matrix is a ‘‘one-point’’ version of the well-known ‘‘two-point‘‘ crossspectral density matrix W(r1, r2; o), as defined, for example, in [5], Section 9.1, that plays a central role in optical coherence theory. [5] E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge University Press, 2007. [6] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995. [7] M. Lahiri, E. Wolf, Optics Communication 281 (2008) 3241. [8] F. Gori, J. Tervo, T. Turunen, Optics Letters 34 (2009) 1447. [9] M. Lahiri, E. Wolf, Optics Letters 34 (2009) 557. [10] It was shown by M. Lahiri, E. Wolf, Optics Communication 281 (324) (2008), that the cross-spectral density matrix of blackbody radiation has the form Eq. (5). [11] M. Lahiri, O. Korotkova, E. Wolf, Optics Communication 281 (2008) 5073. [12] Y. Li, E. Wolf, Journal of the Optical Society of America A, Optics and Image Science 8 (1984) 801.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Coherence properties of Stokes beams Jing-Shyang Horng a, Yajun Li b, Emil Wolf c,n a b c
Department of Electro-Optical Engineering, National United University, No.1, Lien-Da, Kung-Ching Li, Miao-Li, Taiwan 360, R.O.C P.O. Box 975, Great River, NY 11739, USA Department of Physics and Astronomy and the Institute of Optics, University of Rochester, Rochester, NY 14627, USA
a r t i c l e i n f o
abstract
Article history: Received 14 November 2011 Accepted 22 May 2012 Available online 9 June 2012
A Stokes beam is a light beam which is expressible as a superposition of a completely polarized and a completely unpolarized beam. In this note we study coherence properties of a wide class of beams of this kind. The behaviors of the degree of coherence of a Stokes beam produced by a circular planar source are expressed in terms of Lommel functions and theoretical results are illustrated by computed examples. & 2012 Elsevier B.V. All rights reserved.
Keywords: Polarization Beams Coherence theory
1. Introduction The concept of a Stokes beam is a refinement of a notion introduced by G.C. Stokes in a classic paper [1], which may be regarded as the starting point of polarization optics. According to this theory, any light beam may be considered to be the sum of a completely polarized beam and a completely unpolarized one ([1], p. 413). A few years ago it was shown that such decomposition is not possible, in general [2]. In the present note we study coherence properties of a class of Stokes beams. However, it is appropriate to begin with a few words of caution relating to the fact that the concept of completely polarized and completely unpolarized beams is ambiguous, as will become clear later on. This note is organized as follows. In Section 2, a few results from the elementary theory of polarization of light beams are recalled. In Section 3, an example regarding the coherence properties of a Stokes beam from a planar and circular source is given and we found that the degree of coherence of the Stokes beam can be expressed in terms of the Lommel functions originally developed for the investigations of the three-dimensional light distribution near the focus of a converging spherical wave [3].
2. Cross-spectral density matrices of polarized and unpolarized beams Consider a statistically stationary light beam, which propagates into the half-space z4 0, with its axis along the z-direction, n
Corresponding author. Tel.: þ1 585 275 4397; fax: þ 1 585 473 0687. E-mail address: [email protected] (E. Wolf).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.05.039
as illustrated in Fig. 1. The spectral degree of polarization of the beam at any point r may be expressed in terms of two rotational invariants of the correlation matrix [4] " # /Enx ðr, oÞEx ðr, oÞS /Enx ðr, oÞEy ðr, oÞS 2 ð2:1Þ W ðr, oÞ ¼ /En ðr, oÞE ðr, oÞS /En ðr, oÞE ðr, oÞS : x y y y Here Ex and Ey denote the Cartesian components of the members of the statistical ensemble (assumed to be stationary and ergodic) of the electric field at frequency o, in the sense of coherence theory in the space–frequency domain ([5], Chapter. 4), and the asterisk denotes the complex conjugate. The spectral degree of polarization Pðr, oÞ of the beam at the point r, at frequency o, is given by the expression ([5], Eq. (14), p. 179) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u 4Det W ðr, oÞ Pðr, oÞ ¼ u ð2:2Þ t1 l 2 m2 Tr W ðr, oÞ where Det denotes the determinant and Tr the trace. The degree of polarization is bounded by zero and unity ð0 r P r 1Þ. When P ¼ 0 the light beam is said to be unpolarized at the point P(r) at frequency o and in the other extreme case, when P ¼ 1, it is said to be fully polarized at that point and at that frequency. Until fairly recently it was generally assumed that unpolarized light has the same correlation properties as natural light ([3], p. 624; [6], p. 350) and that polarized light has the same behavior as monochromatic light ([3], p. 624; [6], p. 351). If that were so, the correlation matrices of light of this kind would necessarily have the forms 2ðuÞ 1 0 ð2:3Þ W ðr, oÞ ¼ A 0 1
4716
J.-S. Horng et al. / Optics Communications 285 (2012) 4715–4718
Further, as in [2], we will model a light beam that is completely polarized as a beam of monochromatic light. Its cross-spectral density (CDS) matrix in any transverse cross-section of the beam has, therefore, the ‘‘factorized’’ form ([6], Sec. 6.3.2): " n # ex ðr1 , oÞex ðr2 , oÞ enx ðr1 , oÞey ðr2 , oÞ 2ðpÞ ð2:6Þ W ðr1 ,r2 , oÞ ¼ en ðr , oÞe ðr , oÞ en ðr , oÞe ðr , oÞ : x 2 y 2 y 1 y 1 This expression for completely polarized beam may be obtained from Eq. (2.4) by the use of the relationship Enk El ¼ enk el , ðk, l ¼ x, yÞ, where enk and el are components of the complex vectors Enk and El . By the use of Eq. (2.2), one can readily verify that the matrices (2.3) and (2.4) represent beams, at each point, are, indeed, completely unpolarized and completely polarized, respectively. We will refer to the beams represented by the matrices (2.3) and (2.4) as canonical models of unpolarized and polarized light beams, respectively.
3. Coherence properties of Stokes beams
Fig. 1. Notation relating to the arguments of the correlation matrix at points in the far field of Stokes beam.
and " /Enx ðr, oÞEx ðr, oÞS 2ðpÞ W ðr, oÞ ¼ /En ðr, oÞE ðr, oÞS x y
/Enx ðr, oÞEy ðr, oÞS /Ey ðr, oÞEy ðr, oÞS n
# ,
ð2:4Þ
where the superscripts (u) and (p) refer to unpolarized and to polarized light, respectively. In Eqs. (2.3) and (2.4), A is a constant, Ex(r,o) and Ey(r,o) are the Cartesian components of the electric field at the point P(r) and the angular brackets represent the average of the product of two components of the electrical field taken over the statistical ensemble. 2 One can readily confirm that when W matrix has the form of Eq. (2.3), the degree of polarization of the beam at the point P(r) has zero value and that when it has the form given by Eq. (2.4) it has the value unity. It is important to appreciate that Eqs. (2.3) and (2.4) contain information about correlations between transverse, mutually orthogonal components of the electrical field vector at a single point. However, to elucidate the propagation of the correlation matrices, it is necessary to know the values of correlations between the fluctuating field components at pairs of points ([5], Sec. 9.4). The problem of specifying two-point correlation matrices of polarized and unpolarized light has attracted some attention in recent years (see, for example, [7,8,9]) and it was found that there is no unique form of two-point cross-spectral density matrices of such beams. In view of this fact, we will proceed as follows. We will model an unpolarized beam as a beam of natural radiation (see, for example, [6], Sec. 6.3.1). The cross-spectral density of such radiation in any transverse cross-section zQconst of the beam perpendicular to the axis (the z-axis) has the form [10] 2ðuÞ 1 W ðr1 ,r2 , oÞ ¼ Aðq1 , q2 , oÞ 0
0 1
:
ð2:5Þ
where r1 ¼(q1,z), r2 ¼(q2,z), q1 and q2 being the transverse vectorcomponents of r1 and r2, respectively, perpendicular to the beam axis (the z-axis) and z labels the longitudinal components of these vectors (i.e. components along that axis).
We will now examine the spatial coherence properties of some Stokes beams. To do so we need an expression for the CSD matrix of a beam which is a superposition of two beams. Such an expression follows at once from a general formula derived in [11] and one finds that if the two beams propagate along the same direction, which we take to be the z-direction, and if they are independent of each other then the spectral degree of coherence of the beams resulting from their superposition is 2ab 2ba given by Eq. (9) of Ref. [11], with W ¼ W ¼ 0, viz., 2ðaÞ
2ðbÞ
Tr W ðr1 ,r2 , oÞ þ Tr W ðr1 ,r2 , oÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , Zðr1 ,r2 , oÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðaÞ S ðr1 , oÞ þ SðbÞ ðr1 , oÞ SðaÞ ðr2 , oÞ þ SðbÞ ðr2 , oÞ
ð3:1Þ
where 2ðpÞ SðpÞ ðrq , oÞ ¼ Tr W ðrq ,rq , oÞ,
ðp ¼ a, b and q ¼ 1,2Þ:
ð3:2Þ
As an example, consider a Stokes beam produced by a planar, circular, secondary source of radius a in the x,y-plane as shown 0 0 in Fig. 1. In this figure, Q1(q1) and Q2(q2) is a pair of source points and P 1 ðr1 ÞRP1 ðq1 ,zÞ and P2 ðr2 ÞRP2 ðq2 ,zÞ is a pair of field points. Let us choose the CSD matrix of unpolarized light in the source plane z ¼0 to be proportional, in suitable choice of units, to the unit matrix 2ðuÞ 1 0 0 0 , ð3:3Þ W ðq1 , q2 ; z ¼ 0; oÞ ¼ Aðq1 , q2 Þ 0 1 0
0
where A(q1,q2) describes the source in the xy-plane bounded by a circle of radius a, that can be expressed as 0 0 r r ð3:4Þ Aðq01 , q02 Þ ¼ circ 1 circ 2 a a where circ(x) is the circle function and r0 is the distance from the origin O to a variable point in the source plane. The propagation law of CDS matrix from the source plane into the half-space z40 has been investigated by Mandel and Wolf (see, e.g. Eqs. (5.6–14) and (5.16–17) in [6]) and they found that the CSD matrix at a pair of points located in any transverse plane zQcont40 can be expressed in terms of the 2 2 correlation matrix of electric field vector at pairs of points in the source plane
J.-S. Horng et al. / Optics Communications 285 (2012) 4715–4718
zQ0 in the form 2ðuÞ W ðq1 , q2 ,z; oÞ ¼ ZZ 2ðuÞ 0 0 2 0 2 0 0 0 W ðq1 , q2 ; z ¼ 0; oÞKðq1 q1 , q2 q2 ,z; oÞd q1 d q2 , r0 r a
ð3:5Þ where the propagation kernel can be expressed as (see Eq. (3) on p. 182 of [5]) Kðq1 q01 , q2 q02 ,z; oÞ ¼ Gn ðq1 q01 ,z; oÞGðq2 q02 ,z; oÞ,
ð3:6Þ
and G is the Green’s function for paraxial propagation from the 0 source point Q(q ,z ¼0) to the field point P(q,z): ! 2 9qq0 9 ik 0 exp ik ; ð3:7Þ Gðqq ,z; oÞ ¼ 2pz 2z (k¼ o/c, c being the speed of light in vacuum). In the cylindrical coordinates system shown in Fig. 1, the Green function in Eq. (3.7) can be expressed as ik r2 þ r02 2rr0 cosðycÞ : ð3:8Þ exp ik Gðqq0 ,z; oÞ ¼ 2pz 2z On substituting Eq. (3.8) into Eq. (3.7) and then into Eq. (3.6), for the propagation kernel takes the form Kðq1 q01 , q2 q02 ,z; oÞ 2 0 2 02 0 ½r2 þ r02 k 1 2r1 r1 cosðy1 c1 Þ½r2 þ r2 2r2 r2 cosðy2 c2 Þ ¼ exp ik 1 2pz 2z
ð3:9Þ 2
2
The elements d r and d r on the right-hand side of Eq. (3.5) are given by the expression 0 1
2
d r0q ¼ r0q dr0q dyq
0 2
ðq ¼ 1,2Þ:
ð3:10Þ
On substitution of Eqs. (3.9) and (3.10) into Eq. (3.5), one finds, after a straightforward but long calculation, expression for the CSD matrix of the un-polarized beam in any transverse crosssection zQconst in the form 2 2ðuÞ 1 1 0 Y g , ð3:11Þ W ðq1 , q2 ,z; oÞ ¼ 4 0 1 q¼1 q where ( 2
) Z Z 0 0 r þ r02 2 r1 ¼ a y1 ¼ 2p 1 2r1 r1 cosðy1 c1 Þ g1 ¼ exp ik 1 r01 dr01 dy1 2z lz r01 ¼ 0 y1 ¼ 0 " Z 0 # r1 ¼ a r r0 r2 r02 k ¼ expðik 1 Þ 2 J 0 k 1 1 exp ik 1 r01 dr01 ð3:12Þ z 2z z 2z r0 ¼ 0 1
and J0(x) is the zero-th order Bessel function of the first kind. It is useful to introduce the dimensionless parameters u¼
ka2 , z
s0q ¼
r
0 q
a
,
sq ¼
rq a
and
vq ¼ usq ,
ðq ¼ 1,2Þ: ð3:13Þ
# " Z s0 ¼ 1 1 i i g 1 ¼ g 1 ðu,v1 Þ ¼ u exp us21 2 J0 ðv1 s01 Þexp uðs01 Þ2 s01 ds01 : 2 2 s01 ¼ 0
ð3:14Þ Similarly, we may express g2 on the right-hand side of Eq. (3.11) as i g 2 ¼ g 2 ðu,v2 Þ ¼ u exp us22 2
" Z 0 s1 ¼ 1 s02 ¼ 0
# i uðs02 Þ2 s02 ds02 ¼ g n1 ðu,v2 Þ: J 0 ðv2 s02 Þexp 2
The integrals on the right-hand sides of Eqs. (3.14) and (3.15) can be evaluated in terms of the Lommel U- or V-functions [3,12], i.e. Z s¼1 i expðiu=2Þ 2 ½U 1 ðu,vÞ þ iU 2 ðu,vÞ J 0 ðvsÞ exp us2 s ds ¼ 2 u=2 s¼0 expðiu=2Þ i v2 i exp uþ þ iV 0 ðu,vÞ þV 1 ðu,vÞ : ¼ ð3:16Þ u=2 2 u where U1, U2, V0 and V1 are the Lommel U- and V-functions defined by the following series of Bessel functions [4]: 1 u n þ 2s X ð1Þs J n þ 2s ðvÞ, U n ðu,vÞ ¼ ð3:17aÞ v s¼0 and V n ðu,vÞ ¼
1 X
ð1Þs
s¼0
u n2s v
J n2s ðvÞ:
ð3:17bÞ
in which n is an integer. The Lommel U- and V-functions are convenient for computations when 9u=v9 o 1 and 9u/v941, respectively. Next we investigate the fully polarized beam, for which we 2ðpÞ choose the one whose CSD matrix W in the source plane z¼0 has the form 2ðpÞ 1 1 0 0 : ð3:18Þ W ðq1 , q2 ; z ¼ 0; oÞ ¼ Aðq1 , q2 Þ 1 1 Following the same mathematical procedure developed above we 2ðpÞ can express W in the half-space z 40 in the form 2 2ðpÞ 1 1 1 Y g , ð3:19Þ W ðq1 , q2 ,z; oÞ ¼ 4 1 1 q¼1 q Comparing Eqs. (3.11) and (3.19), one readily finds that 2ðuÞ 2ðpÞ Tr W ðq1 , q2 ,z; oÞ ¼ Tr W ðq1 , q2 ,z; oÞ :
ð3:20Þ
On substituting of Eqs. (3.14) and (3.15) into Eqs.(3.11) and (3.19) and then into Eq. (3.20), we find that 2 ðuÞ 2ðpÞ g ðu,v1 Þg n1 ðu,v2 Þ Tr W ðq1 , q2 ,z; oÞ ¼ Tr W ðq1 , q2 ,z; oÞ ¼ 1 : 2 ð3:21Þ The spectral density at the point P(r) can be obtained from the trace of the autocorrelation of the CSD matrix: 2 Sðr, oÞ ¼ Tr W ðr,r, oÞ: It follows that 2
SðpÞ ðqq , oÞ ¼ SðuÞ ðqq , oÞ ¼
9g 1 ðu,vq Þ9 ðq ¼ 1,2Þ : 2
ð3:22Þ
On substituting of Eqs. (3.21) and (3.22) into Eq. (3.1), we are able to write an expression of the spectral degree of coherence for the Stokes beam in the form 2ðuÞ
On substitution of Eq. (3.13) into Eq. (3.12), we obtain
2
4717
ð3:15Þ
Tr W ðq1 , q2 , oÞ g ðu,v1 Þ g n1 ðu,v2 Þ
: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 Zðq1 , q2 , oÞR qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g 1 ðu,v1 Þ g 1 ðu,v2 Þ SðuÞ ðq1 , oÞ SðuÞ ðq2 , oÞ ð3:23Þ It is interesting to note that the modulus of the spectral degree of coherence 9Z(q1,q2,o)9¼1 (see Fig. 2(a)), which implies that the Stokes beam is a spatially fully coherent electromagnetic beam if the CSD matrices of the two sources are expressed in the forms (3.3) and (3.19). It is also interesting to consider the phase distribution of Z(q1,q2,o) over the transverse plane z¼ z0. Assuming that the
4718
J.-S. Horng et al. / Optics Communications 285 (2012) 4715–4718
Fig. 2. Notation relating to the calculations of the modulus and phase distributions of the spectral degree of coherence of the Stokes beam from a circular source in the xOy-plane.
field point P1 is an off-axis point, the field point P2 is on the axis and the two points are separated by a distance d as shown in Fig. 2. The dimensionless parameters defined in Eq. (3.13) now take the following values: u0 ¼
ka2 , z0
v1 ¼ u
d a
and
v2 ¼ 0
ð3:24Þ
On substituting of Eq. (3.24) into Eq. (3.23), we find that the phase variation can be expressed in the form
d Y ¼ arg Zðq1 , q2 , oÞ ¼ arg g 1 ðu0 ,u0 Þg n1 ðu0 ,0Þ a n
d ð3:25Þ ¼ arg½g 1 ðu0 ,u0 Þ þ arg g 1 ðu0 ,0Þ , a where n h i 8 o 2 > arg exp i u0 1 þ d2 ½iV 0 þ V 1 u0 ,u0 d i when d o 1, < 2 a a a d h i n arg g 1 u0 ,u0 ¼ o 2 > a : arg exp 2i u0 1 þ da2 ½U 1 þ iU 2 u0 ,u0 da otherwise:
ð3:26Þ
Fig. 3. Modulus and phase of spectral degree of coherence of Stokes beam from a planar circular source of radius a in the xOy-plane. (a) Modulus of the beam
9Z(q1,q2,o)9, and (b) the phase variationY ¼ arg Zðq1 , q2 , oÞ .
Acknowledgements This research was supported by the US Air Force Office of Scientific Research, under grant No. FA 9550-08-1-0417.
References
and
1J 0 ðu daÞcos u2 : arg g n1 ðu,0Þ ¼ arctan J 0 ðu daÞsin 2u
ð3:27Þ
On substituting of Eqs. (3.26) and (3.27) into (3.25), we are able to plot the three curves in Fig. 3(b) to show the phase of degree of coherence of the Stokes beam as a function of d/a for the three different values of the parameter u0 ¼50, 100 and 200. The curves in Fig. 3(b) show the rapid variations of the phase with the increasing values of u0. In summary, we have derived expressions for the spectral degree of coherence of Stokes beam generated by superposition of a completely unpolarized beam and a completely polarized beam. Results are illustrated by case example regarding the Stokes beam generated by a circular planar source. We found that the coherence properties of such a Stokes beam are rotationally symmetric with respect to the axis of beam and can be expressed analytically in terms of the Lommel functions.
[1] G.C. Stokes, Transactions of the Cambridge Philosophical Society 9 (1852) 399, Reprinted in W. Swindell, Polarized Light, (P. Ross, Dowden, Hutchinson, 1975), pp. 124–141. [2] E. Wolf, Optics Letters 33 (2008) 642. [3] M. Born, E. Wolf, Principles of Optics, 7th ed., Cambridge University Press, 1999, Section 8.8. [4] This matrix is a ‘‘one-point’’ version of the well-known ‘‘two-point‘‘ crossspectral density matrix W(r1, r2; o), as defined, for example, in [5], Section 9.1, that plays a central role in optical coherence theory. [5] E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge University Press, 2007. [6] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995. [7] M. Lahiri, E. Wolf, Optics Communication 281 (2008) 3241. [8] F. Gori, J. Tervo, T. Turunen, Optics Letters 34 (2009) 1447. [9] M. Lahiri, E. Wolf, Optics Letters 34 (2009) 557. [10] It was shown by M. Lahiri, E. Wolf, Optics Communication 281 (324) (2008), that the cross-spectral density matrix of blackbody radiation has the form Eq. (5). [11] M. Lahiri, O. Korotkova, E. Wolf, Optics Communication 281 (2008) 5073. [12] Y. Li, E. Wolf, Journal of the Optical Society of America A, Optics and Image Science 8 (1984) 801.