May 16, 2007 - and College of Optics and Photonics: CREOL and FPCE, University of Central Florida, Orlando, Florida 32816, USA. Received 17 November ...
PHYSICAL REVIEW E 75, 056610 共2007兲
Polarization-induced spectral changes on propagation of stochastic electromagnetic beams Jixiong Pu Department of Electronic Science and Technology, Huaqiao University, Quanzhou, Fujian 362021, China
Olga Korotkova Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
Emil Wolf Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA; Institute of Optics, University of Rochester, Rochester, New York 14627, USA; and College of Optics and Photonics: CREOL and FPCE, University of Central Florida, Orlando, Florida 32816, USA 共Received 17 November 2006; published 16 May 2007兲 It was shown some years ago that the spectrum of a stochastic scalar field depends not only on the source spectrum but also on the degree of coherence of the source. In this paper we show that there are electromagnetic fields for which not only the state of coherence of the source, but also its degree of polarization affect the spectrum of the radiated field. We illustrate the analysis by diagrams which show the far-zone spectra of some stochastic electromagnetic beams generated by sources of different states of coherence and different degrees of polarization. The spectra of the radiated field depend both on coherence properties of the source and its degree of polarization and are found to be different in different directions of observation. DOI: 10.1103/PhysRevE.75.056610
PACS number共s兲: 41.20.Jb, 42.25.Ja, 32.70.Jz, 33.70.Jg
About twenty years ago it was found that spectra of fields generated by stochastic scalar sources generally change on propagation, even in free space 关1兴. Many publications on this subject have appeared since then 关2兴, but until very recently almost all investigations were based on scalar theory 共see, however, 关3兴, 关4兴兲. In recent years there has been an increasing interest in stochastic electromagnetic sources and in the beams which they generate. A unified theory of coherence and polarization formulated several years ago 关5兴 has provided a general description of such beams and also the laws governing their propagation 关6兴. More recently, stochastic beams generated by so-called electromagnetic Gaussian Schell-model sources have been studied 关7–9兴. Not long ago we have discussed far-zone spectra of beams generated by so-called quasihomogeneous stochastic electromagnetic sources with uniform polarization and we derived condition for the spectrum of the radiated field to be the same as the spectrum of the source 关4兴. In the present paper we show that there are stochastic electromagnetic sources whose degree of polarization also affects the spectra of the radiated fields. Let us consider a planar, stochastic, statistically stationary, secondary electromagnetic source, located in the plane z = 0, which radiates into the half space z ⬎ 0 共Fig. 1兲. We assume that the source occupies a finite domain D and that it generates a beam, propagating close to z direction. The crossspectral density matrix of the source at a pair of points, with position vectors 1 = 共x1 , y 1兲 and 2 = 共x2 , y 2兲 is defined as 关5兴
axis 共the z direction兲 at a typical point in the source plane and the asterisk denotes the complex conjugate. Ex and Ey are not Fourier components 共which do not exist for statistically stationary processes兲 of the electric field; they must be interpreted in the sense of coherence theory in the spacefrequency domain 关10兴. Each of the four elements of the cross-spectral density matrix can be expressed in the form W共0兲 ij 共1, 2, 兲
共0兲 共0兲 = 冑S共0兲 i 共1, 兲S j 共2, 兲ij 共1, 2, 兲,
i, j = x,y, 共2兲
共0兲 where S共0兲 x and S y denote the spectral densities 共electric energy densities兲 of the x and y components of the electric field, in the source plane z = 0 and 共0兲 ij denotes the degree of correlation between the two components. The spectral density S共0兲共 ; 兲, the spectral degree of polarization P共0兲共 ; 兲, and the spectral degree of coherence 共0兲共1 , 2 ; 兲 of the source are given by the formulas 关5兴
r2 0
r1 D
r2= r s 2 q2
q1
r1 = r s1
P2 P1 z
J W共0兲共1, 2, 兲 = 关W共0兲 ij 共1, 2, 兲兴 = 关具E*i 共1, 兲E j共2, 兲典兴,
i, j = x,y.
共1兲
Here Ex and Ey are two mutually orthogonal components at frequency of the electric field perpendicular to the beam 1539-3755/2007/75共5兲/056610共6兲
FIG. 1. 共Color online兲 Illustrating the notation relating to propagation of beam. 1 and 2 are position vectors of points in a the source plane and r1 = rs1 and r2 = rs2 共s21 = s22 = 1兲 are position vectors of points in the far zone.
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©2007 The American Physical Society
PHYSICAL REVIEW E 75, 056610 共2007兲
PU, KOROTKOVA, AND WOLF
J共0兲共, ; 兲兴, S共0兲共 ; 兲 = Tr关W 共0兲
P 共 ; 兲 =
共3兲
冑
J共0兲共, ; 兲兴 4 Det关W 1− , J共0兲共, ; 兲兴2 关Tr W
K共r, ; 兲 = − 共4兲
and
共0兲共1, 2 ; 兲 =
J共0兲共 , ; 兲兴 Tr关W 1 2
冑S
共0兲
共1 ; 兲冑S 共2 ; 兲 共0兲
,
共5兲
where Tr and Det denote the trace and the determinant of the matrix, respectively. For simplicity we will assume that the spectral densities 共0兲 S共0兲 x and S y are position independent, i.e., that 共0兲 S共0兲 i 共, 兲 ⬅ Si 共兲,
i = x,y.
where integration extends over the source domain D and
J共r,r; 兲兴. S共r; 兲 = Tr关W
The normalized source spectrum may be defined by the formula SN共0兲共兲 =
冕
S共0兲共兲 ⬁
共8兲
.
共0兲 S共r; 兲 = S共0兲 x 共兲M x共r; 兲 + S y 共兲M y 共r; 兲,
M i共r; 兲 =
i = x,y
may be called spectral modifiers of the spectral density components Sx共兲 and Sy共兲. On making use of Eq. 共9兲 it is seen that the formula 共13兲 for the spectral density may be expressed in terms of the degree of polarization P共0兲共兲 of the source as S共r; 兲 = S共0兲 x 共兲关M x共r, 兲 + ␣共兲M y 共r, 兲兴,
兩Sx共兲 − Sy共兲兩 . S x共 兲 + S y 共 兲
共9兲
* W共0兲 ij 共1, 2, 兲K 共r1, 1, 兲
⫻K共r2, 2, 兲d21d22,
共0兲 ii 共1, 2 ; 兲
共14兲
S 共兲d
Let us now consider the beam generated by the source. We will denote the position vectors of points in a cross section z = z0 = const⬎ 0 of the beam by r 共see Fig. 1兲. Expressions for the elements of the cross-spectral density matrix J W, for the spectral density S, for the degree of polarization P, for the degree of coherence , and for the normalized spectral density SN are given by Eqs. 共1兲, 共3兲–共5兲, and 共8兲 with 1, 2, and replaced by r1, r2, and r, respectively. We will now derive formulas which show how the normalized spectrum changes as the beam propagates. The elements of the cross-spectral density matrix of the electric field at a pair of points r1 and r2 in the half space z ⬎ 0 are given by the formulas 关6兴
冕冕
冕冕
⫻ K*共r, 1 ; 兲K共r, 2 ; 兲d21d22,
共0兲
We will only consider the class of sources for which the two mutually orthogonal components Ex and Ey of the electric vector are uncorrelated at each point of the source, i.e., 共0兲 xy 共 , , 兲 ⬅ 0 关11兴. In this case the cross-spectral density matrix is diagonal and the expression 共4兲 for the degree of polarization of the source reduces to
Wij共r1,r2, 兲 =
共13兲
where 共no summation over repeated indexes implied兲
0
P共0兲共, 兲 =
共12兲
On substituting from Eq. 共2兲 into Eqs. 共10兲 and 共12兲 and recalling Eq. 共6兲 we find that
共6兲
共7兲
共11兲
is the paraxial wave propagation, with k = / c. The spectral density S共r , 兲 of the field is given by the formula of the form of Eq. 共3兲, viz.,
Then Eqs. 共3兲–共5兲 simplify. The spectral density S共0兲 of the source becomes 共0兲 S共0兲共, 兲 ⬅ S共0兲共兲 = S共0兲 x 共兲 + S y 共兲.
ik ik共x2+y2兲/2z e 2z
i = x,y; j = x,y, 共10兲
共15兲
where
␣共兲 ⬅
S共0兲 y 共兲
=
S共0兲 x 共兲
1 − P共0兲共兲 , 1 + P共0兲共兲
共16兲
and we have assumed, without loss of generality, that 共0兲 S共0兲 x 共兲 ⱖ S y 共兲. The normalized spectrum of the electric field at the point r can then be determined on substituting from Eq. 共15兲 into the formula SN共r; 兲 =
冕
S共r; 兲 ⬁
共17兲
,
S共r; 兲d
0
and one finds that SN共r, 兲 =
冕
S共0兲 x 共兲关M x共r, 兲 + ␣共兲M y 共r, 兲兴 ⬁
S共0兲 x 共兲关M x共r, 兲
.
+ ␣共兲M y共r, 兲兴d
0
共18兲 We will now consider how the normalized spectrum SN changes when the beam propagates from the source plane to the far zone. Quantities pertaining to the far zone will be indicated by superscript 共⬁兲. The elements of the crossspectral density matrix of the electric field in the far zone, at points r1 = r1s1, r2 = r2s2 共s21 = s22 = 1兲 are given by the expressions 关Ref. 关3兴, Eq. 共3.1兲兴
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POLARIZATION-INDUCED SPECTRAL CHANGES ON… 1
(a)
(c) |η(|∆ ρ|,ω)|
S N (rs, w)
0.8 0.6 0.4 0.2 0 0
0.2
0.4
0.6
|∆ ρ| [mm]
w/w
0.8
1
1
(b)
(d) |η(|∆ ρ|,ω)|
S N (rs, w)
0.8 0.6 0.4 0.2 0 0
w/w
0.2
0.4
0.6
|∆ ρ| [mm]
0.8
1
FIG. 2. 共Color兲 共a兲, 共b兲 The normalized spectral density of the far field generated by planar unpolarized 关P共0兲共兲 = 0兴 electromagnetic Gaussian Schell-model source, calculated from Eq. 共26兲. The spectra are shown for different directions of observation: = 0°, red curve; = 0.3°, blue curve. The black curve shows the normalized spectral density of the source, calculated from Eq. 共25兲. The parameters of the ¯ / 冑2, Ax = 1, and Ay = 1, ␦yy = 0.05 mm; 共a兲 ␦xx = 0.5 mm, 共b兲 ␦xx = 0.1 mm. 共c兲, 共d兲 The degree of coherence source were taken as = 0.2 共0兲共1 , 2 , 兲 of the sources in 共a兲 and 共b兲, respectively, for symmetrically located points, 2 = −1 = , as function of their separation 兩⌬ 兩 = 2 兩 兩.
2 2 W共⬁兲 ij 共r1s1,r2s2, 兲 = 4 k
冕冕
2 2 2 M 共⬁兲 i 共rs, 兲 = 4 k cos
2
i = x,y; j = x,y,
冕冕
共0兲 ii 共1, 2, 兲
⫻ exp关− iks⬜ · 共2 − 1兲兴d21d22,
W共0兲 ij 共1, 2, 兲exp关− ik共s2⬜ · 2 − s1⬜ · 1兲兴
⫻d 1d 2, 2
exp关ik共r2 − r1兲兴 r 1r 2
i = x,y. 共20兲
共19兲
the integration extending over the source. The vectors s1⬜ and s2⬜ are the projections 共considered as two-dimensional vectors兲 of the unit vectors s1 and s2 onto the source plane z = 0 and 1 and 2 are the angles which s1 and s2 make with the z axis 共see Fig. 1兲. We assume that these angles are small, so that cos p ⬇ 1, 兩s p⬜兩2 ⬅ sin2 p ⬇ 2p 共p = 1 , 2兲. Since the field is assumed to be beamlike, the paraxial approximation in Eq. 共19兲 is implied. The spectral density S共⬁兲共rs ; 兲 of the far field along the direction specified by a unit vector s can then be obtained from Eq. 共15兲 with the spectral modifiers given by the formulas
The normalized spectral density of the far field can then be determined by use of Eqs. 共18兲 and 共20兲, viz., SN共⬁兲共rs, 兲 ⬅
冕
S共⬁兲共rs, 兲 ⬁
共21a兲
共⬁兲
S 共rs, 兲d
0
=
冕
共⬁兲 共⬁兲 S共0兲 x 共兲关M x 共rs, 兲 + ␣共兲M y 共rs, 兲兴 ⬁
共⬁兲 S共0兲 x 共兲关M x 共rs, 兲
+
,
␣共兲M 共⬁兲 y 共rs, 兲兴d
0
共21b兲 where ␣共兲 is given by Eq. 共16兲. In order to illustrate spectral changes of the radiated field as the beam propagates from the source into the far zone, let
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PU, KOROTKOVA, AND WOLF
1
(a)
(c) |η(|∆ ρ|,ω)|
S N (rs, w)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
|∆ ρ| [mm]
w/w
1
1
(b)
(d) |η(|∆ ρ|,ω)|
S N (rs, w)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
|∆ ρ| [mm]
w/w
1
FIG. 3. 共Color兲 The same as Fig. 2 but for a partially polarized source with degree of polarization P共0兲共兲 = 1 / 3. The parameters of the source were chosen to be Ax = 1 and Ay = 2.
us consider a planar, secondary electromagnetic Gaussian Schell-model source 关3,7,9兴, with uncorrelated field components Ex and Ey 关i.e. with 共0兲 ij 共1 , 2 ; 兲 = 0兴 in the chosen set of x, y axes. The elements of the cross-spectral density matrix of such a source have the form 共no summation over repeated indexes is implied兲
plicity we have assumed that the variances of the spectra of the x and the y components are the same. Suppose that the correlation coefficients 共0兲 ii also have Gaussian form with respect to their spatial arguments, i.e., that
冋
共0兲 ii 共2 − 1, 兲 = exp −
W共0兲 ij 共1, 2 ; 兲 共0兲 = S共0兲 i 共, 兲ii 共2 − 1, 兲,
= 0,
if i = j,
if i ⫽ j,
共22兲
共0兲 where S共0兲 x 共 , 兲 and S y 共 , 兲 are the spectra of the electric field components Ex and Ey, in the source plane, and 共0兲 ii 共2 − 1 , 兲 represents the degrees of correlation of the components Ei at points 1 and 2 in that plane. We further 共0兲 assume that the spectra S共0兲 x and S y are position independent and have Gaussian spectral profiles, i.e., that
S共0兲 i 共, 兲
=
A2i
冋
册
¯ 兲2 共 − exp − , 22
共23兲
and that the linear dimensions of the source are large compared to . The quantities Ai and 2 are assumed to be independent of position and of frequency. For the sake of sim-
共 2 − 1兲 2 2␦2ii
册
,
i = x,y,
共24兲
where the parameters ␦ii are assumed to be independent of position. On substituting from Eqs. 共22兲–共24兲 into Eq. 共8兲 one finds that the normalized source spectrum is given by the expression
SN共0兲共, 兲 =
1
冑 冋
exp −
¯ 兲2 共 − 22
册
2 , 1 + erf共 ¯ /冑2兲
共25兲
where erf denotes the error function. An expression for the normalized spectral density of the far field can be readily obtained by first substituting from Eq. 共24兲 for iiinto Eq. 共20兲 and then by substituting the resulting expression 共with r1s1 = r2s2 = rs, and also the right-hand side of Eq. 共23兲 into formula 共21b兲. One then finds, after long but straightforward calculations, that
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POLARIZATION-INDUCED SPECTRAL CHANGES ON… 1
(a)
(c) |η(|∆ ρ|,ω)|
S N (rs, w)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
|∆ ρ| [mm]
w/w
1
1
(b)
(d)
|η(|∆ ρ|,ω)|
SN (rs, w)
0.8
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
|∆ ρ| [mm]
w/w
1
FIG. 4. 共Color兲 The same as Fig. 2 but for a fully polarized source 关P共0兲共兲 = 1兴, with Ax = 1 and Ay = 0.
SN共⬁兲共rs, 兲 =
f共rs, 兲 , g共rs, 兲
冕
共26兲
⬁
冋
0
=
where
册再 冋 册 冋 册冎
冋
¯ 兲2 共 − f共rs, 兲 = A2x 2 exp − 22 + ␣共兲␦2yy exp −
g共rs兲 =
冕
⬁
K2y
册 冋 册 再 冉 冊冋冑
2 exp − 1 4c1c22
¯ 兲2 共 − 2 exp − d 22 K2i
exp −
¯ K2 4c3 4 + 2K2i c3冑2 22 i
¯ + 2c3冑 ¯ 2 + c3冑关44 + 4c12
冋 册册冎
2 2 ␦xx exp − 2 Kx
¯ 2兲兴erf + 2K2i 共2 +
2
,
共27兲
共28兲
冑
1 K2x
1 , 22
+
冋
0
c3 = exp and
K2i =
2c2
2␦2ii
,
i = x,y.
共29兲
The integrals entering in Eq. 共28兲 can be expressed in closed form. One finds with the help of MATHEMATICA that
c2i
,
i = x,y,
共30兲
where c1 =
f共rs, 兲d ,
¯ K2i c1
c2 = K2x + 22 , ¯2 K2x
2K2x 2 + 44
册
.
共31兲
We will now use formulas 共26兲–共28兲 to illustrate how the degree of polarization, and the degree of coherence of the source affect the spectrum of the far field in some typical cases. Figures 2–4 show the spectral densities of the far field generated by an unpolarized source 共P共0兲 = 0兲, by a partially polarized source, with P共0兲 = 1 / 3, and by a completely polarized 共P共0兲 = 1兲 planar, electromagnetic, Gaussian Schellmodel sources calculated by use of Eq. 共26兲. Curves illustrat-
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PU, KOROTKOVA, AND WOLF
ing the behavior of the degrees of coherence of the three sources are also shown in the figures. Although some of the spectral changes are rather small, they are well within the capability of experimental verification. We conclude by saying we have shown that the spectrum of the field in the far zone, radiated by some stochastic, statistically stationary planar electromagnetic sources may depend both on the degree of coherence and on the degree of
polarization of the source. We illustrated this phenomenon by several curves, which also indicate the changes in the spectra of the far field in different directions of observation.
关1兴 E. Wolf, Phys. Rev. Lett. 56, 1370 共1986兲. 关2兴 For review of many publications on this subject see E. Wolf and D. F. V. James, Rep. Prog. Phys. 59, 771 共1996兲. 关3兴 O. Korotkova, B. Hoover, V. Gamiz, and E. Wolf, J. Opt. Soc. Am. A 22共11兲, 2547 共2005兲. 关4兴 J. Pu, O. Korotkova, and E. Wolf, Opt. Lett. 31, 2097 共2006兲. 关5兴 E. Wolf, Phys. Lett. A 312, 263 共2003兲. 关6兴 E. Wolf, Opt. Lett. 28, 1078 共2003兲. 关7兴 F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, J. Opt. A, Pure Appl. Opt. 3, 1 共2001兲. 关8兴 O. Korotkova, M. Salem, and E. Wolf, Opt. Lett. 29, 1173 共2004兲. 关9兴 T. Shirai, O. Korotkova, and E. Wolf, J. Opt. A, Pure Appl. Opt. 7, 232 共2005兲.
关10兴 L. Mandel and E. Wolf, Optical Coherence and Quantum Optics 共Cambridge University Press, Cambridge, 1995兲, Sec. 4.7. 关11兴 The condition 共0兲 xy 共 , , 兲 ⬅ 0 depends, of course, on the choice of the x, y axes. When it is satisfied the axes are along J共0兲共 , , 兲 and the spectra the eigenvectors of the matrix W Sx共 , 兲 and Sy共 , 兲 are the eigenvalues of the matrix. Although the matrix can always be diagonalized by a unitary transformation, the transformation may not be a pure rotation. Physically this implies that, in general, one would need not only rotation but also a phase delay to obtain a transformed field for which the degree of polarization is given by the formula 共9兲. For a fuller discussion of this point, see A. AlQasimi, O. Korotkova, D. James, and E. Wolf, Opt. Lett. 32, 1015 共2007兲.
This research was supported by the U.S. Air Force Office of Scientific Research under Grant No. FA 9550-06-1-0032, by the Air Force Research Laboratory 共AFRC兲 under Contract No. FA 9451-04-C-0296, and by the National Natural Science Foundation of China under Grant No. 60477041.
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