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Radiation from arbitrarily polarized spatially incoherent planar sources
Optics Communications 221 (2003) 257–269 www.elsevier.com/locate/optcom
Radiation from arbitrarily polarized spatially incoherent planar sources Toni Saastamoinen, Jani Tervo *, Jari Turunen Department of Physics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland Received 7 October 2002; received in revised form 9 May 2003; accepted 9 May 2003
Abstract Using the exact electromagnetic theory of partially coherent and partially polarized light, we show by asymptotic techniques that the far-field radiation pattern produced by a spatially incoherent planar source depends significantly on the state and degree of polarization of the source. In general it is different from the cosine-squared law predicted by scalar theory in the non-paraxial domain. The coherence and polarization properties in the far field are examined for incoherent planar sources, which are either linearly, circularly, or azimuthally polarized, or unpolarized. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Polarization; Coherence; Diffraction
1. Introduction It is well known from scalar coherence theory that an incoherent source produces a rotationally symmetric far-field diffraction pattern with a cosine-squared angular distribution regardless of the intensity distribution of the source, which in turn determines the coherence properties of the radiation in the far field [1,2]. However, scalar coherence theory ignores the fact that the electromagnetic character of light can be of major concern whenever large propagation angles are involved, i.e., in the non-paraxial domain of optics. Incoherent sources are prime examples of non-paraxial radiators, delivering a significant proportion of radiant flux outside the paraxial region. Therefore, it is important to consider their radiation properties on the basis of exact three-dimensional models for wave propagation based on MaxwellÕs equations. Some investigations on electromagnetic radiation from incoherent sources are included in [3,4]. In this paper a detailed study is presented for electromagnetic radiation from incoherent planar sources with different polarization properties.
*
Corresponding author. Tel.: +358-13-251-3207; fax: +358-13-251-3290. E-mail address: Jani.Tervo@joensuu.fi (J. Tervo).
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01536-0
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The theory of partially coherent and partially polarized electromagnetic fields involves four coherence tensors, each containing nine correlation functions [2]. The propagation of such fields, which are the most general fields that physical optics can deal with, are governed by sets of either first-order or second-order differential equations that are coupled in a complicated manner [2]. However, field representations based on plane-wave decompositions of three correlation functions are sufficient for the determination of all the remaining correlation functions, since they are connected by MaxwellÕs equations [5]. The angular-spectrum approach together with asymptotic techniques is applied here to determine far-field expressions for radiation from incoherent but arbitrarily polarized (planar, secondary) sources of electromagnetic radiation. In view of scalar coherence theory, all such sources produce a cosine-squared distribution of radiant intensity. However, as we will show, considerations based on electromagnetic theory in general break both the rotational symmetry and the cosine-squared angular distribution of radiant intensity. 2. Far-field diffraction patterns of partially coherent sources By using WolfÕs theory of partially coherent fields in the space-frequency domain [6] we may express the correlations between the cartesian components of the electric field vector Eðr; xÞ and the magnetic field vector Hðr; xÞ by means of four cross-spectral density tensors [2]: h i ðeÞ WðeÞ ðr1 ; r2 ; xÞ ¼ Wij ðr1 ; r2 ; xÞ ¼ Ei ðr1 ; xÞEj ðr2 ; xÞ ; ð1Þ h i ðhÞ WðhÞ ðr1 ; r2 ; xÞ ¼ Wij ðr1 ; r2 ; xÞ ¼ Hi ðr1 ; xÞHj ðr2 ; xÞ ;
ð2Þ
h i ðmÞ WðmÞ ðr1 ; r2 ; xÞ ¼ Wij ðr1 ; r2 ; xÞ ¼ Ei ðr1 ; xÞHj ðr2 ; xÞ ;
ð3Þ
h i ðnÞ WðnÞ ðr1 ; r2 ; xÞ ¼ Wij ðr1 ; r2 ; xÞ ¼ Hi ðr1 ; xÞEj ðr2 ; xÞ ;
ð4Þ
where i; j denote cartesian coordinates x; y; z, the angle brackets represents ensemble averages, and the asterisk denotes complex conjugation. If we assume that no sources exist in the half-space z > 0, then each component of the cross-spectral density tensors (1)–(4) may be expressed in the form of angular spectrum representations of plane waves [3,5] Z Z Z Z 1 ðcÞ ðcÞ Wij ðr1 ; r2 ; xÞ ¼ Aij ðk?1 ; k?2 ; xÞ exp½iðu2 x2 u1 x1 þ v2 y2 v1 y1 Þ
1
exp½iðw2 z2 w1 z1 Þ du1 dv1 du2 dv2 ; where c ¼ e; h; m; n
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 u2b v2b wb ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i u2b þ v2b k 2
if u2b þ v2b < k 2 ; if u2b þ v2b > k 2 ;
ð5Þ
ð6Þ
where b ¼ 1; 2, k ¼ 2p=k is the wave number, and k is the wavelength. The wave number is the same for ðcÞ b ¼ 1 and b ¼ 2 because only a single frequency is considered. The angular correlation functions Aij are obtained by a Fourier-inversion at z1 ¼ z2 ¼ 0 Z Z Z Z 1 1 ðcÞ ðcÞ Aij ðk?1 ; k?2 ; xÞ ¼ Wij ðx1 ; y1 ; 0; x2 ; y2 ; 0; xÞ 4 ð2pÞ 1 exp½iðu2 x2 u1 x1 þ v2 y2 v1 y1 Þ dx1 dy1 dx2 dy2 :
ð7Þ
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259
Fig. 1. Geometry of radiation from an incoherent source into the far field and definition of the angles h and /.
In the theory of radiometry [2] the far-field radiation pattern of the field is of central interest. The expressions of the tensors (1)–(4) in the far-zone can be found by using the method of stationary phase [2,7]. Let us denote the unit position vector by ^s ¼ r=jrj ¼ ðsx ; sy ; sz Þ, see Fig. 1 for notation. Then the asymptotic forms of the cross-spectral density functions are [3–5,8] ðcÞ ðcÞ 2 Wij ðr1^s1 ; r2^s2 ; xÞ ¼ ð2pkÞ sz1 sz2 Aij ðks?1 ; ks?2 ; xÞF ðr1 ; r2 Þ;
ð8Þ
where c ¼ e; h; m; n, s?j ¼ ðsxj ; syj Þ, and F ðr1 ; r2 Þ ¼
exp½ikðr2 r1 Þ
: r1 r2
ð9Þ
Let us now restrict the consideration to planar (secondary) sources, whose intensity and polarization distributions are assumed to be known at the source plane z ¼ 0. Due to the assumed geometry, it is a natural choice to express the components of the cross-spectral density tensors in terms of the angular ðeÞ ðeÞ correlation functions AðeÞ xx , Axy , and Ayy with the help of MaxwellÕs equations. These expressions are given in [5] h i 2 ðeÞ WxzðeÞ ðr1^s1 ; r2^s2 Þ ¼ ð2pkÞ sz1 sx2 AðeÞ ð10Þ xx þ sy2 Axy F ðr1 ; r2 Þ; h i ðeÞ WyzðeÞ ðr1^s1 ; r2^s2 Þ ¼ ð2pkÞ2 sz1 sx2 AðeÞ A þ s y2 yy F ðr1 ; r2 Þ; yx
ð11Þ
n h i h io ðeÞ ðeÞ WzzðeÞ ðr1^s1 ; r2^s2 Þ ¼ ð2pkÞ2 sx1 sx2 AðeÞ þ sy1 sx2 AðeÞ F ðr1 ; r2 Þ; xx þ sy2 Axy yx þ sy2 Ayy
ð12Þ
where
ðeÞ Ajk
ðeÞ Ajk ðks?1 ; ks?2 ; xÞ.
ðeÞ Aij ðks?1 ; ks?2 ; xÞ
The remaining unknown cross-spectral density functions are obtained from h i ðeÞ ¼ Aji ðks?2 ; ks?1 ; xÞ ; ð13Þ
with i; j ¼ x; y, and
h i ðeÞ ðeÞ Wij ðr1^s1 ; r2^s2 ; xÞ ¼ Wji ðr2^s2 ; r1^s1 ; xÞ ;
ð14Þ
with i; j ¼ x; y; z. Here we have assumed that the angular correlation functions satisfy the nonnegative definiteness condition [2,9,10] Z Z X ðeÞ fi ðk?1 Þfj ðk?2 ÞAij ðks?1 ; ks?2 ; xÞd2 k?1 d2 k?2 P 0; ð15Þ i;j
where i; j ¼ x; y and the integration is performed over an arbitrary region. The functions fx and fy may be arbitrary as long as the integrals in Eq. (15) exist. We show in Appendix A that Eq. (15) is a sufficient
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condition for the corresponding nonnegative definiteness conditions to hold for the remaining components of the electric cross-spectral density matrix. The expressions for the magnetic and mixed cross-spectral density functions in the far-zone are presented in the appendix of [5]. In what follows, we consider the properties of the electric field only and hence the superscript ðeÞ is omitted. The asymptotic value of the time-averaged Poynting vector now takes the form [5] rffiffiffiffiffi 2 Z 1 1 0 2p ^s hSðr^s; tÞi ¼ x2 Gð^s; xÞ dx; ð16Þ 2 l0 rc 0 where
Gð^s; xÞ ¼ s2x þ s2z Axx ðks? ; ks? ; xÞ þ s2y þ s2z Ayy ðks? ; ks? ; xÞ þ 2sx sy R Axy ðks? ; ks? ; xÞ :
ð17Þ
If the source is quasimonochromatic, i.e., if the bandwidth Dx x, the integration in Eq. (16) may be omitted. The expression for the radiant intensity J ð^sÞ, which is the central (measurable) quantity in physical radiometry, is immediately obtained from the definition h D E i J ð^sÞ ¼ lim r2 Sðr^s; tÞ : ð18Þ r!1
3. Degree of polarization in the far zone In this section we recall the polarization and coherence properties of the field in the far-zone first examined by Carter and Wolf [4]. Due to the used geometry, it is convenient to express the field in spherical polar coordinates by using the relations Er ðr^s; xÞ ¼ Ex ðr^s; xÞ sin h cos / þ Ey ðr^s; xÞ sin h sin / þ Ez ðr^s; xÞ cos h;
ð19Þ
Eh ðr^s; xÞ ¼ Ex ðr^s; xÞ cos h cos / þ Ey ðr^s; xÞ cos h sin / Ez ðr^s; xÞ sin h;
ð20Þ
E/ ðr^s; xÞ ¼ Ex ðr^s; xÞ sin / þ Ey ðr^s; xÞ cos /:
ð21Þ
It can be easily verified [5], by inserting the far-field radiation patterns of a coherent field into Eq. (19), that the longitudinal component of the field Er vanishes [4,11]. Hence the field is locally two-dimensional and the nonvanishing cross-spectral density functions take the forms Whh ðr1^s1 ; r2^s2 ; xÞ ¼ ð2pkÞ2 ðAxx cos /1 cos /2 þ Axy cos /1 sin /2 þ Ayx sin /1 cos /2 þ Ayy sin /1 sin /2 Þ F ðr1 ; r2 Þ;
ð22Þ
2
Wh/ ðr1^s1 ; r2^s2 ; xÞ ¼ ð2pkÞ cos h2 ðAxx cos /1 sin /2 þ Axy cos /1 cos /2 Ayx sin /1 sin /2 þ Ayy sin /1 cos /2 ÞF ðr1 ; r2 Þ;
ð23Þ
2
W// ðr1^s1 ; r2^s2 ; xÞ ¼ ð2pkÞ cos h1 cos h2 ðAxx sin /1 sin /2 Axy sin /1 cos /2 Ayx cos /1 sin /2 þ Ayy cos /1 cos /2 ÞF ðr1 ; r2 Þ;
ð24Þ
where Ajk Ajk ðks?1 ; ks?2 ; xÞ and we have used Eqs. (8)–(14). The fourth nonvanishing cross-spectral density function W/h is obtained from Eq. (14).
T. Saastamoinen et al. / Optics Communications 221 (2003) 257–269
The two-dimensional coherence matrix in the far zone now takes the form [4] Whh ðr1^s1 ; r2^s2 ; xÞ Wh/ ðr1^s1 ; r2^s2 ; xÞ : Wðr1^s1 ; r2^s2 ; xÞ ¼ W/h ðr1^s1 ; r2^s2 ; xÞ W// ðr1^s1 ; r2^s2 ; xÞ By using Eq. (25) one may define the degree of polarization in the far-field as [4] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 det Wðr^s; r^s; xÞ P ð/; h; xÞ ¼ u t1 h i2 ; tr Wðr^s; r^s; xÞ
261
ð25Þ
ð26Þ
where det and tr stand for the determinant and the trace, respectively, of a matrix. It should be noted that although Eq. (26) was derived for the electric field, it gives the degree of polarization for the magnetic field as well. In the scalar domain of variable-coherence optics, one is interested in the degree of coherence of the field. Unfortunately, the concept of the degree of coherence of electromagnetic fields cannot be extended straightforwardly from the scalar definition. In the past, several authors have defined this quantity by the formula tr Wðr1 ; r2 ; xÞ lðr1 ; r2 ; xÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; tr Wðr1 ; r1 ; xÞ tr Wðr2 ; r2 ; xÞ
ð27Þ
or its space-time equivalent [4,12,13]. The main problem with Eq. (27) is that in certain cases it assumes values below unity even for fully coherent fields [14]. These fundamental problems may be avoided if one restricts to consider correlations between different scalar components of the field. This leads one to define a set of correlation functions Wij ðr1 ; r2 ; xÞ lij ðr1 ; r2 ; xÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Wii ðr1 ; r1 ; xÞWjj ðr2 ; r2 ; xÞ
ð28Þ
which define the degrees of correlation between the components i and j at points r1 and r2 [15]. Also these quantities depend strongly on the chosen coordinate system and hence they must be used with great care. In this article, we characterize the coherence properties by Eq. (28) for the correlations between the cartesian components of the field. These are obtained by inserting Eq. (8) into Eq. (28), which yields Aij ðks?1 ; ks?2 ; xÞ exp½ikðr2 r1 Þ
lij ðr1^s1 ; r2^s2 ; xÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Aii ðks?1 ; ks?1 ; xÞAjj ðks?2 ; ks?2 ; xÞ
ð29Þ
4. Radiation from arbitrarily polarized incoherent sources Let us now turn our attention to radiation patterns of incoherent quasimonochromatic planar sources. By definition, the fluctuations of the incoherent field at two arbitrary points in the source plane z ¼ 0 are completely uncorrelated. This means that the cross-spectral density functions Wxx , Wxy , and Wyy may be expressed in the form Wij ðx1 ; y1 ; 0; x2 ; y2 ; 0Þ ¼ Sij ðx1 ; y1 Þdðx1 x2 Þdðy1 y2 Þ;
ð30Þ
where Sxx and Syy are real functions that define the power spectrum of the source, d is the Dirac delta function, and we have dropped the explicit dependence on x. The function Sxy , which is generally complex, is bounded by the formula 2
jSxy ðx; yÞj 6 Sxx ðx; xÞSyy ðy; yÞ:
ð31Þ
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By inserting Eq. (30) into Eq. (7), we obtain Z Z 1 1 Sij ðx; yÞ exp½iðu1 u2 Þx exp½iðv1 v2 Þy dx dy: Aij ðk?1 ; k?2 Þ ¼ ð2pÞ4 1
ð32Þ
The radiant intensity is then obtained by inserting Eq. (32) into Eq. (16) and using Eq. (18). This yields rffiffiffiffiffi 0 2 2 J ð/; hÞ ¼ 2p k Gð/; hÞ; ð33Þ l0 where
Gð/; hÞ ¼ Ixx cos2 h þ sin2 h cos2 / þ Iyy cos2 h þ sin2 h sin2 / þ sin2 h sinð2/ÞRfIxy g; Iij ¼
Z Z
1 ð2pÞ
4
ð34Þ
1
Sij ðx; yÞ dx dy;
ð35Þ
1
and we have represented the vector ^s in the spherical polar coordinates, i.e., sx ¼ sin h cos /;
ð36Þ
sy ¼ sin h sin /;
ð37Þ
sz ¼ cos h:
ð38Þ
Because of MaxwellÕs divergence equation r E ¼ 0, the z-component of the electric field is determined by its x- and y-components. Hence it is convenient to characterize the local degree of polarization of the field at the source plane z ¼ 0 by a expression analogous to Eq. (26) h i 91=2 8 2 < 4 Sxx ðx; yÞSyy ðx; yÞ jSxy ðx; yÞj = P ðx; yÞ ¼ 1 : ð39Þ 2 : ; Sxx ðx; yÞ þ Syy ðx; yÞ
4.1. Linearly polarized sources For linearly polarized light, the field oscillates, for example, in the xz plane and we have Syy ¼ Sxy ¼ 0. The function Gð/; hÞ now takes the form ð40Þ Gð/; hÞ ¼ Ixx cos2 h þ sin2 h cos2 / ; and thus the radiant intensity takes the form J ð/; hÞ ¼ J ð0; 0Þ cos2 h þ sin2 h cos2 / ;
ð41Þ
which is not azimuthally symmetric. Eq. (41) is illustrated graphically in Fig. 2(a). This result, which was found earlier by Carter [3], differs substantially from the well-known azimuthally symmetric radiant intensity of an incoherent scalar source Js ð/; hÞ ¼ Js ð0; 0Þ cos2 h;
ð42Þ
which is illustrated in Fig. 2(b). It may be easily seen by examining Eq. (41) or Fig. 2(a), that the radiant intensity of the linearly polarized source does not vanish in the limit h ! p=2, except for the values / ¼ p=2. Similar results have been observed in, e.g, rigorous theory of diffraction by an aperture in a perfectly conducting screen [16].
T. Saastamoinen et al. / Optics Communications 221 (2003) 257–269
263
Fig. 2. The normalized radiant intensity distribution of (a) linearly polarized source with Iyy ¼ Ixy ¼ 0 (b) a scalar source (c) a circularly symmetric azimuthally or unpolarized source, and (d) a nonsymmetric azimuthally polarized source with Ixx ¼ 3Iyy and Ixy ¼ 0.
The degree of polarization in the far-field is obtained straightforwardly from Eq. (26) and is equal to unity. The result is clear, since the source is fully polarized with spatially invariant polarization state. The four nonvanishing degrees of correlation take the form lii ðr1^s1 ; r2^s2 Þ ¼ Axx ðks?1 ; ks?2 Þ exp½ikðr2 r1 Þ =Ixx ;
ð43Þ
lij ðr1^s1 ; r2^s2 Þ ¼ Axx ðks?1 ; ks?2 Þ exp½ikðr2 r1 Þ =Ixx ;
ð44Þ
and where i; j ¼ x; z. Hence the absolute value of the degree of correlation depends only on the correlation between the plane waves propagating in the directions defined by the unit vectors ^s1 and ^s2 . Hence the coherence properties of a linearly polarized field closely resemble the coherence properties of an incoherent scalar field. 4.2. Circularly polarized sources Although circular polarization in usually considered only in the case of plane waves, or paraxial beamlike fields, sources with Sxx ¼ Syy ¼ iSxy deserve special attention. Since the function Sxy is imaginary, it has no contribution to the radiant intensity, which takes the simple form 1 J ð/; hÞ ¼ Jð0; 0Þ 1 þ cos2 h ; 2
ð45Þ
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having an azimuthally symmetric distribution. The radiant intensity is illustrated in Fig. 2. As in the case of linearly polarized source, however, the result differs from the scalar case, since the radiant intensity in the limit h ! p=2 does not vanish. This property arises from the z-component of the field because the amplitudes of the plane-wave components with large propagation angles are mainly determined by it. The degree of polarization is, like in the case of linearly polarized fields, equal to unity. The degrees of correlation are best expressed in a compact matrix form 2 3 1 i expði/2 Þ lðr1^s1 ; r2^s2 Þ ¼ 4 i 1 i expði/2 Þ 5Axx ðks?1 ; ks?2 Þ exp½ikðr2 r1 Þ =Ixx : expði/1 Þ i expði/1 Þ exp½ið/2 /1 Þ
ð46Þ As in the previous case, the absolute values of the degrees of coherence depend only on the ratio Axx =Ixx . Hence the only fundamental difference between linearly and circularly polarized fields is the radiant intensity pattern. 4.3. Unpolarized sources The subject of unpolarized three-dimensional electromagnetic sources has been studied creditably by Carter [3], who pointed out, e.g., that if the currents flowing in orthogonal directions are uncorrelated, the results are identical with the scalar theory (the cosine-squared law). However, if we apply the formalism derived above for an incoherent source in the case that Sxx ¼ Syy and Sxy ¼ 0, i.e., the source is unpolarized in the sense of Eq. (39), we immediately obtain the result 1 ð47Þ J ð/; hÞ ¼ J ð0; 0Þ 1 þ cos2 h ; 2 which is identical to Eq. (45). Hence the radiant intensity does not take the form predicted by the scalar theory, e.g., Eq. (42). It is important to notice that the assumption that no sources exist in the positive z-direction immediately leads to the fact that the z-component of the field is strictly dependent on the x- and y-components of the field by MaxwellÕs divergence equation r E ¼ 0 and hence the three-dimensional degree of polarization of the source, in view of [15,17], is generally greater than zero. In addition, one should remember that the definition of unpolarized field is not unique in the case of planar sources and the result depend on the choice of the independent components of the field. The two-dimensional state of polarization in the far-field takes the simple form 1 cos2 h ; ð48Þ 1 þ cos2 h which is illustrated in Fig. 3(a). It is easily seen that the degree of polarization increases from zero to unity as h is increases from zero to p=2. This may be understood if we recall that the x- and y-components propagating in the same direction are completely uncorrelated, whereas the z-component is generally dependent on both of them. Near the z-axis, i.e., if h 0, the amplitudes of the plane waves are almost fully determined by the x- and y-components, which leads to an unpolarized field. On the other hand, as we discussed in the previous subsection, in the limit h ! p=2, the amplitudes are determined by the z-component only. The reason for the fact that the degree of polarization approaches to unity is easily understood when we examine the degrees of correlation 2 3 1 0 cos /2 lðr1^s1 ; r2^s2 Þ ¼ 4 0 1 sin /2 5Axx ðks?1 ; ks?2 Þ exp½ikðr2 r1 Þ =Ixx : ð49Þ cos /1 sin /1 cosð/2 /1 Þ P ð/; hÞ ¼
T. Saastamoinen et al. / Optics Communications 221 (2003) 257–269
265
Fig. 3. Degree of polarization in the far-field of (a) a symmetric azimuthally polarized or an unpolarized source, and (b) a nonsymmetric azimuthally polarized source Ixx ¼ 3Iyy with Ixy ¼ 0.
Now the x- and z-components are completely uncorrelated at /i ¼ p=2, i ¼ 1; 2, whereas the y- and zcomponents do not correlate if /i ¼ 0 or p. This means that the field must be fully polarized when no x- or y-components are present, i.e., in the limit h ! p=2. This property clearly has its roots in the fact that, for example, the z-component of a plane wave with its wave vector perpendicular to the y-axis is determined from the x-component only. This naturally means that the z-component must be completely uncorrelated at /2 /1 ¼ p=2, which is evident also from Eq. (49). 4.4. Azimuthally polarized sources Let us next consider a source which produces an azimuthally polarized field, i.e., the electric field vector in the source plane is parallel to the azimuthal coordinate / [18,19]. We assume the cross-spectral density to be of the form W// ðr1 ; r2 Þ ¼ Sðq1 ; /1 Þdðq1 q2 Þdð/1 /2 Þ; where we have used the relation
x ¼ q cos /; y ¼ q sin /;
ð50Þ
ð51Þ
and Sð0; /Þ ¼ 0. The expressions of the cross-spectral density functions Wxx , Wxy , and Wyy are obtained by using the relations
Ex ðrÞ ¼ Eq ðrÞ cos / E/ ðrÞ sin /; ð52Þ Ey ðrÞ ¼ Eq ðrÞ sin / þ E/ ðrÞ cos /; where Eq and E/ denote the radial and azimuthal components of the field, respectively. Since in the azimuthally polarized case Eq ¼ 0, we obtain Wxx ðr1 ; r2 Þ ¼ W// ðr1 ; r2 Þ sin /1 sin /2 ;
ð53Þ
Wxy ðr1 ; r2 Þ ¼ W// ðr1 ; r2 Þ sin /1 cos /2 ;
ð54Þ
Wyy ðr1 ; r2 Þ ¼ W// ðr1 ; r2 Þ cos /1 cos /2 :
ð55Þ
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By converting also the wave vector into circular cylindrical coordinates, i.e.,
u ¼ a cos w; v ¼ a sin w;
ð56Þ
and inserting Eqs. (50) and (53)–(55) into Eq. (7), we obtain the expressions for the functions Ixx , Ixy , and Iyy Z 1 Z 2p 1 Ijk ¼ Sjk ðq; /Þ q dq d/; ð57Þ 4 ð2pÞ 0 0 where Sxx ðq; /Þ ¼ sin2 /Sðq; /Þ;
ð58Þ
Sxy ðq; /Þ ¼ sin / cos /Sðq; /Þ;
ð59Þ
Syy ðq; /Þ ¼ cos2 /Sðq; /Þ:
ð60Þ
If the function Sðq; /Þ is azimuthally symmetric, i.e., if it is a function of q only, we obtain Ixx ¼ Iyy and Ixy ¼ 0, which leads to a radiant intensity of the form (47). Interestingly, the degree of polarization takes the form identical to Eq. (48), which means that the far-field is unpolarized when h ¼ 0. Hence the axial intensity distribution consists of radially and azimuthally polarized parts, whose amplitudes are equal. This can be easily verified also by using the expressions of Wxx and Wyy and the definition (52). It is evident that, since the field is azimuthally polarized in the source plane, the radial component is generated upon propagation. A similar phenomenon may be found in the domain of coherent optics, where the permitted forms of azimuthally polarized fields are very limited [20,21]. If a field is not of that form, but is azimuthally polarized in a certain plane, the radial component is generated upon propagation [22]. The phenomenon that the degree of polarization in the far-field of an incoherent, but fully polarized, source can be equal to zero was discovered recently by Gori et al. [23], who studied the effect of a polarization-modulating element on an incoherent and unpolarized beam. Also in that case the mean values of Sxx and Syy are equal, whereas the mean value of Sxy is equal to zero. The reason for the phenomenon that radically different local polarization states of the source can lead to essentially similar radiation patterns is that only the ÔglobalÕ degree of polarization of the source, defined by the constants Ixx , Ixy , and Iyy , affect the radiant intensity. Hence there is no practical difference between, e.g., an unpolarized source for which Sxy 0, and an azimuthally polarized source for which Sxy generally does not vanish but Ixy ¼ 0. If the intensity distribution of the source is not symmetric but depends also on the /-coordinate, different radiant intensity and polarization patterns are obtained. If we choose, for example, Sðq; /Þ ¼ SðqÞ cos2 /, we obtain Ixx ¼ 3Iyy , while Ixy still vanishes. This leads to the radiant intensity 1 J ð/; hÞ ¼ J ð0; 0Þ 1 þ 3 cos2 h þ 2 sin2 h cos2 / ; 4
ð61Þ
which is illustrated in Fig. 2(d). The radiant intensity now resembles the radiant intensity of a linearly polarized field which is clear, since the x-component of the field is dominant. The degree of polarization now takes the form n 2 o1=2 P ð/; hÞ ¼ 1 12 cos2 h 1 þ 2 cos2 / þ cos2 h 1 þ 2 sin2 / ;
ð62Þ
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267
illustrated in Fig. 3(b). Now the axial degree on polarization P ð0; 0Þ ¼ 1=2 and the minima, where the pffiffiffi degree of polarization is equal to zero, are found at the points ð/; hÞ ¼ ½p=2; cos1 ð1= 3Þ . Different forms of radiant intensity and polarization patterns can be easily found by varying the azimuthal dependence of the function Sðq; /Þ.
5. Conclusions We have examined radiation from incoherent planar sources by using rigorous electromagnetic theory of partial coherence. The results, which were expressed in terms of the x and y components of the electric field, clearly demonstrate that the scalar theory is not generally applicable to electromagnetic fields in the nonparaxial domain (which is not a surprise). We have shown explicitly that different states of polarization of an incoherent source can lead to similar radiant-intensity patterns and/or degrees of polarization in the farfield. We have also shown, e.g., that an azimuthally symmetric, azimuthally polarized source behaves essentially similarly to a source that is unpolarized in the sense that its electric x and y components are uncorrelated in the source plane.
Acknowledgements The work of J. Tervo is supported by the Academy of Finland and the Nokia Foundation.
Appendix A In this Appendix we show that Eq. (15) holds for i; j ¼ x; y; z if it holds for i; j ¼ x; y. Let us recall the expressions for the angular correlation functions Axz , Ayz , Azx , Azy , and Azz from [5]: AðeÞ xz ðk?1 ; k?2 ; xÞ ¼
i 1 h ðeÞ u2 Axx ðk?1 ; k?2 ; xÞ þ v2 AðeÞ ðk ; k ; xÞ ; ?1 ?2 xy w2
ðA:1Þ
AðeÞ yz ðk?1 ; k?2 ; xÞ ¼
i 1 h ðeÞ u2 Ayx ðk?1 ; k?2 ; xÞ þ v2 AðeÞ yy ðk?1 ; k?2 ; xÞ ; w2
ðA:2Þ
AðeÞ zx ðk?1 ; k?2 ; xÞ ¼
i 1 h ðeÞ ðeÞ u A ðk ; k ; xÞ þ v A ðk ; k ; xÞ ; 1 xx ?1 ?2 1 yx ?1 ?2 w1
ðA:3Þ
AðeÞ zy ðk?1 ; k?2 ; xÞ ¼
i 1 h ðeÞ ðeÞ u A ðk ; k ; xÞ þ v A ðk ; k ; xÞ ; 1 ?1 ?2 1 ?1 ?2 xy yy w1
ðA:4Þ
AðeÞ zz ðk?1 ; k?2 ; xÞ ¼
1 h ðeÞ ðeÞ u1 u2 AðeÞ xx ðk?1 ; k?2 ; xÞ þ u1 v2 Axy ðk?1 ; k?2 ; xÞ þ v1 u2 Ayx ðk?1 ; k?2 ; xÞ w1 w2 i þ v1 v2 AðeÞ yy ðk?1 ; k?2 ; xÞ :
ðA:5Þ
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By inserting Eqs. (A.1)–(A.5) into Eq. (15) with i; j ¼ x; y; z, we obtain Z Z u1 u2 fx ðk?1 Þ fz ðk?1 Þ fx ðk?2 Þ fz ðk?2 Þ AðeÞ xx ðks?1 ; ks?2 ; xÞ w1 w2 u1 v2 þ fx ðk?1 Þ fz ðk?1 Þ fy ðk?2 Þ fz ðk?2 Þ AðeÞ xy ðks?1 ; ks?2 ; xÞ w1 w2 v1 u2 þ fy ðk?1 Þ fz ðk?1 Þ fx ðk?2 Þ fz ðk?2 Þ AðeÞ yx ðks?1 ; ks?2 ; xÞ w1 w2 ) v1 v2 þ fy ðk?1 Þ fz ðk?1 Þ fy ðk?2 Þ fz ðk?2 Þ AðeÞ ðks ; ks ; xÞ d2 k?1 d2 k?2 P 0: ?1 ?2 yy w1 w2 If we now introduce the functions u gx ðk? Þ ¼ fx ðk? Þ fz ðk? Þ; w and v gy ðk? Þ ¼ fy ðk? Þ fz ðk? Þ; w Eq. (A.6) takes the form Z Z h ðeÞ gx ðk?1 Þgx ðk?2 ÞAðeÞ xx ðk?1 ; k?2 ; xÞ þ gx ðk?1 Þgy ðk?2 ÞAxy ðk?1 ; k?2 ; xÞ i ðeÞ þ gy ðk?1 Þgx ðk?2 ÞAðeÞ ðk ; k ; xÞ þ g ðk Þg ðk ÞA ðk ; k ; xÞ d2 k?1 d2 k?2 P 0; ?1 ?2 ?1 y ?2 ?1 ?2 yx y yy
ðA:6Þ
ðA:7Þ
ðA:8Þ
ðA:9Þ
which clearly holds because we assumed that Eq. (15) is valid for i; j ¼ x; y and thus we have obtained the desired result. It should be noticed that the integral in Eq. (A.9) exists in all cases in which the z-component of the field may be determined from the x- and y-components by using the method introduced by Carter [24]. It can be easily shown, with essentially similar steps as we have taken above, that Eq. (15) for i; j ¼ x; y is a sufficient condition also for the equivalent nonnegative definiteness condition including the angular correlation functions of the magnetic and mixed cross-spectral density functions. It is also easily shown by using the result derived in this Appendix that the nonnegative definiteness condition for the electric crossspectral density matrix holds if the corresponding definiteness condition holds for the elements formed by the x- and y-components of the field, of course assuming that no sources exist in the half-space z > 0.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
E.W. Marchand, E. Wolf, J. Opt. Soc. Am. 62 (1972) 379. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995. W.H. Carter, J. Opt. Soc. Am. 9 (1980) 1067. W.H. Carter, E. Wolf, Phys. Rev. A 36 (1987) 1258. J. Tervo, J. Turunen, Opt. Commun. 209 (2002) 7. E. Wolf, J. Opt. Soc. Am. 72 (1982) 343. J.J. Stamnes, Waves in Focal Regions, Adam Hilger, Bristol, 1986. € stlund, A.T. Friberg, Opt. Commun. 197 (2001) 1. P. O J. Perina, Coherence of Light, Reidel, Dordrecht, 1985. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, Pure Appl. Opt. 7 (1998) 941. D.F.V. James, Opt. Commun. 109 (1994) 209. B. Karczewski, Phys. Lett. 5 (1963) 191.
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269
[13] B. Karczewski, Nuovo Cimento 30 (1963) 906. ^ exp½iðkz xtÞ þ ^y exp½iðkx xtÞ and by choosing [14] Such situation is obtained, for example, by using a coherent field with E ¼ x r1 ¼ ð0; 0; 0Þ and r2 ¼ ðp=k; 0; 0Þ. [15] T. Set€ al€ a, A. Shevchenko, M. Kaivola, A.T. Friberg, Phys. Rev. E 66 (2002) 016615. [16] J. Huttunen, A.T. Friberg, J. Turunen, Phys. Rev. E 5 (1995) 3081. [17] T. Set€ al€ a, M. Kaivola, A.T. Friberg, Phys. Rev. Lett. 88 (2002) 123902. [18] R.H. Jordan, D.G. Hall, Opt. Lett. 19 (1994) 427. [19] P.L. Greene, D.G. Hall, J. Opt. Soc. Am. A 13 (1996) 962. [20] J. Tervo, P. Vahimaa, J. Turunen, J. Mod. Opt. 49 (2002) 1537. [21] P. P€ a€ akk€ onen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, Opt. Express 10 (2002) 949. [22] A. Lapucci, M. Ciofini, Opt. Express 9 (2001) 603. [23] F. Gori, M. Santarsiero, R. Borghi, G. Piquero, Opt. Lett. 25 (2000) 1291. [24] W.H. Carter, J. Opt. Soc. Am. 62 (1972) 1195.
Radiation from arbitrarily polarized spatially incoherent planar sources Toni Saastamoinen, Jani Tervo *, Jari Turunen Department of Physics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland Received 7 October 2002; received in revised form 9 May 2003; accepted 9 May 2003
Abstract Using the exact electromagnetic theory of partially coherent and partially polarized light, we show by asymptotic techniques that the far-field radiation pattern produced by a spatially incoherent planar source depends significantly on the state and degree of polarization of the source. In general it is different from the cosine-squared law predicted by scalar theory in the non-paraxial domain. The coherence and polarization properties in the far field are examined for incoherent planar sources, which are either linearly, circularly, or azimuthally polarized, or unpolarized. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Polarization; Coherence; Diffraction
1. Introduction It is well known from scalar coherence theory that an incoherent source produces a rotationally symmetric far-field diffraction pattern with a cosine-squared angular distribution regardless of the intensity distribution of the source, which in turn determines the coherence properties of the radiation in the far field [1,2]. However, scalar coherence theory ignores the fact that the electromagnetic character of light can be of major concern whenever large propagation angles are involved, i.e., in the non-paraxial domain of optics. Incoherent sources are prime examples of non-paraxial radiators, delivering a significant proportion of radiant flux outside the paraxial region. Therefore, it is important to consider their radiation properties on the basis of exact three-dimensional models for wave propagation based on MaxwellÕs equations. Some investigations on electromagnetic radiation from incoherent sources are included in [3,4]. In this paper a detailed study is presented for electromagnetic radiation from incoherent planar sources with different polarization properties.
*
Corresponding author. Tel.: +358-13-251-3207; fax: +358-13-251-3290. E-mail address: Jani.Tervo@joensuu.fi (J. Tervo).
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01536-0
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The theory of partially coherent and partially polarized electromagnetic fields involves four coherence tensors, each containing nine correlation functions [2]. The propagation of such fields, which are the most general fields that physical optics can deal with, are governed by sets of either first-order or second-order differential equations that are coupled in a complicated manner [2]. However, field representations based on plane-wave decompositions of three correlation functions are sufficient for the determination of all the remaining correlation functions, since they are connected by MaxwellÕs equations [5]. The angular-spectrum approach together with asymptotic techniques is applied here to determine far-field expressions for radiation from incoherent but arbitrarily polarized (planar, secondary) sources of electromagnetic radiation. In view of scalar coherence theory, all such sources produce a cosine-squared distribution of radiant intensity. However, as we will show, considerations based on electromagnetic theory in general break both the rotational symmetry and the cosine-squared angular distribution of radiant intensity. 2. Far-field diffraction patterns of partially coherent sources By using WolfÕs theory of partially coherent fields in the space-frequency domain [6] we may express the correlations between the cartesian components of the electric field vector Eðr; xÞ and the magnetic field vector Hðr; xÞ by means of four cross-spectral density tensors [2]: h i ðeÞ WðeÞ ðr1 ; r2 ; xÞ ¼ Wij ðr1 ; r2 ; xÞ ¼ Ei ðr1 ; xÞEj ðr2 ; xÞ ; ð1Þ h i ðhÞ WðhÞ ðr1 ; r2 ; xÞ ¼ Wij ðr1 ; r2 ; xÞ ¼ Hi ðr1 ; xÞHj ðr2 ; xÞ ;
ð2Þ
h i ðmÞ WðmÞ ðr1 ; r2 ; xÞ ¼ Wij ðr1 ; r2 ; xÞ ¼ Ei ðr1 ; xÞHj ðr2 ; xÞ ;
ð3Þ
h i ðnÞ WðnÞ ðr1 ; r2 ; xÞ ¼ Wij ðr1 ; r2 ; xÞ ¼ Hi ðr1 ; xÞEj ðr2 ; xÞ ;
ð4Þ
where i; j denote cartesian coordinates x; y; z, the angle brackets represents ensemble averages, and the asterisk denotes complex conjugation. If we assume that no sources exist in the half-space z > 0, then each component of the cross-spectral density tensors (1)–(4) may be expressed in the form of angular spectrum representations of plane waves [3,5] Z Z Z Z 1 ðcÞ ðcÞ Wij ðr1 ; r2 ; xÞ ¼ Aij ðk?1 ; k?2 ; xÞ exp½iðu2 x2 u1 x1 þ v2 y2 v1 y1 Þ
1
exp½iðw2 z2 w1 z1 Þ du1 dv1 du2 dv2 ; where c ¼ e; h; m; n
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 u2b v2b wb ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i u2b þ v2b k 2
if u2b þ v2b < k 2 ; if u2b þ v2b > k 2 ;
ð5Þ
ð6Þ
where b ¼ 1; 2, k ¼ 2p=k is the wave number, and k is the wavelength. The wave number is the same for ðcÞ b ¼ 1 and b ¼ 2 because only a single frequency is considered. The angular correlation functions Aij are obtained by a Fourier-inversion at z1 ¼ z2 ¼ 0 Z Z Z Z 1 1 ðcÞ ðcÞ Aij ðk?1 ; k?2 ; xÞ ¼ Wij ðx1 ; y1 ; 0; x2 ; y2 ; 0; xÞ 4 ð2pÞ 1 exp½iðu2 x2 u1 x1 þ v2 y2 v1 y1 Þ dx1 dy1 dx2 dy2 :
ð7Þ
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259
Fig. 1. Geometry of radiation from an incoherent source into the far field and definition of the angles h and /.
In the theory of radiometry [2] the far-field radiation pattern of the field is of central interest. The expressions of the tensors (1)–(4) in the far-zone can be found by using the method of stationary phase [2,7]. Let us denote the unit position vector by ^s ¼ r=jrj ¼ ðsx ; sy ; sz Þ, see Fig. 1 for notation. Then the asymptotic forms of the cross-spectral density functions are [3–5,8] ðcÞ ðcÞ 2 Wij ðr1^s1 ; r2^s2 ; xÞ ¼ ð2pkÞ sz1 sz2 Aij ðks?1 ; ks?2 ; xÞF ðr1 ; r2 Þ;
ð8Þ
where c ¼ e; h; m; n, s?j ¼ ðsxj ; syj Þ, and F ðr1 ; r2 Þ ¼
exp½ikðr2 r1 Þ
: r1 r2
ð9Þ
Let us now restrict the consideration to planar (secondary) sources, whose intensity and polarization distributions are assumed to be known at the source plane z ¼ 0. Due to the assumed geometry, it is a natural choice to express the components of the cross-spectral density tensors in terms of the angular ðeÞ ðeÞ correlation functions AðeÞ xx , Axy , and Ayy with the help of MaxwellÕs equations. These expressions are given in [5] h i 2 ðeÞ WxzðeÞ ðr1^s1 ; r2^s2 Þ ¼ ð2pkÞ sz1 sx2 AðeÞ ð10Þ xx þ sy2 Axy F ðr1 ; r2 Þ; h i ðeÞ WyzðeÞ ðr1^s1 ; r2^s2 Þ ¼ ð2pkÞ2 sz1 sx2 AðeÞ A þ s y2 yy F ðr1 ; r2 Þ; yx
ð11Þ
n h i h io ðeÞ ðeÞ WzzðeÞ ðr1^s1 ; r2^s2 Þ ¼ ð2pkÞ2 sx1 sx2 AðeÞ þ sy1 sx2 AðeÞ F ðr1 ; r2 Þ; xx þ sy2 Axy yx þ sy2 Ayy
ð12Þ
where
ðeÞ Ajk
ðeÞ Ajk ðks?1 ; ks?2 ; xÞ.
ðeÞ Aij ðks?1 ; ks?2 ; xÞ
The remaining unknown cross-spectral density functions are obtained from h i ðeÞ ¼ Aji ðks?2 ; ks?1 ; xÞ ; ð13Þ
with i; j ¼ x; y, and
h i ðeÞ ðeÞ Wij ðr1^s1 ; r2^s2 ; xÞ ¼ Wji ðr2^s2 ; r1^s1 ; xÞ ;
ð14Þ
with i; j ¼ x; y; z. Here we have assumed that the angular correlation functions satisfy the nonnegative definiteness condition [2,9,10] Z Z X ðeÞ fi ðk?1 Þfj ðk?2 ÞAij ðks?1 ; ks?2 ; xÞd2 k?1 d2 k?2 P 0; ð15Þ i;j
where i; j ¼ x; y and the integration is performed over an arbitrary region. The functions fx and fy may be arbitrary as long as the integrals in Eq. (15) exist. We show in Appendix A that Eq. (15) is a sufficient
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condition for the corresponding nonnegative definiteness conditions to hold for the remaining components of the electric cross-spectral density matrix. The expressions for the magnetic and mixed cross-spectral density functions in the far-zone are presented in the appendix of [5]. In what follows, we consider the properties of the electric field only and hence the superscript ðeÞ is omitted. The asymptotic value of the time-averaged Poynting vector now takes the form [5] rffiffiffiffiffi 2 Z 1 1 0 2p ^s hSðr^s; tÞi ¼ x2 Gð^s; xÞ dx; ð16Þ 2 l0 rc 0 where
Gð^s; xÞ ¼ s2x þ s2z Axx ðks? ; ks? ; xÞ þ s2y þ s2z Ayy ðks? ; ks? ; xÞ þ 2sx sy R Axy ðks? ; ks? ; xÞ :
ð17Þ
If the source is quasimonochromatic, i.e., if the bandwidth Dx x, the integration in Eq. (16) may be omitted. The expression for the radiant intensity J ð^sÞ, which is the central (measurable) quantity in physical radiometry, is immediately obtained from the definition h D E i J ð^sÞ ¼ lim r2 Sðr^s; tÞ : ð18Þ r!1
3. Degree of polarization in the far zone In this section we recall the polarization and coherence properties of the field in the far-zone first examined by Carter and Wolf [4]. Due to the used geometry, it is convenient to express the field in spherical polar coordinates by using the relations Er ðr^s; xÞ ¼ Ex ðr^s; xÞ sin h cos / þ Ey ðr^s; xÞ sin h sin / þ Ez ðr^s; xÞ cos h;
ð19Þ
Eh ðr^s; xÞ ¼ Ex ðr^s; xÞ cos h cos / þ Ey ðr^s; xÞ cos h sin / Ez ðr^s; xÞ sin h;
ð20Þ
E/ ðr^s; xÞ ¼ Ex ðr^s; xÞ sin / þ Ey ðr^s; xÞ cos /:
ð21Þ
It can be easily verified [5], by inserting the far-field radiation patterns of a coherent field into Eq. (19), that the longitudinal component of the field Er vanishes [4,11]. Hence the field is locally two-dimensional and the nonvanishing cross-spectral density functions take the forms Whh ðr1^s1 ; r2^s2 ; xÞ ¼ ð2pkÞ2 ðAxx cos /1 cos /2 þ Axy cos /1 sin /2 þ Ayx sin /1 cos /2 þ Ayy sin /1 sin /2 Þ F ðr1 ; r2 Þ;
ð22Þ
2
Wh/ ðr1^s1 ; r2^s2 ; xÞ ¼ ð2pkÞ cos h2 ðAxx cos /1 sin /2 þ Axy cos /1 cos /2 Ayx sin /1 sin /2 þ Ayy sin /1 cos /2 ÞF ðr1 ; r2 Þ;
ð23Þ
2
W// ðr1^s1 ; r2^s2 ; xÞ ¼ ð2pkÞ cos h1 cos h2 ðAxx sin /1 sin /2 Axy sin /1 cos /2 Ayx cos /1 sin /2 þ Ayy cos /1 cos /2 ÞF ðr1 ; r2 Þ;
ð24Þ
where Ajk Ajk ðks?1 ; ks?2 ; xÞ and we have used Eqs. (8)–(14). The fourth nonvanishing cross-spectral density function W/h is obtained from Eq. (14).
T. Saastamoinen et al. / Optics Communications 221 (2003) 257–269
The two-dimensional coherence matrix in the far zone now takes the form [4] Whh ðr1^s1 ; r2^s2 ; xÞ Wh/ ðr1^s1 ; r2^s2 ; xÞ : Wðr1^s1 ; r2^s2 ; xÞ ¼ W/h ðr1^s1 ; r2^s2 ; xÞ W// ðr1^s1 ; r2^s2 ; xÞ By using Eq. (25) one may define the degree of polarization in the far-field as [4] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 det Wðr^s; r^s; xÞ P ð/; h; xÞ ¼ u t1 h i2 ; tr Wðr^s; r^s; xÞ
261
ð25Þ
ð26Þ
where det and tr stand for the determinant and the trace, respectively, of a matrix. It should be noted that although Eq. (26) was derived for the electric field, it gives the degree of polarization for the magnetic field as well. In the scalar domain of variable-coherence optics, one is interested in the degree of coherence of the field. Unfortunately, the concept of the degree of coherence of electromagnetic fields cannot be extended straightforwardly from the scalar definition. In the past, several authors have defined this quantity by the formula tr Wðr1 ; r2 ; xÞ lðr1 ; r2 ; xÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; tr Wðr1 ; r1 ; xÞ tr Wðr2 ; r2 ; xÞ
ð27Þ
or its space-time equivalent [4,12,13]. The main problem with Eq. (27) is that in certain cases it assumes values below unity even for fully coherent fields [14]. These fundamental problems may be avoided if one restricts to consider correlations between different scalar components of the field. This leads one to define a set of correlation functions Wij ðr1 ; r2 ; xÞ lij ðr1 ; r2 ; xÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Wii ðr1 ; r1 ; xÞWjj ðr2 ; r2 ; xÞ
ð28Þ
which define the degrees of correlation between the components i and j at points r1 and r2 [15]. Also these quantities depend strongly on the chosen coordinate system and hence they must be used with great care. In this article, we characterize the coherence properties by Eq. (28) for the correlations between the cartesian components of the field. These are obtained by inserting Eq. (8) into Eq. (28), which yields Aij ðks?1 ; ks?2 ; xÞ exp½ikðr2 r1 Þ
lij ðr1^s1 ; r2^s2 ; xÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Aii ðks?1 ; ks?1 ; xÞAjj ðks?2 ; ks?2 ; xÞ
ð29Þ
4. Radiation from arbitrarily polarized incoherent sources Let us now turn our attention to radiation patterns of incoherent quasimonochromatic planar sources. By definition, the fluctuations of the incoherent field at two arbitrary points in the source plane z ¼ 0 are completely uncorrelated. This means that the cross-spectral density functions Wxx , Wxy , and Wyy may be expressed in the form Wij ðx1 ; y1 ; 0; x2 ; y2 ; 0Þ ¼ Sij ðx1 ; y1 Þdðx1 x2 Þdðy1 y2 Þ;
ð30Þ
where Sxx and Syy are real functions that define the power spectrum of the source, d is the Dirac delta function, and we have dropped the explicit dependence on x. The function Sxy , which is generally complex, is bounded by the formula 2
jSxy ðx; yÞj 6 Sxx ðx; xÞSyy ðy; yÞ:
ð31Þ
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By inserting Eq. (30) into Eq. (7), we obtain Z Z 1 1 Sij ðx; yÞ exp½iðu1 u2 Þx exp½iðv1 v2 Þy dx dy: Aij ðk?1 ; k?2 Þ ¼ ð2pÞ4 1
ð32Þ
The radiant intensity is then obtained by inserting Eq. (32) into Eq. (16) and using Eq. (18). This yields rffiffiffiffiffi 0 2 2 J ð/; hÞ ¼ 2p k Gð/; hÞ; ð33Þ l0 where
Gð/; hÞ ¼ Ixx cos2 h þ sin2 h cos2 / þ Iyy cos2 h þ sin2 h sin2 / þ sin2 h sinð2/ÞRfIxy g; Iij ¼
Z Z
1 ð2pÞ
4
ð34Þ
1
Sij ðx; yÞ dx dy;
ð35Þ
1
and we have represented the vector ^s in the spherical polar coordinates, i.e., sx ¼ sin h cos /;
ð36Þ
sy ¼ sin h sin /;
ð37Þ
sz ¼ cos h:
ð38Þ
Because of MaxwellÕs divergence equation r E ¼ 0, the z-component of the electric field is determined by its x- and y-components. Hence it is convenient to characterize the local degree of polarization of the field at the source plane z ¼ 0 by a expression analogous to Eq. (26) h i 91=2 8 2 < 4 Sxx ðx; yÞSyy ðx; yÞ jSxy ðx; yÞj = P ðx; yÞ ¼ 1 : ð39Þ 2 : ; Sxx ðx; yÞ þ Syy ðx; yÞ
4.1. Linearly polarized sources For linearly polarized light, the field oscillates, for example, in the xz plane and we have Syy ¼ Sxy ¼ 0. The function Gð/; hÞ now takes the form ð40Þ Gð/; hÞ ¼ Ixx cos2 h þ sin2 h cos2 / ; and thus the radiant intensity takes the form J ð/; hÞ ¼ J ð0; 0Þ cos2 h þ sin2 h cos2 / ;
ð41Þ
which is not azimuthally symmetric. Eq. (41) is illustrated graphically in Fig. 2(a). This result, which was found earlier by Carter [3], differs substantially from the well-known azimuthally symmetric radiant intensity of an incoherent scalar source Js ð/; hÞ ¼ Js ð0; 0Þ cos2 h;
ð42Þ
which is illustrated in Fig. 2(b). It may be easily seen by examining Eq. (41) or Fig. 2(a), that the radiant intensity of the linearly polarized source does not vanish in the limit h ! p=2, except for the values / ¼ p=2. Similar results have been observed in, e.g, rigorous theory of diffraction by an aperture in a perfectly conducting screen [16].
T. Saastamoinen et al. / Optics Communications 221 (2003) 257–269
263
Fig. 2. The normalized radiant intensity distribution of (a) linearly polarized source with Iyy ¼ Ixy ¼ 0 (b) a scalar source (c) a circularly symmetric azimuthally or unpolarized source, and (d) a nonsymmetric azimuthally polarized source with Ixx ¼ 3Iyy and Ixy ¼ 0.
The degree of polarization in the far-field is obtained straightforwardly from Eq. (26) and is equal to unity. The result is clear, since the source is fully polarized with spatially invariant polarization state. The four nonvanishing degrees of correlation take the form lii ðr1^s1 ; r2^s2 Þ ¼ Axx ðks?1 ; ks?2 Þ exp½ikðr2 r1 Þ =Ixx ;
ð43Þ
lij ðr1^s1 ; r2^s2 Þ ¼ Axx ðks?1 ; ks?2 Þ exp½ikðr2 r1 Þ =Ixx ;
ð44Þ
and where i; j ¼ x; z. Hence the absolute value of the degree of correlation depends only on the correlation between the plane waves propagating in the directions defined by the unit vectors ^s1 and ^s2 . Hence the coherence properties of a linearly polarized field closely resemble the coherence properties of an incoherent scalar field. 4.2. Circularly polarized sources Although circular polarization in usually considered only in the case of plane waves, or paraxial beamlike fields, sources with Sxx ¼ Syy ¼ iSxy deserve special attention. Since the function Sxy is imaginary, it has no contribution to the radiant intensity, which takes the simple form 1 J ð/; hÞ ¼ Jð0; 0Þ 1 þ cos2 h ; 2
ð45Þ
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having an azimuthally symmetric distribution. The radiant intensity is illustrated in Fig. 2. As in the case of linearly polarized source, however, the result differs from the scalar case, since the radiant intensity in the limit h ! p=2 does not vanish. This property arises from the z-component of the field because the amplitudes of the plane-wave components with large propagation angles are mainly determined by it. The degree of polarization is, like in the case of linearly polarized fields, equal to unity. The degrees of correlation are best expressed in a compact matrix form 2 3 1 i expði/2 Þ lðr1^s1 ; r2^s2 Þ ¼ 4 i 1 i expði/2 Þ 5Axx ðks?1 ; ks?2 Þ exp½ikðr2 r1 Þ =Ixx : expði/1 Þ i expði/1 Þ exp½ið/2 /1 Þ
ð46Þ As in the previous case, the absolute values of the degrees of coherence depend only on the ratio Axx =Ixx . Hence the only fundamental difference between linearly and circularly polarized fields is the radiant intensity pattern. 4.3. Unpolarized sources The subject of unpolarized three-dimensional electromagnetic sources has been studied creditably by Carter [3], who pointed out, e.g., that if the currents flowing in orthogonal directions are uncorrelated, the results are identical with the scalar theory (the cosine-squared law). However, if we apply the formalism derived above for an incoherent source in the case that Sxx ¼ Syy and Sxy ¼ 0, i.e., the source is unpolarized in the sense of Eq. (39), we immediately obtain the result 1 ð47Þ J ð/; hÞ ¼ J ð0; 0Þ 1 þ cos2 h ; 2 which is identical to Eq. (45). Hence the radiant intensity does not take the form predicted by the scalar theory, e.g., Eq. (42). It is important to notice that the assumption that no sources exist in the positive z-direction immediately leads to the fact that the z-component of the field is strictly dependent on the x- and y-components of the field by MaxwellÕs divergence equation r E ¼ 0 and hence the three-dimensional degree of polarization of the source, in view of [15,17], is generally greater than zero. In addition, one should remember that the definition of unpolarized field is not unique in the case of planar sources and the result depend on the choice of the independent components of the field. The two-dimensional state of polarization in the far-field takes the simple form 1 cos2 h ; ð48Þ 1 þ cos2 h which is illustrated in Fig. 3(a). It is easily seen that the degree of polarization increases from zero to unity as h is increases from zero to p=2. This may be understood if we recall that the x- and y-components propagating in the same direction are completely uncorrelated, whereas the z-component is generally dependent on both of them. Near the z-axis, i.e., if h 0, the amplitudes of the plane waves are almost fully determined by the x- and y-components, which leads to an unpolarized field. On the other hand, as we discussed in the previous subsection, in the limit h ! p=2, the amplitudes are determined by the z-component only. The reason for the fact that the degree of polarization approaches to unity is easily understood when we examine the degrees of correlation 2 3 1 0 cos /2 lðr1^s1 ; r2^s2 Þ ¼ 4 0 1 sin /2 5Axx ðks?1 ; ks?2 Þ exp½ikðr2 r1 Þ =Ixx : ð49Þ cos /1 sin /1 cosð/2 /1 Þ P ð/; hÞ ¼
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Fig. 3. Degree of polarization in the far-field of (a) a symmetric azimuthally polarized or an unpolarized source, and (b) a nonsymmetric azimuthally polarized source Ixx ¼ 3Iyy with Ixy ¼ 0.
Now the x- and z-components are completely uncorrelated at /i ¼ p=2, i ¼ 1; 2, whereas the y- and zcomponents do not correlate if /i ¼ 0 or p. This means that the field must be fully polarized when no x- or y-components are present, i.e., in the limit h ! p=2. This property clearly has its roots in the fact that, for example, the z-component of a plane wave with its wave vector perpendicular to the y-axis is determined from the x-component only. This naturally means that the z-component must be completely uncorrelated at /2 /1 ¼ p=2, which is evident also from Eq. (49). 4.4. Azimuthally polarized sources Let us next consider a source which produces an azimuthally polarized field, i.e., the electric field vector in the source plane is parallel to the azimuthal coordinate / [18,19]. We assume the cross-spectral density to be of the form W// ðr1 ; r2 Þ ¼ Sðq1 ; /1 Þdðq1 q2 Þdð/1 /2 Þ; where we have used the relation
x ¼ q cos /; y ¼ q sin /;
ð50Þ
ð51Þ
and Sð0; /Þ ¼ 0. The expressions of the cross-spectral density functions Wxx , Wxy , and Wyy are obtained by using the relations
Ex ðrÞ ¼ Eq ðrÞ cos / E/ ðrÞ sin /; ð52Þ Ey ðrÞ ¼ Eq ðrÞ sin / þ E/ ðrÞ cos /; where Eq and E/ denote the radial and azimuthal components of the field, respectively. Since in the azimuthally polarized case Eq ¼ 0, we obtain Wxx ðr1 ; r2 Þ ¼ W// ðr1 ; r2 Þ sin /1 sin /2 ;
ð53Þ
Wxy ðr1 ; r2 Þ ¼ W// ðr1 ; r2 Þ sin /1 cos /2 ;
ð54Þ
Wyy ðr1 ; r2 Þ ¼ W// ðr1 ; r2 Þ cos /1 cos /2 :
ð55Þ
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By converting also the wave vector into circular cylindrical coordinates, i.e.,
u ¼ a cos w; v ¼ a sin w;
ð56Þ
and inserting Eqs. (50) and (53)–(55) into Eq. (7), we obtain the expressions for the functions Ixx , Ixy , and Iyy Z 1 Z 2p 1 Ijk ¼ Sjk ðq; /Þ q dq d/; ð57Þ 4 ð2pÞ 0 0 where Sxx ðq; /Þ ¼ sin2 /Sðq; /Þ;
ð58Þ
Sxy ðq; /Þ ¼ sin / cos /Sðq; /Þ;
ð59Þ
Syy ðq; /Þ ¼ cos2 /Sðq; /Þ:
ð60Þ
If the function Sðq; /Þ is azimuthally symmetric, i.e., if it is a function of q only, we obtain Ixx ¼ Iyy and Ixy ¼ 0, which leads to a radiant intensity of the form (47). Interestingly, the degree of polarization takes the form identical to Eq. (48), which means that the far-field is unpolarized when h ¼ 0. Hence the axial intensity distribution consists of radially and azimuthally polarized parts, whose amplitudes are equal. This can be easily verified also by using the expressions of Wxx and Wyy and the definition (52). It is evident that, since the field is azimuthally polarized in the source plane, the radial component is generated upon propagation. A similar phenomenon may be found in the domain of coherent optics, where the permitted forms of azimuthally polarized fields are very limited [20,21]. If a field is not of that form, but is azimuthally polarized in a certain plane, the radial component is generated upon propagation [22]. The phenomenon that the degree of polarization in the far-field of an incoherent, but fully polarized, source can be equal to zero was discovered recently by Gori et al. [23], who studied the effect of a polarization-modulating element on an incoherent and unpolarized beam. Also in that case the mean values of Sxx and Syy are equal, whereas the mean value of Sxy is equal to zero. The reason for the phenomenon that radically different local polarization states of the source can lead to essentially similar radiation patterns is that only the ÔglobalÕ degree of polarization of the source, defined by the constants Ixx , Ixy , and Iyy , affect the radiant intensity. Hence there is no practical difference between, e.g., an unpolarized source for which Sxy 0, and an azimuthally polarized source for which Sxy generally does not vanish but Ixy ¼ 0. If the intensity distribution of the source is not symmetric but depends also on the /-coordinate, different radiant intensity and polarization patterns are obtained. If we choose, for example, Sðq; /Þ ¼ SðqÞ cos2 /, we obtain Ixx ¼ 3Iyy , while Ixy still vanishes. This leads to the radiant intensity 1 J ð/; hÞ ¼ J ð0; 0Þ 1 þ 3 cos2 h þ 2 sin2 h cos2 / ; 4
ð61Þ
which is illustrated in Fig. 2(d). The radiant intensity now resembles the radiant intensity of a linearly polarized field which is clear, since the x-component of the field is dominant. The degree of polarization now takes the form n 2 o1=2 P ð/; hÞ ¼ 1 12 cos2 h 1 þ 2 cos2 / þ cos2 h 1 þ 2 sin2 / ;
ð62Þ
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illustrated in Fig. 3(b). Now the axial degree on polarization P ð0; 0Þ ¼ 1=2 and the minima, where the pffiffiffi degree of polarization is equal to zero, are found at the points ð/; hÞ ¼ ½p=2; cos1 ð1= 3Þ . Different forms of radiant intensity and polarization patterns can be easily found by varying the azimuthal dependence of the function Sðq; /Þ.
5. Conclusions We have examined radiation from incoherent planar sources by using rigorous electromagnetic theory of partial coherence. The results, which were expressed in terms of the x and y components of the electric field, clearly demonstrate that the scalar theory is not generally applicable to electromagnetic fields in the nonparaxial domain (which is not a surprise). We have shown explicitly that different states of polarization of an incoherent source can lead to similar radiant-intensity patterns and/or degrees of polarization in the farfield. We have also shown, e.g., that an azimuthally symmetric, azimuthally polarized source behaves essentially similarly to a source that is unpolarized in the sense that its electric x and y components are uncorrelated in the source plane.
Acknowledgements The work of J. Tervo is supported by the Academy of Finland and the Nokia Foundation.
Appendix A In this Appendix we show that Eq. (15) holds for i; j ¼ x; y; z if it holds for i; j ¼ x; y. Let us recall the expressions for the angular correlation functions Axz , Ayz , Azx , Azy , and Azz from [5]: AðeÞ xz ðk?1 ; k?2 ; xÞ ¼
i 1 h ðeÞ u2 Axx ðk?1 ; k?2 ; xÞ þ v2 AðeÞ ðk ; k ; xÞ ; ?1 ?2 xy w2
ðA:1Þ
AðeÞ yz ðk?1 ; k?2 ; xÞ ¼
i 1 h ðeÞ u2 Ayx ðk?1 ; k?2 ; xÞ þ v2 AðeÞ yy ðk?1 ; k?2 ; xÞ ; w2
ðA:2Þ
AðeÞ zx ðk?1 ; k?2 ; xÞ ¼
i 1 h ðeÞ ðeÞ u A ðk ; k ; xÞ þ v A ðk ; k ; xÞ ; 1 xx ?1 ?2 1 yx ?1 ?2 w1
ðA:3Þ
AðeÞ zy ðk?1 ; k?2 ; xÞ ¼
i 1 h ðeÞ ðeÞ u A ðk ; k ; xÞ þ v A ðk ; k ; xÞ ; 1 ?1 ?2 1 ?1 ?2 xy yy w1
ðA:4Þ
AðeÞ zz ðk?1 ; k?2 ; xÞ ¼
1 h ðeÞ ðeÞ u1 u2 AðeÞ xx ðk?1 ; k?2 ; xÞ þ u1 v2 Axy ðk?1 ; k?2 ; xÞ þ v1 u2 Ayx ðk?1 ; k?2 ; xÞ w1 w2 i þ v1 v2 AðeÞ yy ðk?1 ; k?2 ; xÞ :
ðA:5Þ
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By inserting Eqs. (A.1)–(A.5) into Eq. (15) with i; j ¼ x; y; z, we obtain Z Z u1 u2 fx ðk?1 Þ fz ðk?1 Þ fx ðk?2 Þ fz ðk?2 Þ AðeÞ xx ðks?1 ; ks?2 ; xÞ w1 w2 u1 v2 þ fx ðk?1 Þ fz ðk?1 Þ fy ðk?2 Þ fz ðk?2 Þ AðeÞ xy ðks?1 ; ks?2 ; xÞ w1 w2 v1 u2 þ fy ðk?1 Þ fz ðk?1 Þ fx ðk?2 Þ fz ðk?2 Þ AðeÞ yx ðks?1 ; ks?2 ; xÞ w1 w2 ) v1 v2 þ fy ðk?1 Þ fz ðk?1 Þ fy ðk?2 Þ fz ðk?2 Þ AðeÞ ðks ; ks ; xÞ d2 k?1 d2 k?2 P 0: ?1 ?2 yy w1 w2 If we now introduce the functions u gx ðk? Þ ¼ fx ðk? Þ fz ðk? Þ; w and v gy ðk? Þ ¼ fy ðk? Þ fz ðk? Þ; w Eq. (A.6) takes the form Z Z h ðeÞ gx ðk?1 Þgx ðk?2 ÞAðeÞ xx ðk?1 ; k?2 ; xÞ þ gx ðk?1 Þgy ðk?2 ÞAxy ðk?1 ; k?2 ; xÞ i ðeÞ þ gy ðk?1 Þgx ðk?2 ÞAðeÞ ðk ; k ; xÞ þ g ðk Þg ðk ÞA ðk ; k ; xÞ d2 k?1 d2 k?2 P 0; ?1 ?2 ?1 y ?2 ?1 ?2 yx y yy
ðA:6Þ
ðA:7Þ
ðA:8Þ
ðA:9Þ
which clearly holds because we assumed that Eq. (15) is valid for i; j ¼ x; y and thus we have obtained the desired result. It should be noticed that the integral in Eq. (A.9) exists in all cases in which the z-component of the field may be determined from the x- and y-components by using the method introduced by Carter [24]. It can be easily shown, with essentially similar steps as we have taken above, that Eq. (15) for i; j ¼ x; y is a sufficient condition also for the equivalent nonnegative definiteness condition including the angular correlation functions of the magnetic and mixed cross-spectral density functions. It is also easily shown by using the result derived in this Appendix that the nonnegative definiteness condition for the electric crossspectral density matrix holds if the corresponding definiteness condition holds for the elements formed by the x- and y-components of the field, of course assuming that no sources exist in the half-space z > 0.
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[13] B. Karczewski, Nuovo Cimento 30 (1963) 906. ^ exp½iðkz xtÞ þ ^y exp½iðkx xtÞ and by choosing [14] Such situation is obtained, for example, by using a coherent field with E ¼ x r1 ¼ ð0; 0; 0Þ and r2 ¼ ðp=k; 0; 0Þ. [15] T. Set€ al€ a, A. Shevchenko, M. Kaivola, A.T. Friberg, Phys. Rev. E 66 (2002) 016615. [16] J. Huttunen, A.T. Friberg, J. Turunen, Phys. Rev. E 5 (1995) 3081. [17] T. Set€ al€ a, M. Kaivola, A.T. Friberg, Phys. Rev. Lett. 88 (2002) 123902. [18] R.H. Jordan, D.G. Hall, Opt. Lett. 19 (1994) 427. [19] P.L. Greene, D.G. Hall, J. Opt. Soc. Am. A 13 (1996) 962. [20] J. Tervo, P. Vahimaa, J. Turunen, J. Mod. Opt. 49 (2002) 1537. [21] P. P€ a€ akk€ onen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, Opt. Express 10 (2002) 949. [22] A. Lapucci, M. Ciofini, Opt. Express 9 (2001) 603. [23] F. Gori, M. Santarsiero, R. Borghi, G. Piquero, Opt. Lett. 25 (2000) 1291. [24] W.H. Carter, J. Opt. Soc. Am. 62 (1972) 1195.