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On the vectorial structure of non-paraxial radially polarized light fields
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Optics Communications 281 (2008) 3046–3050 www.elsevier.com/locate/optcom
On the vectorial structure of non-paraxial radially polarized light fields R. Martı´nez-Herrero a, P.M. Mejı´as a,*, S. Bosch b a b
´ ptica, Facultad de Ciencias Fı´sicas, Universidad Complutense de Madrid, 28040 Madrid, Spain Departamento de O ´ ptica, Facultad de Fı´sica, Universidad de Barcelona, 08028 Barcelona, Spain Departamento de Fı´sica, Aplicada y O Received 7 November 2007; received in revised form 1 February 2008; accepted 4 February 2008
Abstract To describe in a detailed way the polarization features of non-paraxial radially polarized fields, a simple analytical propagation law is provided in terms of their angular plane-wave spectrum. A comparison between non-paraxial results and paraxial predictions is investigated by introducing a parameter describing the width of the angular spectrum. Two particular examples of radially polarized light fields are also considered: Bessel-type fields whose angular spectrum is confined on a Montgomery’s ring, and certain radially polarized Laguerre–Gauss beams. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction As is well known, radially polarized beams are receiving increasing attention in the literature [1–13]. Apart from the methods for generating such fields and their multiple applications, with the advent of micro- and nano-devices the research into the vectorial structure of these fields in the non-paraxial regime has revealed to be an interesting goal, which involves diameters of the spot region smaller than the light wavelength. In such cases, widely employed analytical functions used to model light beams do not satisfy the Maxwell equations in an exact way, and the paraxial approach becomes inappropriate. In this sense, different vectorial formulations of electromagnetic beams have been explored (see, for example, Refs. [14–21]). Among the treatments used in the literature (for instance, on the basis of the Hertz vectors [15,18], or by using the complex-source-point model [17]) we will employ here a representation of the general solution of the Maxwell equations based on the plane-wave spectrum of a electromagnetic field [20,21]. Within this framework, *
Corresponding author. E-mail address: pmmejias@fis.ucm.es (P.M. Mejı´as).
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.02.005
the main purpose of the present work is to study the propagation and the polarization distribution across the transverse beam profile of freely-propagating non-paraxial radially polarized fields (RPFs). The paper is organized as follows. In the next section, the formalism and the key definitions to be used in the rest of the paper are introduced. In Section 3, in terms of the angular plane-wave spectrum, the general free-propagation law is given for this class of fields. In the same section, a number of simple but general properties are also inferred. Section 4 defines a parameter in order to compare the non-paraxial results (valid for strongly focused beams) with paraxial predictions. In Section 5, the analytical results are applied to certain cases of current interest, namely, Bessel-type beams and Laguerre–Gauss type fields. Finally, the main conclusions are summarized in Section 6. 2. Formalism and key definitions For simplicity, we will consider the time-harmonic electric and magnetic fields, E and H, respectively, fulfilling the Maxwell equations in a homogeneous isotropic medium. In terms of their respective angular spectrum, E can be written in the form
R. Martı´nez-Herrero et al. / Optics Communications 281 (2008) 3046–3050
Eðx; y; zÞ ¼ ðEs ; Ep ; Ez Þ ¼
Z
e ðu; v; zÞ exp½ikðxu þ yvÞdudv; E ð1Þ
and the analogous expression for H. In Eq. (1) (x, y, z) denotes the Cartesian variables, k is the wavenumber of the light, the subscripts s and p refer to the Cartesian coordinates transverse to the z-axis, which has been chosen to e represents be the propagation direction of the beam, and E the spatial Fourier-transform of E (the same applies for f e and f H ). In terms of E H , the Maxwell equations become: f e e þ ik H e ¼ L e ¼ 0; and L E L H þ ik E ¼ 0; L E e H ¼ 0, where L = (iku, ikv, o/oz). For convenience, we will use planar polar coordinates (q, /) related to the Cartesian Fourier-transform variables through the equations u = q cos/, v = q sin/. For propagation distances long enough to neglect the contribution of the evanescent waves, we can consider that q takes values on the interval [0, 1], and we get [20] pffiffiffiffiffiffiffiffiffiffiffiffiffi e ðq; /; zÞ ¼ E e 0 ðq; /Þ exp ikz 1 q2 ; E ð2:aÞ f e ðq; /; zÞ; H ðq; /; zÞ ¼ sðq; /Þ E
ð2:bÞ
e 0 s ¼ 0 and s is the unit vector in the direction of where E pffiffiffiffiffiffiffiffiffiffiffiffiffi each plane wave, i.e., sðq; /Þ ¼ ðq cos /; q sin /; 1 q2 Þ. It can be shown [20,21] that the electric field solution of the Maxwell equations can be written in cylindrical coordinates R, h and z as follows: Z 1 Z 2p EðR; h; zÞ ¼ ½aðq; /Þe1 ð/Þ þ bðq; /Þe2 ðq; /Þ 0
0
exp½ikRq cosð/ hÞ h pffiffiffiffiffiffiffiffiffiffiffiffiffii exp ikz 1 q2 qdqd/
ð3Þ
Z
Z
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i pffiffiffiffiffiffiffiffiffiffiffiffiffi aðq; /Þ cosð/ hÞ þ 1 q2 bðq; /Þ sinð/ hÞ 0 0 h pffiffiffiffiffiffiffiffiffiffiffiffiffii exp ½ikRq cosð/ hÞ exp ikz 1 q2 qdqd/ ¼ 0 1
2p
h
ð5Þ By following a similar procedure to that used in Ref. [22], it can be shown that functions a(q, /) and b(q, /) become for RPFs aðq; /Þ ¼ 0; bðq; /Þ ¼ bðqÞ: ð6Þ Consequently, we conclude that a RPF propagates according with the general law ð7Þ EðR; h; zÞ ¼ 2piG1 ðR; zÞuR 2pG0 ðR; zÞuz ; where uR is the radial unitary vector (cosh, sinh, 0), uz is the unit vector along z, and Z 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffii G0 ðR; zÞ ¼ bðqÞqJ 0 ðkRqÞ exp ikz 1 q2 qdq; ð8:aÞ 0 Z 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffi bðqÞ 1 q2 J 1 ðkRqÞ exp ikz 1 q2 qdq; G1 ðR; zÞ ¼ 0
ð8:bÞ J0 and J1 being the Bessel functions of the first kind of order 0 and 1, respectively. Eqs. (6)–(8) are, in a sense, the main result of this paper. They enable us to determine the vector amplitude of a RPF at any transverse plane. Note also that the functions G0(R, z) and G1(R, z) are given in terms of the planar-wave spectrum b(q) that defines the particular RPF. In addition, from Eqs. (7) and (8), several simple but general consequences can be inferred at once for RPFs: (i) The transverse part of any RPF is radially polarized at any plane z = constant (as expected). (ii) On the axis z (R = 0), any RPF only has longitudinal component (i.e., the field is linearly polarized along z). This is consistent with the rotationally-symmetric polarization structure of the RPFs. (iii) When R differs from zero, the ellipse that describes the electric field vector associated to a RPF is contained in the plane defined by uR and the z axis. (iv) The characteristics of the polarization ellipse depend on R and z, but are independent of h.
e 0 e2 are integrae 0 e1 ; bðq; /Þ ¼ E where aðq; /Þ ¼ E ble functions that define the plane-wave of pspectrum ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos /; the field, and e = (sin/,cos/,0); e ¼ 1 q 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2 sin /; qÞ are unitary vectors. Recall that s, e1 and e2 form a triad of mutually orthogonal system of unit vectors. For convenience, instead of using the Cartesian transverse components Es and Ep of the field (defined with respect to some laboratory axes), we will consider in the present paper its radial and azimuthal components, ER and Eh, at each point of a transverse plane z = constant, namely,
It is interesting to notice that the general propagation law (7) is valid for non-paraxial fields as well as for paraxial beams. We will next compare both cases.
ER ðR; h; zÞ ¼ cos hEs ðR; h; zÞ þ sin hEp ðR; h; zÞ;
ð4:aÞ
4. Comparison with the paraxial case
Eh ðR; h; zÞ ¼ sin hEs ðR; h; zÞ þ cos hEp ðR; h; zÞ;
ð4:bÞ
and Ez remains unaltered. The radially polarized fields are then defined as those fields for which Eh equals zero everywhere. 3. Propagation law and general properties Application of Eqs. (3) and (4) to RPFs gives
Strictly speaking, the paraxial approximation would analytically follow from the substitutions (Fresnel approach) exp[ikz(1 q2)1/2] ffi exp(ikz) exp[ikz(q2/2)] and e2 ffi (cos/ , sin/, 0) in Eq. (3). However, to get deeper insight into the physical meaning of this approach, it would be better to handle directly the plane-wave spectrum that characterizes the RPF: Thus, let us define the so-called (squared) width of the angular spectrum, w2, in the form
R. Martı´nez-Herrero et al. / Optics Communications 281 (2008) 3046–3050
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R
e ðu; v; zÞj2 dudv ðu2 þ v2 Þj E ; w ¼ R e ðu; v; zÞj2 dudv jE 2
ð9Þ
e represents again the spatial Fourier transform of where E the field E (cf. Eq. (1)). Note that w2 does not depend on z. Eq. (9) is valid for general beams. In the particular case of RPFs, it follows from Eqs. (3), (4) and (6) that w2 becomes R1 2 j bðqÞj q3 dq 2 w ¼ R0 1 : ð10Þ 2 jbðqÞj qdq 0 In other words, w2 is given by the (squared) width (second-order moment) of the function jb(q)j2. This parameter w2 could then be understood as a measure of the size of the angular spectrum (far field). Accordingly, its value is inversely proportional to the size r (measured in k-units) of the region at a transverse (near field) plane z where jEj2 is significant. Consequently, paraxiality is related with small values of w2, whereas greater w2 refers to the non-paraxial case. Let us now evaluate the importance of the longitudinal field component Ez with respect to the total field E. This can be done by calculating the ratio kEzk2/kEk2, where R 1 R 2p R 1 R 2p 2 2 2 kEz k 0 0 jEz j2 RdRdh, and kEk 0 0 jEj RdRd h. We get for RPFs w2 ¼
kEz k
2
kEk2
ð11Þ
: 2
Moreover, kEk2 = kERk2 + kEzk2, where kER k 2 jER j RdRdh. It then follows for RPFs kEz k
2
kER k
2
¼
w2 : 1 w2
R 1 R 2p 0
0
ð12Þ
From the above equations we immediately conclude that, for RPFs, the parameter w2 can be considered as a measure of the significance of the longitudinal component of the field (quantitatively evaluated by Eqs. (11) and (12)). As a consequence, a paraxial RPF would exhibit small values of Ez. Moreover, Eq. (12) can also be understood as a measure (averaged over the transverse cross-section) of the squared ratio between the major and the minor axes of the polarization ellipse. Therefore, since w2 drastically reduces in the paraxial regime, such ellipse would resemble a segment along the radial (transverse) direction. This behavior will be illustrated in the examples. 5. Application to several illustrative examples We will next apply the propagation Eq. (7) to some particular cases of current interest. We first consider those radially polarized Bessel-type beams whose angular plane-wave spectrum (confined on a ring) is defined by the functions [22,23] a(q,h) = 0 and b(q) = B d(q q), where B and q are constants (q < 1), and d() denotes the Dirac delta function. In this case, G0 and G1 reduce to
h pffiffiffiffiffiffiffiffiffiffiffiffiffii G0 ðR; zÞ ¼ Bq2 J 0 ðkRqÞ exp ikz 1 q2 ; h pffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffi G1 ðR; zÞ ¼ Bq 1 q2 J 1 ðkRqÞ exp ikz 1 q2 :
ð13:aÞ ð13:bÞ
After substituting these expressions into Eq. (7), we get at once several general features concerning the polarization structure of these fields: (i) The polarization does not depend on z, so that, for a fixed radial distance, it remains invariant under free propagation. This result is consistent with the invariance properties of this class of fields [22,23]. (ii) The field only has longitudinal component for radial distances Rn = jn1/qk, where jn1 are the zeros of J1. (iii) The field is linearly polarized in the radial direction for distances Rm = jm0/qk, where jm0 denotes the zeros of J0. Let us now focus our attention on those transverse fields e 0 (see Eq. (2.a)) is given by whose vector E 2 2 e 0 ¼ qL1 2q exp q ðcos /; sin /; 0Þ; E ð14Þ n D2 D2 where L1n refers to generalized Laguerre polynomials [1,9– 11,24,25]. D = 1/kx0 is a constant proportional to the beam divergence at the far field, and x0 is the 1/e intensity beam radius of the Gaussian factor at the near field. It should be remarked that RPFs whose angular spectrum takes the form (14) have recently been analysed [11]. Differences with the present example arise from the fact of that we consider RPFs whose plane-wave spectrum b(q) is ob~ 0 e2 , which is, from an tained from the inner product E algebraic point of view [20,21], the projection of vector ~ 0 onto the subspace generated by e2 (note that, for these E ~ 0 e1 ¼ 0). In this sense, the resulting non-paraxial fields, E RPF studied in the present paper should be understood as the vector beam closest (best fitted) to the field defined by Eq. (14). We get pffiffiffiffiffiffiffiffiffiffiffiffiffi 2q2 q2 1 2 e bðqÞ ¼ E 0 e2 ¼ q 1 q Ln exp 2 ; ð15Þ D2 D The freely-propagating RPF can then be computed at different transverse planes by using Eq. (7). Fig. 1 illustrates the behavior of the vectorial structure of this kind of fields in the non-paraxial case (x0 less than k), which can be summarized as follows: (i) At radial distances close enough to the z-axis, the longitudinal field component Ez predominates. (ii) At radial distances far enough from the z-axis, the polarization is essentially longitudinal. (iii) Compared with the case of uniformly linearly-polarized Gaussian-model non-paraxial beams [20,21], the significance of the longitudinal component Ez has revealed to be much more important for these RPFs. (iv) The above general characteristics also apply for longer propagation distances z.
R. Martı´nez-Herrero et al. / Optics Communications 281 (2008) 3046–3050
Fig. 1. Squared modulus of the electric field at a plane z = 10k for the beam defined by Eq. (15), with x0 = 0.3k and n = 1. In abcissae, the radial distance R is given in units of k. Ordinates are given in logarithmic scale (the value on the z-axis is normalised to unity). The continuous red line refers to the global field (cf. Eq. (7)), the dashed blue line refers to the radial component (along uR), and the dotted green line refers to the longitudinal component (along uz). We see that Ez predominates for two ranges of radial distances, R k and R k. However, for intermediate distances, R 10k, the modulus of the radial and the longitudinal components of the field are almost identical. In the inset of this figure it is represented the polarization ellipse that describes the electric field at a radial distance R = 0.2k: Abcissae and ordinates show the radial and the longitudinal components, respectively. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)
3049
Fig. 3. The same as in Fig. 1 but now with x0 = 0.1k. In the inset the polarization ellipse has been represented for R = 10k.
Fig. 4. The same as in Fig. 1 but now with x0 = k. Blue and red lines are indistinguishable. In the inset the polarization ellipse has been represented for R = 10 k. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. The same as in Fig. 1 but now with z = k. In the inset the polarization ellipse has been represented for a radial distance R = 0.3 k.
An example of the polarization ellipse close to the z-axis (R 0) is shown in the inset of Fig. 1. To clarify and complete the above results, Figs. 2 and 3 have also been included: Fig. 2 (very short propagation distances) shows a similar polarization behavior to that
observed in the case computed in Fig. 1. Fig. 3 considers the highly non-paraxial case (x0 = 0.1 k). We see that the general characteristics pointed out in Fig. 1 are also preserved, in a qualitative way, in the present case. Fig. 4 illustrates the behavior in the paraxial regime (x0 = 5k). At each point of the transverse plane, the polarization ellipse essentially behaves as a line along the radial direction, orthogonal to the z-axis. Thus, compared with the non-paraxial case, jEzj is now negligible (as expected). 6. Conclusions A general but analytically simple propagation law has been given (Eq. (7)) that allows to describe and compute in a detailed way the polarization features of a non-paraxial
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R. Martı´nez-Herrero et al. / Optics Communications 281 (2008) 3046–3050
RPF at any transverse plane. Within this framework, since the field amplitude has been separated into its radial (transverse) and longitudinal components, one can infer at once a number of general polarization properties. The analytical results are valid for non-paraxial as well as for paraxial beams. Moreover, the importance of the longitudinal component of the field has been evaluated by means of a parameter whose physical meaning can be understood as the size of the angular spectrum of the field. In addition, such parameter allows us to compare the characteristics of both, the paraxial and non-paraxial regimes, and is closely related with the ratio (averaged over the transverse beam cross-section) between the major and minor axes of the polarization ellipse. The above formalism has been shown to be easy to use by applying it to the study of the polarization structure of two kind of RPFs of current importance, namely, Besseltype beams and Laguerre–Gauss type fields. Acknowledgements This work has been supported by the MEC of Spain, under Projects FIS2007-63396 and DPI2005-08443-C02-01, and by CM-UCM, Research Group 910335 (2008). We also thank Dr. Artur Carnicer his valuable help. References [1] S.C. Tidwell, D.H. Ford, W.D. Kimura, Appl. Opt. 29 (1990) 2234. [2] A.A. Tovar, J. Opt. Soc. Am. A 15 (1998) 2705.
[3] A.V. Nesterov, V.G. Niziev, J. Phys. D 33 (2000) 1817. [4] R. Oron, S. Blit, N. Davidson, A.A. Fiesem, Appl. Phys. Lett. 77 (2000) 3322. [5] D.J. Armstrong, M.C. Philips, A.V. Smith, Appl. Opt. 42 (2003) 3550. [6] R. Dorn, S. Quabis, G. Leuchs, Phys. Rev. Lett. 91 (2003) 233901-1. [7] N. Pasilly, R. de S. Denis, K. Aı¨t-Ameur, F. Treussart, R. Hierle, J.-F Roch, J. Opt. Soc. Am. A 22 (2005) 984. [8] M.S. Roth, E.W. Wyss, H. Glur, H.P. Weber, Opt. Lett. 30 (2005) 1665. [9] D.M. Deng, J. Opt. Soc. Am. B 23 (2006) 1228. [10] D. Deng, Q. Guo, L. Wu, X. Yang, J. Opt. Soc. Am. B 24 (2007) 636. [11] D. Deng, Q. Guo, Opt. Lett. 32 (2007) 2711. [12] R. Martı´nez-Herrero, G. Piquero, P.M. Mejı´as, Opt. Commun. 281 (2008) 756. [13] R. Martı´nez-Herrero, P.M. Mejı´as, G. Piquero, V. Ramı´rez-Sa´nchez, Opt. Commun. (2008), doi:10.1016/j.optcom.2007.12.023. [14] D.G. Hall, Opt. Lett. 21 (1996) 9. [15] P. Varga, P. To¨ro¨k, Opt. Lett. 21 (1996) 1523. [16] S.R. Seshadri, J. Opt. Soc. Am. A 15 (1998) 2712. [17] C.J.R. Sheppard, S. Saghafi, J. Opt. Soc. Am. A 16 (1999) 1381. [18] C.J.R. Sheppard, J. Opt. Soc. Am. A 17 (2000) 335. [19] R. Borghi, A. Ciattoni, M. Santarsiero, J. Opt. Soc. Am. A 19 (2002) 1207. [20] R. Martı´nez-Herrero, P.M. Mejı´as, S. Bosch, A. Carnicer, J. Opt. Soc. Am. A 18 (2001) 1678. [21] P.M. Mejı´as, R. Martı´nez-Herrero, G. Piquero, J.M. Movilla, Prog. Quantum Electron. 26 (2002) 65. [22] P. Pa¨a¨kko¨nen, J. Tervo, P. Vahimaa, J. Turunen, F. Gori, Opt. Express 10 (2002) 949. [23] J. Tervo, J. Turunen, Opt. Express 9 (2001) 622. [24] G. Zhou, Opt. Lett. 31 (2006) 2616. [25] M. Abramowitz, I. Stegun (Eds.), Handbook of Mathematical Functions, Dover, 1972.
Optics Communications 281 (2008) 3046–3050 www.elsevier.com/locate/optcom
On the vectorial structure of non-paraxial radially polarized light fields R. Martı´nez-Herrero a, P.M. Mejı´as a,*, S. Bosch b a b
´ ptica, Facultad de Ciencias Fı´sicas, Universidad Complutense de Madrid, 28040 Madrid, Spain Departamento de O ´ ptica, Facultad de Fı´sica, Universidad de Barcelona, 08028 Barcelona, Spain Departamento de Fı´sica, Aplicada y O Received 7 November 2007; received in revised form 1 February 2008; accepted 4 February 2008
Abstract To describe in a detailed way the polarization features of non-paraxial radially polarized fields, a simple analytical propagation law is provided in terms of their angular plane-wave spectrum. A comparison between non-paraxial results and paraxial predictions is investigated by introducing a parameter describing the width of the angular spectrum. Two particular examples of radially polarized light fields are also considered: Bessel-type fields whose angular spectrum is confined on a Montgomery’s ring, and certain radially polarized Laguerre–Gauss beams. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction As is well known, radially polarized beams are receiving increasing attention in the literature [1–13]. Apart from the methods for generating such fields and their multiple applications, with the advent of micro- and nano-devices the research into the vectorial structure of these fields in the non-paraxial regime has revealed to be an interesting goal, which involves diameters of the spot region smaller than the light wavelength. In such cases, widely employed analytical functions used to model light beams do not satisfy the Maxwell equations in an exact way, and the paraxial approach becomes inappropriate. In this sense, different vectorial formulations of electromagnetic beams have been explored (see, for example, Refs. [14–21]). Among the treatments used in the literature (for instance, on the basis of the Hertz vectors [15,18], or by using the complex-source-point model [17]) we will employ here a representation of the general solution of the Maxwell equations based on the plane-wave spectrum of a electromagnetic field [20,21]. Within this framework, *
Corresponding author. E-mail address: pmmejias@fis.ucm.es (P.M. Mejı´as).
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.02.005
the main purpose of the present work is to study the propagation and the polarization distribution across the transverse beam profile of freely-propagating non-paraxial radially polarized fields (RPFs). The paper is organized as follows. In the next section, the formalism and the key definitions to be used in the rest of the paper are introduced. In Section 3, in terms of the angular plane-wave spectrum, the general free-propagation law is given for this class of fields. In the same section, a number of simple but general properties are also inferred. Section 4 defines a parameter in order to compare the non-paraxial results (valid for strongly focused beams) with paraxial predictions. In Section 5, the analytical results are applied to certain cases of current interest, namely, Bessel-type beams and Laguerre–Gauss type fields. Finally, the main conclusions are summarized in Section 6. 2. Formalism and key definitions For simplicity, we will consider the time-harmonic electric and magnetic fields, E and H, respectively, fulfilling the Maxwell equations in a homogeneous isotropic medium. In terms of their respective angular spectrum, E can be written in the form
R. Martı´nez-Herrero et al. / Optics Communications 281 (2008) 3046–3050
Eðx; y; zÞ ¼ ðEs ; Ep ; Ez Þ ¼
Z
e ðu; v; zÞ exp½ikðxu þ yvÞdudv; E ð1Þ
and the analogous expression for H. In Eq. (1) (x, y, z) denotes the Cartesian variables, k is the wavenumber of the light, the subscripts s and p refer to the Cartesian coordinates transverse to the z-axis, which has been chosen to e represents be the propagation direction of the beam, and E the spatial Fourier-transform of E (the same applies for f e and f H ). In terms of E H , the Maxwell equations become: f e e þ ik H e ¼ L e ¼ 0; and L E L H þ ik E ¼ 0; L E e H ¼ 0, where L = (iku, ikv, o/oz). For convenience, we will use planar polar coordinates (q, /) related to the Cartesian Fourier-transform variables through the equations u = q cos/, v = q sin/. For propagation distances long enough to neglect the contribution of the evanescent waves, we can consider that q takes values on the interval [0, 1], and we get [20] pffiffiffiffiffiffiffiffiffiffiffiffiffi e ðq; /; zÞ ¼ E e 0 ðq; /Þ exp ikz 1 q2 ; E ð2:aÞ f e ðq; /; zÞ; H ðq; /; zÞ ¼ sðq; /Þ E
ð2:bÞ
e 0 s ¼ 0 and s is the unit vector in the direction of where E pffiffiffiffiffiffiffiffiffiffiffiffiffi each plane wave, i.e., sðq; /Þ ¼ ðq cos /; q sin /; 1 q2 Þ. It can be shown [20,21] that the electric field solution of the Maxwell equations can be written in cylindrical coordinates R, h and z as follows: Z 1 Z 2p EðR; h; zÞ ¼ ½aðq; /Þe1 ð/Þ þ bðq; /Þe2 ðq; /Þ 0
0
exp½ikRq cosð/ hÞ h pffiffiffiffiffiffiffiffiffiffiffiffiffii exp ikz 1 q2 qdqd/
ð3Þ
Z
Z
3047
i pffiffiffiffiffiffiffiffiffiffiffiffiffi aðq; /Þ cosð/ hÞ þ 1 q2 bðq; /Þ sinð/ hÞ 0 0 h pffiffiffiffiffiffiffiffiffiffiffiffiffii exp ½ikRq cosð/ hÞ exp ikz 1 q2 qdqd/ ¼ 0 1
2p
h
ð5Þ By following a similar procedure to that used in Ref. [22], it can be shown that functions a(q, /) and b(q, /) become for RPFs aðq; /Þ ¼ 0; bðq; /Þ ¼ bðqÞ: ð6Þ Consequently, we conclude that a RPF propagates according with the general law ð7Þ EðR; h; zÞ ¼ 2piG1 ðR; zÞuR 2pG0 ðR; zÞuz ; where uR is the radial unitary vector (cosh, sinh, 0), uz is the unit vector along z, and Z 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffii G0 ðR; zÞ ¼ bðqÞqJ 0 ðkRqÞ exp ikz 1 q2 qdq; ð8:aÞ 0 Z 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffi bðqÞ 1 q2 J 1 ðkRqÞ exp ikz 1 q2 qdq; G1 ðR; zÞ ¼ 0
ð8:bÞ J0 and J1 being the Bessel functions of the first kind of order 0 and 1, respectively. Eqs. (6)–(8) are, in a sense, the main result of this paper. They enable us to determine the vector amplitude of a RPF at any transverse plane. Note also that the functions G0(R, z) and G1(R, z) are given in terms of the planar-wave spectrum b(q) that defines the particular RPF. In addition, from Eqs. (7) and (8), several simple but general consequences can be inferred at once for RPFs: (i) The transverse part of any RPF is radially polarized at any plane z = constant (as expected). (ii) On the axis z (R = 0), any RPF only has longitudinal component (i.e., the field is linearly polarized along z). This is consistent with the rotationally-symmetric polarization structure of the RPFs. (iii) When R differs from zero, the ellipse that describes the electric field vector associated to a RPF is contained in the plane defined by uR and the z axis. (iv) The characteristics of the polarization ellipse depend on R and z, but are independent of h.
e 0 e2 are integrae 0 e1 ; bðq; /Þ ¼ E where aðq; /Þ ¼ E ble functions that define the plane-wave of pspectrum ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos /; the field, and e = (sin/,cos/,0); e ¼ 1 q 2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2 sin /; qÞ are unitary vectors. Recall that s, e1 and e2 form a triad of mutually orthogonal system of unit vectors. For convenience, instead of using the Cartesian transverse components Es and Ep of the field (defined with respect to some laboratory axes), we will consider in the present paper its radial and azimuthal components, ER and Eh, at each point of a transverse plane z = constant, namely,
It is interesting to notice that the general propagation law (7) is valid for non-paraxial fields as well as for paraxial beams. We will next compare both cases.
ER ðR; h; zÞ ¼ cos hEs ðR; h; zÞ þ sin hEp ðR; h; zÞ;
ð4:aÞ
4. Comparison with the paraxial case
Eh ðR; h; zÞ ¼ sin hEs ðR; h; zÞ þ cos hEp ðR; h; zÞ;
ð4:bÞ
and Ez remains unaltered. The radially polarized fields are then defined as those fields for which Eh equals zero everywhere. 3. Propagation law and general properties Application of Eqs. (3) and (4) to RPFs gives
Strictly speaking, the paraxial approximation would analytically follow from the substitutions (Fresnel approach) exp[ikz(1 q2)1/2] ffi exp(ikz) exp[ikz(q2/2)] and e2 ffi (cos/ , sin/, 0) in Eq. (3). However, to get deeper insight into the physical meaning of this approach, it would be better to handle directly the plane-wave spectrum that characterizes the RPF: Thus, let us define the so-called (squared) width of the angular spectrum, w2, in the form
R. Martı´nez-Herrero et al. / Optics Communications 281 (2008) 3046–3050
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R
e ðu; v; zÞj2 dudv ðu2 þ v2 Þj E ; w ¼ R e ðu; v; zÞj2 dudv jE 2
ð9Þ
e represents again the spatial Fourier transform of where E the field E (cf. Eq. (1)). Note that w2 does not depend on z. Eq. (9) is valid for general beams. In the particular case of RPFs, it follows from Eqs. (3), (4) and (6) that w2 becomes R1 2 j bðqÞj q3 dq 2 w ¼ R0 1 : ð10Þ 2 jbðqÞj qdq 0 In other words, w2 is given by the (squared) width (second-order moment) of the function jb(q)j2. This parameter w2 could then be understood as a measure of the size of the angular spectrum (far field). Accordingly, its value is inversely proportional to the size r (measured in k-units) of the region at a transverse (near field) plane z where jEj2 is significant. Consequently, paraxiality is related with small values of w2, whereas greater w2 refers to the non-paraxial case. Let us now evaluate the importance of the longitudinal field component Ez with respect to the total field E. This can be done by calculating the ratio kEzk2/kEk2, where R 1 R 2p R 1 R 2p 2 2 2 kEz k 0 0 jEz j2 RdRdh, and kEk 0 0 jEj RdRd h. We get for RPFs w2 ¼
kEz k
2
kEk2
ð11Þ
: 2
Moreover, kEk2 = kERk2 + kEzk2, where kER k 2 jER j RdRdh. It then follows for RPFs kEz k
2
kER k
2
¼
w2 : 1 w2
R 1 R 2p 0
0
ð12Þ
From the above equations we immediately conclude that, for RPFs, the parameter w2 can be considered as a measure of the significance of the longitudinal component of the field (quantitatively evaluated by Eqs. (11) and (12)). As a consequence, a paraxial RPF would exhibit small values of Ez. Moreover, Eq. (12) can also be understood as a measure (averaged over the transverse cross-section) of the squared ratio between the major and the minor axes of the polarization ellipse. Therefore, since w2 drastically reduces in the paraxial regime, such ellipse would resemble a segment along the radial (transverse) direction. This behavior will be illustrated in the examples. 5. Application to several illustrative examples We will next apply the propagation Eq. (7) to some particular cases of current interest. We first consider those radially polarized Bessel-type beams whose angular plane-wave spectrum (confined on a ring) is defined by the functions [22,23] a(q,h) = 0 and b(q) = B d(q q), where B and q are constants (q < 1), and d() denotes the Dirac delta function. In this case, G0 and G1 reduce to
h pffiffiffiffiffiffiffiffiffiffiffiffiffii G0 ðR; zÞ ¼ Bq2 J 0 ðkRqÞ exp ikz 1 q2 ; h pffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffi G1 ðR; zÞ ¼ Bq 1 q2 J 1 ðkRqÞ exp ikz 1 q2 :
ð13:aÞ ð13:bÞ
After substituting these expressions into Eq. (7), we get at once several general features concerning the polarization structure of these fields: (i) The polarization does not depend on z, so that, for a fixed radial distance, it remains invariant under free propagation. This result is consistent with the invariance properties of this class of fields [22,23]. (ii) The field only has longitudinal component for radial distances Rn = jn1/qk, where jn1 are the zeros of J1. (iii) The field is linearly polarized in the radial direction for distances Rm = jm0/qk, where jm0 denotes the zeros of J0. Let us now focus our attention on those transverse fields e 0 (see Eq. (2.a)) is given by whose vector E 2 2 e 0 ¼ qL1 2q exp q ðcos /; sin /; 0Þ; E ð14Þ n D2 D2 where L1n refers to generalized Laguerre polynomials [1,9– 11,24,25]. D = 1/kx0 is a constant proportional to the beam divergence at the far field, and x0 is the 1/e intensity beam radius of the Gaussian factor at the near field. It should be remarked that RPFs whose angular spectrum takes the form (14) have recently been analysed [11]. Differences with the present example arise from the fact of that we consider RPFs whose plane-wave spectrum b(q) is ob~ 0 e2 , which is, from an tained from the inner product E algebraic point of view [20,21], the projection of vector ~ 0 onto the subspace generated by e2 (note that, for these E ~ 0 e1 ¼ 0). In this sense, the resulting non-paraxial fields, E RPF studied in the present paper should be understood as the vector beam closest (best fitted) to the field defined by Eq. (14). We get pffiffiffiffiffiffiffiffiffiffiffiffiffi 2q2 q2 1 2 e bðqÞ ¼ E 0 e2 ¼ q 1 q Ln exp 2 ; ð15Þ D2 D The freely-propagating RPF can then be computed at different transverse planes by using Eq. (7). Fig. 1 illustrates the behavior of the vectorial structure of this kind of fields in the non-paraxial case (x0 less than k), which can be summarized as follows: (i) At radial distances close enough to the z-axis, the longitudinal field component Ez predominates. (ii) At radial distances far enough from the z-axis, the polarization is essentially longitudinal. (iii) Compared with the case of uniformly linearly-polarized Gaussian-model non-paraxial beams [20,21], the significance of the longitudinal component Ez has revealed to be much more important for these RPFs. (iv) The above general characteristics also apply for longer propagation distances z.
R. Martı´nez-Herrero et al. / Optics Communications 281 (2008) 3046–3050
Fig. 1. Squared modulus of the electric field at a plane z = 10k for the beam defined by Eq. (15), with x0 = 0.3k and n = 1. In abcissae, the radial distance R is given in units of k. Ordinates are given in logarithmic scale (the value on the z-axis is normalised to unity). The continuous red line refers to the global field (cf. Eq. (7)), the dashed blue line refers to the radial component (along uR), and the dotted green line refers to the longitudinal component (along uz). We see that Ez predominates for two ranges of radial distances, R k and R k. However, for intermediate distances, R 10k, the modulus of the radial and the longitudinal components of the field are almost identical. In the inset of this figure it is represented the polarization ellipse that describes the electric field at a radial distance R = 0.2k: Abcissae and ordinates show the radial and the longitudinal components, respectively. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 3. The same as in Fig. 1 but now with x0 = 0.1k. In the inset the polarization ellipse has been represented for R = 10k.
Fig. 4. The same as in Fig. 1 but now with x0 = k. Blue and red lines are indistinguishable. In the inset the polarization ellipse has been represented for R = 10 k. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. The same as in Fig. 1 but now with z = k. In the inset the polarization ellipse has been represented for a radial distance R = 0.3 k.
An example of the polarization ellipse close to the z-axis (R 0) is shown in the inset of Fig. 1. To clarify and complete the above results, Figs. 2 and 3 have also been included: Fig. 2 (very short propagation distances) shows a similar polarization behavior to that
observed in the case computed in Fig. 1. Fig. 3 considers the highly non-paraxial case (x0 = 0.1 k). We see that the general characteristics pointed out in Fig. 1 are also preserved, in a qualitative way, in the present case. Fig. 4 illustrates the behavior in the paraxial regime (x0 = 5k). At each point of the transverse plane, the polarization ellipse essentially behaves as a line along the radial direction, orthogonal to the z-axis. Thus, compared with the non-paraxial case, jEzj is now negligible (as expected). 6. Conclusions A general but analytically simple propagation law has been given (Eq. (7)) that allows to describe and compute in a detailed way the polarization features of a non-paraxial
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RPF at any transverse plane. Within this framework, since the field amplitude has been separated into its radial (transverse) and longitudinal components, one can infer at once a number of general polarization properties. The analytical results are valid for non-paraxial as well as for paraxial beams. Moreover, the importance of the longitudinal component of the field has been evaluated by means of a parameter whose physical meaning can be understood as the size of the angular spectrum of the field. In addition, such parameter allows us to compare the characteristics of both, the paraxial and non-paraxial regimes, and is closely related with the ratio (averaged over the transverse beam cross-section) between the major and minor axes of the polarization ellipse. The above formalism has been shown to be easy to use by applying it to the study of the polarization structure of two kind of RPFs of current importance, namely, Besseltype beams and Laguerre–Gauss type fields. Acknowledgements This work has been supported by the MEC of Spain, under Projects FIS2007-63396 and DPI2005-08443-C02-01, and by CM-UCM, Research Group 910335 (2008). We also thank Dr. Artur Carnicer his valuable help. References [1] S.C. Tidwell, D.H. Ford, W.D. Kimura, Appl. Opt. 29 (1990) 2234. [2] A.A. Tovar, J. Opt. Soc. Am. A 15 (1998) 2705.
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