Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ilc.phys.msu.ru/upload/iblock/588/y2l.pdf Äàòà èçìåíåíèÿ: Tue Oct 26 10:45:37 2010 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:39:16 2012 Êîäèðîâêà: ISO8859-5

Author's personal copy

Available online at www.sciencedirect.com

Optics Communications 281 (2008) 3906-3912 www.elsevier.com/locate/optcom

Generation of reflected second-harmonic light beam with inhomogeneous transversal distribution of polarization from the surface of chiral medium by normally incident Gaussian beam
V.A. Makarov
b

a,b

, I.A. Perezhogin

b,*

a Physics Department, M.V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119992, Russia International Laser Center, M.V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119992, Russia

Received 18 December 2007; accepted 29 March 2008

Abstract A principle possibility of second harmonic generation (SHG) from the surface of a chiral medium by normally incident focused fundamental beam has been shown earlier, and the key features of this phenomenon (forbidden in a planewave approximation) have been outlined in [N.I. Koroteev, V.A. Makarov, S.N. Volkov, Laser Phys. 8 (1998) 532-535]. In our work we have obtained analytical expressions, which describe the distributions of intensity and polarization in the cross-section of a second-harmonic (SH) light beam. It is found that the polarization state drastically changes along the cross-section of the signal light beam. The polarization effects concerning the transversal inhomogeneous polarization distribution were studied in detail. It is shown that the measurement of the polarization state in certain areas of the SH beam cross-section gives us quantitative information directly about the medium material constants. ã 2008 Elsevier B.V. All rights reserved.
Keywords: Second harmonic generation; Elliptic polarization; Spatial dispersion; Gyrotropy; Chirality; Surface; Inhomogeneous polarization distribution

1. Introduction Surface second harmonic generation (SHG) is very often used as a tool for the investigation of complex organic molecules, which are located in surface layers or in thin films. The conditions of the appearance of second harmonic (SH) signal and the methods of detection of the response provided by the chiral properties of the molecules are well known and described in the literature [2-7]. Various schemes of SHG and ellipsometry of light at double frequency are continuously being improved. These are used for various applications such as surface diagnostic and spectroscopy studies [8-12], and in the control of film growth [13]. Modern spectroscopic studies analyze the rotation of the polarization plane of SH, the difference between the intensities of its linearly (or circularly) polar*

Corresponding author. Tel.: +7 495 939 31 47; fax: +7 495 939 31 13. E-mail address: iap1@mail.ru (I.A. Perezhogin).

ized components [14-16], and develop and improve the methods of separation of the signal at double frequency provided by the medium chirality [17]. In theoretical studies carried out within the framework of planewave approximation the authors made attempts by different ways to take into account the spatial dispersion of a nonlinear optical response of a chiral medium [18-21] as well as the surface layer inhomogeneity of the medium optical properties [22], and also to find ways of experimental separation of bulk and surface contributions in the SH signal [23]. The interconnection between the orientation of the molecules at the surface and the characteristics of the generated SH was also studied [24]. It is necessary to remark the importance of considering the spatial finiteness of light beams in their reflection from the border between two media. Already a long time ago it has been shown, that the effects having the first and the second order on the incident beam divergence angle give small (but essential) corrections to the laws of geometric optics

0030-4018/$ - see front matter ã 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.064


Author's personal copy

V.A. Makarov, I.A. Perezhogin / Optics Communications 281 (2008) 3906-3912

3907

for the reflection and refraction of light beams (both for 2D-beams [25] and 3D-beams [26]; see also Refs. 1-7 in [25] and experimental work [27]). Also these effects are responsible for the changes of the intensity and polarization distributions in the reflected and refracted beam cross-sections. Intense research in this area is also continuing presently [28-30]. In particular, these effects play a special role when the reflected signal is absent within the planewave approximation (when the incident beam is polarized in the plane of incidence and the incidence angle is equal to the Brewster angle). For the first time the authors of [1,31] have taken into account the spatial finiteness of fundamental and signal light beams in SHG from the surface of a nonlinear medium with a spatial dispersion of quadratic nonlinearity in their studies of intensity and polarization of the signal wave at double frequency. In [31] the oblique incidence of two-dimensional Gaussian beam (slit beam) has been considered, and the normal incidence of three-dimensional Gaussian beam has been studied in [1]. The optical inhomogeneity of the surface layer of a nonlinear medium has been taken into account by means of modified border conditions for electromagnetic fields. The substantiation of reasonableness of such an approach has been presented in [32,33]. It is necessary to remark that in the case of the normal incidence of a fundamental beam, SHG is forbidden within the planewave approximation. That is why in [1] the main attention had been paid to the question of possibility of SHG and calculation of characteristics of its intensity and power, using the formula obtained for the spatial Fourier-image of the reflected wave on double frequency in the far field zone. In general case this Fourier-image depends on the nonlocal optical quadratic susceptibility ^?2î of the medium c (because local susceptibility ^?2î of the medium bulk v becomes zero in the case of SHG (unlike in sum-frequency generation)) and the surface optical quadratic sus^ ^ ceptibility j?2î . Tensors ^?2î and j?2î describe the spatial c dispersion of quadratic optical response of the medium and the nonlinearity of its surface correspondingly. The spatial transversal distribution of polarization of SH radiation has not been studied neither in this work nor later. The only exception is the illustration for the simplest case of linear polarization of the incident wave, which is considered in [1]. The transversal polarization distribution in the signal beam in the three-wave mixing processes can be very specific and deserves to be explored in detail. This is already confirmed by the results of [34], where the investigation of the transversal distribution of polarization in a sumfrequency beam has been performed for the process of sum-frequency generation in a bulk of isotropic gyrotropic medium by two collinear elliptically polarized Gaussian beams. It has been shown [34] that all the parameters of light field, namely, the intensity I(r, u, z), the ellipticity degree M(r, u, z) and the angle of rotation w(r, u, z) of the main axis of the polarization ellipse, the angle

U(r, u, z) of orientation of the electric field vector at fixed timing (indicating the phase difference along the beam cross-section) essentially change in the transversal dimension of the signal beam, depending on the polar angle coordinate u. The appearance of the last parameter U(r, z) for spatially finite elliptically polarized light beams is caused by the inhomogeneity of oscillation phase distribution for the electric field vector, which is not the case, of course, for the plane wave. The other reason promoting these studies of inhomogeneous polarization distributions is the possibility of their spectroscopic applications. It is generally recognized that the values of the components of material tensors characterizing the quadratic optical nonlinearity of the surface and spatial dispersion of optical nonlinearity of medium bulk are more easily obtained if one is analyzing the dependencies of the SH polarization on the parameters of the polarization of the incident wave and on the nonlinear medium constants rather than the dependence of the SH intensity on the same quantities [10, 15-17]. But even precise analysis of the polarization dependence performed in [9] for an achiral medium can give values of relation of certain components only in the absence of spatial dispersion of bulk nonlinearity. The measurements of the relative intensities of circularly or linearly polarized components of the SH signal in a number of cases lead to a complex system of nonlinear algebraic equations, which does not always allow the convenient extraction of the information about the ^ components of ^?2î and j?2î . c The analysis of the transversal inhomogeneous polarization distributions and their dependence on the parameters of incident light and the nonlinear medium constants in terms of M(r) and w(r) (and, possibly, even U(r)) provides an alternative and effective method of extraction of the information about the nonlinear susceptibilities of the bulk and the surface of the medium. In this work we present the results of the studies of the formation of inhomogeneously polarized light beam at double frequency generated in reflection from the surface of a chiral medium in the case of a normally incident focused fundamental elliptically polarized light beam with a Gaussian intensity profile. The graphics represented by means of the analytical formulas obtained in this work illustrate the drastic changes of the polarization state along the cross-section of the signal beam. 2. Analytic solution of the problem Let us consider (as in [1]) that an elliptically polarized broad Gaussian beam (with effective waist size w >> k, where k is the light wavelength) normally falls on the surface of the medium (z = 0) E1 ?r; z Ì 0î Ì eE0 expfÐixt Ð r2 =w2 g: ? 1î

Here E0 is the scalar amplitude of the light field, e is the unit complex vector defining the polarization state of the 2 incident radiation, in general case, jej Ì 1. Its specific form


Author's personal copy

3908

V.A. Makarov, I.A. Perezhogin / Optics Communications 281 (2008) 3906-3912

depends on the orthogonal basis, which is used in a given task. We assume that the waist of the beam is located at the surface of the medium. The phase and the polarization of the incident beam are uniformly distributed in its crosssection. As the source for our further considerations we use the formula for the transversal component of the Fourierimage of the reflected light field at double frequency found in [1] (if the incident wave is given as (1)) ~ E2x ?k Ì
2x?

; z Ì 0î w2 E
2 2 0

x?1 ? n1 î ?1 ? n2x î

fÐ?b1 n

2x

? ic0 =n2x î?e Ñ eîk2x?
2 2x?

? 2n1 ?k2x? Ñ eîÍb3 e Ð b5 Íez Ò eg expfÐ2ixt Ð w2 k

=8 g; ?2î

where k2x? is the transversal part (lying in the xy-plane) of the wave vector k2x of the SH, n1 and n2x are the refraction ?2î indexes at frequencies x and 2x, respectively, b1 Ì jzxx , ?2î ?2î b3 Ì jxxz and b5 Ì jxyz are three of the four independent ^ components of 3rd rank tensor j?2î , which characterizes the quadratic optical response of the surface with symmetry 1. This tensor has permutation symmetry in its last two indexes as a consequence of pump wave frequency ?2î degeneration. c0 Ì xcxxyy is proportional to one of three independent components of the 4th rank tensor ^?2î , which c is responsible for the linear spatial dispersion of the quadratic optical bulk susceptibility of chiral medium (symmetry 11) [1,31-33]. In order to obtain the spatial distribution of light field in the reflected beam at double frequency, it is sufficient to a perform reverse Fourier-transform of (2). Taking into account the dependence of the SH light field on the propagation coordinate z within the framework of parabolic approximation, this spatial distribution is given as following: ZZ ~ E2x ?r; zî Ì E2x ?k 2x? ; z Ì 0î expfik 2x? r Ð ik 2x z Ð izk
2 2x?

show that the same studies can be performed as well for absorptive media with complex material constants. It is convenient to describe the SH light field considered 2 in this work by the normalized intensity I Ì ?jE? j ? 2 2 2 jEÐ j î=2, the ellipticity degree M Ì ?jE? j ÐjEÐ j î= 2 2 ?jE? j ?jEÐ j î and the rotation angle W Ì 0:5 arg?E? EÓ î Ð of the polarization ellipse, and also by the angle U Ì arg?E? ? EÓ î between the electric field vector and Ð the axis x of the coordinate system at fixed timing t = k2xz/2x. Of course, consideration of this parameter makes sense only if we are considering not a plane wave, but finite light beams. The expressions for these four parameters defined above contain EÁ = Ex Á iEy, which are the complex circularly polarized amplitudes of the light wave. Remarkably, Ð1 6 M 6 1. M = Á1 corresponds to the circular polarization states (with left or right rotation, depending on the sign of M), and M = 0 corresponds to the linear polarization. With these definitions of the ellipticity degree and the angle of rotation of the polarization ellipse it is reasonable to define the polarization vector of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi al fundament wave ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as the following: p e Ì ?1 ? M 0 î=2ffiffiffiÑ eiW0 eÐ ? ?1 Ð M 0 î=2 Ñ eÐiW0 e? , where p eÖ Ì ?ex Ö iey î= 2, where ex,y are the unit vectors in x- and y-axes positive directions. In this case M0 is the ellitpicity degree of the polarization ellipse of the incident wave (correspondingly to our definition) and can be varied in the range of Ð1 6 M0 6 1. Since the medium is symmetric with respect to any rotation around the axis of the incident beam, the angle of rotation w0 of the polarization ellipse of the incident wave can be taken equal to 0. At that, the polarization vector of theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi radiation will look as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi incident e Ì ? 1 ? M 0 î =2 e Ð ? ? 1 Ð M 0 î =2 e ? . By using (4) it is not difficult to obtain the following cumbersome expressions for normalized intensity, ellipticity degree, angle U in the cylindrical coordinate system with coordinates r, u and z I ?r; u; zî Ì 0:5 jD?r; zîj2 Í4n2 ?b2 ? b2 î??1 Ð M 2 î 5 13 0 Ò?2?n2x b2 ? c2 =n2x îÐ 4n1 n2x b1 b3 î 1 2 0 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ? 1 Ð M 2 fÍ4n2 ?b2 ? b2 îÐ 4n1 n2x b1 b 0 5 13 ? 4n1 b5 M 0 c0 =n2x cos 2u ?Í4n1 n2x b1 b ? 4n1 b3 M 0 c0 =n2x sin 2ug M ?uî Ì ÍM ?b ? b î??1 Ð M în1 b5 c0 =n2x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ? 1 Ð M 2 fÍM 0 ?n2 ?b2 ? b2 îÐ n1 n2x b1 b3 î 5 13 0
2 0 n1 2 3 2 5 2 0

=2k 2x gdk

2x?

?3î

Constituting (2) in (3) and carrying out double integration, we obtain E2x ?r; zî Ì 1 32piE 2 x?1 ? n î2 ?1 ? n î b ?zî w 1 2x Ò fÐ?b1 n2x ? ic0 =n2x î?e Ñ eîr
2 2 0

3

5

? 5î

? 2n1 ?r Ñ eîÍb3 e Ð b5 Íez Ò eg Ò expfÐ2ixt Ð ik 2x z Ð 2r2 =w2 b?zîg; ?4î where b?zî Ì ?1 Ð 4iz=k 2x w2 î. In principle, formula (4) describes the electric field transversal distribution in the SH beam for both absorptive and non-absorptive media (for example, in far-off-resonance and in near-resonance cases). But further, for simplicity, we consider only non-absorptive media with real values of n1, n2x, b1, b3, b5 and c0, though results of our work

? n1 b5 c0 =n2x cos 2u ?ÍM 0 n1 n2x b1 b ? n1 b3 c0 =n2x sin 2ug Ñ
2 0 2 2 2 x b1

5

=2 ? c n1 n2x b1 b3 î ??1 Ð M î?n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ? 1 Ð M 2 fÍn2 ?b2 ? b2 îÐ n1 n2x b1 b3 5 13 0 ? n1 b5 M 0 c0 =n2x cos 2u ?Ín1 n2x b1 b ? n1 b3 M 0 c0 =n2x sin 2ug
Ð1 5

Í n ? b2 ? b2 3 5 2 =?2n2x îÐ 0 2
2 1

î

? 6î


Author's personal copy

V.A. Makarov, I.A. Perezhogin / Optics Communications 281 (2008) 3906-3912

3909

W?uî Ì 0:5Arg ?Ín1 ?b3 Ð ib5 îÐ n2x b1 2 ? c2 =n2x îÑ exp?2iuî 0 2 > : ?n2 ?b3 Ð ib5 î Ñ exp?Ð2iuî 1
2

8 > <

2n1 ?b3 Ð ib5 î ? qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ín1 ?b3 Ð ib5 îÐ n2x b1 ? iM 0 c0 =n2x > ; 1 Ð M2 0 ?7î In formulas (5), (6) pffiffiffi 1 16 2 p iE2 exp?Ð2ixt Ð ik 2 zî 0 D?r; zî Ì 2 r 2 b ?zî x w2 ?1 ? n1 î ?1 ? nSH î Ò expfÐ2r2 =w2 b?zîg; x = rcosu, y = rsinu. We should notice, that formula (7) makes any sense only if M 0 6Ì Ö1 . It can be seen from (5)-(7), that M and w do not depend on the propagation coordinate z and the transversal coordinate r. The expression for the angle U between the electric field vector and the x-axis of the coordinate system at fixed timing for arbitrary values of z is very cumbersome and non-informative. This fact is circumstanced by the parabolic phase accumulation due to the traveling in the linear medium. At z = 0 this expression has the following form: U?u; z Ì 0î qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ì Argf 1 Ð M 2 Ñ?c0 =n2x îÑ exp?iuî 0 ? iM 0 n1 ?b3 Ð ib5 î exp?Ðiuîg 3. Discussion of results The expressions (5)-(8) give full information about the distributions of intensity and polarization in the transversal section of the SH beam. It can be seen from these, that I(u), M(u), w(u) U(u) are, of course, the periodic func?8î

9 > =

tions, but I(u), M(u), and w(u) have a period of p, and U(u) has a period of 2p. It is easy to show that the directions of the electric field vector in any two diametrically opposite points of the cross-section of the beam at fixed timing are opposite to each other. That is why the polarization structure of the SH beam is symmetric to 180À-rotation. Taking this in mind, we analyze I(u), M(u), w(u) and U(u) only for 0 6 u 6 p. All the four quantities (5)-(8) essentially depend on the constants characterizing the spatial dispersion of quadratic optical response of the nonlinear medium and the surface nonlinearity, and also on the polarization of the incident radiation. Fig. 1a shows the example of inhomogeneous distribution of the intensity and the polarization of the SH light field (in dimensionless units x1 = 2x/w and y1 =2y/w) in terms of the polarization ellipses in separate points of the cross-section of the SH beam at z = 0 (nearby the surface) in case of elliptic polarization of the incident radiation. The sum of squared axes of each ellipse is proportional to the intensity of light in the point, given by the center of the ellipse (at radius-vector r), the relation of the axes of the ellipse can be unambiguously expressed through the M(r), and the inclination angle of the main axis of the ellipse is equal to w(r). The orientation of the electric field vector at fixed timing in the point given by r is shown by the small circle at the boundary of the ellipse. Shaded ellipses correspond to the clockwise rotation of the polarization vector and opened ellipses correspond to the counterclockwise rotation. It can be seen from the figure that the polarization state essentially changes along the beam cross-section, and one can find regions with linear (M(r, u) = 0), elliptical (Ð1 < M(r, u) < 1) and circular (M(r, u) = Á1) polarizations. For comparison, the transversal polarization distribution of the fundamental beam is shown in Fig. 1b. Unlike in Fig. 1a, all the ellipses are oriented in the same way and have the same ellipticity degree, and U(r, z) 0.

Fig. 1. The transversal spatial distributions of light polarization in the SH beam (a) and in the fundamental beam (b). M 0 Ì 0:3, n1 Ì 1:33, n2x Ì 1:35, b3 =b1 Ì 1:2, b5 =b1 Ì 0:2, c0 =b1 Ì 2.


Author's personal copy

3910

V.A. Makarov, I.A. Perezhogin / Optics Communications 281 (2008) 3906-3912

Using (5)-(8) it is not difficult to find that in the case of the absence of a nonlocal bulk response of the medium (c0 = 0), dependencies I(M0), w(M0) (for arbitrary fixed value of u) become even functions, and M(M0) becomes an odd function. The value of U(M0) suffers a shift on p if the sign of M0 is changed. It is worth to notice that if the fundamental wave is linearly polarized in this case (i.e. M0 = 0, c0 = 0), then the SH radiation is also linearly polarized in each point of the SH beam cross-section. At that, the phase difference between the oscillations of the electric field vector in each of the two points of the crosssection can be equal either to zero or to p. Also if c0 =0, the I(u) and M(u) dependencies have the same coordinates umax and umin for minimum and maximum. If there exists a nonlocal response of the medium bulk, the dependence of the polarization state of the SH radiation becomes more complex. In this case the functions I(M0), w(M0) and M(M0) do not possess even or odd properties, but the simultaneous change of the sign of M0 b c0 preserves the values of I(M0, c0) and w(M0, c0) and their sign, and changes the sign of M(M0, c0) (preserving its absolute value). The value of U(M0, c0) suffers a shift on p under this condition. The coordinates of maxima and minima of I(u) and M(u) are different. Even if the fundamental wave is linearly polarized, the SH radiation will have an elliptical polarization in general case, and the polarization state will be inhomogeneously distributed along the SH beam cross-section. Fig. 2 demonstrates the behavior of the spatial distribution of the polarization of light at double frequency near the surface of the medium for different values of the ellipticity degree of the incident wave polarization ellipse and for fixed values of medium parameters: n1 = 1.33, n2x = 1.35, b3/b1 = 1.2, b5/b1 = 0.2, c0/b1 =2. In Fig. 2a each ellipse in the horizontal line corresponding to the given M0 (marked at ordinate-axis) shows the polarization state and the electric field vector orientation at fixed timing at the radial direction corresponding to the polar angle u (marked at abscissa axis) in the beam cross-section. Fig. 2b-d shows w(u), U(u) and M(u) for different values of M0. The range of variation of the angle of rotation w(u) (Fig. 2b) achieves maximum size, when M0 is close to zero. M(u) changes very slightly if |M0| is close to 1 (Fig. 2d). Apart from this, it can be seen from Fig 2a and d that both intensity extrema as well as both ellipticity degree extrema are located at mutually perpendicular radial directions in the beam cross-section (this also directly follows from (5) and (6)). When the incident beam is circularly polarized, the reflected beam at double frequency is also circularly polarized with the same direction of polarization rotation as in the incident beam. In this case the transversal distribution of intensity of the SH radiation has radial symmetry and is 2 2 2 proportional to ?jb3 j ?jb5 j î=jb1 j , i.e. the signal wave in this case depends only on the surface nonlinearity. In general case, when M0 6Ì 0, the equation M(u)= 0 may have two roots, one root or no real roots. In the first

case the cross-section of the SH beam is divided by two direct lines u = u1,2 (roots of the equation M(u) = 0) crossing its center into four sectors in such a way that the directions of rotation of the polarization vector in each of the two neighboring sectors are opposite, because the ellipticity degree changes its sign at these lines (such case is illustrated at Fig. 1a). These lines are given by the following expressions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1;2 Ì arctgÍ?ÐB Ö B2 Ð AC î=Aî; ? 9î

Fig. 2. The dependencies of (a) the shape and orientation of the polarization ellipse, (b) the angle of polarization ellipse rotation, (c) the orientation of the electric field vector at fixed timing and (d) the ellipticity degree of the polarization ellipse on the polar angle coordinate in the cross-section of the SH beam. Curves 1-5 correspond to M 0 Ì Ð0:8; Ð0:4; 0; 0:4; 0:8 (curves at (c) almost coincide with each other). n1 Ì 1:33, n2x Ì 1:35, b3 =b1 Ì 1:2, b5 =b1 Ì 0:2, c0 =b1 Ì 2.


Author's personal copy

V.A. Makarov, I.A. Perezhogin / Optics Communications 281 (2008) 3906-3912

3911

Fig. 3. Radially polarized SH beam, generated in the medium with b3 Ì 0, b5 Ì 0 irrespective of the fundamental wave polarization. n2x Ì 1:35, n1 Ì 1:33; c0 =b1 Ì 1.

where

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A Ì ?1 Ð 1 Ð M 2 î?M 0 n1 ?b2 ? b2 î? b5 c0 =n2x îÐ 0 3 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 M 0 ?M 0 b5 c0 =n2x ? 1 Ð M 0 Ñ n2x b1 b3 î, B Ì 1 Ð M 2 ?b3 c0 = 0 n
2x

Fig. 4. Polarization distribution in the SH beam cross-section in the case of linear polarization of fundamental wave M 0 Ì 0 and the following values of medium parameters: b3 =b1 Ì b5 =b1 Ì 2; n1 Ì 1:33; n2x Ì 1:35; c0 =b1 Ì 0:5. Lines marked at the figure correspond to roots of the equation M ?uî Ì 0. The SH radiation is linearly polarized at these lines, and the polarization plane is oriented along them.

? n2x b1 b5 M 0 î, C Ì M 0 n1 ?b2 ? b2 î??1 Ð M 2 îb5 c0 =n2x . 0 3 5 If B2 Ð AC Ì 0 then the SH radiation is linearly polarized at the direct line u = u0, where u0 = arctg[ÐB/A], and the direction of rotation of the polarization vector does not change in the beam cross-section. And, finally, in the third case, when B2 Ð AC < 0, there is only an elliptically polarized radiation at double frequency and the direction of the polarization rotation does not change in the SH beam. The following two particular cases of the considered problem are of special interest because these are connected with possible spectroscopic applications. If the nonlinear optical response of the surface can be ^ neglected j?2î 0, then for any values of M0 the SH radiation is linearly polarized in each point of the reflected beam and the polarization plane is oriented along the radius of the beam in each point: W?uîj^?2î Ì0 Ì u. Such a k radial polarization distribution in the SH beam is shown at Fig. 3. The length of each arrow at the figure is proportional to the amplitude of the light field in the corresponding point of the beam cross-section. The orientation of the arrow (equal to the orientation of the polarization plane) coincides with the polar angle u in each point. It can be shown from (9), that if the fundamental beam is linearly polarized, then there are the lines u1 = p/2 and u2 = Ðarctg(b5/b3) in the SH beam cross-section, where the radiation is linearly polarized (these lines u1,2 correspond to the roots of the equation M(u) = 0), and the plane of polarization at these lines is oriented along them, i.e. w(u = p/2; M0 = 0) = p/2, w(u = Ðarctg(b5/b3); M0 = 0) = Ðarctg(b5/b3). Fig. 4 illustrates such a case and the corresponding lines are marked there; b5/b3 = 1 and the inclination angle of the second line is equal to Ð45À. Therefore, detecting the part of the SH beam, which is close to

the certain line crossing the center of the beam (for example, by means of slit aperture), and choosing this line in such a way that the polarization of the detected light is linear and the polarization plane is oriented along the selected line, we measure the positions of these lines and the angle between them. The result of our measurement will give us the value of b5/b3. Further measurement of the intensity of the SH radiation in the case of circularly polarized pump wave will give us a combination ?jb3 j2 ?jb5 j2 î=jb1 j2 , which together with b5/b3 allows to simply find b5/b1 and b3/b1. For experimental observation of the polarization effects described above in the cross-section of the reflected beam at double frequency (for example, the search of directions in the beam, where light is linearly polarized, and the polarization plane is oriented along the radius) it is sufficient to use the method described in [35], where the polarization effects in the beam cross-section were investigated for the problem of light self-action: the radiation reflected from the nonlinear medium passes through the polarizer, and the resulting distribution of light intensity is measured on the CCD-matrix. By rotating the polarizer with sufficiently small angle step and by measuring the intensity distribution at CCD-matrix for all the positions of the polarizer, we can find the polarization of light in every point of the beam cross-section, recording the values of maximum and minimum intensity detected in the given point for all the orientations of the polarizer and the angle of polarizer orientation corresponding to maximum. If necessary, the beam falling on the CCD-matrix can be resized by means of lens or system of lenses. If one needs to measure the intensity distributions of the circularly polarized light components with either right or left rotation directly, the birefringent plate ``k/4" can be installed before the polarizer, as in [35]. In this case the intensity distribution of the corre-


Author's personal copy

3912

V.A. Makarov, I.A. Perezhogin / Optics Communications 281 (2008) 3906-3912 [3] M.J. Crawford, S. Haslam, J.M. Probert, Yu.A. Gruzdkov, J.G. Frey, Chem. Phys. Lett. 230 (1994) 260. [4] T. Petralli-Mallow, T.M. Wong, J.D. Byers, H.I. Yee, J.M. Hicks, J. Phys. Chem. 97 (1993) 1383. [5] J.J. Maki, T. Verbiest, M. Kauranen, S. Van Elshocht, A. Persoons, J. Chem. Phys. 105 (1996) 767. [6] M. Kauranen, T. Verbiest, J.J. Maki, A. Persoons, Nonlinear Opt. Princ. Mater. Phenom. Dev. 15 (1996) 13. [7] M. Kauranen, T. Verbiest, A. Persoons, J. Mod. Opt. 45 (1998) 403. [8] X. Quelin, A. Bourdon, Opt. Commun. 208 (2002) 197. [9] R.M. Plocinik, G.J. Simpson, Anal. Chim. Acta 496 (2003) 133. [10] A.J. Timson, R.D. Spencer-Smith, A.K. Alexander, R. Greef, J.G. Frey, Meas. Sci. Technol. 14 (2003) 508. [11] S. Yamada, Y. Niidome, T. Nakano, Y. Tanaka, Y. Shimada, S.I.Yin Lee, T. Matsuo, Thin Solid Films 352 (1999) 1. [12] G. Martin, E. Toussaere, L. Soulier, J. Zyss, Synth. Met. 127 (2002) 105. [13] L. Mantese, K. Selinidis, P.T. Wilson, D. Lim, Y.Y. Jiang, J.G. Ekerdt, M.C. Downer, Appl. Surf. Sci. 154-155 (2000) 229. [14] M.C. Schanne-Klein, F. Hache, T. Brotin, C. Andraud, A. Collet, Chem. Phys. Lett. 338 (2001) 159. [15] M.C. Schanne-Klein, T. Boulesteix, F. Hache, M. Alexandre, G. Lemercier, C. Andraud, Chem. Phys. Lett. 362 (2002) 103. [16] J.M. Hicks, T. Petralli-Mallow, Appl. Phys. B 68 (1999) 589. [17] J.C. Conboy, M.A. Kriech, Anal. Chim. Acta 496 (2003) 143. [18] J.J. Maki, M. Kauranen, A. Persoons, Phys. Rev. B 51 (1995) 1425. [19] M. Kauranen, T. Verbiest, A. Persoons, Nonlinear Opt. Princ. Mater. Phenom. Dev. 8 (1994) 243. [20] S. Van Elshocht, T. Verbiest, M. Kauranen, A. Persoons, B.M.W. Langeveld-Voss, E.W. Meijer, J. Chem. Phys. 107 (1997) 8201. [21] M. Kauranen, J.J. Maki, T. Verbiest, S. Van Elshocht, A. Persoons, Phys. Rev. B 55 (1997) R1985. [22] R. Stolle, M. Loddoch, G. Marowsky, Nonlinear Opt. Princ. Mater. Phenom. Dev. 8 (1994) 79. [23] Y.R. Shen, Appl. Phys. B 68 (1999) 295. [24] J.G. Frey, Chem. Phys. Lett. 323 (2000) 454. [25] M.A. Porras, Opt. Commun. 131 (1996) 13. [26] J.-J. Greffet, C. Baylard, Opt. Commun. 93 (1992) 271. [27] M. Greena, P. Kirkbya, R.S. Timsita, Phys. Lett. A 45 (1973) 259. }Ä [28] A. Kohazi-Kis, Opt. Commun. 253 (2005) 28. [29] K.Yu. Bliokh, Yu.P. Bliokh, Phys. Rev. E 75 (2007) 066609-1. [30] W. Nasalski, Phys. Rev. E 74 (2006) 056613-1. [31] N.I. Koroteev, V.A. Makarov, S.N. Volkov, Opt. Commun. 138 (1997) 113. [32] A.A. Golubkov, V.A. Makarov, J. Russ. Laser Res. 17 (1996) 480; A.A. Golubkov, V.A. Makarov, Izv. RAN, ser. fiz. (in Russian) 59 (1995) 93. [33] A.A. Golubkov, V.A. Makarov, Laser Phys. 6 (1996) 1013; A.A. Golubkov, V.A. Makarov, UFN (in Russian) 165 (1995) 339. [34] S.N. Volkov, V.A. Makarov, I.A. Perezhogin, Quantum Electron. 36 (2006) 860. [35] A. Gahl, A. Aumann, M. Moeller, W. Lange, J. Opt. B: Quantum Semiclass. Opt. 1 (1999) 121.

sponding component of light can be obtained by orienting the polarizer at either +45À or Ð45À for detecting the right circularly polarized or the left circularly polarized component. This provides an easy calculation of the ellipticity degree distribution. 4. Conclusion In this work we present the detailed analytical study of the polarization effects in the cross-section of the SH beam generated from the surface of a chiral medium by a normally incident focused Gaussian beam of a fundamental radiation. The polarization properties of SH beam depend only on the polar angle coordinate in the cross-section of the beam, and the coordinate dependencies of the intensities of the circularly polarized light components are the products of polar radius dependence and polar angle dependence. This fact essentially simplifies the analysis of these distributions and allows one to acquire the information about the medium nonlinearity by selective detection of the radiation from the certain areas of the SH beam cross-section (along certain lines crossing the center of the beam) and by subsequent measurement of the polarization state of light in these areas and also by search of the radial directions in the beam cross-section, which correspond to maximum and minimum of intensity. It is shown that in the case of the absence of surface contribution to the signal at double frequency or its negligibility compared to bulk contribution, the SH beam becomes radially polarized for any polarization state of the incident beam, and the intensity of SH is maximum if the incident beam is linearly polarized. The authors believe that the results obtained in this work are promising for future applications for spectroscopy analysis of surfaces of chiral medium or thin chiral layers and for the methods of formation of inhomogeneously polarized light beams. The authors are grateful to Dr. S.N. Volkov for valuable discussions. References
[1] N.I. Koroteev, V.A. Makarov, S.N. Volkov, Laser Phys. 8 (1998) 532. [2] J.D. Byers, H.I. Yee, T. Petralli-Mallow, J.M. Hicks, Phys. Rev. B 49 (1994) 14643.