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Magnetic-dipole susceptibilities in electric-field induced second-harmonic generation
Optical Materials 21 (2002) 7–10 www.elsevier.com/locate/optmat
Magnetic-dipole susceptibilities in electric-field induced second-harmonic generation Sonja Sioncke *, Thierry Verbiest, Andre Persoons Laboratory of Chemical and Biological Dynamics, University of Leuven, Celestijnenlaan 200 D, B-3001 Heverlee, Leuven, Belgium
Abstract A new measurement technique has been developed to determine the magnitude of magnetic-dipole contributions to the second-order nonlinearity of oriented solutions of chiral molecules. The method is based on a modified electric-field induced second-harmonic generation technique. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.65.K; 11.30.R Keywords: Chirality; Magnetic-dipole contributions; Second-order nonlinear optics
Chiral materials possess no reflection symmetry and as a consequence occur in two enantiomers that are mirror images of each other. In linear optics, chiral materials are known for their optical activity, e.g. circular dichroism and polarization rotation [1,2]. For isotropic chiral materials, optical activity effects are due to contributions of magnetic-dipole interactions to the linear optical properties of the medium. Recently, it has been shown that chirality can also lead to optical activity in the nonlinear response. For example, the secondharmonic efficiency of Langmuir–Blodgett films of chiral polymers depends on the handedness of incoming circularly polarized fundamental light [3,4]. However, the origin of this nonlinear optical activity effect is more complex than in linear optics. From general arguments, it is known that nonlin* Corresponding author. Tel.: +32-16-32-71-69; fax: +32-1632-79-82. E-mail addresses: [email protected] (S. Sioncke), [email protected] (A. Persoons).
ear optical activity can occur solely from electricdipole transitions, but it is known that magneticdipole transitions can also contribute [5]. In fact, recent SHG measurements on Langmuir–Blodgett films indicate that magnetic-dipole contributions to the nonlinearity can be of the same order of magnitude than the usual electric-dipole contributions [3,4]. Hence, the nonlinear optical (NLO) properties of chiral materials can be significantly enhanced by optimizing magnetic-dipole contributions to the nonlinearity, in addition to the usual electric-dipole contributions. In addition, the symmetry properties of magnetic-dipole interactions are different from those of electric-dipole interactions. As a consequence, they can support electricdipole forbidden second-order NLO processes in highly symmetric media [5]. Hence, the importance of magnetic interactions in chiral media could therefore provide a new approach to the development of second-order NLO materials. For this reason, measurement techniques that specifically probe magnetic contributions to the nonlinearity
0925-3467/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 3 4 6 7 ( 0 2 ) 0 0 1 0 5 - 2
8
S. Sioncke et al. / Optical Materials 21 (2002) 7–10
are needed. Here, we propose a new measurement procedure based on the detection of elliptically polarized second-harmonic light that can be used to study the role of magnetic contributions in chiral materials. The interaction between the second-order nonlinear (chiral) material and the incident electromagnetic field can be treated to various degrees of detail. In most materials, higher multi-pole contributions to the nonlinearity can be neglected with respect to the electric-dipole interaction. However, for chiral materials, the importance of magnetic contributions is well established [6], and one can argue that chirality favors magnetic interactions. Therefore, we will consider both electric-dipole and magnetic-dipole contributions to treat the second-order nonlinear optical properties of chiral materials. Hence, for the case of second-harmonic generation we can define a second-order polarization at frequency 2x as [3,4]: X eem Pi ð2xÞ ¼ bveee ijk Ej ðxÞEk ðxÞ þ vijk Ej ðxÞBk ðxÞc
by orienting chiral dipolar molecules in solution by an external static electric field. For a C1 symmetry and for the process of second-harmonic generation, there are four independent components of the veee and vmee tensors and seven independent components for the veem tensor: eee eee eee eee eee eee veee zzz ; vzxx ¼ vzyy ; vxxz ¼ vyyz ¼ vxzx ¼ vyzy ; eee eee eee veee xyz ¼ vxzy ¼ vyxz ¼ vyzx mee mee mee mee mee mee vmee zzz ; vzxx ¼ vzyy ; vxxz ¼ vyyz ¼ vxzx ¼ vyzy ; mee mee mee vmee xyz ¼ vxzy ¼ vyxz ¼ vyzx eem eem eem eem eem eem veem zzz ; vzxx ¼ vzyy ; vxxz ¼ vyyz ; vxzx ¼ vyzy ; eem eem eem eem eem veem xyz ¼ vyxz ; vxzy ¼ vyzx ; vzxy ¼ vzyx
Next if we take a fundamental laser beam that is traveling along the x-direction and polarized along the z-direction, the only components of the electric and magnetic field that have to be considered are Ez ðxÞ and By ðxÞ. Hence, the polarization and magnetization components that are able to radiate are
j;k
ð1Þ where we have included magnetic contributions up to first order of the magnetic interaction. The subscripts ijk refer to Cartesian coordinates in the laboratory (macroscopic) frame. The superscripts in the susceptibility components of the tensors v associate the respective subscripts with electricdipole (e) or magnetic-dipole (m) interactions. In addition, we also have to define a secondorder magnetization at frequency 2x: X Mi ð2xÞ ¼ vmee ð2Þ ijk Ej ðxÞEk ðxÞ j;k
Since both polarization and magnetization act as sources of radiation, we can define an effective second-order polarization Peff ð2xÞ, that combines both contributions, as [7,8]: c r Mð2xÞ ð3Þ Peff ð2xÞ ¼ Pð2xÞ þ i2x Let us consider a chiral medium with net orientation along the z-direction and with an orientational distribution of the molecules that is isotropic in the xy-plane. The symmetry of such a system is C1 and this type of medium can for example be prepared
Pz ð2xÞ ¼ veee zzz Ez ðxÞEz ðxÞ; Py ð2xÞ ¼ veem yzy Ez ðxÞBy ðxÞ; Mz ð2xÞ ¼ vmee zzz Ez ðxÞEz ðxÞ
ð4Þ
and the effective polarization becomes Peff ð2xÞ ¼ veee zzz Ez ðxÞEz ðxÞz mee þ ðveem yzy þ vzzz ÞEz ðxÞEz ðxÞy
ð5Þ
This expression can now be used to describe the second-harmonic response of the chiral medium with C1 -symmetry. For off-resonance excitation, the components of veem and vmee have been shown theoretically to be imaginary-values quantities, while the components of veee are essentially real [9,10]. In general however (i.e. closer to resonance), there will be an arbitrary phase difference between both contributions. Hence, if we take a closer look at Eq. (5), it is clear that there will generally be a phase difference between the z- and y-component of the effective polarization. The effective polarization will generate two secondharmonic fields, one z-component Ez ð2xÞ and one y-component Ey ð2xÞ that differ in phase, and thus give rise to elliptically polarized SH-light. Hence,
S. Sioncke et al. / Optical Materials 21 (2002) 7–10
the presence of ellipticity in the second-harmonic light is evidence of the presence of magnetic-dipole contributions to the nonlinearity. Moreover, the mee combination veem yzy þ vzzz in Eq. (5) only exists in chiral samples and the presence of elliptically polarized light is therefore also an indication for the presence of chirality in the material system. The materials system and experimental geometry considered in the previous paragraph can easily be accomplished from a practical point of view. The sample consists of a solution of a chiral dipolar molecule (or polymer) in a traditional EFISHG (electric-field induced second-harmonic generation)-cell. An external electric field (oriented along the z-direction) aligns the molecules and creates a medium with the desired C1 symmetry. Fundamental light from a pulsed Nd-YAG laser (1064 nm) is polarized along the z-direction and is directed through the EFISHG-cell. Whereas for an achiral medium the SHG-signal emanating from the cell is vertically polarized, the SHG-signal for a chiral medium with magnetic contributions will be elliptically polarized. The ellipticity of the secondharmonic light is detected by using a quarter-wave plate that continuously modulates the polarization of the SHG-signal. The horizontally (Ey ð2xÞ) and vertically (Ez ð2xÞ) polarized components of the second-harmonic light (after being modulated by the quarter-wave plate) are detected by two photomultipliers. The difference between both signals DI is then plotted as a function of the angle of the quarter-wave plate. If we write Ez ð2x) en Ey ð2xÞ as: Ez ð2xÞ ¼ E1 eixt
9
E1 is related toveee zzz and E2 is proportional to mee eem mee veem þ v . Hence, veee can be yzy zzz zzz and vyzy þ vzzz obtained by fitting the experimental data to Eq. (7). In our experiments we used a solution of p-nitroaniline as an achiral reference and a chloroform solution of poly-c-benzyl-L -glutamate as the chiral sample. Poly-c-benzyl-L -glutamate is a synthetic protein and adopts a helical configuration in certain solvents (e.g. chloroform). The helical structure is expected to enhance the magnetic contributions. Second-harmonic generation from a Poly-c-benzyl-L -glutamate solution in an EFISHG experiment has been reported by Levine et al., but the role of magnetic-dipole contributions was not discussed [11]. The rotation patterns of both products are shown in Figs. 1 and 2. The experimental data points were fitted to Eq. (7). The most important aspect of the polarization line shapes is the symmetry of the plot with respect to the 180° rotation angle. Whereas for an achiral sample the polarization line shape is fully symmetric with respect to the 180°, the line shape for a chiral sample becomes asymmetric in the presence of magnetic contributions. This can be clearly seen in Fig. 1 (p-nitroaniline) and 2(poly-c-benzyl-L -glutamate). Hence, from Fig. 2 we can conclude that magnetic contributions to the nonlinearity are present in the poly-c-benzyl-L -glutamate sample. At this point it
ð6Þ
Ey ð2xÞ ¼ E2 eiðxtþ/Þ
we can express the intensity difference DI as a function of E1 , E2 , / the phase difference between Ez ð2xÞ and Ey ð2xÞ and the angle of the quarter wave plate h: DI ¼ ððE1 sinð2hÞ sinð/ÞE2 Þ2 þ ðcosð2hÞE1 þ sinð2hÞ cosð/ÞE2 Þ2 2
ððcosð/Þ þ cosð2hÞ sinð/ÞÞ ÞE22 2
þ ðsinð2hÞE1 þ E2 ðsinð/Þ cosð2hÞ cosð/ÞÞ Þ ð7Þ
Fig. 1. DI as a function of the rotation angle h of the quarterwave plate for a solution of p-nitroaniline.
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S. Sioncke et al. / Optical Materials 21 (2002) 7–10
Acknowledgements This work was supported by research grants from the Fund for Scientific Research-Flanders (FWO-V, no. G.0338.98 and 9.0407.98), from the Belgian government (No. IUAP P4/11), and from the University of Leuven (No. GOA/2000/03). T.V. is a Postdoctoral Fellow and S.S. a Research Assistant of the Fund for Scientific ResearchFlanders. References
Fig. 2. DI as a function of the rotation angle h of the quarterwave plate for a solution of poly-c -benzyl-L -glutamate.
was not possible to quantitatively calculate the magnitude of the magnetic contributions: mechanical stress in the windows of the EFISHG-cell considerably complicates the measurements and only qualitative conclusions can be drawn. However, we are quite confident in our experimental results. The achiral reference shows no asymmetry, while the chiral solution is clearly asymmetric. Asymmetry caused by experimental error (e.g. by mechanical stress in the cell windows) would show up in both the chiral as well as in the achiral sample.
[1] L.D. Barron, Molecular Light Scattering and Optical Activity, Cambridge, Cambridge, 1982. [2] G. Wagniere, Linear and Nonlinear Optical Properties of Molecules, VCH, Weinheim, 1993. [3] M. Kauranen, T. Verbiest, E.W. Meijer, E.E. Havinga, M.N. Teerenstra, A.J. Schouten, R.J.M. Nolte, A. Persoons, Adv. Mater. 7 (1995) 641. [4] S. Van Elshocht, T. Verbiest, M. Kauranen, A. Persoons, B.M.W. Langeveld-Voss, E.W. Meijer, J. Chem. Phys. 107 (1997) 8201. [5] M. Kauranen, T. Verbiest, A. Persoons, J. Nonlinear Opt. Phys. Mater. 8 (1999) 171. [6] E.W. Meijer, E.E. Havinga, G.L.J.A. Rikken, Phys. Rev. Lett. 65 (1990) 37. [7] J.D. Jackson, Classical Electrodynamics, third ed., WileyInterscience, New York, 1999. [8] Y.R. Shen, The Principles of Nonlinear Optics, WileyInterscience, New York, 1984. [9] P.S. Pershan, Phys. Rev. 130 (1963) 919. [10] E. Adler, Phys. Rev. 134 (1964) A728. [11] B.F. Levine, C.G. Bethea, J. Chem. Phys. 65 (1976) 1989.
Magnetic-dipole susceptibilities in electric-field induced second-harmonic generation Sonja Sioncke *, Thierry Verbiest, Andre Persoons Laboratory of Chemical and Biological Dynamics, University of Leuven, Celestijnenlaan 200 D, B-3001 Heverlee, Leuven, Belgium
Abstract A new measurement technique has been developed to determine the magnitude of magnetic-dipole contributions to the second-order nonlinearity of oriented solutions of chiral molecules. The method is based on a modified electric-field induced second-harmonic generation technique. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.65.K; 11.30.R Keywords: Chirality; Magnetic-dipole contributions; Second-order nonlinear optics
Chiral materials possess no reflection symmetry and as a consequence occur in two enantiomers that are mirror images of each other. In linear optics, chiral materials are known for their optical activity, e.g. circular dichroism and polarization rotation [1,2]. For isotropic chiral materials, optical activity effects are due to contributions of magnetic-dipole interactions to the linear optical properties of the medium. Recently, it has been shown that chirality can also lead to optical activity in the nonlinear response. For example, the secondharmonic efficiency of Langmuir–Blodgett films of chiral polymers depends on the handedness of incoming circularly polarized fundamental light [3,4]. However, the origin of this nonlinear optical activity effect is more complex than in linear optics. From general arguments, it is known that nonlin* Corresponding author. Tel.: +32-16-32-71-69; fax: +32-1632-79-82. E-mail addresses: [email protected] (S. Sioncke), [email protected] (A. Persoons).
ear optical activity can occur solely from electricdipole transitions, but it is known that magneticdipole transitions can also contribute [5]. In fact, recent SHG measurements on Langmuir–Blodgett films indicate that magnetic-dipole contributions to the nonlinearity can be of the same order of magnitude than the usual electric-dipole contributions [3,4]. Hence, the nonlinear optical (NLO) properties of chiral materials can be significantly enhanced by optimizing magnetic-dipole contributions to the nonlinearity, in addition to the usual electric-dipole contributions. In addition, the symmetry properties of magnetic-dipole interactions are different from those of electric-dipole interactions. As a consequence, they can support electricdipole forbidden second-order NLO processes in highly symmetric media [5]. Hence, the importance of magnetic interactions in chiral media could therefore provide a new approach to the development of second-order NLO materials. For this reason, measurement techniques that specifically probe magnetic contributions to the nonlinearity
0925-3467/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 3 4 6 7 ( 0 2 ) 0 0 1 0 5 - 2
8
S. Sioncke et al. / Optical Materials 21 (2002) 7–10
are needed. Here, we propose a new measurement procedure based on the detection of elliptically polarized second-harmonic light that can be used to study the role of magnetic contributions in chiral materials. The interaction between the second-order nonlinear (chiral) material and the incident electromagnetic field can be treated to various degrees of detail. In most materials, higher multi-pole contributions to the nonlinearity can be neglected with respect to the electric-dipole interaction. However, for chiral materials, the importance of magnetic contributions is well established [6], and one can argue that chirality favors magnetic interactions. Therefore, we will consider both electric-dipole and magnetic-dipole contributions to treat the second-order nonlinear optical properties of chiral materials. Hence, for the case of second-harmonic generation we can define a second-order polarization at frequency 2x as [3,4]: X eem Pi ð2xÞ ¼ bveee ijk Ej ðxÞEk ðxÞ þ vijk Ej ðxÞBk ðxÞc
by orienting chiral dipolar molecules in solution by an external static electric field. For a C1 symmetry and for the process of second-harmonic generation, there are four independent components of the veee and vmee tensors and seven independent components for the veem tensor: eee eee eee eee eee eee veee zzz ; vzxx ¼ vzyy ; vxxz ¼ vyyz ¼ vxzx ¼ vyzy ; eee eee eee veee xyz ¼ vxzy ¼ vyxz ¼ vyzx mee mee mee mee mee mee vmee zzz ; vzxx ¼ vzyy ; vxxz ¼ vyyz ¼ vxzx ¼ vyzy ; mee mee mee vmee xyz ¼ vxzy ¼ vyxz ¼ vyzx eem eem eem eem eem eem veem zzz ; vzxx ¼ vzyy ; vxxz ¼ vyyz ; vxzx ¼ vyzy ; eem eem eem eem eem veem xyz ¼ vyxz ; vxzy ¼ vyzx ; vzxy ¼ vzyx
Next if we take a fundamental laser beam that is traveling along the x-direction and polarized along the z-direction, the only components of the electric and magnetic field that have to be considered are Ez ðxÞ and By ðxÞ. Hence, the polarization and magnetization components that are able to radiate are
j;k
ð1Þ where we have included magnetic contributions up to first order of the magnetic interaction. The subscripts ijk refer to Cartesian coordinates in the laboratory (macroscopic) frame. The superscripts in the susceptibility components of the tensors v associate the respective subscripts with electricdipole (e) or magnetic-dipole (m) interactions. In addition, we also have to define a secondorder magnetization at frequency 2x: X Mi ð2xÞ ¼ vmee ð2Þ ijk Ej ðxÞEk ðxÞ j;k
Since both polarization and magnetization act as sources of radiation, we can define an effective second-order polarization Peff ð2xÞ, that combines both contributions, as [7,8]: c r Mð2xÞ ð3Þ Peff ð2xÞ ¼ Pð2xÞ þ i2x Let us consider a chiral medium with net orientation along the z-direction and with an orientational distribution of the molecules that is isotropic in the xy-plane. The symmetry of such a system is C1 and this type of medium can for example be prepared
Pz ð2xÞ ¼ veee zzz Ez ðxÞEz ðxÞ; Py ð2xÞ ¼ veem yzy Ez ðxÞBy ðxÞ; Mz ð2xÞ ¼ vmee zzz Ez ðxÞEz ðxÞ
ð4Þ
and the effective polarization becomes Peff ð2xÞ ¼ veee zzz Ez ðxÞEz ðxÞz mee þ ðveem yzy þ vzzz ÞEz ðxÞEz ðxÞy
ð5Þ
This expression can now be used to describe the second-harmonic response of the chiral medium with C1 -symmetry. For off-resonance excitation, the components of veem and vmee have been shown theoretically to be imaginary-values quantities, while the components of veee are essentially real [9,10]. In general however (i.e. closer to resonance), there will be an arbitrary phase difference between both contributions. Hence, if we take a closer look at Eq. (5), it is clear that there will generally be a phase difference between the z- and y-component of the effective polarization. The effective polarization will generate two secondharmonic fields, one z-component Ez ð2xÞ and one y-component Ey ð2xÞ that differ in phase, and thus give rise to elliptically polarized SH-light. Hence,
S. Sioncke et al. / Optical Materials 21 (2002) 7–10
the presence of ellipticity in the second-harmonic light is evidence of the presence of magnetic-dipole contributions to the nonlinearity. Moreover, the mee combination veem yzy þ vzzz in Eq. (5) only exists in chiral samples and the presence of elliptically polarized light is therefore also an indication for the presence of chirality in the material system. The materials system and experimental geometry considered in the previous paragraph can easily be accomplished from a practical point of view. The sample consists of a solution of a chiral dipolar molecule (or polymer) in a traditional EFISHG (electric-field induced second-harmonic generation)-cell. An external electric field (oriented along the z-direction) aligns the molecules and creates a medium with the desired C1 symmetry. Fundamental light from a pulsed Nd-YAG laser (1064 nm) is polarized along the z-direction and is directed through the EFISHG-cell. Whereas for an achiral medium the SHG-signal emanating from the cell is vertically polarized, the SHG-signal for a chiral medium with magnetic contributions will be elliptically polarized. The ellipticity of the secondharmonic light is detected by using a quarter-wave plate that continuously modulates the polarization of the SHG-signal. The horizontally (Ey ð2xÞ) and vertically (Ez ð2xÞ) polarized components of the second-harmonic light (after being modulated by the quarter-wave plate) are detected by two photomultipliers. The difference between both signals DI is then plotted as a function of the angle of the quarter-wave plate. If we write Ez ð2x) en Ey ð2xÞ as: Ez ð2xÞ ¼ E1 eixt
9
E1 is related toveee zzz and E2 is proportional to mee eem mee veem þ v . Hence, veee can be yzy zzz zzz and vyzy þ vzzz obtained by fitting the experimental data to Eq. (7). In our experiments we used a solution of p-nitroaniline as an achiral reference and a chloroform solution of poly-c-benzyl-L -glutamate as the chiral sample. Poly-c-benzyl-L -glutamate is a synthetic protein and adopts a helical configuration in certain solvents (e.g. chloroform). The helical structure is expected to enhance the magnetic contributions. Second-harmonic generation from a Poly-c-benzyl-L -glutamate solution in an EFISHG experiment has been reported by Levine et al., but the role of magnetic-dipole contributions was not discussed [11]. The rotation patterns of both products are shown in Figs. 1 and 2. The experimental data points were fitted to Eq. (7). The most important aspect of the polarization line shapes is the symmetry of the plot with respect to the 180° rotation angle. Whereas for an achiral sample the polarization line shape is fully symmetric with respect to the 180°, the line shape for a chiral sample becomes asymmetric in the presence of magnetic contributions. This can be clearly seen in Fig. 1 (p-nitroaniline) and 2(poly-c-benzyl-L -glutamate). Hence, from Fig. 2 we can conclude that magnetic contributions to the nonlinearity are present in the poly-c-benzyl-L -glutamate sample. At this point it
ð6Þ
Ey ð2xÞ ¼ E2 eiðxtþ/Þ
we can express the intensity difference DI as a function of E1 , E2 , / the phase difference between Ez ð2xÞ and Ey ð2xÞ and the angle of the quarter wave plate h: DI ¼ ððE1 sinð2hÞ sinð/ÞE2 Þ2 þ ðcosð2hÞE1 þ sinð2hÞ cosð/ÞE2 Þ2 2
ððcosð/Þ þ cosð2hÞ sinð/ÞÞ ÞE22 2
þ ðsinð2hÞE1 þ E2 ðsinð/Þ cosð2hÞ cosð/ÞÞ Þ ð7Þ
Fig. 1. DI as a function of the rotation angle h of the quarterwave plate for a solution of p-nitroaniline.
10
S. Sioncke et al. / Optical Materials 21 (2002) 7–10
Acknowledgements This work was supported by research grants from the Fund for Scientific Research-Flanders (FWO-V, no. G.0338.98 and 9.0407.98), from the Belgian government (No. IUAP P4/11), and from the University of Leuven (No. GOA/2000/03). T.V. is a Postdoctoral Fellow and S.S. a Research Assistant of the Fund for Scientific ResearchFlanders. References
Fig. 2. DI as a function of the rotation angle h of the quarterwave plate for a solution of poly-c -benzyl-L -glutamate.
was not possible to quantitatively calculate the magnitude of the magnetic contributions: mechanical stress in the windows of the EFISHG-cell considerably complicates the measurements and only qualitative conclusions can be drawn. However, we are quite confident in our experimental results. The achiral reference shows no asymmetry, while the chiral solution is clearly asymmetric. Asymmetry caused by experimental error (e.g. by mechanical stress in the cell windows) would show up in both the chiral as well as in the achiral sample.
[1] L.D. Barron, Molecular Light Scattering and Optical Activity, Cambridge, Cambridge, 1982. [2] G. Wagniere, Linear and Nonlinear Optical Properties of Molecules, VCH, Weinheim, 1993. [3] M. Kauranen, T. Verbiest, E.W. Meijer, E.E. Havinga, M.N. Teerenstra, A.J. Schouten, R.J.M. Nolte, A. Persoons, Adv. Mater. 7 (1995) 641. [4] S. Van Elshocht, T. Verbiest, M. Kauranen, A. Persoons, B.M.W. Langeveld-Voss, E.W. Meijer, J. Chem. Phys. 107 (1997) 8201. [5] M. Kauranen, T. Verbiest, A. Persoons, J. Nonlinear Opt. Phys. Mater. 8 (1999) 171. [6] E.W. Meijer, E.E. Havinga, G.L.J.A. Rikken, Phys. Rev. Lett. 65 (1990) 37. [7] J.D. Jackson, Classical Electrodynamics, third ed., WileyInterscience, New York, 1999. [8] Y.R. Shen, The Principles of Nonlinear Optics, WileyInterscience, New York, 1984. [9] P.S. Pershan, Phys. Rev. 130 (1963) 919. [10] E. Adler, Phys. Rev. 134 (1964) A728. [11] B.F. Levine, C.G. Bethea, J. Chem. Phys. 65 (1976) 1989.