- Email: [email protected]
Magneto-optical effects of saturating light for arbitrary field direction
15 May 1998
Optics Communications 151 Ž1998. 40–45
Magneto-optical effects of saturating light for arbitrary field direction G. Nienhuis a
a,)
, F. Schuller
b
Huygens Laboratorium, RijksuniÕersiteit Leiden, Postbus 9504, 2300 RA Leiden, Netherlands b Laboratoire de Physique des Lasers, UniÕersite´ Paris-Nord, 93430 Villetaneuse, France Received 30 October 1997; accepted 14 January 1998
Abstract We study magneto-optically induced rotation and ellipticity of resonant linearly polarized light incident on an atomic vapour. We allow for an arbitrary strength and direction of the magnetic field, and arbitrary degree of saturation. For magneto-optical rotation, the result is the sum of an absorptive and a dispersive contribution, which represent a generalized Voigt and Faraday effect respectively. These terms are not independent, since both depend on the steady-state density matrix of the atoms in the field. A similar separation is found for the induced ellipticity, with dispersive and absorptive terms interchanged. The rotation angle is numerically evaluated for a simple atomic transition. The symmetry properties of the dipole polarization are discussed. q 1998 Elsevier Science B.V. All rights reserved. PACS: 07.60.Fs ; 32.60.q i
1. Introduction When a light beam traverses a macroscopic medium with an anisotropic dielectric susceptibility, its polarization properties are modified. This modification is due to a differential dispersion or absorption for opposite polarization vectors, which reflects birefringence or dichroism. When the anisotropy is induced by an external constant magnetic field B, the two opposite circular polarizations in the plane normal to B see a different refractive index. This circular birefringence leads to a rotation of the direction of linear polarization, where the rotation angle is proportional to the propagation length. This is the Faraday effect. In its pure form, it occurs for a linearly polarized light beam propagating parallel to B. In addition, the two opposite circular polarizations also undergo a different absorption coefficient. The resulting circular dichroism induces an ellipticity in the polarization.
)
E-mail: [email protected]
The difference in absorption of the linear polarizations along and normal to the magnetic field direction will also induce an effective rotation of the linear polarization, provided that the components of the polarization parallel and normal to B are both non-vanishing. The maximal rotation occurs when one of these polarization components has been absorbed completely, so that the rotation angle can never surpass pr2. This is the Voigt effect, which results from a magnetically induced linear dichroism. Its standard configuration requires a magnetic field normal to the propagation direction. In this case, an ellipticity in the polarization is caused by the magnetically induced linear birefringence. The Faraday and the Voigt effects constitute a classic and sensitive spectroscopic tool for molecular systems w1x. In this case, the light beam is usually non-saturating, and the magnetic field is sufficiently weak, so that only the lowest non-vanishing order has to be accounted for. In the case of a nearly resonant light beam traversing atomic vapours, saturation effects are easily produced, so that the rotation angle and the ellipticity depend on the light inten-
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 0 3 4 - 0
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
sity. Several recent experiments have studied non-linear magneto-optical rotation in atomic vapours w2–5x. The interest in the Faraday effect is also stimulated by experiments aiming at the observation of parity violation w6–8x or time-reversal violation w5x in optical rotation. On the theoretical side, recent work on the Faraday effect has been focusing on atomic transitions between states with specific angular momenta: J s 0 ™ J s 1 w9x, 1 ™ 0 w10x and 1r2 ™ 1r2 w11x. Also the saturated Voigt effect has been analyzed for the 0 ™ 1 transition w12x. These works consider pure Faraday or Voigt geometries, with the wave vector k parallel or perpendicular to the magnetic field B. Because of the quite different signatures of these two effects and the required high sensitivity in the experimental signals, it is important to understand the characteristics of magneto-optical rotation in cases where the relative orientation of k and B is not perfectly controlled. In fact, optical rotation in a magnetic field with an arbitrary direction has also received attention recently w13,14x. The optical rotation is expressed in terms of the real and imaginary Ždispersive and absorptive. part of the susceptibility of the system. In an earlier treatment w15x, a number of special configurations was analyzed in the case of a 0 ™ 1 or a 1 ™ 0 transition. In the present paper we consider the magneto-optical change of polarization for an arbitrary geometry and for arbitrary values of the light intensity and the magnetic field. We study both the rotation angle and the induced ellipticity angle for an incident linearly polarized light beam. A distinctive feature of our discussion is that the description of these angles in terms of the macroscopic Maxwell equations is strictly separated from the microscopic description of the atomic vapour. This allows us to clarify the symmetry properties of the various contributions. The separation relies on the dipole nature of the atomic transition. In this case, the density matrix of the atoms, which determines the dipole polarization P of the medium, does not depend on the direction of the wave vector k. The optical activity that gives rise to rotation and to ellipticity is fully determined by the component of P normal to the polarization plane, which is spanned by the wave vector and the polarization direction. This component of the dipole polarization P is expressed in terms of generalized susceptibilities, that account for saturation effects. These susceptibilities depend exclusively on the magnetic-field strength, the light intensity, and the angle between B and the polarization direction. Their evaluation requires the calculation of the steady-state optical coherences of the atoms. The advantage of the analysis is that the information contained in magneto-optical rotation and ellipticity is more clearly identified. It is applicable to arbitrary transitions J ™ J X. As an illustration, we display the dispersive and absorptive contribution to the rotation angle for various relative orientations of the wave vector, the magnetic field and the polarization direction for the simplest atomic transition 0 ™ 1.
41
2. Optical rotation of polarization for given dipole density The amplitude, polarization and phase of a light wave propagating through a polarized medium is affected by the dipole density P. This effect is determined by Maxwell’s equation
v2 = = Ž= = E . s
c2
ž
1 Eq
´0
/
P .
Ž1.
We consider a coordinate frame Ojhz , and a monochromatic light beam with frequency v propagating in the j-direction. The electric field E and the dipole density P are then expressed as E Ž r ,t . s Re E Ž j . e i k jyi v t , P Ž r ,t . s Re P Ž j . e i k jyi v t , Ž2. Ž v s ck .. The amplitudes E and P are practically constant over a wavelength, but they may vary slowly in the propagation direction, due to absorption and dispersion effects. The left-hand side of Eq. Ž1. is fully transverse, so that it has a vanishing j-component 1 Ej q Pj s 0. Ž3. ´0 This demonstrates that the j-component of P must be compensated by a small j-component of the electric field E. The components of the field E and the polarization P in the hz-plane are denoted as Et and Pt . Neglecting the second derivative of Et , we find the propagation equation for these transverse components in the form d iv Et s P. Ž4. dj 2´0c t We assume that the beam has linear polarization, so that Et has a specific direction in space. The component of P that is parallel to Et has an in-phase and an out-of-phase part, which give rise to dispersion and absorption respectively. Likewise, the component of P that is normal to the polarization plane has an in-phase and an out-of-phase part. The in-phase part induces an ellipticity in the polarization, and the optical rotation is due to the out-of-phase part. In order to specific the rotation angle c , we choose the polarization plane as the jz-plane, so that Et s Ezˆ. Then the optical rotation is determined by the imaginary part of Ph and we find from Eq. Ž4. dc v Im PhrE. Ž 5a . dj 2´0c The ellipticity in the polarization can be described by the ellipticity angle ´ , so that tan ´ is the ratio of the short and the long axis of the polarization ellipse. For small ellipticities we find from Eq. Ž4. v d´ sy Re PhrE. Ž 5b. dj 2´0c
42
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
This represents an effective circular dichroism of the system. A non-vanishing component of P normal to the polarization plane can only occur when the symmetry of the sample for inversion through this plane is broken. In the case of natural optical activity, the symmetry breaking results from the fact that the molecules in the sample differ from their mirror image. Magneto-optical rotation occurs when the symmetry breaking is due to a magnetic field.
3. Symmetry properties of dipole density in presence of magnetic field We consider the case that the dipole density component Ph is induced by a static magnetic field B. In the frame Ojhz , the direction of B is specified by the polar angle Q and the azimuthal angle F , so that Q is the angle between E and B, and F is the angle between the polarization plane and the EB-plane. It will be convenient to decompose the dipole density P in the component P V inside the EB-plane and normal to E, the component P F normal to the EB-plane, and the component P E parallel to E. This geometry is sketched in Fig. 1. The components P V and P F will be shown to represent a generalized Voigt and Faraday effect, respectively. The optical rotation Eq. Ž5a. and the ellipticity Eq. Ž5b. are determined by the component Ph s P V sin F q PF cos F .
Ž6.
For the standard case of a vapour with an isotropic velocity distribution, the dipole density P is independent of the direction of the wave vector k. In the presence of a static magnetic field B, the evaluation of the dipole density P is best performed in a coordinate frame O xyz with the z-axis along B, illustrated in Fig. 2. The frame is fixed
Fig. 2. Coordinate frame for the evaluation of the steady-state density matrix and the generalized susceptibilities.
by the further requirement that E lies in the yz-plane, so that E y s E sin Q ,
E z s E cos Q .
Ž7.
From a comparison between Figs. 1 and 2, it is obvious that the relevant components of the dipole density P are P V s yPy cos Q q Pz sin Q ,
P F s Px .
Ž8.
For a given sample, the components Px , Py and Pz are functions of E, B and the angle Q only, and the same is true for the components P F and P V . In the absence of natural optical activity, rotation of the configuration over p about the B-axis shows that the components Px and Py are odd functions of Q , whereas the component Pz is even. This implies that both P V and P F are odd in Q . When Q is replaced by p y Q , which corresponds to inversion through the xy-plane, only Pz is inverted, and Px and Py are unchanged. Eq. Ž8. then shows that P V is inverted, and P F is invariant under this transformation. When E and B are parallel, or Q s 0, the components P V and P F both vanish, and no optical rotation or ellipticity arises. When E and B are orthogonal, so that Q s pr2, P V is zero.
4. Magneto-optical effects in terms of effective susceptibilities
Fig. 1. General configuration for magneto-optical rotation and dichroism. PV and PF are components of the dipole polarization normal to the polarization direction E, with PV in the EB-plane, and PF normal to the EB-plane.
Explicit expressions for the components of the dipole density will depend upon the specific properties of the medium, such as the values of the angular momenta of the atomic states coupling the driven transition, and radiative and collisional relaxation rates. In the absence of saturation, it is convenient to express the dipole density P in terms of a susceptibility tensor w13,14x. Because of the axial symmetry of the configuration, the eigenvectors of this tensor are given by the set of the spherical polarization vectors u "1 s .
1
'2 Ž xˆ " i yˆ . ,
u 0 s zˆ ,
Ž9.
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
and the relation between P and E is determined by the three susceptibility eigenvalues. In fact, also for an arbitrary intensity of the radiation field, the dipole polarization can be parametrized in terms of effective spherical susceptibilities Kl Ž l s y1,0,1., which we define by the ratios Kl s Plr Ž ´ 0 El . ,
Ž 10 .
with Pl s P P ul) , El s E P ul). The Cartesian components of P are expressed in terms of Pl as Px s
1
yi
'2
Ž Py1 y P1 . , Py s
'2 Ž P1 q Py1 . ,
Pz y P0 ,
Ž 11.
and the electric field ŽEq. Ž7.. has the spherical components E "1 s
iE
'2
sin Q ,
E0 s E cos Q .
Ž 12.
It is important to notice that the effective susceptibilities Kl are functions of the field amplitude E and the magnetic-field strength B, and of the angle Q between E and B. At low intensities, when saturation effects can be ignored, the susceptibilities Kl depend on the magneticfield strength B only. An expression for the magneto-optical rotation ŽEq. Ž5a.. and the circular dichroism ŽEq. Ž5b.. in terms of the effective susceptibilities Kl can now be obtained by using Eqs. Ž6., Ž8., Ž10. – Ž12.. The physical significance of the various terms is more transparent if we separate the susceptibilities as Kl s Dl q iAl ,
Ž 13.
in terms of their real Ždispersive. and imaginary Žabsorptive. parts. The result for the magneto-optical rotation is dc
v s
dj
2c
sin Q cos Q sin F
A 0y 12 Ž A1q Ay1 .
q 12 sin Q cos F Ž Dy1 y D 1 . 4 .
Ž 14a.
Likewise, we find for the circular dichroism d´
v s
dj
2c
ysin Q cos Q sin F
D 0y 12 Ž D 1q D y 1 .
q 12 sin Q cos F Ž Ay1 y A1 . .
4
Ž 14b.
The first terms in Eqs. Ž14a. and Ž14b. are due to the component P V of the dipole density, which is the component of the transverse part of P in the EB-plane. This term has an absorptive nature in Eq. Ž14a., and it is dispersive in Eq. Ž14b.. Eqs. Ž14a. and Ž14b. give the differential rotation angle and the change in ellipticity angle per unit propagation length. For vapours the total change in polarization direction usually remains small, so that the rotation angle and the ellipticity angle are well represented by the
43
product of the right-hand sides of Ž14. with the propagation length. At low intensities, the functions Al are just Zeemanshifted absorption profiles, and Dl is a Zeeman-shifted dispersion profile. When also the magnetic field is weak, the first terms in Eq. Ž14a. and Eq. Ž14b. have the spectral signature of the second derivative of an absorption profile or a dispersion profile. The second terms, which result from the component P F of P normal to the EB-plane, get the signature of the first derivative of a dispersion curve or absorption curve at low intensities and for weak magnetic fields. This demonstrates that the two terms in Eq. Ž14a. may be viewed as generalized Voigt and Faraday contributions to the optical rotation. The two terms in Eq. Ž14b. are the analogues of these contributions to magnetically induced ellipticity. The structure of Eq. Ž14a. is similar to that of eq. Ž20. of Ref. w13x, and to eq. Ž22. of Ref. w14x. These papers discuss only the case of Faraday rotation at low intensity, where the functions Al and Dl are independent of the intensity, and the polarization direction. Our result ŽEq. Ž14a.. is not restricted to low intensity. Arbitrary values both of the intensity and the magnetic-field strength are allowed. The effective absorption and dispersion profiles Al and Dl are functions of the strengths E and B, and of the angle Q between E and B. The trigonometric factors in Eqs. Ž14a. and Ž14b. have a simple significance. Fig. 1 shows directly that sin Q cos F s sin b ,
Ž 15.
with b the angle between B and the plane normal to k. Hence, the Faraday term in Eq. Ž14a. scale as the component Bj of B in the propagation direction, and the same is true for the last term in Eq. Ž14b.. Likewise, one easily checks that sin Q cos Q sin F s cos 2b sin u cos u ,
Ž 16 .
with Q the angle between E and the normal to the Bk-plane. Hence, the Voigt contribution to Eq. Ž14a., just as the first term in Eq. Ž14b., scales as the product Bz Bh of the components of B in the polarization direction and normal to the plane of polarization. For weak magnetic fields, the Voigt contribution to Eq. Ž14a. is second-order in the field, and it is roughly proportional to the second derivative of the Žpossibly saturated. absorption profile. Conversely, the lowest-order term in B of the Faraday contribution to Eq. Ž14a. is first-order, and proportional to the first derivative of the dispersion profile. Therefore, for weak magnetic fields, the Voigt contribution tends to be smaller than the Faraday term by a factor of the order of the ratio of the Zeeman shift v Z and the Žpossibly Doppler-broadened. linewidth. A similar conclusion holds
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
44
for the two contributions to the circular dichroism in Eq. Ž14b..
5. Numerical examples All that is needed to evaluate the magneto-optical rotation angle explicitly is the set of effective susceptibilities Kl. In the case of a resonant field driving a specific atomic transition, these quantities are determined by the steady-state density matrix. This density matrix depends on the beam amplitude E, the magnetic-field strength B and the angle Q between them, but not on the propagation direction. In Ref. w12x, the saturated Voigt effect is evaluated for an arbitrary angle Q between the magnetic field and the polarization direction and for an atomic transition 0 ™ 1. The analysis in the present paper shows that the same density matrix as a function of Q can also be used to evaluate the combined Faraday and Voigt rotation for an arbitrary orientation of the magnetic field. As an illustration, we display the Faraday and the Voigt contribution to the differential rotation in the simple case of a J s 0 to J s 1 transition of an atom. We introduce the dimensionless quantities RVsy RF s y
2 G´ 0 Nm2
G´ 0 Nm2
A 0 y 12 Ž A1 q Ay1 . ,
Ž Dy1 y D1 . ,
Ž 17 .
Fig. 4. Plot of the Faraday factor R F . The values of the parameters is the same as in Fig. 3.
light field, and for various values of the angle Q between the polarization direction and B. Likewise, Fig. 4 shows the corresponding Faraday term R F . One should recall that in the absence of saturation, these terms would not depend on the angle Q . Fig. 5 displays both the Voigt and the Faraday term, for a fixed value of the angle Q s pr4, but for various degrees of saturation. These curves display the simple case that collisional effects and Doppler broadening are ignored. For the transition 0 ™ 1, collisional effects are not very dramatic, since optical pumping plays no role. The effect of Doppler broadening is expressed as a standard convolution with a Maxwellian distribution of Doppler
with G the spontaneous decay rate, m the transition dipole between the ground state and any one of the substates of the excited level, and N the density of atoms. The magnetic field is chosen to have a strength that gives a Larmor precession frequency v Z s Gr2. According to Eq. Ž14. the rotation angle is a linear combination of these quantities, with angle-dependent factors. Fig. 3 displays the Voigt term R V as a function of the detuning, for a saturating
Fig. 3. Plot of the Voigt factor R V , as a function of the frequency detuning D, for a light intensity corresponding to m2 E 2 r G 2 s 2, and a Zeeman shift v Z s G r2. Solid curve: Q sp r6, dashed curve: Q sp r4, dotted curve: Q sp r3.
Fig. 5. Plot of the Voigt factor R V and the Faraday factor R F , for Q sp r4, v Z s G r2, and for various values of the intensity. Ža. No saturation; Žb. m2 E 2 r G 2 s 2; Žc. m2 E 2 r G 2 s8.
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
shifts. In the absence of saturation, the terms Al and Dl are just Zeeman-shifted absorption and dispersion profiles, so that the terms R V and R F give zero when integrated over the frequency detuning D. This is in accordance with Fig. 5a. A Doppler width that is large compared with the Zeeman shift then leads to a Faraday term proportional to the first derivative of the Doppler profile, and a Voigt term proportional to the second derivative. Hence at low intensity, the Voigt term tends to be small compared with the Faraday effect. The other curves in Fig. 5 show that this argument no longer holds for saturating intensities.
6. Conclusions We have derived general symmetry properties of magneto-optical rotation and circular dichroism for an arbitrary magnetic-field direction and arbitrary intensity. The main results Eqs. Ž14a. and Ž14b. expresses the rotation angle and the ellipticity angle in terms of generalized effective absorption and dispersion curves, that are defined in the frame with the magnetic field as quantization axis, for arbitrary values both of the magnetic-field strength and the intensity. The absorptive and dispersive terms represent a Voigt and a Faraday effect, respectively. For an atomic vapour, these terms are independent only when the Zeeman shift is small compared with the homogeneous linewidth, and the intensity is non-saturating. In general, the effective absorption and dispersion curves are functions of the light amplitude, the magnetic field strength, and the angle between the magnetic field and the linear light polarization. The result is illustrated by numerical results valid for the simplest atomic transition.
45
Acknowledgements This work is supported in part by the European Commission under contract no. CHRX-CT93-0366. References w1x L.D. Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, London, 1982. w2x K.H. Drake, W. Lange, J. Mlynek, Optics Comm. 66 Ž1988. 315. w3x L.M. Barkov, D.A. Melik-Pachayev, M.S. Zolotorev, Optics Comm. 70 Ž1989. 467. w4x X. Chen, V.L. Telegdi, A. Weis, Optics Comm. 78 Ž1990. 337. w5x B. Schuh, S.I. Kanorsky, A. Weis, T.W. Hansch, Optics ¨ Comm. 100 Ž1993. 451. w6x I.O.G. Davies, P.E.G. Baird, J.L. Nicol, J. Phys. B 20 Ž1987. 5371. w7x M.J.D. Macpherson, K.P. Zetie, R.B. Warrington, D.N. Stacey, J.P. Hoare, Phys. Rev. Lett. 67 Ž1991. 2784. w8x N.H. Edwards, S.J. Phipp, P.E.G. Baird, S. Nakayama, Phys. Rev. Lett. 74 Ž1995. 2654. w9x F. Schuller, M.J.D. Macpherson, D.N. Stacey, Physica C 147 Ž1987. 321. w10x F. Schuller, M.J.D. Macpherson, D.N. Stacey, Optics Comm. 71 Ž1989. 61. w11x F. Schuller, D.N. Stacey, R.B. Warrington, K.P. Zetie, J. Phys. B 28 Ž1995. 3783. w12x F. Schuller, M.J.D. Macpherson, D.N. Stacey, R.B. Warrington, K.P. Zetie, Optics Comm. 86 Ž1991. 123. w13x F. Schuller, R.B. Warrington, K.P. Zetie, M.J.D. Macpherson, D.N. Stacey, Optics Comm. 93 Ž1992. 169. w14x N.H. Edwards, S.J. Phipp, P.E.G. Baird, J. Phys. B 28 Ž1995. 4041. w15x S.V. Fomichev, J. Phys. B 24 Ž1991. 4695.
Optics Communications 151 Ž1998. 40–45
Magneto-optical effects of saturating light for arbitrary field direction G. Nienhuis a
a,)
, F. Schuller
b
Huygens Laboratorium, RijksuniÕersiteit Leiden, Postbus 9504, 2300 RA Leiden, Netherlands b Laboratoire de Physique des Lasers, UniÕersite´ Paris-Nord, 93430 Villetaneuse, France Received 30 October 1997; accepted 14 January 1998
Abstract We study magneto-optically induced rotation and ellipticity of resonant linearly polarized light incident on an atomic vapour. We allow for an arbitrary strength and direction of the magnetic field, and arbitrary degree of saturation. For magneto-optical rotation, the result is the sum of an absorptive and a dispersive contribution, which represent a generalized Voigt and Faraday effect respectively. These terms are not independent, since both depend on the steady-state density matrix of the atoms in the field. A similar separation is found for the induced ellipticity, with dispersive and absorptive terms interchanged. The rotation angle is numerically evaluated for a simple atomic transition. The symmetry properties of the dipole polarization are discussed. q 1998 Elsevier Science B.V. All rights reserved. PACS: 07.60.Fs ; 32.60.q i
1. Introduction When a light beam traverses a macroscopic medium with an anisotropic dielectric susceptibility, its polarization properties are modified. This modification is due to a differential dispersion or absorption for opposite polarization vectors, which reflects birefringence or dichroism. When the anisotropy is induced by an external constant magnetic field B, the two opposite circular polarizations in the plane normal to B see a different refractive index. This circular birefringence leads to a rotation of the direction of linear polarization, where the rotation angle is proportional to the propagation length. This is the Faraday effect. In its pure form, it occurs for a linearly polarized light beam propagating parallel to B. In addition, the two opposite circular polarizations also undergo a different absorption coefficient. The resulting circular dichroism induces an ellipticity in the polarization.
)
E-mail: [email protected]
The difference in absorption of the linear polarizations along and normal to the magnetic field direction will also induce an effective rotation of the linear polarization, provided that the components of the polarization parallel and normal to B are both non-vanishing. The maximal rotation occurs when one of these polarization components has been absorbed completely, so that the rotation angle can never surpass pr2. This is the Voigt effect, which results from a magnetically induced linear dichroism. Its standard configuration requires a magnetic field normal to the propagation direction. In this case, an ellipticity in the polarization is caused by the magnetically induced linear birefringence. The Faraday and the Voigt effects constitute a classic and sensitive spectroscopic tool for molecular systems w1x. In this case, the light beam is usually non-saturating, and the magnetic field is sufficiently weak, so that only the lowest non-vanishing order has to be accounted for. In the case of a nearly resonant light beam traversing atomic vapours, saturation effects are easily produced, so that the rotation angle and the ellipticity depend on the light inten-
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 0 3 4 - 0
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
sity. Several recent experiments have studied non-linear magneto-optical rotation in atomic vapours w2–5x. The interest in the Faraday effect is also stimulated by experiments aiming at the observation of parity violation w6–8x or time-reversal violation w5x in optical rotation. On the theoretical side, recent work on the Faraday effect has been focusing on atomic transitions between states with specific angular momenta: J s 0 ™ J s 1 w9x, 1 ™ 0 w10x and 1r2 ™ 1r2 w11x. Also the saturated Voigt effect has been analyzed for the 0 ™ 1 transition w12x. These works consider pure Faraday or Voigt geometries, with the wave vector k parallel or perpendicular to the magnetic field B. Because of the quite different signatures of these two effects and the required high sensitivity in the experimental signals, it is important to understand the characteristics of magneto-optical rotation in cases where the relative orientation of k and B is not perfectly controlled. In fact, optical rotation in a magnetic field with an arbitrary direction has also received attention recently w13,14x. The optical rotation is expressed in terms of the real and imaginary Ždispersive and absorptive. part of the susceptibility of the system. In an earlier treatment w15x, a number of special configurations was analyzed in the case of a 0 ™ 1 or a 1 ™ 0 transition. In the present paper we consider the magneto-optical change of polarization for an arbitrary geometry and for arbitrary values of the light intensity and the magnetic field. We study both the rotation angle and the induced ellipticity angle for an incident linearly polarized light beam. A distinctive feature of our discussion is that the description of these angles in terms of the macroscopic Maxwell equations is strictly separated from the microscopic description of the atomic vapour. This allows us to clarify the symmetry properties of the various contributions. The separation relies on the dipole nature of the atomic transition. In this case, the density matrix of the atoms, which determines the dipole polarization P of the medium, does not depend on the direction of the wave vector k. The optical activity that gives rise to rotation and to ellipticity is fully determined by the component of P normal to the polarization plane, which is spanned by the wave vector and the polarization direction. This component of the dipole polarization P is expressed in terms of generalized susceptibilities, that account for saturation effects. These susceptibilities depend exclusively on the magnetic-field strength, the light intensity, and the angle between B and the polarization direction. Their evaluation requires the calculation of the steady-state optical coherences of the atoms. The advantage of the analysis is that the information contained in magneto-optical rotation and ellipticity is more clearly identified. It is applicable to arbitrary transitions J ™ J X. As an illustration, we display the dispersive and absorptive contribution to the rotation angle for various relative orientations of the wave vector, the magnetic field and the polarization direction for the simplest atomic transition 0 ™ 1.
41
2. Optical rotation of polarization for given dipole density The amplitude, polarization and phase of a light wave propagating through a polarized medium is affected by the dipole density P. This effect is determined by Maxwell’s equation
v2 = = Ž= = E . s
c2
ž
1 Eq
´0
/
P .
Ž1.
We consider a coordinate frame Ojhz , and a monochromatic light beam with frequency v propagating in the j-direction. The electric field E and the dipole density P are then expressed as E Ž r ,t . s Re E Ž j . e i k jyi v t , P Ž r ,t . s Re P Ž j . e i k jyi v t , Ž2. Ž v s ck .. The amplitudes E and P are practically constant over a wavelength, but they may vary slowly in the propagation direction, due to absorption and dispersion effects. The left-hand side of Eq. Ž1. is fully transverse, so that it has a vanishing j-component 1 Ej q Pj s 0. Ž3. ´0 This demonstrates that the j-component of P must be compensated by a small j-component of the electric field E. The components of the field E and the polarization P in the hz-plane are denoted as Et and Pt . Neglecting the second derivative of Et , we find the propagation equation for these transverse components in the form d iv Et s P. Ž4. dj 2´0c t We assume that the beam has linear polarization, so that Et has a specific direction in space. The component of P that is parallel to Et has an in-phase and an out-of-phase part, which give rise to dispersion and absorption respectively. Likewise, the component of P that is normal to the polarization plane has an in-phase and an out-of-phase part. The in-phase part induces an ellipticity in the polarization, and the optical rotation is due to the out-of-phase part. In order to specific the rotation angle c , we choose the polarization plane as the jz-plane, so that Et s Ezˆ. Then the optical rotation is determined by the imaginary part of Ph and we find from Eq. Ž4. dc v Im PhrE. Ž 5a . dj 2´0c The ellipticity in the polarization can be described by the ellipticity angle ´ , so that tan ´ is the ratio of the short and the long axis of the polarization ellipse. For small ellipticities we find from Eq. Ž4. v d´ sy Re PhrE. Ž 5b. dj 2´0c
42
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
This represents an effective circular dichroism of the system. A non-vanishing component of P normal to the polarization plane can only occur when the symmetry of the sample for inversion through this plane is broken. In the case of natural optical activity, the symmetry breaking results from the fact that the molecules in the sample differ from their mirror image. Magneto-optical rotation occurs when the symmetry breaking is due to a magnetic field.
3. Symmetry properties of dipole density in presence of magnetic field We consider the case that the dipole density component Ph is induced by a static magnetic field B. In the frame Ojhz , the direction of B is specified by the polar angle Q and the azimuthal angle F , so that Q is the angle between E and B, and F is the angle between the polarization plane and the EB-plane. It will be convenient to decompose the dipole density P in the component P V inside the EB-plane and normal to E, the component P F normal to the EB-plane, and the component P E parallel to E. This geometry is sketched in Fig. 1. The components P V and P F will be shown to represent a generalized Voigt and Faraday effect, respectively. The optical rotation Eq. Ž5a. and the ellipticity Eq. Ž5b. are determined by the component Ph s P V sin F q PF cos F .
Ž6.
For the standard case of a vapour with an isotropic velocity distribution, the dipole density P is independent of the direction of the wave vector k. In the presence of a static magnetic field B, the evaluation of the dipole density P is best performed in a coordinate frame O xyz with the z-axis along B, illustrated in Fig. 2. The frame is fixed
Fig. 2. Coordinate frame for the evaluation of the steady-state density matrix and the generalized susceptibilities.
by the further requirement that E lies in the yz-plane, so that E y s E sin Q ,
E z s E cos Q .
Ž7.
From a comparison between Figs. 1 and 2, it is obvious that the relevant components of the dipole density P are P V s yPy cos Q q Pz sin Q ,
P F s Px .
Ž8.
For a given sample, the components Px , Py and Pz are functions of E, B and the angle Q only, and the same is true for the components P F and P V . In the absence of natural optical activity, rotation of the configuration over p about the B-axis shows that the components Px and Py are odd functions of Q , whereas the component Pz is even. This implies that both P V and P F are odd in Q . When Q is replaced by p y Q , which corresponds to inversion through the xy-plane, only Pz is inverted, and Px and Py are unchanged. Eq. Ž8. then shows that P V is inverted, and P F is invariant under this transformation. When E and B are parallel, or Q s 0, the components P V and P F both vanish, and no optical rotation or ellipticity arises. When E and B are orthogonal, so that Q s pr2, P V is zero.
4. Magneto-optical effects in terms of effective susceptibilities
Fig. 1. General configuration for magneto-optical rotation and dichroism. PV and PF are components of the dipole polarization normal to the polarization direction E, with PV in the EB-plane, and PF normal to the EB-plane.
Explicit expressions for the components of the dipole density will depend upon the specific properties of the medium, such as the values of the angular momenta of the atomic states coupling the driven transition, and radiative and collisional relaxation rates. In the absence of saturation, it is convenient to express the dipole density P in terms of a susceptibility tensor w13,14x. Because of the axial symmetry of the configuration, the eigenvectors of this tensor are given by the set of the spherical polarization vectors u "1 s .
1
'2 Ž xˆ " i yˆ . ,
u 0 s zˆ ,
Ž9.
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
and the relation between P and E is determined by the three susceptibility eigenvalues. In fact, also for an arbitrary intensity of the radiation field, the dipole polarization can be parametrized in terms of effective spherical susceptibilities Kl Ž l s y1,0,1., which we define by the ratios Kl s Plr Ž ´ 0 El . ,
Ž 10 .
with Pl s P P ul) , El s E P ul). The Cartesian components of P are expressed in terms of Pl as Px s
1
yi
'2
Ž Py1 y P1 . , Py s
'2 Ž P1 q Py1 . ,
Pz y P0 ,
Ž 11.
and the electric field ŽEq. Ž7.. has the spherical components E "1 s
iE
'2
sin Q ,
E0 s E cos Q .
Ž 12.
It is important to notice that the effective susceptibilities Kl are functions of the field amplitude E and the magnetic-field strength B, and of the angle Q between E and B. At low intensities, when saturation effects can be ignored, the susceptibilities Kl depend on the magneticfield strength B only. An expression for the magneto-optical rotation ŽEq. Ž5a.. and the circular dichroism ŽEq. Ž5b.. in terms of the effective susceptibilities Kl can now be obtained by using Eqs. Ž6., Ž8., Ž10. – Ž12.. The physical significance of the various terms is more transparent if we separate the susceptibilities as Kl s Dl q iAl ,
Ž 13.
in terms of their real Ždispersive. and imaginary Žabsorptive. parts. The result for the magneto-optical rotation is dc
v s
dj
2c
sin Q cos Q sin F
A 0y 12 Ž A1q Ay1 .
q 12 sin Q cos F Ž Dy1 y D 1 . 4 .
Ž 14a.
Likewise, we find for the circular dichroism d´
v s
dj
2c
ysin Q cos Q sin F
D 0y 12 Ž D 1q D y 1 .
q 12 sin Q cos F Ž Ay1 y A1 . .
4
Ž 14b.
The first terms in Eqs. Ž14a. and Ž14b. are due to the component P V of the dipole density, which is the component of the transverse part of P in the EB-plane. This term has an absorptive nature in Eq. Ž14a., and it is dispersive in Eq. Ž14b.. Eqs. Ž14a. and Ž14b. give the differential rotation angle and the change in ellipticity angle per unit propagation length. For vapours the total change in polarization direction usually remains small, so that the rotation angle and the ellipticity angle are well represented by the
43
product of the right-hand sides of Ž14. with the propagation length. At low intensities, the functions Al are just Zeemanshifted absorption profiles, and Dl is a Zeeman-shifted dispersion profile. When also the magnetic field is weak, the first terms in Eq. Ž14a. and Eq. Ž14b. have the spectral signature of the second derivative of an absorption profile or a dispersion profile. The second terms, which result from the component P F of P normal to the EB-plane, get the signature of the first derivative of a dispersion curve or absorption curve at low intensities and for weak magnetic fields. This demonstrates that the two terms in Eq. Ž14a. may be viewed as generalized Voigt and Faraday contributions to the optical rotation. The two terms in Eq. Ž14b. are the analogues of these contributions to magnetically induced ellipticity. The structure of Eq. Ž14a. is similar to that of eq. Ž20. of Ref. w13x, and to eq. Ž22. of Ref. w14x. These papers discuss only the case of Faraday rotation at low intensity, where the functions Al and Dl are independent of the intensity, and the polarization direction. Our result ŽEq. Ž14a.. is not restricted to low intensity. Arbitrary values both of the intensity and the magnetic-field strength are allowed. The effective absorption and dispersion profiles Al and Dl are functions of the strengths E and B, and of the angle Q between E and B. The trigonometric factors in Eqs. Ž14a. and Ž14b. have a simple significance. Fig. 1 shows directly that sin Q cos F s sin b ,
Ž 15.
with b the angle between B and the plane normal to k. Hence, the Faraday term in Eq. Ž14a. scale as the component Bj of B in the propagation direction, and the same is true for the last term in Eq. Ž14b.. Likewise, one easily checks that sin Q cos Q sin F s cos 2b sin u cos u ,
Ž 16 .
with Q the angle between E and the normal to the Bk-plane. Hence, the Voigt contribution to Eq. Ž14a., just as the first term in Eq. Ž14b., scales as the product Bz Bh of the components of B in the polarization direction and normal to the plane of polarization. For weak magnetic fields, the Voigt contribution to Eq. Ž14a. is second-order in the field, and it is roughly proportional to the second derivative of the Žpossibly saturated. absorption profile. Conversely, the lowest-order term in B of the Faraday contribution to Eq. Ž14a. is first-order, and proportional to the first derivative of the dispersion profile. Therefore, for weak magnetic fields, the Voigt contribution tends to be smaller than the Faraday term by a factor of the order of the ratio of the Zeeman shift v Z and the Žpossibly Doppler-broadened. linewidth. A similar conclusion holds
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
44
for the two contributions to the circular dichroism in Eq. Ž14b..
5. Numerical examples All that is needed to evaluate the magneto-optical rotation angle explicitly is the set of effective susceptibilities Kl. In the case of a resonant field driving a specific atomic transition, these quantities are determined by the steady-state density matrix. This density matrix depends on the beam amplitude E, the magnetic-field strength B and the angle Q between them, but not on the propagation direction. In Ref. w12x, the saturated Voigt effect is evaluated for an arbitrary angle Q between the magnetic field and the polarization direction and for an atomic transition 0 ™ 1. The analysis in the present paper shows that the same density matrix as a function of Q can also be used to evaluate the combined Faraday and Voigt rotation for an arbitrary orientation of the magnetic field. As an illustration, we display the Faraday and the Voigt contribution to the differential rotation in the simple case of a J s 0 to J s 1 transition of an atom. We introduce the dimensionless quantities RVsy RF s y
2 G´ 0 Nm2
G´ 0 Nm2
A 0 y 12 Ž A1 q Ay1 . ,
Ž Dy1 y D1 . ,
Ž 17 .
Fig. 4. Plot of the Faraday factor R F . The values of the parameters is the same as in Fig. 3.
light field, and for various values of the angle Q between the polarization direction and B. Likewise, Fig. 4 shows the corresponding Faraday term R F . One should recall that in the absence of saturation, these terms would not depend on the angle Q . Fig. 5 displays both the Voigt and the Faraday term, for a fixed value of the angle Q s pr4, but for various degrees of saturation. These curves display the simple case that collisional effects and Doppler broadening are ignored. For the transition 0 ™ 1, collisional effects are not very dramatic, since optical pumping plays no role. The effect of Doppler broadening is expressed as a standard convolution with a Maxwellian distribution of Doppler
with G the spontaneous decay rate, m the transition dipole between the ground state and any one of the substates of the excited level, and N the density of atoms. The magnetic field is chosen to have a strength that gives a Larmor precession frequency v Z s Gr2. According to Eq. Ž14. the rotation angle is a linear combination of these quantities, with angle-dependent factors. Fig. 3 displays the Voigt term R V as a function of the detuning, for a saturating
Fig. 3. Plot of the Voigt factor R V , as a function of the frequency detuning D, for a light intensity corresponding to m2 E 2 r G 2 s 2, and a Zeeman shift v Z s G r2. Solid curve: Q sp r6, dashed curve: Q sp r4, dotted curve: Q sp r3.
Fig. 5. Plot of the Voigt factor R V and the Faraday factor R F , for Q sp r4, v Z s G r2, and for various values of the intensity. Ža. No saturation; Žb. m2 E 2 r G 2 s 2; Žc. m2 E 2 r G 2 s8.
G. Nienhuis, F. Schullerr Optics Communications 151 (1998) 40–45
shifts. In the absence of saturation, the terms Al and Dl are just Zeeman-shifted absorption and dispersion profiles, so that the terms R V and R F give zero when integrated over the frequency detuning D. This is in accordance with Fig. 5a. A Doppler width that is large compared with the Zeeman shift then leads to a Faraday term proportional to the first derivative of the Doppler profile, and a Voigt term proportional to the second derivative. Hence at low intensity, the Voigt term tends to be small compared with the Faraday effect. The other curves in Fig. 5 show that this argument no longer holds for saturating intensities.
6. Conclusions We have derived general symmetry properties of magneto-optical rotation and circular dichroism for an arbitrary magnetic-field direction and arbitrary intensity. The main results Eqs. Ž14a. and Ž14b. expresses the rotation angle and the ellipticity angle in terms of generalized effective absorption and dispersion curves, that are defined in the frame with the magnetic field as quantization axis, for arbitrary values both of the magnetic-field strength and the intensity. The absorptive and dispersive terms represent a Voigt and a Faraday effect, respectively. For an atomic vapour, these terms are independent only when the Zeeman shift is small compared with the homogeneous linewidth, and the intensity is non-saturating. In general, the effective absorption and dispersion curves are functions of the light amplitude, the magnetic field strength, and the angle between the magnetic field and the linear light polarization. The result is illustrated by numerical results valid for the simplest atomic transition.
45
Acknowledgements This work is supported in part by the European Commission under contract no. CHRX-CT93-0366. References w1x L.D. Barron, Molecular Light Scattering and Optical Activity, Cambridge University Press, London, 1982. w2x K.H. Drake, W. Lange, J. Mlynek, Optics Comm. 66 Ž1988. 315. w3x L.M. Barkov, D.A. Melik-Pachayev, M.S. Zolotorev, Optics Comm. 70 Ž1989. 467. w4x X. Chen, V.L. Telegdi, A. Weis, Optics Comm. 78 Ž1990. 337. w5x B. Schuh, S.I. Kanorsky, A. Weis, T.W. Hansch, Optics ¨ Comm. 100 Ž1993. 451. w6x I.O.G. Davies, P.E.G. Baird, J.L. Nicol, J. Phys. B 20 Ž1987. 5371. w7x M.J.D. Macpherson, K.P. Zetie, R.B. Warrington, D.N. Stacey, J.P. Hoare, Phys. Rev. Lett. 67 Ž1991. 2784. w8x N.H. Edwards, S.J. Phipp, P.E.G. Baird, S. Nakayama, Phys. Rev. Lett. 74 Ž1995. 2654. w9x F. Schuller, M.J.D. Macpherson, D.N. Stacey, Physica C 147 Ž1987. 321. w10x F. Schuller, M.J.D. Macpherson, D.N. Stacey, Optics Comm. 71 Ž1989. 61. w11x F. Schuller, D.N. Stacey, R.B. Warrington, K.P. Zetie, J. Phys. B 28 Ž1995. 3783. w12x F. Schuller, M.J.D. Macpherson, D.N. Stacey, R.B. Warrington, K.P. Zetie, Optics Comm. 86 Ž1991. 123. w13x F. Schuller, R.B. Warrington, K.P. Zetie, M.J.D. Macpherson, D.N. Stacey, Optics Comm. 93 Ž1992. 169. w14x N.H. Edwards, S.J. Phipp, P.E.G. Baird, J. Phys. B 28 Ž1995. 4041. w15x S.V. Fomichev, J. Phys. B 24 Ž1991. 4695.