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Raman scattering using vortex light
Author’s Accepted Manuscript Raman Scattering Using Vortex Light Jian Li, Jiufeng J. Tu, Joseph L. Birman
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To appear in: Journal of Physical and Chemistry of Solids Received date: 5 March 2014 Revised date: 10 July 2014 Accepted date: 16 October 2014 Cite this article as: Jian Li, Jiufeng J. Tu and Joseph L. Birman, Raman Scattering Using Vortex Light, Journal of Physical and Chemistry of Solids, http://dx.doi.org/10.1016/j.jpcs.2014.10.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Raman Scattering Using Vortex Light Jian Lia,∗, Jiufeng J. Tua , Joseph L. Birmana a
Physics Department, The City College of the City University of New York, NY, USA 10031
Abstract We re-examine the theory of Raman scattering in cubic crystals. The unconventional vector potential of vortex light leads to new selection rules. We show that in this novel optical process, (a) silent phonon modes become active and (b) scattering tensors change for ordinary Raman active phonon modes. Calculation based on a simplified model shows that the vortex Raman scattering intensity can be comparable with that of ordinary Raman process. Keywords: Raman spectroscopy, phonons PACS: 03.65.Fd, 61.50.Ah, 42.50.Ct, 42.50.Tx
1. Introduction Light can carry both spin angular momentum (SAM) and orbital angular momentum (OAM) [1]. SAM determines the polarization direction of the beam while OAM describes its spatial profile. Laguerre-Gaussian (LG) beams are one family of light beams with non-zero OAM. LG beams have helical phase fronts, and a phase singularity near the beam center leads to vanishing intensities. Today, LG beams can be readily produced [2], and similar helical profiles have been realized in other forms of waves [3, 4, 5, 6]. The existing studies of the interaction between vortex light and matter include optical tweezers [7, 8, 9, 10], the rotational Doppler effect [11, 12], and others [13]. As suggested by many authors [14], OAM is not just a property of the beam but rather an intrinsic property of photons. This opens the possibility of spectroscopic applications. However, due to the smallness of single atoms, atomic transitions have not been realized. The lack of experimental confirmation of atomic transitions may also be explained by the fact that [15] the interactions between vortex light and electronic degrees of freedom are higher order effects. In solids, the electronic wave functions extend over the entire interaction volume and electrons can “see” the whole profile of the vortex light. In this manuscript, we present the first study of Raman processes using vortex light. The discussion is focused on high symmetry cubic crystals. We analyze and ∗
Corresponding author Email address: [email protected] (Jian Li )
Preprint submitted to Elsevier
October 17, 2014
tabulate the Raman tensors for the forward scattering geometry with vortex light and show that two effects are expected: (a) additional Γ2 phonons and (b) a new scattering cross section dependence on polarization for Γ3 phonons (Fig. 2). The relative intensity of vortex Raman is also studied. The scattering intensity is shown to be comparable with that of the ordinary Raman process. This manuscript is arranged as follows: section 2 gives the general theory of light scattering. Parallel discussions of Raman scattering using both ordinary light and vortex light are presented in sections 3 and 4. The results of Raman tensor analysis are tabulated for cubic crystals. A calculation of the scattering intensity is also presented. Section 5 shows the experimental implementations. Discussions are given in section 6.
2. General Discussion of Light Scattering Intensity In a light scattering process, the total scattering intensity is given by Imn =
27 π 5 I0 (ν0 + νmn )4 ∑ ∣(αρσ )mn ∣2 3 2 c4 ρ,σ
(1)
m, n are indices for vibrational states and ρ, σ are indices for the incident and scattered photons. The polarizability α is realted to the initial, final and intermediate vibrational states through the following relation: (αρσ )mn =
(Mρ )rn (Mσ )mr (Mρ )mr (Mσ )rn 1 + ] ∑[ ̵ 2π h r νrm − ν0 νrn + ν0
(2)
(Mρ )mn is the transition matrix between vibrational levels m and n, in the ˆ ρ: presence of the radiation operator m ˆ ρ Ψm dτ (Mρ )mn = ∫ Ψ∗r m
(3)
The vibrational wave function Ψm , under Born-Oppenheimer approximation, can be expressed as Ψm = Ψgi = Θg (ξ, Q)φgi (Q) ≡ ∣g > ∣gi > where Θ and φ are the electronic and vibrational states, respectively. Therefore, the transition matrix (Mρ )mn can now be written explicitly as: ˆ ρ (Q)]g,e (φg )dQ ˆ ρ (Θg φgi )dξdQ ≡ ∫ (φev )∗ [M (Mρ )mn = (Mρ )gi,ev = ∫ (Θe φev )∗ m i (4) where the electronic transition moment M is defined. Θ depends on the vibrational states Q thus M can be expressed in Q using Taylor expansion: a
ˆ ρ )0g,e + ∑ hes Qa (M ˆ ρ )0g,e ˆ ρ )g,e = (M (M ΔE e,s a,s
(5)
ˆ ρ )0g,e corresponds to the processes without phonon The zeroth order term (M generation or annihilation. The summation is over all vibrational states a and 2
all excited electronic states s but not for states e. Putting all the terms together, the polarizability α becomes: (αρσ )gi,gj = A + B (6a) 1 1 1 ˆ ρ )0g,e (M ˆ σ )0g,e < gj∣ev >< ev∣gi >} (6b) + ){(M A = ∑( h ev νev,gi − ν0 νev,gi + ν0 1 ha 1 ˆ ρ )0 (M ˆ σ )0 < gj∣ev >< ev∣Qa ∣gi > +⋯} B = ∑( ) ∑ es {(M g,e g,s h ev νev,gi ± ν0 s,a ΔEes (6c) The term A does not involve changes in vibrational states while term B does. They correspond to Rayleigh and Raman scatterings respectively. The Raman term includes both Stokes and anti-Stokes components. Both ordinary and resonant Raman processes can be discussed in this framework but only the non-resonant case is shown in this work. In the following sections, comparisons between Raman scattering using ordinary light and vortex light will be presented.
3. Raman Effect for Ordinary Light Ordinary radiation from lasers can be described as plane waves. The vector potential of a plane wave propagating in the z direction can be written as: ex + βˆ ey )e−iωt+ikz + c.c. A = A0 (αˆ α or β determines the polarization of the photon with α2 + β 2 = 1. They are real numbers for linearly polarized light. The light-matter interaction can be put in e (A ⋅ p) and this leads to the explicit form of radiation operator the form of m ˆ ρ: m ˆ = [A0 (αˆ m ex + βˆ ey ) ⋅ p] e−iωt+ikz + c.c. (7) The first component that is associated with e−iωt corresponds to photon absorption and its complex conjugate corresponds to photon emission. In the dipole approximation, the factor eikz is dropped. The non-vanishing terms in equation (6(c)) correspond to Raman active phonon modes. To have a non-vanishing intensity requires hae,s , (Mρ )g,e and (Mσ )g,s to be non-zero, a condition that can be satisfied when: Γa ⊗ Γρ ⊗ Γσ ∋ Γ1 Γa , Γρ and Γσ are irreducible representations of the phonon, the incident photon and the scattering photon, respectively. Raman tensors give the relative intensities of the same phonon in different scattering geometries. It has been shown that the Raman tensors are Clebsch-Gordon coefficients [16].
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4. Raman Effect for Vortex Light Laguerre-Gaussian (LG) functions are a set of solutions of the Maxwell’s equations in the paraxial approximation [14]. Mathematically, a LG beam is described by Laguerre polynomials with a Gaussian envelop. The vector potential of a LG beam in the Lorentz-gauge is [17, 18]: √ Al,p = A0 (αˆ ex + βˆ ey )
√ w0 ∣l∣ 2ρ2 2p! 2ρ ∣l∣ ilφ−iωt+ikz Lp ( 2 )( ) e + c.c. π(∣l∣ + p)! w(z) w (z) w(z)
∣l∣
Lp (x) are generalized Laguerre polynomials; l is the azimuthal index and it is sometimes referred to as the “topological charge”; p is the radial index. ρ is the radial distance from the center axis of the beam; φ is the azimuthal angle; z is the axial distance from the waist. w0 is the waist size; w(z) is the radius at which the field amplitude drop to 1/e of their axial values. p = 0 is chosen to ˆ ρ for a LG beam is therefore: simplify the analysis. The radiation operator m √ √ 1 w0 ∣l∣ 2ρ2 2ρ ∣l∣ ilφ−iωt+ikz ˆ = [A0 (αˆ m L ( 2 )( ) e ex + βˆ ey ) ] ⋅ p + c.c. (8) π∣l∣! w(z) w (z) w(z) The term eikz can be neglected in the dipole approximation. In what is remaining, the part related to e−iωt is associated with photon absorption and it transforms according to (αx + βy)ρl eilφ ≡ (ρ ⋅ ˆ I )ρl eilφ ; the complex conjugate part related to eiωt is associated with photon emission and it transforms according to (αx + βy)ρl e−ilφ ≡ (ρ ⋅ ˆS )ρl e−ilφ . 4.1. Selection rules Same as in the case for ordinary Raman process, the non-vanishing matrix elements for Raman scattering with a LG beam requires: Γa ⊗ Γρ ⊗ Γσ ∋ Γ1 Γa , Γρ and Γσ are the irreducible representations of the phonon, incident photon and scattered photon, respectively. The Raman tensor is written in the form of Pαβγδ (Γσj ). Full index Γσj is needed to denote the jth branch of the σth phonon instead of single index a. Similarly, the single indices ρ and σ are replaced by (α, β) and (γ, δ) to represent wave vector and polarization direction of incident and scattered photons. In ordinary Raman scattering process, the interaction between light and matter does not change the OAM of photon: l remains zero after the scattering. We make the assumption here that the Raman scattering of a LG beam does not change the OAM of photons: topological charge l remains the same. Consequently, we express incident and scattered photons in the form of (ρ⋅ I )ρl eilφ and (ρ⋅ s )ρl e−ilφ . The Raman tensor Pαβγδ (Γσj ) is determined by
4
OAM l=0 l=1 l≥2
Γ11 a 0 ( ) 0 a a 0 ( ) 0 a a 0 ( ) 0 a
Γ12 0 0 ( ) 0 0 0 0 ( ) 0 0 b 0 ( ) 0 −b
Γ13 b 0 ( ) 0 b c 0 ( ) 0 c c 0 ( ) 0 c
Γ2 √ 3 − 3b ( 0 −d ( 0 −d ( 0
√0 ) 3b 0 ) d 0 ) d
Γ41,2,3 0 0 ( ) 0 0 0 0 ( ) 0 0 0 0 ( ) 0 0
Γ51,2 0 0 ( ) 0 0 0 0 ( ) 0 0 0 0 ( ) 0 0
Γ35 0 c ( ) c 0 0 e ( ) e 0 0 e ( ) e 0
Table 1: Raman tensors Pz,s ,I ,z (Γσ j ) for the σth branch of the Γj phonon mode in the forward scattering geometry using vortex light for crystals with O, Td and Oh space groups symmetry. Special case of l = 0 corresponds to Raman tensors of ordinary light [19].
the Clebsch-Gordan coefficients of three representations [16]: scattered photon, incident photon and phonon: Pz, s , I ,z (Γσj ) = (ρ ⋅ s )ρl e−ilφ ⊗ (ρ ⋅ I )ρl eilφ ⊗ φjσ We consider only the case of forward scattering in which the scattering geometry is z( s , I )z and the 3 by 3 Raman tensors are reduced to 2 by 2. The Raman tensor analysis is given in appendix A and the results are tabulated in table 1. Quite surprisingly, the Raman tensors for l ≥ 2 excitations have the same form therefore no difference would l make from symmetry point of view when l ≥ 2. The constants a, b, c, d, e depend on winding number l and the symmetry of the crystal. Compared with l = 0 ordinary Raman process, non-zero OAM brings new features: the appearance of the Γ2 phonon for l ≥ 2 photon excitation and the decoupling of the two Raman tensors for the Γ3 phonon for l ≥ 1 photon excitation. 4.2. Relative intensity of LG Raman compared with ordinary Raman Due to the periodic potentials induced by the crystal lattice, the electrons in solids can be expressed as Bloch waves: ψn,k (r) = eik⋅r unk (r). Taking into account of the vanishing photon wave vectors, the electronic transition moment M connecting the ground ψgk (r) and the excited state ψek (r) is: ∗ (r)[A0 (ρl eilφ ) ⋅ p]ψg,k (r) dr Mg,e = ∑ ∫ ψe,k k
(9)
The normal Raman scattering case corresponds to l = 0. The Taylor expansion form of the electronic transition moment M gives a
ˆ ρ )0 + ∑ hes Qa (M ˆ ρ )0 ˆ ρ )g,e = (M (M g,e g,e a,s ΔEe,s The first order expansion coefficient haes does not depend on topological charge l. haes depends on the property of the material: it is the perturbation energy 5
due to the mixing of electronic states Θe and Θs induced by vibrational mode Qa . Similarly, other parameters such as νev,gi and ΔEes , depend only on the properties of the material and they are independent of the winding number l. This leads to the conclusion that the use of vortex light would affect the scattering intensities through the transition moment M only. The electronic wave function and the winding number l will determine the relative values of M(l ≠ 0) with respect to M(l = 0). The actual number is estimated using the zinc-blende model whose electronic structure has long been studied. Due to Van-Hove singularities, in equation (9), dominant contributions in momentum space are from near the zone center. The electronic transition moment M is therefore simplified to: ∗ (r)[A0 (ρl eilφ ) ⋅ p]ψg,k=0 (r) dr Mg,e = ∫ ψe,k=0
(10)
The analytical expression of the electronic wave functions of zinc-blende and diamond in nearly free electron approximation can be found in standard text(±1, ±1, ±1) are taken books [20]. Only electronic states with wave vector 2π a into account because they make up most of the electronic bands at Γ point. To further simplify the calculation, the face-center-cubic lattice is reduced to simple cubic. The modified electronic wave functions are listed with symmetry labeling: √ (11a) φ11,1 =( 8) cos(πx/a) cos(πy/a) cos(πz/a) √ (11b) φ14,1 =( 8) sin(πx/a) cos(πy/a) cos(πz/a) √ 1 (11c) φ4,2 =( 8) cos(πx/a) sin(πy/a) cos(πz/a) √ 1 φ4,3 =( 8) cos(πx/a) cos(πy/a) sin(πz/a) (11d) √ (11e) φ24,1 =( 8) sin(πx/a) sin(πy/a) cos(πz/a) √ 2 (11f) φ4,2 =( 8) sin(πx/a) cos(πy/a) sin(πz/a) √ 2 φ4,3 =( 8) cos(πx/a) sin(πy/a) sin(πz/a) (11g) φλσ,i is the wave function for the ith branch of the σth representation at its λth appearance. Let’s consider vortex light propagating in the z direction whose polarization is in x. The electronic transition moment becomes: Mg,e =
a
∗ ψe,k=0 (r)[A0 (ρl eilφ )
−a
̵ −ih∂ ]ψg,k=0 (r) dr ∂x
(12)
Only transitions from φ11,1 to φλ4,i are considered. Because of orthogonality, only φ14,1 , φ14,2 and φ24,1 are non-vanishing. The relative intensities of the electronic transition moment M in those transitions are shown in figure 1 up to l = 50. A potentially enhanced scattering intensity compared with l = 0 ordinary Raman process is expected for all three transitions under study with a proper choice of winding number l. The maxima of the three transitions locate 6
Figure 1: Relative intensity of electronic transition moments: (∎) transitions from φ11,1 to φ14,1 ; (▲) transitions from φ11,1 to φ14,2 ; (●) transitions from φ11,1 to φ24,1 . The maxima of the three transitions are located at l = 21, 35, 27.
7
at l = 21, 35, 27 respectively. Our simple calculation shows that the intensity of Raman scattering with vortex light can be comparable to that of the ordinary light, if not larger. The experimentally observed scattering intensities are summed over intermediate energy levels. This may reduce the enhancement factor obtained for those three particular transitions.
5. Experimental Implementations From table 1, two new effects are expected for vortex Raman: (1) the appearance of the Γ2 phonon modes and (b) the change in the polarization dependence of the cross section for the Γ3 phonon. 5.1. Additional Γ2 phonon mode “Additional” means that although this phonon mode exists in the crystal regardless of the measurement technique used, it is silent in an ordinary Raman process, and it becomes Raman active with vortex light. Compared with l = 0, Γ2 phonon is active for photons with l ≥ 2 OAM. The Raman tensors for Γ2 phonon are diagonal. This mode is observed as long as the polarization directions of the incident and scattered photons are not perpendicular. Highest intensity is expected when incident photons and scattered photons are polarized in the same direction. 5.2. Change of cross section dependence of polarization geometry for Γ3 phonon Γ3 phonon is active in both ordinary Raman (l = 0) and vortex Raman (l ≥ 1). In a l = 0 ordinary Raman process, two Raman tensors share the same coupling constant b while for l ≥ 1 vortex Raman scattering process the two Raman tensors have two different coupling constants c and d. As a result, the two Raman tensors for Γ3 phonon are no longer connected by symmetry when vortex light is used. This leads to different cross section dependence on polarization orientations. The scattering cross section for Γ3 phonon is I(Γ3 ) = [ S ⋅ P (Γ13 ) ⋅ I ]2 + [ S ⋅ P (Γ23 ) ⋅ I ]2 . Consider a special scattering geometry where the incident light is polarized in the I = √12 eˆx + √12 eˆy direction and the scattered ex + sin θˆ ey direction. The Raman intensity light is measured in the S = cos θˆ for ordinary light is now Il=0 (Γ3 ) = 2b2 (1 − cos θ sin θ) and the angular dependence of Raman intensity for vortex light is Il≥1 (Γ3 ) =
c2 + d2 + (c2 − d2 ) cos θ sin θ. 2
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Figure 2: Angular dependence of Raman scattering intensity for (a) a ordinary √ √ Raman process ex + (1/ 2)ˆ ey and (b) a vortex Raman process, with incident light polarized in the (1/ 2)ˆ √ direction. The intensities are in arbitrary units. A special value of c = 3d is used for (b). Polarization dependence with l=0 light
Polarization dependence with l>0 light 90
90 120
120
60
150
150
30
180
330
0
330
210
300
240
30
180
0
210
60
300
240 270
270
The intensity plots for Il=0 (Γ3 ) and Il≥1 (Γ3 ) are given in√Figure 2a and 2b. The plots are in arbitrary unit and a special value of c = 3d is used for figure 2b. A clear difference in angular dependence of scattering intensities is observed. Other ratios between c and d will have less dramatic effect but in general the angular √ distributions of intensity have different patterns unless accidentally d = ± 3c. 5.3. Bismuth Germanate Bi4 Ge3 O12 To find Γ2 or Γ3 phonons in cubic crystals, we conducted a systematic investigation of phonon structure with cubic symmetry. Cubic bismuth germanate Bi4 Ge3 O12 is found to satisfy the experimental requirements. It belongs to space group I ¯ 42d (Td6 ). The phonon modes are 4Γ1 +5Γ2 +9Γ3 +14Γ4 +15Γ5 [21]. Both Γ2 and Γ3 can be used to test vortex Raman effect. Bi4 Ge3 O12 crystal is transparent therefore the experiment can be carried out in the forward scattering geometry. Pure Bi4 Ge3 O12 is not fluorescent. Moderate photo refractive effect has been observed in the UV range [22] and non-linear effect can be ignored for visible light excitation. We therefore consider pure Bi4 Ge3 O12 single crystal suitable for vortex Raman experiment. Only part of phonon frequencies have been identified [23]. For those phonons, symmetry assignment can be done by considering polarization-dependent differential cross section.
6. Discussion The fact that topological charge of a light beam remains the same in an ordinary light scattering process leads us to make the assumption that the scattering 9
of photon does not change the OAM of vortex light. We want to emphasis that exclusion of this assumption does not invalidate the symmetry analysis: the analysis presented above is merely one of the many possibilities. Analysis of other possibilities can be conducted in the same framework. The relative intensity of vortex Raman scattering is studied using a simplified model. We focus on quantities that depend only on winding number l. The electronic transition moment between ground state and excited states is calculated using wave functions in nearly free electron model. A potential enhancement in scattering intensity can be achieved by choosing proper winding number. In our simple model, the maxima locate at l ∼ 30. Laser focusing does not significantly change the enhancement factor for electronic transition moment M. Vortex beam focusing would affect scattering intensities the same way focusing does in ordinary Raman scattering process. To summarize, the theory of light scattering is re-visited. Parallel comparisons of Raman processes with ordinary light and vortex light are presented. The discussions are kept general: our analysis applies to Rayleigh scattering, ordinary Raman scattering and resonant Raman scattering. We focus on both Raman scattering tensors and scattering intensities. New selection rules are predicted and vortex Raman scattering intensity is shown to be comparable with that of ordinary Raman process. We identify a suitable crystal system for the realization of vortex Raman scattering.
Appendix A. Analysis on Raman Scattering Tensors The analysis is carried out in O point group. Td is isomorphic to O and Oh is the direct product of O and the inversion group. As a result, the Raman tensors of Td and Oh can be obtained from that of O point group, with a slight modification of notations: the Γ1 , Γ2 , Γ3 , Γ4 , Γ5 representations of O point group are the Γ1 , Γ2 , Γ3 , Γ4 , Γ5 representations of Td point group and the Γ1+ , Γ2+ , Γ3+ , Γ4+ , Γ5+ representations of Oh point group. Only even representations are of interest in a Raman process. Diagonal elements in Raman tensors are for situations in which both incident and scattered light are polarized in x (or y) direction. (x)ρl e−ilφ ⊗ (x)ρl eilφ = x2 (x2 + y 2 )l and (y)ρl e−ilφ ⊗ (y)ρl eilφ = y 2 (x2 + y 2 )l . The off diagonal elements in the Raman tensor correspond to situations in which the incident photon is polarized in the x (or y) direction while the scattered photon is polarized in the y (or x) direction. (x)ρl e−ilφ ⊗ (y)ρl eilφ = (y)ρl e−ilφ ⊗ (x)ρl eilφ = xy(x2 + y 2 )l . Using projection operators, x2 (x2 + y 2 )l and y 2 (x2 + y 2 )l are decomposed into basis functions that transform according to irreducible representations: x2 (x2 + 1 2 1 2 + φ2,γ and y 2 (x2 + y 2 )l = φ11 − φ12 + φ1,γ − φ2,γ . The y 2 )l = φ11 + φ12 + φ1,γ 3 3 3 3
10
terms in the expansion are defined as the following: φ11 φ12 1 φ1,γ 3 2 φ2,γ 3
1 ≡ [(x2 + y 2 )l+1 + (y 2 + z 2 )l+1 + (z 2 + x2 )l+1 ]; 6 1 ≡ [(x2 − y 2 )(x2 + y 2 )l + (y 2 − z 2 )(y 2 + z 2 )l + (z 2 − x2 )(z 2 + x2 )l ]; 6 1 1 1 ≡ [(x2 + y 2 )l+1 − (y 2 + z 2 )l+1 − (z 2 + x2 )l+1 ]; 3 2 2 1 1 2 1 2 2 2 2 l 2 ≡ [(x − y )(x + y ) − (y − z )(y 2 + z 2 )l − (z 2 − x2 )(z 2 + x2 )l ]. 3 2 2
Applying the same method, it is shown that xy(x2 + y 2 )l contains only Γ35 part of the irreducible representation: φ35 = xy(x2 + y 2 )l 2 1 One subtlety lies in φ1,γ and φ2,γ functions. It can be shown that γ1 ≠ γ2 3 3 for any l ≥ 1. In other words, the two Γ13 and Γ23 functions contained in x2 (x2 + y 2 )l and y 2 (x2 + y 2 )l belong to two in-equivalent sets of Γ3 representations except for l = 0. An intuitive example can be found in point group C3v , where (x, y) and (xz, yz) are two in-equivalent sets of basis functions for the E representation. x is connected with y and xz is connected with yz. But x is not connected to xz or yz by symmetry operations. The result of γ1 ≠ γ2 is that the two Raman tensors for Γ3 phonon have different coupling coefficients (see table 1). This is in contrast with the l = 0 case where the same coupling constant b is shared by the two Raman tensors for the Γ3 phonon. Another point to be mentioned is the decomposition of φ12 function: φ12 = 1 [(x2 − y 2 )(x2 + y 2 )l + (y 2 − z 2 )(y 2 + z 2 )l + (z 2 − x2 )(z 2 + x2 )l ]. It becomes 6 zero for l = 0 and l = 1. This means that the Raman tensors for Γ2 phonon exist for only l ≥ 2.
References [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Phys. Rev. A 45 (1992) 8185. [2] V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, JETP Lett. 52 (1990) 429. [3] G. A. Turnbull, D. A. Roberson, G. M. Smith, L. Allen, M. J. Padgett, Opt. Commun. 127 (1990) 183. [4] B. Thide, H. Then, J. Sjoholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, R. Khamitova, Phys. Rev. Lett. 99 (2007) 087701. [5] S. Sasaki, I. McNulty, Phys. Rev. Lett. 100 (2008) 124801.
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[6] K. Volke-Sepulveda, A. O. Santillan, R. R. Boullosa, Phys. Rev. Lett. 100 (2008) 024302. [7] H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75 (1995) 826. [8] N. Simpson, K. Dholakia, L. Allen, M. Padgett, Opt. Lett. 22 (1997) 52. [9] L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. Bryant, K. Dholakia, Science 292 (2001) 912. [10] A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, Phys. Rev. Lett. 88 (2002) 053601. [11] G. Nienhuis, Opt. Commun. 132 (1996) 8. [12] J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, M. J. Padgett, Phys. Rev. Lett. 81 (1998) 4828. [13] A. M. Yao, M. J. Padgett, Adv. Opt. Photon. 3 (2011) 161. [14] R. Loudon, Phys. Rev. A 68 (2003) 013806. [15] M. Babiker, C. R. Bennett, D. L. Andrews, L. C. D. Romero, Phys. Rev. Lett. 89 (2002) 143601. [16] J. L. Birman, R. Berenson, Phys. Rev. B 9 (1974) 4512. [17] A. Alexandrescu, D. Cojoc, E. DiFabrizio, Phys. Rev. Lett. 96 (2006) 243001. [18] A. Picon, A. Benseny, J. Mompart, J. R. V. de Aldana, L. Plaja, G. F. Calvo, L. Roso, New J. Phys. 12 (2010) 083053. [19] R. Loudon, Adv. Phys. 13 (1964) 423. [20] P. Y. Yu, M. Cardona, Fundamentals of Semiconductors Physics and Material Properties, Springer-Verlag Berlin Heidelberg New York, 2001. [21] A. A. Kaminskii, S. N. Bagayev, N. V. Kravtsov, S. N. Chekina, Y. V. Vasilievn, N. I. Ivannikova, K. Ueda, J. Lu, H. J. Eichler, G. M. A. Gad, J. Hanuza, J. Fernandez, P. Reiche, Laser Physics 11 (2001) 897. [22] B. Briat, A. Watterich, F. Ramaz, L. Kovacs, B. Forget, N. Romanov, Optical Materials 20 (2002) 253. [23] R. Titorenkova, B. Mihailova, R. Petrova, M. Gospodinov, L. Konstantinov, AIP Conf. Proc. 1203 (2010) 289.
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To appear in: Journal of Physical and Chemistry of Solids Received date: 5 March 2014 Revised date: 10 July 2014 Accepted date: 16 October 2014 Cite this article as: Jian Li, Jiufeng J. Tu and Joseph L. Birman, Raman Scattering Using Vortex Light, Journal of Physical and Chemistry of Solids, http://dx.doi.org/10.1016/j.jpcs.2014.10.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Raman Scattering Using Vortex Light Jian Lia,∗, Jiufeng J. Tua , Joseph L. Birmana a
Physics Department, The City College of the City University of New York, NY, USA 10031
Abstract We re-examine the theory of Raman scattering in cubic crystals. The unconventional vector potential of vortex light leads to new selection rules. We show that in this novel optical process, (a) silent phonon modes become active and (b) scattering tensors change for ordinary Raman active phonon modes. Calculation based on a simplified model shows that the vortex Raman scattering intensity can be comparable with that of ordinary Raman process. Keywords: Raman spectroscopy, phonons PACS: 03.65.Fd, 61.50.Ah, 42.50.Ct, 42.50.Tx
1. Introduction Light can carry both spin angular momentum (SAM) and orbital angular momentum (OAM) [1]. SAM determines the polarization direction of the beam while OAM describes its spatial profile. Laguerre-Gaussian (LG) beams are one family of light beams with non-zero OAM. LG beams have helical phase fronts, and a phase singularity near the beam center leads to vanishing intensities. Today, LG beams can be readily produced [2], and similar helical profiles have been realized in other forms of waves [3, 4, 5, 6]. The existing studies of the interaction between vortex light and matter include optical tweezers [7, 8, 9, 10], the rotational Doppler effect [11, 12], and others [13]. As suggested by many authors [14], OAM is not just a property of the beam but rather an intrinsic property of photons. This opens the possibility of spectroscopic applications. However, due to the smallness of single atoms, atomic transitions have not been realized. The lack of experimental confirmation of atomic transitions may also be explained by the fact that [15] the interactions between vortex light and electronic degrees of freedom are higher order effects. In solids, the electronic wave functions extend over the entire interaction volume and electrons can “see” the whole profile of the vortex light. In this manuscript, we present the first study of Raman processes using vortex light. The discussion is focused on high symmetry cubic crystals. We analyze and ∗
Corresponding author Email address: [email protected] (Jian Li )
Preprint submitted to Elsevier
October 17, 2014
tabulate the Raman tensors for the forward scattering geometry with vortex light and show that two effects are expected: (a) additional Γ2 phonons and (b) a new scattering cross section dependence on polarization for Γ3 phonons (Fig. 2). The relative intensity of vortex Raman is also studied. The scattering intensity is shown to be comparable with that of the ordinary Raman process. This manuscript is arranged as follows: section 2 gives the general theory of light scattering. Parallel discussions of Raman scattering using both ordinary light and vortex light are presented in sections 3 and 4. The results of Raman tensor analysis are tabulated for cubic crystals. A calculation of the scattering intensity is also presented. Section 5 shows the experimental implementations. Discussions are given in section 6.
2. General Discussion of Light Scattering Intensity In a light scattering process, the total scattering intensity is given by Imn =
27 π 5 I0 (ν0 + νmn )4 ∑ ∣(αρσ )mn ∣2 3 2 c4 ρ,σ
(1)
m, n are indices for vibrational states and ρ, σ are indices for the incident and scattered photons. The polarizability α is realted to the initial, final and intermediate vibrational states through the following relation: (αρσ )mn =
(Mρ )rn (Mσ )mr (Mρ )mr (Mσ )rn 1 + ] ∑[ ̵ 2π h r νrm − ν0 νrn + ν0
(2)
(Mρ )mn is the transition matrix between vibrational levels m and n, in the ˆ ρ: presence of the radiation operator m ˆ ρ Ψm dτ (Mρ )mn = ∫ Ψ∗r m
(3)
The vibrational wave function Ψm , under Born-Oppenheimer approximation, can be expressed as Ψm = Ψgi = Θg (ξ, Q)φgi (Q) ≡ ∣g > ∣gi > where Θ and φ are the electronic and vibrational states, respectively. Therefore, the transition matrix (Mρ )mn can now be written explicitly as: ˆ ρ (Q)]g,e (φg )dQ ˆ ρ (Θg φgi )dξdQ ≡ ∫ (φev )∗ [M (Mρ )mn = (Mρ )gi,ev = ∫ (Θe φev )∗ m i (4) where the electronic transition moment M is defined. Θ depends on the vibrational states Q thus M can be expressed in Q using Taylor expansion: a
ˆ ρ )0g,e + ∑ hes Qa (M ˆ ρ )0g,e ˆ ρ )g,e = (M (M ΔE e,s a,s
(5)
ˆ ρ )0g,e corresponds to the processes without phonon The zeroth order term (M generation or annihilation. The summation is over all vibrational states a and 2
all excited electronic states s but not for states e. Putting all the terms together, the polarizability α becomes: (αρσ )gi,gj = A + B (6a) 1 1 1 ˆ ρ )0g,e (M ˆ σ )0g,e < gj∣ev >< ev∣gi >} (6b) + ){(M A = ∑( h ev νev,gi − ν0 νev,gi + ν0 1 ha 1 ˆ ρ )0 (M ˆ σ )0 < gj∣ev >< ev∣Qa ∣gi > +⋯} B = ∑( ) ∑ es {(M g,e g,s h ev νev,gi ± ν0 s,a ΔEes (6c) The term A does not involve changes in vibrational states while term B does. They correspond to Rayleigh and Raman scatterings respectively. The Raman term includes both Stokes and anti-Stokes components. Both ordinary and resonant Raman processes can be discussed in this framework but only the non-resonant case is shown in this work. In the following sections, comparisons between Raman scattering using ordinary light and vortex light will be presented.
3. Raman Effect for Ordinary Light Ordinary radiation from lasers can be described as plane waves. The vector potential of a plane wave propagating in the z direction can be written as: ex + βˆ ey )e−iωt+ikz + c.c. A = A0 (αˆ α or β determines the polarization of the photon with α2 + β 2 = 1. They are real numbers for linearly polarized light. The light-matter interaction can be put in e (A ⋅ p) and this leads to the explicit form of radiation operator the form of m ˆ ρ: m ˆ = [A0 (αˆ m ex + βˆ ey ) ⋅ p] e−iωt+ikz + c.c. (7) The first component that is associated with e−iωt corresponds to photon absorption and its complex conjugate corresponds to photon emission. In the dipole approximation, the factor eikz is dropped. The non-vanishing terms in equation (6(c)) correspond to Raman active phonon modes. To have a non-vanishing intensity requires hae,s , (Mρ )g,e and (Mσ )g,s to be non-zero, a condition that can be satisfied when: Γa ⊗ Γρ ⊗ Γσ ∋ Γ1 Γa , Γρ and Γσ are irreducible representations of the phonon, the incident photon and the scattering photon, respectively. Raman tensors give the relative intensities of the same phonon in different scattering geometries. It has been shown that the Raman tensors are Clebsch-Gordon coefficients [16].
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4. Raman Effect for Vortex Light Laguerre-Gaussian (LG) functions are a set of solutions of the Maxwell’s equations in the paraxial approximation [14]. Mathematically, a LG beam is described by Laguerre polynomials with a Gaussian envelop. The vector potential of a LG beam in the Lorentz-gauge is [17, 18]: √ Al,p = A0 (αˆ ex + βˆ ey )
√ w0 ∣l∣ 2ρ2 2p! 2ρ ∣l∣ ilφ−iωt+ikz Lp ( 2 )( ) e + c.c. π(∣l∣ + p)! w(z) w (z) w(z)
∣l∣
Lp (x) are generalized Laguerre polynomials; l is the azimuthal index and it is sometimes referred to as the “topological charge”; p is the radial index. ρ is the radial distance from the center axis of the beam; φ is the azimuthal angle; z is the axial distance from the waist. w0 is the waist size; w(z) is the radius at which the field amplitude drop to 1/e of their axial values. p = 0 is chosen to ˆ ρ for a LG beam is therefore: simplify the analysis. The radiation operator m √ √ 1 w0 ∣l∣ 2ρ2 2ρ ∣l∣ ilφ−iωt+ikz ˆ = [A0 (αˆ m L ( 2 )( ) e ex + βˆ ey ) ] ⋅ p + c.c. (8) π∣l∣! w(z) w (z) w(z) The term eikz can be neglected in the dipole approximation. In what is remaining, the part related to e−iωt is associated with photon absorption and it transforms according to (αx + βy)ρl eilφ ≡ (ρ ⋅ ˆ I )ρl eilφ ; the complex conjugate part related to eiωt is associated with photon emission and it transforms according to (αx + βy)ρl e−ilφ ≡ (ρ ⋅ ˆS )ρl e−ilφ . 4.1. Selection rules Same as in the case for ordinary Raman process, the non-vanishing matrix elements for Raman scattering with a LG beam requires: Γa ⊗ Γρ ⊗ Γσ ∋ Γ1 Γa , Γρ and Γσ are the irreducible representations of the phonon, incident photon and scattered photon, respectively. The Raman tensor is written in the form of Pαβγδ (Γσj ). Full index Γσj is needed to denote the jth branch of the σth phonon instead of single index a. Similarly, the single indices ρ and σ are replaced by (α, β) and (γ, δ) to represent wave vector and polarization direction of incident and scattered photons. In ordinary Raman scattering process, the interaction between light and matter does not change the OAM of photon: l remains zero after the scattering. We make the assumption here that the Raman scattering of a LG beam does not change the OAM of photons: topological charge l remains the same. Consequently, we express incident and scattered photons in the form of (ρ⋅ I )ρl eilφ and (ρ⋅ s )ρl e−ilφ . The Raman tensor Pαβγδ (Γσj ) is determined by
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OAM l=0 l=1 l≥2
Γ11 a 0 ( ) 0 a a 0 ( ) 0 a a 0 ( ) 0 a
Γ12 0 0 ( ) 0 0 0 0 ( ) 0 0 b 0 ( ) 0 −b
Γ13 b 0 ( ) 0 b c 0 ( ) 0 c c 0 ( ) 0 c
Γ2 √ 3 − 3b ( 0 −d ( 0 −d ( 0
√0 ) 3b 0 ) d 0 ) d
Γ41,2,3 0 0 ( ) 0 0 0 0 ( ) 0 0 0 0 ( ) 0 0
Γ51,2 0 0 ( ) 0 0 0 0 ( ) 0 0 0 0 ( ) 0 0
Γ35 0 c ( ) c 0 0 e ( ) e 0 0 e ( ) e 0
Table 1: Raman tensors Pz,s ,I ,z (Γσ j ) for the σth branch of the Γj phonon mode in the forward scattering geometry using vortex light for crystals with O, Td and Oh space groups symmetry. Special case of l = 0 corresponds to Raman tensors of ordinary light [19].
the Clebsch-Gordan coefficients of three representations [16]: scattered photon, incident photon and phonon: Pz, s , I ,z (Γσj ) = (ρ ⋅ s )ρl e−ilφ ⊗ (ρ ⋅ I )ρl eilφ ⊗ φjσ We consider only the case of forward scattering in which the scattering geometry is z( s , I )z and the 3 by 3 Raman tensors are reduced to 2 by 2. The Raman tensor analysis is given in appendix A and the results are tabulated in table 1. Quite surprisingly, the Raman tensors for l ≥ 2 excitations have the same form therefore no difference would l make from symmetry point of view when l ≥ 2. The constants a, b, c, d, e depend on winding number l and the symmetry of the crystal. Compared with l = 0 ordinary Raman process, non-zero OAM brings new features: the appearance of the Γ2 phonon for l ≥ 2 photon excitation and the decoupling of the two Raman tensors for the Γ3 phonon for l ≥ 1 photon excitation. 4.2. Relative intensity of LG Raman compared with ordinary Raman Due to the periodic potentials induced by the crystal lattice, the electrons in solids can be expressed as Bloch waves: ψn,k (r) = eik⋅r unk (r). Taking into account of the vanishing photon wave vectors, the electronic transition moment M connecting the ground ψgk (r) and the excited state ψek (r) is: ∗ (r)[A0 (ρl eilφ ) ⋅ p]ψg,k (r) dr Mg,e = ∑ ∫ ψe,k k
(9)
The normal Raman scattering case corresponds to l = 0. The Taylor expansion form of the electronic transition moment M gives a
ˆ ρ )0 + ∑ hes Qa (M ˆ ρ )0 ˆ ρ )g,e = (M (M g,e g,e a,s ΔEe,s The first order expansion coefficient haes does not depend on topological charge l. haes depends on the property of the material: it is the perturbation energy 5
due to the mixing of electronic states Θe and Θs induced by vibrational mode Qa . Similarly, other parameters such as νev,gi and ΔEes , depend only on the properties of the material and they are independent of the winding number l. This leads to the conclusion that the use of vortex light would affect the scattering intensities through the transition moment M only. The electronic wave function and the winding number l will determine the relative values of M(l ≠ 0) with respect to M(l = 0). The actual number is estimated using the zinc-blende model whose electronic structure has long been studied. Due to Van-Hove singularities, in equation (9), dominant contributions in momentum space are from near the zone center. The electronic transition moment M is therefore simplified to: ∗ (r)[A0 (ρl eilφ ) ⋅ p]ψg,k=0 (r) dr Mg,e = ∫ ψe,k=0
(10)
The analytical expression of the electronic wave functions of zinc-blende and diamond in nearly free electron approximation can be found in standard text(±1, ±1, ±1) are taken books [20]. Only electronic states with wave vector 2π a into account because they make up most of the electronic bands at Γ point. To further simplify the calculation, the face-center-cubic lattice is reduced to simple cubic. The modified electronic wave functions are listed with symmetry labeling: √ (11a) φ11,1 =( 8) cos(πx/a) cos(πy/a) cos(πz/a) √ (11b) φ14,1 =( 8) sin(πx/a) cos(πy/a) cos(πz/a) √ 1 (11c) φ4,2 =( 8) cos(πx/a) sin(πy/a) cos(πz/a) √ 1 φ4,3 =( 8) cos(πx/a) cos(πy/a) sin(πz/a) (11d) √ (11e) φ24,1 =( 8) sin(πx/a) sin(πy/a) cos(πz/a) √ 2 (11f) φ4,2 =( 8) sin(πx/a) cos(πy/a) sin(πz/a) √ 2 φ4,3 =( 8) cos(πx/a) sin(πy/a) sin(πz/a) (11g) φλσ,i is the wave function for the ith branch of the σth representation at its λth appearance. Let’s consider vortex light propagating in the z direction whose polarization is in x. The electronic transition moment becomes: Mg,e =
a
∗ ψe,k=0 (r)[A0 (ρl eilφ )
−a
̵ −ih∂ ]ψg,k=0 (r) dr ∂x
(12)
Only transitions from φ11,1 to φλ4,i are considered. Because of orthogonality, only φ14,1 , φ14,2 and φ24,1 are non-vanishing. The relative intensities of the electronic transition moment M in those transitions are shown in figure 1 up to l = 50. A potentially enhanced scattering intensity compared with l = 0 ordinary Raman process is expected for all three transitions under study with a proper choice of winding number l. The maxima of the three transitions locate 6
Figure 1: Relative intensity of electronic transition moments: (∎) transitions from φ11,1 to φ14,1 ; (▲) transitions from φ11,1 to φ14,2 ; (●) transitions from φ11,1 to φ24,1 . The maxima of the three transitions are located at l = 21, 35, 27.
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at l = 21, 35, 27 respectively. Our simple calculation shows that the intensity of Raman scattering with vortex light can be comparable to that of the ordinary light, if not larger. The experimentally observed scattering intensities are summed over intermediate energy levels. This may reduce the enhancement factor obtained for those three particular transitions.
5. Experimental Implementations From table 1, two new effects are expected for vortex Raman: (1) the appearance of the Γ2 phonon modes and (b) the change in the polarization dependence of the cross section for the Γ3 phonon. 5.1. Additional Γ2 phonon mode “Additional” means that although this phonon mode exists in the crystal regardless of the measurement technique used, it is silent in an ordinary Raman process, and it becomes Raman active with vortex light. Compared with l = 0, Γ2 phonon is active for photons with l ≥ 2 OAM. The Raman tensors for Γ2 phonon are diagonal. This mode is observed as long as the polarization directions of the incident and scattered photons are not perpendicular. Highest intensity is expected when incident photons and scattered photons are polarized in the same direction. 5.2. Change of cross section dependence of polarization geometry for Γ3 phonon Γ3 phonon is active in both ordinary Raman (l = 0) and vortex Raman (l ≥ 1). In a l = 0 ordinary Raman process, two Raman tensors share the same coupling constant b while for l ≥ 1 vortex Raman scattering process the two Raman tensors have two different coupling constants c and d. As a result, the two Raman tensors for Γ3 phonon are no longer connected by symmetry when vortex light is used. This leads to different cross section dependence on polarization orientations. The scattering cross section for Γ3 phonon is I(Γ3 ) = [ S ⋅ P (Γ13 ) ⋅ I ]2 + [ S ⋅ P (Γ23 ) ⋅ I ]2 . Consider a special scattering geometry where the incident light is polarized in the I = √12 eˆx + √12 eˆy direction and the scattered ex + sin θˆ ey direction. The Raman intensity light is measured in the S = cos θˆ for ordinary light is now Il=0 (Γ3 ) = 2b2 (1 − cos θ sin θ) and the angular dependence of Raman intensity for vortex light is Il≥1 (Γ3 ) =
c2 + d2 + (c2 − d2 ) cos θ sin θ. 2
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Figure 2: Angular dependence of Raman scattering intensity for (a) a ordinary √ √ Raman process ex + (1/ 2)ˆ ey and (b) a vortex Raman process, with incident light polarized in the (1/ 2)ˆ √ direction. The intensities are in arbitrary units. A special value of c = 3d is used for (b). Polarization dependence with l=0 light
Polarization dependence with l>0 light 90
90 120
120
60
150
150
30
180
330
0
330
210
300
240
30
180
0
210
60
300
240 270
270
The intensity plots for Il=0 (Γ3 ) and Il≥1 (Γ3 ) are given in√Figure 2a and 2b. The plots are in arbitrary unit and a special value of c = 3d is used for figure 2b. A clear difference in angular dependence of scattering intensities is observed. Other ratios between c and d will have less dramatic effect but in general the angular √ distributions of intensity have different patterns unless accidentally d = ± 3c. 5.3. Bismuth Germanate Bi4 Ge3 O12 To find Γ2 or Γ3 phonons in cubic crystals, we conducted a systematic investigation of phonon structure with cubic symmetry. Cubic bismuth germanate Bi4 Ge3 O12 is found to satisfy the experimental requirements. It belongs to space group I ¯ 42d (Td6 ). The phonon modes are 4Γ1 +5Γ2 +9Γ3 +14Γ4 +15Γ5 [21]. Both Γ2 and Γ3 can be used to test vortex Raman effect. Bi4 Ge3 O12 crystal is transparent therefore the experiment can be carried out in the forward scattering geometry. Pure Bi4 Ge3 O12 is not fluorescent. Moderate photo refractive effect has been observed in the UV range [22] and non-linear effect can be ignored for visible light excitation. We therefore consider pure Bi4 Ge3 O12 single crystal suitable for vortex Raman experiment. Only part of phonon frequencies have been identified [23]. For those phonons, symmetry assignment can be done by considering polarization-dependent differential cross section.
6. Discussion The fact that topological charge of a light beam remains the same in an ordinary light scattering process leads us to make the assumption that the scattering 9
of photon does not change the OAM of vortex light. We want to emphasis that exclusion of this assumption does not invalidate the symmetry analysis: the analysis presented above is merely one of the many possibilities. Analysis of other possibilities can be conducted in the same framework. The relative intensity of vortex Raman scattering is studied using a simplified model. We focus on quantities that depend only on winding number l. The electronic transition moment between ground state and excited states is calculated using wave functions in nearly free electron model. A potential enhancement in scattering intensity can be achieved by choosing proper winding number. In our simple model, the maxima locate at l ∼ 30. Laser focusing does not significantly change the enhancement factor for electronic transition moment M. Vortex beam focusing would affect scattering intensities the same way focusing does in ordinary Raman scattering process. To summarize, the theory of light scattering is re-visited. Parallel comparisons of Raman processes with ordinary light and vortex light are presented. The discussions are kept general: our analysis applies to Rayleigh scattering, ordinary Raman scattering and resonant Raman scattering. We focus on both Raman scattering tensors and scattering intensities. New selection rules are predicted and vortex Raman scattering intensity is shown to be comparable with that of ordinary Raman process. We identify a suitable crystal system for the realization of vortex Raman scattering.
Appendix A. Analysis on Raman Scattering Tensors The analysis is carried out in O point group. Td is isomorphic to O and Oh is the direct product of O and the inversion group. As a result, the Raman tensors of Td and Oh can be obtained from that of O point group, with a slight modification of notations: the Γ1 , Γ2 , Γ3 , Γ4 , Γ5 representations of O point group are the Γ1 , Γ2 , Γ3 , Γ4 , Γ5 representations of Td point group and the Γ1+ , Γ2+ , Γ3+ , Γ4+ , Γ5+ representations of Oh point group. Only even representations are of interest in a Raman process. Diagonal elements in Raman tensors are for situations in which both incident and scattered light are polarized in x (or y) direction. (x)ρl e−ilφ ⊗ (x)ρl eilφ = x2 (x2 + y 2 )l and (y)ρl e−ilφ ⊗ (y)ρl eilφ = y 2 (x2 + y 2 )l . The off diagonal elements in the Raman tensor correspond to situations in which the incident photon is polarized in the x (or y) direction while the scattered photon is polarized in the y (or x) direction. (x)ρl e−ilφ ⊗ (y)ρl eilφ = (y)ρl e−ilφ ⊗ (x)ρl eilφ = xy(x2 + y 2 )l . Using projection operators, x2 (x2 + y 2 )l and y 2 (x2 + y 2 )l are decomposed into basis functions that transform according to irreducible representations: x2 (x2 + 1 2 1 2 + φ2,γ and y 2 (x2 + y 2 )l = φ11 − φ12 + φ1,γ − φ2,γ . The y 2 )l = φ11 + φ12 + φ1,γ 3 3 3 3
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terms in the expansion are defined as the following: φ11 φ12 1 φ1,γ 3 2 φ2,γ 3
1 ≡ [(x2 + y 2 )l+1 + (y 2 + z 2 )l+1 + (z 2 + x2 )l+1 ]; 6 1 ≡ [(x2 − y 2 )(x2 + y 2 )l + (y 2 − z 2 )(y 2 + z 2 )l + (z 2 − x2 )(z 2 + x2 )l ]; 6 1 1 1 ≡ [(x2 + y 2 )l+1 − (y 2 + z 2 )l+1 − (z 2 + x2 )l+1 ]; 3 2 2 1 1 2 1 2 2 2 2 l 2 ≡ [(x − y )(x + y ) − (y − z )(y 2 + z 2 )l − (z 2 − x2 )(z 2 + x2 )l ]. 3 2 2
Applying the same method, it is shown that xy(x2 + y 2 )l contains only Γ35 part of the irreducible representation: φ35 = xy(x2 + y 2 )l 2 1 One subtlety lies in φ1,γ and φ2,γ functions. It can be shown that γ1 ≠ γ2 3 3 for any l ≥ 1. In other words, the two Γ13 and Γ23 functions contained in x2 (x2 + y 2 )l and y 2 (x2 + y 2 )l belong to two in-equivalent sets of Γ3 representations except for l = 0. An intuitive example can be found in point group C3v , where (x, y) and (xz, yz) are two in-equivalent sets of basis functions for the E representation. x is connected with y and xz is connected with yz. But x is not connected to xz or yz by symmetry operations. The result of γ1 ≠ γ2 is that the two Raman tensors for Γ3 phonon have different coupling coefficients (see table 1). This is in contrast with the l = 0 case where the same coupling constant b is shared by the two Raman tensors for the Γ3 phonon. Another point to be mentioned is the decomposition of φ12 function: φ12 = 1 [(x2 − y 2 )(x2 + y 2 )l + (y 2 − z 2 )(y 2 + z 2 )l + (z 2 − x2 )(z 2 + x2 )l ]. It becomes 6 zero for l = 0 and l = 1. This means that the Raman tensors for Γ2 phonon exist for only l ≥ 2.
References [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, Phys. Rev. A 45 (1992) 8185. [2] V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, JETP Lett. 52 (1990) 429. [3] G. A. Turnbull, D. A. Roberson, G. M. Smith, L. Allen, M. J. Padgett, Opt. Commun. 127 (1990) 183. [4] B. Thide, H. Then, J. Sjoholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, R. Khamitova, Phys. Rev. Lett. 99 (2007) 087701. [5] S. Sasaki, I. McNulty, Phys. Rev. Lett. 100 (2008) 124801.
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[6] K. Volke-Sepulveda, A. O. Santillan, R. R. Boullosa, Phys. Rev. Lett. 100 (2008) 024302. [7] H. He, M. E. J. Friese, N. R. Heckenberg, H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75 (1995) 826. [8] N. Simpson, K. Dholakia, L. Allen, M. Padgett, Opt. Lett. 22 (1997) 52. [9] L. Paterson, M. MacDonald, J. Arlt, W. Sibbett, P. Bryant, K. Dholakia, Science 292 (2001) 912. [10] A. T. O’Neil, I. MacVicar, L. Allen, M. J. Padgett, Phys. Rev. Lett. 88 (2002) 053601. [11] G. Nienhuis, Opt. Commun. 132 (1996) 8. [12] J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, M. J. Padgett, Phys. Rev. Lett. 81 (1998) 4828. [13] A. M. Yao, M. J. Padgett, Adv. Opt. Photon. 3 (2011) 161. [14] R. Loudon, Phys. Rev. A 68 (2003) 013806. [15] M. Babiker, C. R. Bennett, D. L. Andrews, L. C. D. Romero, Phys. Rev. Lett. 89 (2002) 143601. [16] J. L. Birman, R. Berenson, Phys. Rev. B 9 (1974) 4512. [17] A. Alexandrescu, D. Cojoc, E. DiFabrizio, Phys. Rev. Lett. 96 (2006) 243001. [18] A. Picon, A. Benseny, J. Mompart, J. R. V. de Aldana, L. Plaja, G. F. Calvo, L. Roso, New J. Phys. 12 (2010) 083053. [19] R. Loudon, Adv. Phys. 13 (1964) 423. [20] P. Y. Yu, M. Cardona, Fundamentals of Semiconductors Physics and Material Properties, Springer-Verlag Berlin Heidelberg New York, 2001. [21] A. A. Kaminskii, S. N. Bagayev, N. V. Kravtsov, S. N. Chekina, Y. V. Vasilievn, N. I. Ivannikova, K. Ueda, J. Lu, H. J. Eichler, G. M. A. Gad, J. Hanuza, J. Fernandez, P. Reiche, Laser Physics 11 (2001) 897. [22] B. Briat, A. Watterich, F. Ramaz, L. Kovacs, B. Forget, N. Romanov, Optical Materials 20 (2002) 253. [23] R. Titorenkova, B. Mihailova, R. Petrova, M. Gospodinov, L. Konstantinov, AIP Conf. Proc. 1203 (2010) 289.
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