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Laser-induced circular dichroism
CHEIIUCAL PHYSICS LETTERS
Volume 49. number 3
LASER-INDUCED
CIRCULAR
1 August 1977
DICHROISM
T. THIRUNAMACH4NDIWN Department
of Chemistry,
University
Collzge London.
London
WC 1, UK
Received 27 April 1977 An achiral molecule can exhibit circular dichroism in the presence of an intense beam of circularly polarized light. Differential absorption rates, based on quantum electrodynamial calculations, are reported for two types of transiticns: those which are (a) eIec:ric-dipoIe aIlowed, and (b) elec~ric-dipole forbidden, but two-photon allowed. In both cases, the differential rate is shown to be linearly dependent on the Intensity of the laser beam.
An achiral molecule, in the presence of an intense beam of circularly polarized light, can exhibit optical activity which can be detected by using another beam of light as a probe. Recently, Liao and Bjorklund [l] have observed the rotation of plane of polarization of light by passing a polarized beam through sodium vapour under the influence of circularly polarized radiation. In this note, the complementary effect, circular dichroism induced by a laser field, is examined using quantum electrodynamics. Two types are distinguished: (a) the transition showing induced circular dichroism is allowed by one-photon absorption, and (b) the transition is one-photon forbidden, but allowed by two-photon absorption. The molecule can acquire chirality by coupling to the circularly polarized laser radiation field either by scattering (case a) or by virtual absorption (case b); it can then show differential absorption with respect to the second incident beam. In the present work, the initial and final molecular states are assumed to be non-degenerate and can therefore be described by real wavefunctions. For case (a), let n2 t 0 be the particular transition of interest_ Due to the coupling with the laser field of mode k (frequency w) and polarization e CL)(k), circular dichroism is induced in the transition nz + 0. By extending the theory of natural circular dichroism [Z] ) it is possible to calculate the induced circular dichroism. For a randomly oriented molecular system, the result, expressed as difference in absorption rates for left- and right-circularly polarized light of frequency w’ [= @In -&Jfi] , is . (1) where repeated suffices imply summation over cartesian components. In (1) I is the irradiance (power per unit area) of the laser beam and 9’ is the intensity, expressed as energy density per Hz, of the second beam. The tensor PhPv is given by
(2) Since the transition is electric-dipole allowed, the laser-induced circular dichroism may be viewed as resulting from the interference of the amplitude for the first-order graph for direct absorption with the amplitudes for the thirdorder graphs of the type shown in fig. 1. A similar interference occurs in the theory of induced circular dichroism [3,4], where the chirality is induced 536
CHEMICAL PHYSICS LETTERS
Volume 49, number 3
1 August 1977
Fig. 1. Typical dichroism.
time-ordered
diagrams
for laser-induced
circular
by the presence of chiral molecules. The third-order graphs represent the scattering of a photon from the laser beam and the absorption of a photon of frequency o’ from the second beam. The matrix elements, evaluated in the usual manner, together with the Fermi ruIe Iead to the result (1). It should be noted that in the calculation of the third-order matrix elements, allowance must be made for the possibility of the intermediate states being the same as either the initial or final state. However, these terms vanish on rotational averaging_ The induced circular dichroism may also be expressed as an asymmetry factor,
(WLL> - WLR>)/(WLL) + WLR))= (d/c)(k’t’)(/py3g
- pp*phn:~)/(~‘J*ol*,
(3)
which is found to depend on the intensity of the laser beam and on the angle between the directions of propagation of the two beams. The second type of laser-induced circular dichroism is associated with a molecular transition which cannot take place by one-photon absorption, but can occur by a two-photon mechanism corresponding to the absorption of one photon from each beam subject to the energy conservation condition (Em -E&/h = o + w’. The matrix element for the two-photon absorption is l/2 &L)(k) r
MLsLIR 7 - (27r?$~w/v)~~~ (27&iW’/v)
e,l.LlR’(k’)
x’
[;‘F;;,
r
+ ;z;;e],
(4)
Or
where n and n’ are photon occupation numbers. In contrast to type (a) discussed earIier, the first-order matrix element is zero for this case, and the contribution to the circuIar dichroism arises from the square of (4). Thus we find, after rotational averaging, +L>
- (rLR)=-(2=21~‘/3~2c)(k^.k^‘)
x
c ’ (p”’
+
(pm’ x PSO)(Prox PSO)
J-.s
x jP)
- (Frox
1
PSO)
1
(EOr+h’)(EOs+jZ~)
1
1
(Eo,+fiw’)(Eos+~w’)
+ (Eor+~w)(Eos+fiw) 1
+ (Eor+frw)(Eo,+fiw’)
-
(5)
By a suitable choice of frequencies of the two beams, it is possible to achieve near-resonance with a particular intermediate state thus leading to enhanced differential absorption. Under such conditions the differential rate (5) becomes (rLL)
- WLR>=
(~~19’/3h~c)(k^.k^‘)
l~mrX~‘012/(Eo,+fiw)(Eor+~w’).
Details of laser-induced circular dichroism and other aspects of laser-induced elsewhere.
(6) optical activity will be discussed
537
Volume 49, number 3
CHEMICAL PHYSICS LETTERS
References [l] P.F. I-iao and G-C. Bjorklund, Phys. Rev_ Letters 36 (1976) 584. [2] E.A. Power and T. Thirunamachandnn, J. Chcm. Phys. 60 (1974) 3695. [ 3 ] D-P. Craig, E-A. Power and T. Thirunamachandran, Chem. Phys. Letters 27 (1974) 149. [4] D-P. Craig, E.A. Power andT. Thirunamachandran, Proc. Roy. Sot. A348 (1976) 19.
1 August
1977
Volume 49. number 3
LASER-INDUCED
CIRCULAR
1 August 1977
DICHROISM
T. THIRUNAMACH4NDIWN Department
of Chemistry,
University
Collzge London.
London
WC 1, UK
Received 27 April 1977 An achiral molecule can exhibit circular dichroism in the presence of an intense beam of circularly polarized light. Differential absorption rates, based on quantum electrodynamial calculations, are reported for two types of transiticns: those which are (a) eIec:ric-dipoIe aIlowed, and (b) elec~ric-dipole forbidden, but two-photon allowed. In both cases, the differential rate is shown to be linearly dependent on the Intensity of the laser beam.
An achiral molecule, in the presence of an intense beam of circularly polarized light, can exhibit optical activity which can be detected by using another beam of light as a probe. Recently, Liao and Bjorklund [l] have observed the rotation of plane of polarization of light by passing a polarized beam through sodium vapour under the influence of circularly polarized radiation. In this note, the complementary effect, circular dichroism induced by a laser field, is examined using quantum electrodynamics. Two types are distinguished: (a) the transition showing induced circular dichroism is allowed by one-photon absorption, and (b) the transition is one-photon forbidden, but allowed by two-photon absorption. The molecule can acquire chirality by coupling to the circularly polarized laser radiation field either by scattering (case a) or by virtual absorption (case b); it can then show differential absorption with respect to the second incident beam. In the present work, the initial and final molecular states are assumed to be non-degenerate and can therefore be described by real wavefunctions. For case (a), let n2 t 0 be the particular transition of interest_ Due to the coupling with the laser field of mode k (frequency w) and polarization e CL)(k), circular dichroism is induced in the transition nz + 0. By extending the theory of natural circular dichroism [Z] ) it is possible to calculate the induced circular dichroism. For a randomly oriented molecular system, the result, expressed as difference in absorption rates for left- and right-circularly polarized light of frequency w’ [= @In -&Jfi] , is . (1) where repeated suffices imply summation over cartesian components. In (1) I is the irradiance (power per unit area) of the laser beam and 9’ is the intensity, expressed as energy density per Hz, of the second beam. The tensor PhPv is given by
(2) Since the transition is electric-dipole allowed, the laser-induced circular dichroism may be viewed as resulting from the interference of the amplitude for the first-order graph for direct absorption with the amplitudes for the thirdorder graphs of the type shown in fig. 1. A similar interference occurs in the theory of induced circular dichroism [3,4], where the chirality is induced 536
CHEMICAL PHYSICS LETTERS
Volume 49, number 3
1 August 1977
Fig. 1. Typical dichroism.
time-ordered
diagrams
for laser-induced
circular
by the presence of chiral molecules. The third-order graphs represent the scattering of a photon from the laser beam and the absorption of a photon of frequency o’ from the second beam. The matrix elements, evaluated in the usual manner, together with the Fermi ruIe Iead to the result (1). It should be noted that in the calculation of the third-order matrix elements, allowance must be made for the possibility of the intermediate states being the same as either the initial or final state. However, these terms vanish on rotational averaging_ The induced circular dichroism may also be expressed as an asymmetry factor,
(WLL> - WLR>)/(WLL) + WLR))= (d/c)(k’t’)(/py3g
- pp*phn:~)/(~‘J*ol*,
(3)
which is found to depend on the intensity of the laser beam and on the angle between the directions of propagation of the two beams. The second type of laser-induced circular dichroism is associated with a molecular transition which cannot take place by one-photon absorption, but can occur by a two-photon mechanism corresponding to the absorption of one photon from each beam subject to the energy conservation condition (Em -E&/h = o + w’. The matrix element for the two-photon absorption is l/2 &L)(k) r
MLsLIR 7 - (27r?$~w/v)~~~ (27&iW’/v)
e,l.LlR’(k’)
x’
[;‘F;;,
r
+ ;z;;e],
(4)
Or
where n and n’ are photon occupation numbers. In contrast to type (a) discussed earIier, the first-order matrix element is zero for this case, and the contribution to the circuIar dichroism arises from the square of (4). Thus we find, after rotational averaging, +L>
- (rLR)=-(2=21~‘/3~2c)(k^.k^‘)
x
c ’ (p”’
+
(pm’ x PSO)(Prox PSO)
J-.s
x jP)
- (Frox
1
PSO)
1
(EOr+h’)(EOs+jZ~)
1
1
(Eo,+fiw’)(Eos+~w’)
+ (Eor+~w)(Eos+fiw) 1
+ (Eor+frw)(Eo,+fiw’)
-
(5)
By a suitable choice of frequencies of the two beams, it is possible to achieve near-resonance with a particular intermediate state thus leading to enhanced differential absorption. Under such conditions the differential rate (5) becomes (rLL)
- WLR>=
(~~19’/3h~c)(k^.k^‘)
l~mrX~‘012/(Eo,+fiw)(Eor+~w’).
Details of laser-induced circular dichroism and other aspects of laser-induced elsewhere.
(6) optical activity will be discussed
537
Volume 49, number 3
CHEMICAL PHYSICS LETTERS
References [l] P.F. I-iao and G-C. Bjorklund, Phys. Rev_ Letters 36 (1976) 584. [2] E.A. Power and T. Thirunamachandnn, J. Chcm. Phys. 60 (1974) 3695. [ 3 ] D-P. Craig, E-A. Power and T. Thirunamachandran, Chem. Phys. Letters 27 (1974) 149. [4] D-P. Craig, E.A. Power andT. Thirunamachandran, Proc. Roy. Sot. A348 (1976) 19.
1 August
1977