- Email: [email protected]
Radiation force induced by resonant light: from atom to nanoparticle
ARTICLE IN PRESS
Journal of Luminescence 108 (2004) 351–354
Radiation force induced by resonant light: from atom to nanoparticle Takuya Iidaa,*, Hajime Ishiharaa,b a
Department of Materials Engineering Science, Graduate school of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka 560-8531, Japan b CREST, Japan Science and Technology Corporation, Japan
Abstract We derive analytical expressions for the radiation force induced by electronically resonant light which apply over the size range from atoms to nanoparticles. These are based on the method of Maxwell stress tensor and microscopic response theory. The radiation force is expressed as a sum of a scattering force, an absorbing force and additional terms arising from the asymmetric spatial distribution of the internal field from light-exciton coupled modes. In the case of a particle with a radius of few tens of nm, a particular interesting point is that, under certain conditions, substantially large contributions from these terms appear as forces in opposite direction to the direction of propagation of plane wave light. r 2004 Elsevier B.V. All rights reserved. PACS: 32.80.t; 71.35.y; 78.67.n; 78.90.+t Keywords: Atom trapping; Optical manipulation; Radiation force; Resonance; Exciton; Quantum dot
1. Introduction Since the first experiment of mechanical managing of microparticles with a laser-induced radiation force (RF) [1], various fundamental studies and applications of the optical manipulation of micrometre objects have been demonstrated. This technique is utilized in a variety of fields including material engineering, biochemistry and micromachining. In many studies of the optical manipulation of micrometre objects, electronically non*Corresponding author. Tel.: +81-6-6850-6401; fax: +81-66850-6401. E-mail address: [email protected] (T. Iida).
resonant laser light with high power is utilized to generate sufficiently strong and heat-free forces (see article in Ref. [2]). However, the handling of nanoscale objects by using this technique is still challenging. On the other hand, resonant light is utilized for trapping of atoms in order to enhance the induced polarization [3]. Recently, we have theoretically proposed an enhancement mechanism of RF by the electronically resonant light [4] and a new type of optical manipulation that enables us to sort nanoparticles according to their size, and to arrange them in ways, where the individual characteristics of nanoparticles based on quantum mechanical effects is explicitly utilized [5]. In this previous work, the RF was obtained by
0022-2313/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2004.01.074
ARTICLE IN PRESS T. Iida, H. Ishihara / Journal of Luminescence 108 (2004) 351–354
352
numerical calculation with surface integration of the Maxwell stress tensor (MST) obtained with the microscopic response field. In this present contribution, we derive analytical expressions for the RF exerted on a spherical nanoparticle, with confining of the excitonic center-of-mass (CM) motion, in order to clarify the connection between the properties of RFs on atomic systems with those on nanoscale condensed matter.
2. Theory For the steady state, we obtain the RF on the object by performing the integration of the MST over the surface surrounding the object [6], namely Z 2 /FS ¼ T n ds ; ð1Þ s 2
where / S is the time average, T is the MST, i.e., 2 2 T ¼ EE þ HH ð1=2ÞðjEj2 þ jHj2 ÞI ; and n is the normal vector at the surface surrounding object. In this expression, E and H are the2self-consistently calculated Maxwell fields, and I is the unit tensor. Eq. (1) can be derived regardless of whether the fields are microscopic or macroscopic. Hence, we can obtain the RF reflecting the quantum mechanical properties of objects by using the2 microscopic fields as E and H in the calculation of T : The response field is calculated by solving a selfconsistent system of Maxwell equations and the constitutive equations on the microscopic level. This means that the exciton-induced polarization is described by the non-local susceptibility and that the fields contain microscopic spatial distributions. By solving these equations within the linear response regime, we obtain the microscopic response field as [7], Eðr; oÞ ¼ EðiÞ ðr; oÞ X Z 2 þ ½Fz dr0 G ðr; r0 ; oÞ~ r 0z ðr0 Þ ; z ðiÞ
ð2Þ
from the nth excitonic level to the mth one. If we use bases diagonalizing the kinetic energy part in the Hamiltonian, off-diagonal components of the light-exciton interaction remain. Neglecting these off-diagonal components, Fz is described as Fz ¼ Fzð0Þ =½E% z _o ig ; where Fzð0Þ is the interaction between the transitionRdipole and the incident field expressed as Fzð0Þ ¼ v dr ~ r z0 ðrÞ EðiÞ ðr; oÞ (corresponding to the Rabi frequency of atomic systems in Ref. [3]) and g is a non-radiative damping parameter. We substitute the exact solution of det j½Ez dzz0 þ Azz0 _odzz0 j ¼ 0 into each eigen-energy of zth light-exciton coupled states E% z ¼ Ez þ Dz iGz in the later numerical calculation. Here Ez is the resonance energy of the bare exciton as a sum of the energy of the bulk transverse exciton and the quantized kinetic energy of excitonic CM motion, Dz the radiative shift from the bare excitonic energy, Gz the radiative width of the light-exciton coupled state and Azz0 ðoÞ the interaction between induced polarizations via electromagnetic fields [8].
3. Results and discussion In the following calculations, we employ the propagating plane wave EðiÞ ðzÞ ¼ Eiþ ðzÞ and the standing wave EðiÞ ðzÞ ¼ Eiþ ðzÞ þ Ei ðzÞ ¼ E0 ð0; 2 cos½k0 ðz z1 Þ ; 0Þ exp½iot as incident light, where Ei7 ðzÞ¼E0 ð0; exp½7ik0 ðzz1 Þ ; 0Þ exp½iot ; k0 ¼ o=c is the wavenumber in the vacuum and z1 is the anti-node position of the standing wave. Assuming the center of the particle is located at z ¼ 0; we expand the incident field into spherical surface harmonics and solve the microscopic Maxwell equation to determine the response fields. Substituting the response field Eq. (2) into Eq. (1), we obtain the following explicit expressions of the RF. For the plane wave propagating along z-axis (Fig. 1(a)), the expression becomes
V 2
0
where E ðr; oÞ is the incident field, G ðr; r ; oÞ the Green’s function including the effect of the back# ground dielectrics and ~ r mn ðrÞ ¼ /mjPðrÞjnS is a matrix element of the transition-dipole density
/Fz S ¼
X z
þ
k0
X zz0
ðjFzð0Þ j2 =2Þ½Gz þ g
ðEz þ Dz _oÞ2 þ ðGz þ gÞ2 azz0 ðR; oÞ;
ð3Þ
ARTICLE IN PRESS T. Iida, H. Ishihara / Journal of Luminescence 108 (2004) 351–354
Fig. 1. Geometry of the calculation. (a) For the case of irradiating a spherical nanoparticle with a propagating plane wave. (b) A nanoparticle is placed in the standing-wave field comprising two plane waves propagating in opposite directions along the z-axis with respect to each other.
353
ð4Þ
Fig. 2. Spectra of acceleration (radiation force/particle mass) for the case of irradiating a spherical nanoparticle with a propagating plane wave. The parameters for the CuCl Z3 exciton are used: eb ¼ 5:59; ET ¼ 3:2022 ðeVÞ; DLT ¼ 5:7 ðmeVÞ; Mex ¼ 2:3m0 ; where m0 is the free electron mass. (a) A comparison between the total radiation force calculated by using the method in Ref. [5] (exact) and that calculated by using Eq. (3) (approx.) for a small particle of radius 30 nm, including its gravitational acceleration. (b) Total force on a particle of radius of 50 nm, together with each force component, namely the scattering force, absorbing force and an additional term. All these forces are calculated by using the method in Ref. [5].
In the case of R-0; azz0 ðR; oÞ and bzz0 ðR; oÞ can be neglected because jFzð0Þ j2 pR3 and azz0 ðR; oÞ; bzz0 ðR; oÞpR6 where R is the radius of nanoparticle. Numerically evaluating Eq. (3) without azz0 ðR; oÞ for a radius of 30 nm, we have good agreement with the result of exact calculation (Fig. 2(a)). The remaining terms become closer to the atomic ones within the linear response regime of Ref. [3]. In Eq. (3), the terms proportional to Gz and proportional to g are called the scattering and absorbing force (dissipative force) terms, respectively, because the scattering cross section sscat and the absorbing cross section sabs are, respectively, proportional to Gz and g in the small radius limit (these two forces push the particle only toward the propagation direction of an incident plane-wave field). On the other hand, the first term of the
right-hand side of Eq. (4) is called as the gradient force (reactive force) because the magnitude of the force is proportional to the intensity gradient of the incident light. The sign of this force changes at the resonance energy Ez þ Dz : However, for a particle with a radius of several tens of nm, the magnitude of azz0 ðR; oÞand bzz0 ðR; oÞ greatly increases because of enhancement of the induced polarizations with an asymmetric spatial pattern from non-locality of the response. Particularly, for this particle with a radius of tens of nm, this force greatly contributes as negative force under certain conditions even if the incident light is a propagating plane wave with no gradient of intensity (arrow /IS in Fig. 2(b)). On the other hand, the additional term behaves as positive force (arrow /IIIS in Fig. 2(b)) from the superposition of
and the expression for the standing wave (Fig. 1(b)) is /Fz S ¼
X 1 ðEz þ Dz _oÞðjFzð0Þ j=E0 Þ2 ðrjEðiÞ j2 Þjz¼0 z
þ
4 X
ðEz þ Dz _ðoÞÞ2 þ ðGz þ gÞ2
bzz0 ðR; oÞ:
zz0
ARTICLE IN PRESS 354
T. Iida, H. Ishihara / Journal of Luminescence 108 (2004) 351–354
In conclusion, analytical expressions for radiation forces induced by resonant light are obtained, which are applicable over the size range from atoms to nanoparticles. Furthermore, we have clarified the presence of the condition that the radiation force has negative components even when the incident light is homogeneous. The future aim is to investigate condition under which this peculiar contribution of the radiation force is more prominent.
Acknowledgements The authors are grateful to Prof. K. Cho for fruitful discussions and support. They also thank Dr. H. Ajiki for useful discussions. This work was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Fig. 3. Spatial distributions of the absolute value of electric field corresponding to arrows /IS–/IVS in Fig. 2(b) (for unit incident intensity). A sphere is enclosed within the white circle. (a) For strong forward scattering so that the negative force is enhanced at /IS. This is from the anisotropy of the superposition of incident light and radiation arising from the TE mode exciton. (b), (d) Exactly at the energies of the lowest TE-mode and TM-mode excitons, respectively, where the additional term becomes 0. (c) At an energy between the lowest TE-mode and TM-mode excitons, where the additional term behaves as a positive force.
light-exciton coupled modes with different parities (Fig. 3(c)). We can understand that the origin of this force is similar to a gradient force arising from the asymmetric spatial distribution of the internal field.
References [1] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Opt. Lett. 11 (1986) 288. [2] H.-J. Guntherodt, D. Anselmetti, E. Meyer (Eds.), Forces in Scanning Probe Methods, NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 1995. [3] C. Cohen-Tanoudji, in: J. Dalibard, J. Raimond, J. ZinnJustion (Eds.), Fundamental Systems in Quantum Optics, Proceedings of the Les Houches Summer School, Session LIII, Amsterdam, North-Holland, 1992. [4] T. Iida, H. Ishihara, Opt. Lett. 27 (2002) 754. [5] T. Iida, H. Ishihara, Phys. Rev. Lett. 90 (2003) 057403. [6] J.D. Jackson, Classical Electrodynamics, 3rd Edition, Wiley, New York, 1999. [7] K. Cho, Prog. Theor. Phys. 106 (Suppl.) (1991) 225; K. Cho, Optical Response of Nanostructures, Springer, 2003. [8] H. Ajiki, J. Lumin. 94–95 (2001) 173.
Journal of Luminescence 108 (2004) 351–354
Radiation force induced by resonant light: from atom to nanoparticle Takuya Iidaa,*, Hajime Ishiharaa,b a
Department of Materials Engineering Science, Graduate school of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka 560-8531, Japan b CREST, Japan Science and Technology Corporation, Japan
Abstract We derive analytical expressions for the radiation force induced by electronically resonant light which apply over the size range from atoms to nanoparticles. These are based on the method of Maxwell stress tensor and microscopic response theory. The radiation force is expressed as a sum of a scattering force, an absorbing force and additional terms arising from the asymmetric spatial distribution of the internal field from light-exciton coupled modes. In the case of a particle with a radius of few tens of nm, a particular interesting point is that, under certain conditions, substantially large contributions from these terms appear as forces in opposite direction to the direction of propagation of plane wave light. r 2004 Elsevier B.V. All rights reserved. PACS: 32.80.t; 71.35.y; 78.67.n; 78.90.+t Keywords: Atom trapping; Optical manipulation; Radiation force; Resonance; Exciton; Quantum dot
1. Introduction Since the first experiment of mechanical managing of microparticles with a laser-induced radiation force (RF) [1], various fundamental studies and applications of the optical manipulation of micrometre objects have been demonstrated. This technique is utilized in a variety of fields including material engineering, biochemistry and micromachining. In many studies of the optical manipulation of micrometre objects, electronically non*Corresponding author. Tel.: +81-6-6850-6401; fax: +81-66850-6401. E-mail address: [email protected] (T. Iida).
resonant laser light with high power is utilized to generate sufficiently strong and heat-free forces (see article in Ref. [2]). However, the handling of nanoscale objects by using this technique is still challenging. On the other hand, resonant light is utilized for trapping of atoms in order to enhance the induced polarization [3]. Recently, we have theoretically proposed an enhancement mechanism of RF by the electronically resonant light [4] and a new type of optical manipulation that enables us to sort nanoparticles according to their size, and to arrange them in ways, where the individual characteristics of nanoparticles based on quantum mechanical effects is explicitly utilized [5]. In this previous work, the RF was obtained by
0022-2313/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2004.01.074
ARTICLE IN PRESS T. Iida, H. Ishihara / Journal of Luminescence 108 (2004) 351–354
352
numerical calculation with surface integration of the Maxwell stress tensor (MST) obtained with the microscopic response field. In this present contribution, we derive analytical expressions for the RF exerted on a spherical nanoparticle, with confining of the excitonic center-of-mass (CM) motion, in order to clarify the connection between the properties of RFs on atomic systems with those on nanoscale condensed matter.
2. Theory For the steady state, we obtain the RF on the object by performing the integration of the MST over the surface surrounding the object [6], namely Z 2 /FS ¼ T n ds ; ð1Þ s 2
where / S is the time average, T is the MST, i.e., 2 2 T ¼ EE þ HH ð1=2ÞðjEj2 þ jHj2 ÞI ; and n is the normal vector at the surface surrounding object. In this expression, E and H are the2self-consistently calculated Maxwell fields, and I is the unit tensor. Eq. (1) can be derived regardless of whether the fields are microscopic or macroscopic. Hence, we can obtain the RF reflecting the quantum mechanical properties of objects by using the2 microscopic fields as E and H in the calculation of T : The response field is calculated by solving a selfconsistent system of Maxwell equations and the constitutive equations on the microscopic level. This means that the exciton-induced polarization is described by the non-local susceptibility and that the fields contain microscopic spatial distributions. By solving these equations within the linear response regime, we obtain the microscopic response field as [7], Eðr; oÞ ¼ EðiÞ ðr; oÞ X Z 2 þ ½Fz dr0 G ðr; r0 ; oÞ~ r 0z ðr0 Þ ; z ðiÞ
ð2Þ
from the nth excitonic level to the mth one. If we use bases diagonalizing the kinetic energy part in the Hamiltonian, off-diagonal components of the light-exciton interaction remain. Neglecting these off-diagonal components, Fz is described as Fz ¼ Fzð0Þ =½E% z _o ig ; where Fzð0Þ is the interaction between the transitionRdipole and the incident field expressed as Fzð0Þ ¼ v dr ~ r z0 ðrÞ EðiÞ ðr; oÞ (corresponding to the Rabi frequency of atomic systems in Ref. [3]) and g is a non-radiative damping parameter. We substitute the exact solution of det j½Ez dzz0 þ Azz0 _odzz0 j ¼ 0 into each eigen-energy of zth light-exciton coupled states E% z ¼ Ez þ Dz iGz in the later numerical calculation. Here Ez is the resonance energy of the bare exciton as a sum of the energy of the bulk transverse exciton and the quantized kinetic energy of excitonic CM motion, Dz the radiative shift from the bare excitonic energy, Gz the radiative width of the light-exciton coupled state and Azz0 ðoÞ the interaction between induced polarizations via electromagnetic fields [8].
3. Results and discussion In the following calculations, we employ the propagating plane wave EðiÞ ðzÞ ¼ Eiþ ðzÞ and the standing wave EðiÞ ðzÞ ¼ Eiþ ðzÞ þ Ei ðzÞ ¼ E0 ð0; 2 cos½k0 ðz z1 Þ ; 0Þ exp½iot as incident light, where Ei7 ðzÞ¼E0 ð0; exp½7ik0 ðzz1 Þ ; 0Þ exp½iot ; k0 ¼ o=c is the wavenumber in the vacuum and z1 is the anti-node position of the standing wave. Assuming the center of the particle is located at z ¼ 0; we expand the incident field into spherical surface harmonics and solve the microscopic Maxwell equation to determine the response fields. Substituting the response field Eq. (2) into Eq. (1), we obtain the following explicit expressions of the RF. For the plane wave propagating along z-axis (Fig. 1(a)), the expression becomes
V 2
0
where E ðr; oÞ is the incident field, G ðr; r ; oÞ the Green’s function including the effect of the back# ground dielectrics and ~ r mn ðrÞ ¼ /mjPðrÞjnS is a matrix element of the transition-dipole density
/Fz S ¼
X z
þ
k0
X zz0
ðjFzð0Þ j2 =2Þ½Gz þ g
ðEz þ Dz _oÞ2 þ ðGz þ gÞ2 azz0 ðR; oÞ;
ð3Þ
ARTICLE IN PRESS T. Iida, H. Ishihara / Journal of Luminescence 108 (2004) 351–354
Fig. 1. Geometry of the calculation. (a) For the case of irradiating a spherical nanoparticle with a propagating plane wave. (b) A nanoparticle is placed in the standing-wave field comprising two plane waves propagating in opposite directions along the z-axis with respect to each other.
353
ð4Þ
Fig. 2. Spectra of acceleration (radiation force/particle mass) for the case of irradiating a spherical nanoparticle with a propagating plane wave. The parameters for the CuCl Z3 exciton are used: eb ¼ 5:59; ET ¼ 3:2022 ðeVÞ; DLT ¼ 5:7 ðmeVÞ; Mex ¼ 2:3m0 ; where m0 is the free electron mass. (a) A comparison between the total radiation force calculated by using the method in Ref. [5] (exact) and that calculated by using Eq. (3) (approx.) for a small particle of radius 30 nm, including its gravitational acceleration. (b) Total force on a particle of radius of 50 nm, together with each force component, namely the scattering force, absorbing force and an additional term. All these forces are calculated by using the method in Ref. [5].
In the case of R-0; azz0 ðR; oÞ and bzz0 ðR; oÞ can be neglected because jFzð0Þ j2 pR3 and azz0 ðR; oÞ; bzz0 ðR; oÞpR6 where R is the radius of nanoparticle. Numerically evaluating Eq. (3) without azz0 ðR; oÞ for a radius of 30 nm, we have good agreement with the result of exact calculation (Fig. 2(a)). The remaining terms become closer to the atomic ones within the linear response regime of Ref. [3]. In Eq. (3), the terms proportional to Gz and proportional to g are called the scattering and absorbing force (dissipative force) terms, respectively, because the scattering cross section sscat and the absorbing cross section sabs are, respectively, proportional to Gz and g in the small radius limit (these two forces push the particle only toward the propagation direction of an incident plane-wave field). On the other hand, the first term of the
right-hand side of Eq. (4) is called as the gradient force (reactive force) because the magnitude of the force is proportional to the intensity gradient of the incident light. The sign of this force changes at the resonance energy Ez þ Dz : However, for a particle with a radius of several tens of nm, the magnitude of azz0 ðR; oÞand bzz0 ðR; oÞ greatly increases because of enhancement of the induced polarizations with an asymmetric spatial pattern from non-locality of the response. Particularly, for this particle with a radius of tens of nm, this force greatly contributes as negative force under certain conditions even if the incident light is a propagating plane wave with no gradient of intensity (arrow /IS in Fig. 2(b)). On the other hand, the additional term behaves as positive force (arrow /IIIS in Fig. 2(b)) from the superposition of
and the expression for the standing wave (Fig. 1(b)) is /Fz S ¼
X 1 ðEz þ Dz _oÞðjFzð0Þ j=E0 Þ2 ðrjEðiÞ j2 Þjz¼0 z
þ
4 X
ðEz þ Dz _ðoÞÞ2 þ ðGz þ gÞ2
bzz0 ðR; oÞ:
zz0
ARTICLE IN PRESS 354
T. Iida, H. Ishihara / Journal of Luminescence 108 (2004) 351–354
In conclusion, analytical expressions for radiation forces induced by resonant light are obtained, which are applicable over the size range from atoms to nanoparticles. Furthermore, we have clarified the presence of the condition that the radiation force has negative components even when the incident light is homogeneous. The future aim is to investigate condition under which this peculiar contribution of the radiation force is more prominent.
Acknowledgements The authors are grateful to Prof. K. Cho for fruitful discussions and support. They also thank Dr. H. Ajiki for useful discussions. This work was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Fig. 3. Spatial distributions of the absolute value of electric field corresponding to arrows /IS–/IVS in Fig. 2(b) (for unit incident intensity). A sphere is enclosed within the white circle. (a) For strong forward scattering so that the negative force is enhanced at /IS. This is from the anisotropy of the superposition of incident light and radiation arising from the TE mode exciton. (b), (d) Exactly at the energies of the lowest TE-mode and TM-mode excitons, respectively, where the additional term becomes 0. (c) At an energy between the lowest TE-mode and TM-mode excitons, where the additional term behaves as a positive force.
light-exciton coupled modes with different parities (Fig. 3(c)). We can understand that the origin of this force is similar to a gradient force arising from the asymmetric spatial distribution of the internal field.
References [1] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Opt. Lett. 11 (1986) 288. [2] H.-J. Guntherodt, D. Anselmetti, E. Meyer (Eds.), Forces in Scanning Probe Methods, NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 1995. [3] C. Cohen-Tanoudji, in: J. Dalibard, J. Raimond, J. ZinnJustion (Eds.), Fundamental Systems in Quantum Optics, Proceedings of the Les Houches Summer School, Session LIII, Amsterdam, North-Holland, 1992. [4] T. Iida, H. Ishihara, Opt. Lett. 27 (2002) 754. [5] T. Iida, H. Ishihara, Phys. Rev. Lett. 90 (2003) 057403. [6] J.D. Jackson, Classical Electrodynamics, 3rd Edition, Wiley, New York, 1999. [7] K. Cho, Prog. Theor. Phys. 106 (Suppl.) (1991) 225; K. Cho, Optical Response of Nanostructures, Springer, 2003. [8] H. Ajiki, J. Lumin. 94–95 (2001) 173.