- Email: [email protected]
Collective Rayleigh scattering from dielectric particles: A classical theory of the Collective Atomic Recoil Laser
1 March 1998
Optics Communications 148 Ž1998. 54–58
Collective Rayleigh scattering from dielectric particles: a classical theory of the Collective Atomic Recoil Laser B.W.J. McNeil 1, G.R.M. Robb Department of Physics and Applied Physics, UniÕersity of Strathclyde, Glasgow G4 0NG, Scotland, UK Received 17 September 1997; accepted 23 October 1997
Abstract A classical theory of Rayleigh scattering of electromagnetic radiation by dielectric particles is presented, in which the particles may interact cooperatively via common pump and probe radiation fields. The collective nature of the process is manifest both in the exponential growth of the bunching of the particles at the radiation wavelength to form a particle grating, and in the exponential growth of the counterpropagating probe radiation intensity. We call this ‘Collective Rayleigh Scattering’. This process arises from a fully classical description of the Collective Atomic Recoil Laser ŽCARL. with linear dielectric particles. q 1998 Elsevier Science B.V. PACS: 33.20.Fb; 94.10.Gb; 78.35.q c; 72.20.Dp
1. Introduction Rayleigh scattering may occur when radiation is incident upon a dielectric particle which has dimensions that are small with respect to the radiation wavelength. The electric field of the radiation induces an oscillatory dipole upon the dielectric particle which in turn re-radiates radiation in directions different from the incident radiation. Previous work has shown that it is possible to bunch such particles at the radiation wavelength to form a refractive grating w1x. This process relies upon the temperature dependence of the particle refractive index via the absorption of radiation and has been called ‘Forced Rayleigh Scattering’ ŽFRS.. Typically this effect may be observed in a system of particles within two counterpropagating coherent radiation fields, the grating being formed along the axis of copropagation. This Letter presents the theory of a new classical phenomenon which we call ‘Collective Rayleigh Scattering’ ŽCRS.. This phenomenon originates from a fully
1
E-mail: [email protected]
classical analysis of the Collective Atomic Recoil Laser ŽCARL. w2x. As with FRS, CRS may generate a refractive grating of particles at the radiation wavelength, but relies upon the dispersive rather than the absorptive properties of the particles, and so is physically distinct from FRS. Further, CRS is a collectiÕe effect resulting in the formation of a grating and counterpropagating radiation field which have exponential growth rates. Thus the grating may be generated from an electromagnetic pump field with either a small counterpropagating Žprobe. field or fluctuations due to noise in the particle density.
2. Theory In order to demonstrate the principle of CRS we use a simple 1-D model consisting of a strong plane wave pump field which is scattered by an initially uniform spatial distribution of dielectric particles. The scattered radiation is modelled by an initially very weak counterpropagating plane wave probe field. The collective dynamic behaviour of the dielectric particles will occur via interaction with the common radiation
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 7 . 0 0 6 2 9 - 9
B.W.J. McNeil, G.R.M. Robb r Optics Communications 148 (1998) 54–58
fields, the electric field of which is described as follows: E Ž z ,t . s E1Ž z ,t . q E2 Ž z ,t . ,
Ž1.
iŽ k zy v t .
where E1Ž z,t . s Ž A1Ž z,t . e q c.c.. xˆ and E2Ž z,t . s Ž A 2 eyiŽ k zq v t . q c.c.. xˆ represent two counterpropagating plane waves in "z. ˆ The real constant A 2 defines the strong pump field and the complex variable A1Ž z,t . defines the envelope of the counterpropagating probe field which we assume obeys the slowly varying envelope approximation ŽSVEA.. In what follows we assume that the particles are small and are made from an uncharged dielectric material with a constant scalar susceptibility, x , describing a linear isotropic medium. By small we mean that the characteristic dimensions of the particles are much smaller than the radiation wavelength i.e. r < l, where l s 2prk. In this way the radiation fields within a particle may be assumed to be approximately constant, allowing the particle position within the fields to be approximated by the Dirac delta function d Ž r y r j ., where r j is the position vector of the jth particle. So far no assumptions have been made regarding the origin of the dielectric properties or the actual number of atoms or molecules constituting the particles.
55
where the complex susceptibility has been written as x s x 1 q i x 2 . In deriving Eq. Ž4. we have used the SVEA approximation by neglecting terms E A1rE t < v A1. The non-resonant ‘fast’ terms A e " i2 v t and the dipole–dipole forces have also been omitted. By integrating Eq. Ž4. over the particle volume Vp and introducing the particle phase u j s 2 kz j the z-component of the equations of motion of the jth particle are obtained: duj dt d pj dt
2k s M
pj ,
Ž5.
s 2 e 0 kVp i x 1 A 2 A1 Ž z ,t . e iu j y c.c.
qx 2 < A1Ž z ,t . < 2 y A22
4.
Ž6.
Here p is the z-component of the particle momentum and M is the mass of the particle. The first term of Ž6. represents a Ždispersive. ponderomotive force while the second term describes the Žabsorptive. radiation pressure force exerted upon the particle. The x-component of the particle force is zero as seen by application of the divergence theorem in the integral over the particle volume. ŽThe particle has zero net charge..
2.1. Particle equations of motion
2.2. Field eÕolution equation
We begin by considering the forces exerted on a single particle by the pump and probe radiation fields defined by Eq. Ž1.. From the Lorentz force equation, the force per unit particle volume will be
The evolution of the probe radiation field is obtained from the Maxwell wave equation with a polarisation current source term:
F s qE q j = B,
ž
Ž2.
where q is the electric charge density and j is the current density. As the particles are assumed neutral dielectrics, then q is the polarisation charge density and j is the polarisation current so that q s y= P P ,
j s E PrE t ,
Ž3.
where P is the polarisation. We assume that the polarisation is induced by the radiation fields and so is determined only by these fields and the complex susceptibility x via the linear relation: P Ž z ,t . s e 0 x A1Ž z ,t . e iŽ k zy v t .
c
zˆ = Ž E1 y E2 . ,
and q and j from Ž3. into Eq. Ž2. yields the force per unit particle volume, the z-component of which is given by: Fz s 2 e 0 k i x 1 A 2 A1 Ž z ,t . e i2 k z y c.c.
qx 2 < A1Ž z ,t . < 2 y A22
4,
Ž4.
1 E2
2
y
c2 E t 2
/
E Ž z ,t . s ym 0 c 2 = Ž = P P . q m0
E 2 P Ž z ,t . E t2
.
Ž7.
Note that the polarisation vector here arises from the distribution of many dielectric particles and is denoted P Ž z,t . to distinguish it from the polarisation P Ž z,t . of the particle medium as defined by x . The first term of Ž7. will be zero as = P P s 0, P having no x- or y-dependence and, as we are neglecting dipole–dipole interactions, Pz s 0. Substituting for the field Ž1. into the wave equation Ž7. we obtain
Ez
Substitution for P Ž z,t ., the radiation magnetic fields B Ž z ,t . s
Ez
E A1
qA 2 eyiŽ k zq v t . q c.c. 4 x. ˆ
1
E2
q
1 E A1 c Et
ik s 2 e0
eyiŽ k zy v t . P P x. ˆ
Ž8.
In deriving Ž8. the amplitude of the pump field E2 Ž z,t . is assumed constant and the SVEA approximation has been applied. Further, in the spirit of this approximation, it has been assumed that E 2 P Ž z,t .rE t 2 f yv 2 P Ž z,t .. We now calculate the polarisation P Ž z,t . as a sum over the contributions of the individual particle dipole moments, ie. P Ž z ,t . s Ý d Ž z ,t . d Ž r y r j Ž t . . , j
Ž9.
B.W.J. McNeil, G.R.M. Robb r Optics Communications 148 (1998) 54–58
56
where d Ž z,t . is the dipole moment of a particle induced by the radiation fields and r j Ž t . are the particle position vectors. The dipole moment of a particle is given by d Ž z ,t . s
HV P Ž z ,t . dV f e V Ž x A 0 p
yiŽ k zq v t .
2
e
q c.c. . xˆ ,
where the latter approximation occurs as we have assumed that the polarisation is induced by the strong pump field E2 Ž z,t . alone and, as the dimensions of the particle are assumed small, the spatial variation of the radiation field throughout the particle may be neglected. Substituting for Ž9. into the wave equation Ž8. we obtain q
Ez
1 E A1
ikVp A 2 x
f
c Et
2
Ý eyi ud Ž r y r j Ž t . . ,
Ž 10.
j
where again we have neglected fast terms A e " i2 v t . We now average this equation over a sample volume V with z-dimensions sufficiently small so as to allow the wave envelope A1Ž z,t . to be approximately constant. We then obtain
E A1 Ez
q
1 E A1 c Et
ikVp A 2 x n p
s
2
²eyi u : ,
Ž 11.
p Ž.... is an average over the Np where ²Ž....: s Ž1rNp .Ý Njs1 particles contained within the volume V, and n p is the number of particles per unit volume. We assume that Np 4 1 in order to avoid large statistical variations in the averaging process.
2.3. Scaling the coupled equations Eqs. Ž5., Ž6. and Ž11. now form a closed system of coupled partial differential equations describing the selfconsistent evolution of both the particle dynamics and the probe radiation. We now assume that x 1 4 x 2 and scale these coupled equations by defining the scaled variables ts
ct lc
,
zs
z
p
,
ps
(
rc n p Mc 2
lc
A Ž z ,t . s yi2
rc Mc
e0
,
l A1Ž z ,t . ,
lc s
, Ž 12 .
4prc
and
rc s
ž
e 0 x 12 A22 f 4m p c 2
1r3
/
f 1.67 = 10y9
x 12 I2 f
ž / mp
1r3
,
Ž 13.
where m p is the mass density of a particle, f is a fractional filling factor equal to the total volume occupied by the particles per unit volume and I2 is the pump radiation intensity. With this scaling the particle equations of motion and the wave equation reduce to d u jrd t s pj ,
EA Et
EA q
Ez
s²eyi u :.
Ž 15. Ž 16 .
Here
p
E A1
d p jrd t s y Ž A e iu j q c.c. . y G ,
Ž 14.
Gs
e 0 x 2 A22 rc2 m p c 2
f 1.85 = 10y26
x 2 I2 rc2 m p
Ž 17.
is the radiation pressure parameter due to absorption of the pump radiation. These equations are identical in form to those derived for the CARL interaction for a system of two-level atoms w3x. For times t < Gy1, the radiation pressure term G may be neglected and the equations become identical in form to the equations describing the Free Electron Laser ŽFEL. interaction w4x. Under certain realisable conditions both of these interactions give rise to an exponential instability in both the probe radiation intensity and also the density modulation of the particles as described by the bunching parameter b s ²eyi u :w4x. The solutions to these equations fall into two limiting cases as determined by the length of the interaction region containing the particles with respect to the cooperation length l c Ž12.. The cooperation length is that length over which particles may interact cooperatively via the radiation fields. For the case of particles which are, on average, stationary with respect to the laboratory frame of reference, the cooperation length is equal to the gain length of the system, l g , this being the characteristic length for the exponential instability w3x. The first limiting case is the steady state limit where the length of the interaction region occupied by the particles, L, is much larger than a cooperation length l c . In this limit propagation effects may be neglected except in a small region of the sample where ‘strong superradiance’ may occur w3,4x. The second limit is where the length of the interaction region is much less than a cooperation length and propagation effects dominate the interaction. This regime gives rise to ‘weak superradiance’ w3,4x. All previous discussions of the CARL mechanism have discussed the pump driven polarisation in terms of a quantum mechanical atomic two-level system w2,3x. For atoms the source of the polarisation is purely electronic in nature and has been described by the so-called MaxwellBloch equations w5x and so may include non-linear and saturation effects. Eqs. Ž14. – Ž16. however are purely classical in origin being derived from Maxwells’ equations and the Lorentz force equation. The pump driven polarisation, as determined by x , need not be electronic in nature but may also result from ionic andror dipolar contributions. Further, the particles may be considered, not as single atoms, but as a grouping of many bound atoms or molecules, so long as the sizes of the particles are much smaller than the radiation wavelength. As we will now show, this has important consequences for the type of gain
B.W.J. McNeil, G.R.M. Robb r Optics Communications 148 (1998) 54–58
57
mechanism of the probe intensity that may occur during the interaction between the particles and fields.
particle size Ž19. gives:
2.4. The effect of temperature
allowing for a wide range of particle sizes. The radiation pressure term Ž17. is G f 7.48 = 10y8 , and as in what follows we only consider values of t < Gy1, radiation pressure effects may be neglected. The cooperation length Ž12. l c s 3.25 = 10 3 m. Clearly interaction regions within the laboratory are going to be much less than this length which puts the system firmly in the weak superradiant regime. We will assume an interaction length of L s 10 cm which in scaled units is L s Lrl c s 3.08 = 10y5. For this weak superradiant regime a good approximation to the wave equation Ž16. is given by w3x:
For a ‘cold’ resonant distribution of particles the initial distribution of scaled momenta pj is centred on zero and has a width s < 1 w3x. In this case the evolution of both the bunching and the probe radiation intensity is exponential up to saturation. On increasing the width of the distribution of pj the exponential gain decreases as does the saturation intensity. For larger values of s ) 1 the gain mechanism changes to a Raman type gain w3x with much smaller saturation intensities than that of the cold distribution case. The spread in p which gives a cold distribution type evolution typically occurs for s Q st f 0.1 which we henceforth define as the cold distribution limit. If we consider the system of particles to be Maxwellian at temperature T then the Žone-dimensional. spread in p of the particles is easily shown to be
ss
1
rc
(
k BT
.
Mc 2
For a cold distribution Ž s Q 0.1. we then require that T ŽK. Q
rc2 Mc 2st 2 kB
f 1.81 = 10
20
M
x 12 I2 f
ž / mp
2r3
, Ž 18 .
where we have used the definition of rc Ž13.. Hence the larger the mass of the particle, the higher the equilibrium temperature may be for the particle distribution to be considered ‘cold’. The requirement that the particles be small with respect to the radiation wavelength and the cold distribution limit Ž18. are combined to describe the range of particle sizes as follows:
ž
3k BT 4pst rc2 m p c 2
1.92 = 10y8 m Q r < 10y4 m,
d Ardt s²eyi u :y K Ž A y A 0 . ,
Ž 20.
where K s 1rL is a damping parameter and A 0 s AŽ z s 0,t .. In Fig. 1 we show the solution of Eqs. Ž14., Ž15. and Ž20. for an initial scaled probe field of A 0 s 10y6 and scaled particle momentum of p j Ž t s 0. s 0 ; j. The peak of the scaled probe intensity is < A < 2 f 6 = 10y10 corresponding to an intensity of I1 f 31 mW my2 . This is very small with respect to the steady state value at saturation Ž< A < 2 f 1.4. and is typical of the weak superradiant regime w4x. We stress that the bunching of the particles however, as measured by < b < s <²eyi u :<, attains the significant value of f 0.75 indicating a large spatial modulation of the particle density. As we are considering macroscopic particles the effect of Joule heating via pump absorption on the bulk properties of each dielectric particle may become important. We now justify the neglect of such heating in the above
1r3
/
Q r < l.
Ž 19.
The requirement of having many particles within the averaged volume Ž Np 4 1. may further reduce the upper limit depending upon the geometry of the system.
3. Example We now give a simple example. We consider a system of solid glass spheres as the particles, being illuminated by a pump laser of wavelength l s 100 mm and intensity I2 s 1 MW my2 . We use the following values for the material properties of the glass w6,7x: m p s 2.6 = 10 3 kg my3 and x Ž l s 100 mm. s 2.87 q i0.063, and let the fractional filling factor f s 10y3. With these values then rc s 2.45 = 10y9 and for T s 300 K, the limits on the
Fig. 1. The scaled probe intensity < A < 2 and the bunching parameter < b < as a function of the scaled time t.
B.W.J. McNeil, G.R.M. Robb r Optics Communications 148 (1998) 54–58
58
example. As the rate of Joule heating per unit particle volume by the pump is Q s k x 2 I2 , then the rate of change of particle temperature is given by dT
Q s
dt
s cÕ mp
k x 2 I2
mp cÕ
f 2.27 = 10 3 K sy1 ,
where c Õ s 670 J kgy1 Ky1 is the specific heat capacity for glass. For the simulation of Fig. 1 the total time duration to saturation of the probe is f 10 ms which gives a temperature rise in a glass particle of f 25 K. The system as described should then show significant bunching of the dielectric particles without any catastrophic heating of the particles.
4. Conclusion It has been shown that a system of small dielectric particles initially homogeneously distributed in space may collectively scatter an incident coherent electromagnetic wave. As the particles are small with respect to the radiation wavelength, and they interact via the common radiation fields to form a spatially modulated Žbunched. distribution, we call this ‘Collective Rayleigh Scattering’. As in the CARL, this effect is due solely to the dispersive properties of the particles unlike Forced Rayleigh Scattering which relies upon the particles’ absorptive properties. An example was given where the interaction was of the weak superradiant type. Although the scattered field for this example is very weak, the predicted modulation of the particle density is large. This suggests that it should be possible to produce density gratings of macroscopic dielectric particles via CRS at room temperature. Collective Rayleigh Scattering should also be observable in ring cavity systems analogous with that as described for the CARL and described in detail in Ref. w3x. It
may also occur in a system of particles suspended in a fluid if the bunching forces are sufficient to overcome viscous effects and could also arise in very large systems. A tentative example would be inter-stellar dust clouds pumped by stellar masers where cosmic distances may allow the interaction to take place over many gain lengths resulting in a much higher intensity of scattered radiation than the weak superradiant example described above. However, other effects such as plasma interactions within the dust clouds and the non-neutrality of the particles would have to be taken into account.
Acknowledgements We would like to thank the EPSRC and the DRA for supporting this work, and R. Bonifacio, A.D.R. Phelps and P.J.M. van der Slot for useful discussions and helpful suggestions.
References w1x D.W. Phol, S.E. Schwarz, V. Irniger, Phys. Rev. Lett. 31 Ž1973. 32. w2x R. Bonifacio, L. De Salvo, Nucl. Instrum. Methods Phys. Res. A 341 Ž1994. 360; R. Bonifacio, L. De Salvo, L.M. Narducci, E.J. D’Angelo, Phys. Rev. A 50 Ž1994. 1716. w3x R. Bonifacio, G.R.M. Robb, B.W.J. McNeil, Phys. Rev. A 56 Ž1997. 912. w4x R. Bonifacio, F. Casagrande, G. Cerchioni, L. De Salvo Souza, P. Pierini, N. Piovella, Riv. Nuovo Cimento 13 Ž1990. 9. w5x F.T. Arecchi, R. Bonifacio, IEEE J. Quantum Electron. QE-1 Ž1965. 169; L. Allen, J.H. Eberly, Optical Resonance and Two Level Atoms, Dover, New York, 1987. w6x E.D. Palik ŽEd.., Handbook of Optical Constants of Solids, Academic Press, London, 1985. w7x R.M. Tennent ŽEd.., Science Data Book, 8th ed., Oliver and Boyd, 1983.
Optics Communications 148 Ž1998. 54–58
Collective Rayleigh scattering from dielectric particles: a classical theory of the Collective Atomic Recoil Laser B.W.J. McNeil 1, G.R.M. Robb Department of Physics and Applied Physics, UniÕersity of Strathclyde, Glasgow G4 0NG, Scotland, UK Received 17 September 1997; accepted 23 October 1997
Abstract A classical theory of Rayleigh scattering of electromagnetic radiation by dielectric particles is presented, in which the particles may interact cooperatively via common pump and probe radiation fields. The collective nature of the process is manifest both in the exponential growth of the bunching of the particles at the radiation wavelength to form a particle grating, and in the exponential growth of the counterpropagating probe radiation intensity. We call this ‘Collective Rayleigh Scattering’. This process arises from a fully classical description of the Collective Atomic Recoil Laser ŽCARL. with linear dielectric particles. q 1998 Elsevier Science B.V. PACS: 33.20.Fb; 94.10.Gb; 78.35.q c; 72.20.Dp
1. Introduction Rayleigh scattering may occur when radiation is incident upon a dielectric particle which has dimensions that are small with respect to the radiation wavelength. The electric field of the radiation induces an oscillatory dipole upon the dielectric particle which in turn re-radiates radiation in directions different from the incident radiation. Previous work has shown that it is possible to bunch such particles at the radiation wavelength to form a refractive grating w1x. This process relies upon the temperature dependence of the particle refractive index via the absorption of radiation and has been called ‘Forced Rayleigh Scattering’ ŽFRS.. Typically this effect may be observed in a system of particles within two counterpropagating coherent radiation fields, the grating being formed along the axis of copropagation. This Letter presents the theory of a new classical phenomenon which we call ‘Collective Rayleigh Scattering’ ŽCRS.. This phenomenon originates from a fully
1
E-mail: [email protected]
classical analysis of the Collective Atomic Recoil Laser ŽCARL. w2x. As with FRS, CRS may generate a refractive grating of particles at the radiation wavelength, but relies upon the dispersive rather than the absorptive properties of the particles, and so is physically distinct from FRS. Further, CRS is a collectiÕe effect resulting in the formation of a grating and counterpropagating radiation field which have exponential growth rates. Thus the grating may be generated from an electromagnetic pump field with either a small counterpropagating Žprobe. field or fluctuations due to noise in the particle density.
2. Theory In order to demonstrate the principle of CRS we use a simple 1-D model consisting of a strong plane wave pump field which is scattered by an initially uniform spatial distribution of dielectric particles. The scattered radiation is modelled by an initially very weak counterpropagating plane wave probe field. The collective dynamic behaviour of the dielectric particles will occur via interaction with the common radiation
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 7 . 0 0 6 2 9 - 9
B.W.J. McNeil, G.R.M. Robb r Optics Communications 148 (1998) 54–58
fields, the electric field of which is described as follows: E Ž z ,t . s E1Ž z ,t . q E2 Ž z ,t . ,
Ž1.
iŽ k zy v t .
where E1Ž z,t . s Ž A1Ž z,t . e q c.c.. xˆ and E2Ž z,t . s Ž A 2 eyiŽ k zq v t . q c.c.. xˆ represent two counterpropagating plane waves in "z. ˆ The real constant A 2 defines the strong pump field and the complex variable A1Ž z,t . defines the envelope of the counterpropagating probe field which we assume obeys the slowly varying envelope approximation ŽSVEA.. In what follows we assume that the particles are small and are made from an uncharged dielectric material with a constant scalar susceptibility, x , describing a linear isotropic medium. By small we mean that the characteristic dimensions of the particles are much smaller than the radiation wavelength i.e. r < l, where l s 2prk. In this way the radiation fields within a particle may be assumed to be approximately constant, allowing the particle position within the fields to be approximated by the Dirac delta function d Ž r y r j ., where r j is the position vector of the jth particle. So far no assumptions have been made regarding the origin of the dielectric properties or the actual number of atoms or molecules constituting the particles.
55
where the complex susceptibility has been written as x s x 1 q i x 2 . In deriving Eq. Ž4. we have used the SVEA approximation by neglecting terms E A1rE t < v A1. The non-resonant ‘fast’ terms A e " i2 v t and the dipole–dipole forces have also been omitted. By integrating Eq. Ž4. over the particle volume Vp and introducing the particle phase u j s 2 kz j the z-component of the equations of motion of the jth particle are obtained: duj dt d pj dt
2k s M
pj ,
Ž5.
s 2 e 0 kVp i x 1 A 2 A1 Ž z ,t . e iu j y c.c.
qx 2 < A1Ž z ,t . < 2 y A22
4.
Ž6.
Here p is the z-component of the particle momentum and M is the mass of the particle. The first term of Ž6. represents a Ždispersive. ponderomotive force while the second term describes the Žabsorptive. radiation pressure force exerted upon the particle. The x-component of the particle force is zero as seen by application of the divergence theorem in the integral over the particle volume. ŽThe particle has zero net charge..
2.1. Particle equations of motion
2.2. Field eÕolution equation
We begin by considering the forces exerted on a single particle by the pump and probe radiation fields defined by Eq. Ž1.. From the Lorentz force equation, the force per unit particle volume will be
The evolution of the probe radiation field is obtained from the Maxwell wave equation with a polarisation current source term:
F s qE q j = B,
ž
Ž2.
where q is the electric charge density and j is the current density. As the particles are assumed neutral dielectrics, then q is the polarisation charge density and j is the polarisation current so that q s y= P P ,
j s E PrE t ,
Ž3.
where P is the polarisation. We assume that the polarisation is induced by the radiation fields and so is determined only by these fields and the complex susceptibility x via the linear relation: P Ž z ,t . s e 0 x A1Ž z ,t . e iŽ k zy v t .
c
zˆ = Ž E1 y E2 . ,
and q and j from Ž3. into Eq. Ž2. yields the force per unit particle volume, the z-component of which is given by: Fz s 2 e 0 k i x 1 A 2 A1 Ž z ,t . e i2 k z y c.c.
qx 2 < A1Ž z ,t . < 2 y A22
4,
Ž4.
1 E2
2
y
c2 E t 2
/
E Ž z ,t . s ym 0 c 2 = Ž = P P . q m0
E 2 P Ž z ,t . E t2
.
Ž7.
Note that the polarisation vector here arises from the distribution of many dielectric particles and is denoted P Ž z,t . to distinguish it from the polarisation P Ž z,t . of the particle medium as defined by x . The first term of Ž7. will be zero as = P P s 0, P having no x- or y-dependence and, as we are neglecting dipole–dipole interactions, Pz s 0. Substituting for the field Ž1. into the wave equation Ž7. we obtain
Ez
Substitution for P Ž z,t ., the radiation magnetic fields B Ž z ,t . s
Ez
E A1
qA 2 eyiŽ k zq v t . q c.c. 4 x. ˆ
1
E2
q
1 E A1 c Et
ik s 2 e0
eyiŽ k zy v t . P P x. ˆ
Ž8.
In deriving Ž8. the amplitude of the pump field E2 Ž z,t . is assumed constant and the SVEA approximation has been applied. Further, in the spirit of this approximation, it has been assumed that E 2 P Ž z,t .rE t 2 f yv 2 P Ž z,t .. We now calculate the polarisation P Ž z,t . as a sum over the contributions of the individual particle dipole moments, ie. P Ž z ,t . s Ý d Ž z ,t . d Ž r y r j Ž t . . , j
Ž9.
B.W.J. McNeil, G.R.M. Robb r Optics Communications 148 (1998) 54–58
56
where d Ž z,t . is the dipole moment of a particle induced by the radiation fields and r j Ž t . are the particle position vectors. The dipole moment of a particle is given by d Ž z ,t . s
HV P Ž z ,t . dV f e V Ž x A 0 p
yiŽ k zq v t .
2
e
q c.c. . xˆ ,
where the latter approximation occurs as we have assumed that the polarisation is induced by the strong pump field E2 Ž z,t . alone and, as the dimensions of the particle are assumed small, the spatial variation of the radiation field throughout the particle may be neglected. Substituting for Ž9. into the wave equation Ž8. we obtain q
Ez
1 E A1
ikVp A 2 x
f
c Et
2
Ý eyi ud Ž r y r j Ž t . . ,
Ž 10.
j
where again we have neglected fast terms A e " i2 v t . We now average this equation over a sample volume V with z-dimensions sufficiently small so as to allow the wave envelope A1Ž z,t . to be approximately constant. We then obtain
E A1 Ez
q
1 E A1 c Et
ikVp A 2 x n p
s
2
²eyi u : ,
Ž 11.
p Ž.... is an average over the Np where ²Ž....: s Ž1rNp .Ý Njs1 particles contained within the volume V, and n p is the number of particles per unit volume. We assume that Np 4 1 in order to avoid large statistical variations in the averaging process.
2.3. Scaling the coupled equations Eqs. Ž5., Ž6. and Ž11. now form a closed system of coupled partial differential equations describing the selfconsistent evolution of both the particle dynamics and the probe radiation. We now assume that x 1 4 x 2 and scale these coupled equations by defining the scaled variables ts
ct lc
,
zs
z
p
,
ps
(
rc n p Mc 2
lc
A Ž z ,t . s yi2
rc Mc
e0
,
l A1Ž z ,t . ,
lc s
, Ž 12 .
4prc
and
rc s
ž
e 0 x 12 A22 f 4m p c 2
1r3
/
f 1.67 = 10y9
x 12 I2 f
ž / mp
1r3
,
Ž 13.
where m p is the mass density of a particle, f is a fractional filling factor equal to the total volume occupied by the particles per unit volume and I2 is the pump radiation intensity. With this scaling the particle equations of motion and the wave equation reduce to d u jrd t s pj ,
EA Et
EA q
Ez
s²eyi u :.
Ž 15. Ž 16 .
Here
p
E A1
d p jrd t s y Ž A e iu j q c.c. . y G ,
Ž 14.
Gs
e 0 x 2 A22 rc2 m p c 2
f 1.85 = 10y26
x 2 I2 rc2 m p
Ž 17.
is the radiation pressure parameter due to absorption of the pump radiation. These equations are identical in form to those derived for the CARL interaction for a system of two-level atoms w3x. For times t < Gy1, the radiation pressure term G may be neglected and the equations become identical in form to the equations describing the Free Electron Laser ŽFEL. interaction w4x. Under certain realisable conditions both of these interactions give rise to an exponential instability in both the probe radiation intensity and also the density modulation of the particles as described by the bunching parameter b s ²eyi u :w4x. The solutions to these equations fall into two limiting cases as determined by the length of the interaction region containing the particles with respect to the cooperation length l c Ž12.. The cooperation length is that length over which particles may interact cooperatively via the radiation fields. For the case of particles which are, on average, stationary with respect to the laboratory frame of reference, the cooperation length is equal to the gain length of the system, l g , this being the characteristic length for the exponential instability w3x. The first limiting case is the steady state limit where the length of the interaction region occupied by the particles, L, is much larger than a cooperation length l c . In this limit propagation effects may be neglected except in a small region of the sample where ‘strong superradiance’ may occur w3,4x. The second limit is where the length of the interaction region is much less than a cooperation length and propagation effects dominate the interaction. This regime gives rise to ‘weak superradiance’ w3,4x. All previous discussions of the CARL mechanism have discussed the pump driven polarisation in terms of a quantum mechanical atomic two-level system w2,3x. For atoms the source of the polarisation is purely electronic in nature and has been described by the so-called MaxwellBloch equations w5x and so may include non-linear and saturation effects. Eqs. Ž14. – Ž16. however are purely classical in origin being derived from Maxwells’ equations and the Lorentz force equation. The pump driven polarisation, as determined by x , need not be electronic in nature but may also result from ionic andror dipolar contributions. Further, the particles may be considered, not as single atoms, but as a grouping of many bound atoms or molecules, so long as the sizes of the particles are much smaller than the radiation wavelength. As we will now show, this has important consequences for the type of gain
B.W.J. McNeil, G.R.M. Robb r Optics Communications 148 (1998) 54–58
57
mechanism of the probe intensity that may occur during the interaction between the particles and fields.
particle size Ž19. gives:
2.4. The effect of temperature
allowing for a wide range of particle sizes. The radiation pressure term Ž17. is G f 7.48 = 10y8 , and as in what follows we only consider values of t < Gy1, radiation pressure effects may be neglected. The cooperation length Ž12. l c s 3.25 = 10 3 m. Clearly interaction regions within the laboratory are going to be much less than this length which puts the system firmly in the weak superradiant regime. We will assume an interaction length of L s 10 cm which in scaled units is L s Lrl c s 3.08 = 10y5. For this weak superradiant regime a good approximation to the wave equation Ž16. is given by w3x:
For a ‘cold’ resonant distribution of particles the initial distribution of scaled momenta pj is centred on zero and has a width s < 1 w3x. In this case the evolution of both the bunching and the probe radiation intensity is exponential up to saturation. On increasing the width of the distribution of pj the exponential gain decreases as does the saturation intensity. For larger values of s ) 1 the gain mechanism changes to a Raman type gain w3x with much smaller saturation intensities than that of the cold distribution case. The spread in p which gives a cold distribution type evolution typically occurs for s Q st f 0.1 which we henceforth define as the cold distribution limit. If we consider the system of particles to be Maxwellian at temperature T then the Žone-dimensional. spread in p of the particles is easily shown to be
ss
1
rc
(
k BT
.
Mc 2
For a cold distribution Ž s Q 0.1. we then require that T ŽK. Q
rc2 Mc 2st 2 kB
f 1.81 = 10
20
M
x 12 I2 f
ž / mp
2r3
, Ž 18 .
where we have used the definition of rc Ž13.. Hence the larger the mass of the particle, the higher the equilibrium temperature may be for the particle distribution to be considered ‘cold’. The requirement that the particles be small with respect to the radiation wavelength and the cold distribution limit Ž18. are combined to describe the range of particle sizes as follows:
ž
3k BT 4pst rc2 m p c 2
1.92 = 10y8 m Q r < 10y4 m,
d Ardt s²eyi u :y K Ž A y A 0 . ,
Ž 20.
where K s 1rL is a damping parameter and A 0 s AŽ z s 0,t .. In Fig. 1 we show the solution of Eqs. Ž14., Ž15. and Ž20. for an initial scaled probe field of A 0 s 10y6 and scaled particle momentum of p j Ž t s 0. s 0 ; j. The peak of the scaled probe intensity is < A < 2 f 6 = 10y10 corresponding to an intensity of I1 f 31 mW my2 . This is very small with respect to the steady state value at saturation Ž< A < 2 f 1.4. and is typical of the weak superradiant regime w4x. We stress that the bunching of the particles however, as measured by < b < s <²eyi u :<, attains the significant value of f 0.75 indicating a large spatial modulation of the particle density. As we are considering macroscopic particles the effect of Joule heating via pump absorption on the bulk properties of each dielectric particle may become important. We now justify the neglect of such heating in the above
1r3
/
Q r < l.
Ž 19.
The requirement of having many particles within the averaged volume Ž Np 4 1. may further reduce the upper limit depending upon the geometry of the system.
3. Example We now give a simple example. We consider a system of solid glass spheres as the particles, being illuminated by a pump laser of wavelength l s 100 mm and intensity I2 s 1 MW my2 . We use the following values for the material properties of the glass w6,7x: m p s 2.6 = 10 3 kg my3 and x Ž l s 100 mm. s 2.87 q i0.063, and let the fractional filling factor f s 10y3. With these values then rc s 2.45 = 10y9 and for T s 300 K, the limits on the
Fig. 1. The scaled probe intensity < A < 2 and the bunching parameter < b < as a function of the scaled time t.
B.W.J. McNeil, G.R.M. Robb r Optics Communications 148 (1998) 54–58
58
example. As the rate of Joule heating per unit particle volume by the pump is Q s k x 2 I2 , then the rate of change of particle temperature is given by dT
Q s
dt
s cÕ mp
k x 2 I2
mp cÕ
f 2.27 = 10 3 K sy1 ,
where c Õ s 670 J kgy1 Ky1 is the specific heat capacity for glass. For the simulation of Fig. 1 the total time duration to saturation of the probe is f 10 ms which gives a temperature rise in a glass particle of f 25 K. The system as described should then show significant bunching of the dielectric particles without any catastrophic heating of the particles.
4. Conclusion It has been shown that a system of small dielectric particles initially homogeneously distributed in space may collectively scatter an incident coherent electromagnetic wave. As the particles are small with respect to the radiation wavelength, and they interact via the common radiation fields to form a spatially modulated Žbunched. distribution, we call this ‘Collective Rayleigh Scattering’. As in the CARL, this effect is due solely to the dispersive properties of the particles unlike Forced Rayleigh Scattering which relies upon the particles’ absorptive properties. An example was given where the interaction was of the weak superradiant type. Although the scattered field for this example is very weak, the predicted modulation of the particle density is large. This suggests that it should be possible to produce density gratings of macroscopic dielectric particles via CRS at room temperature. Collective Rayleigh Scattering should also be observable in ring cavity systems analogous with that as described for the CARL and described in detail in Ref. w3x. It
may also occur in a system of particles suspended in a fluid if the bunching forces are sufficient to overcome viscous effects and could also arise in very large systems. A tentative example would be inter-stellar dust clouds pumped by stellar masers where cosmic distances may allow the interaction to take place over many gain lengths resulting in a much higher intensity of scattered radiation than the weak superradiant example described above. However, other effects such as plasma interactions within the dust clouds and the non-neutrality of the particles would have to be taken into account.
Acknowledgements We would like to thank the EPSRC and the DRA for supporting this work, and R. Bonifacio, A.D.R. Phelps and P.J.M. van der Slot for useful discussions and helpful suggestions.
References w1x D.W. Phol, S.E. Schwarz, V. Irniger, Phys. Rev. Lett. 31 Ž1973. 32. w2x R. Bonifacio, L. De Salvo, Nucl. Instrum. Methods Phys. Res. A 341 Ž1994. 360; R. Bonifacio, L. De Salvo, L.M. Narducci, E.J. D’Angelo, Phys. Rev. A 50 Ž1994. 1716. w3x R. Bonifacio, G.R.M. Robb, B.W.J. McNeil, Phys. Rev. A 56 Ž1997. 912. w4x R. Bonifacio, F. Casagrande, G. Cerchioni, L. De Salvo Souza, P. Pierini, N. Piovella, Riv. Nuovo Cimento 13 Ž1990. 9. w5x F.T. Arecchi, R. Bonifacio, IEEE J. Quantum Electron. QE-1 Ž1965. 169; L. Allen, J.H. Eberly, Optical Resonance and Two Level Atoms, Dover, New York, 1987. w6x E.D. Palik ŽEd.., Handbook of Optical Constants of Solids, Academic Press, London, 1985. w7x R.M. Tennent ŽEd.., Science Data Book, 8th ed., Oliver and Boyd, 1983.