Download Magnetic field contribution to the Lorentz model

K. E. Oughstun and R. A. Albanese
Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. A
1751
Magnetic field contribution to the Lorentz model
Kurt E. Oughstun
College of Engineering and Mathematical Sciences, University of Vermont, Burlington, Vermont 05405-0156
Richard A. Albanese
Information Operations and Special Projects Division, U. S. Air Force Research Laboratory,
Brooks City-Base, Texas 78235
Received November 10, 2005; accepted December 8, 2005; posted January 24, 2006 (Doc. ID 65838)
The classical Lorentz model of dielectric dispersion is based on the microscopic Lorentz force relation and Newton’s second law of motion for an ensemble of harmonically bound electrons. The magnetic field contribution in
the Lorentz force relation is neglected because it is typically small in comparison with the electric field contribution. Inclusion of this term leads to a microscopic polarization density that contains both perpendicular
and parallel components relative to the plane wave propagation vector. The modified parallel and perpendicular polarizabilities are both nonlinear in the local electric field strength. © 2006 Optical Society of America
OCIS codes: 260.2030, 190.4400, 160.4760, 160.4330.
1. INTRODUCTION
1–3
The Lorentz model
of resonant dispersion phenomena
in dielectric materials is a classical model of central importance in optics4,5 as well as in the broader discipline of
electromagnetics.6 It is a causal model6,7 that describes
both normal and anomalous dispersion phenomena from
the infrared through the optical regions of the electromagnetic spectrum. This model is based on the microscopic Lorentz force relation
f共r,t兲 = ␳共r,t兲e共r,t兲 +
1
储c储
j共r,t兲 ⫻ b共r,t兲,
共1兲
where ␳共r , t兲 is the microscopic change density and j共r , t兲
= ␳共r , t兲dr / dt the microscopic current density of the
charged particle at the space–time point 共r , t兲 and where
f共r , t兲 is the microscopic force density exerted on the
charged particle by the electromagnetic field with microscopic electric intensity vector e共r , t兲 and magnetic induction vector b共r , t兲. The quantity ⴱ appearing in the double
brackets 储ⴱ储 in any equation here is used as a conversion
factor between cgs and MKS units.6,8 If that factor is included in that equation, then the equation is in cgs units,
while if the factor is omitted from that equation, then it is
in MKS units. If no such factor appears in a given equation, then that equation is correct in either system of
units.
The magnetic field contribution appearing on the righthand side of Eq. (1) is typically assumed to be negligible
in comparison with the electric field contribution for sufficiently small field strengths and values of the relative
velocity v / c of the charged particle, where v = 兩dr / dt兩. This
assumption then leads to the classical Lorentz theory of
dielectric dispersion in which electric field effects alone
are considered. The question then remains as to what effects are introduced by the inclusion of the magnetic field
contribution in the Lorentz theory. The solution of this
important problem has been partially addressed9–11 for
1084-7529/06/071751-6/$15.00
the special case of a chiral molecule, where it has been
shown11 that the magnitude of the magnetic response of
the induced second-order optical activity can be comparable to the magnitude of the electric response alone.
However, as we have been unable to find any previously
published general solution of this problem, it is addressed
in this paper.
2. MODIFIED LORENTZ MODEL OF
DIELECTRIC DISPERSION
With the complete Lorentz force relation given in Eq. (1)
as the driving force, the equation of motion of a harmonically bound electron is given by
d 2r j
dt
2
+ 2␦j
drj
dt
+ ␻j2rj = −
qe
m
冉
Eeff共r,t兲 +
1 drj
储c储 dt
冊
⫻ Beff共r,t兲 ,
共2兲
where Eeff共r , t兲 is the effective local electric field intensity
and Beff共r , t兲 is the effective local magnetic induction field
at the space–time point 共r , t兲 and where rj = rj共r , t兲 describes the displacement of the electron from its equilibrium position. Here qe denotes the magnitude of the
charge and m the mass of the harmonically bound electron with undamped resonance frequency ␻j and phenomenological damping constant ␦j. The temporal Fourier integral representation of the electric and magnetic field
vectors of the effective local plane electromagnetic wave is
given by8
Ẽeff共r, ␻兲 =
冕
⬁
−⬁
© 2006 Optical Society of America
Eeff共r,t兲ei␻tdt,
共3兲
1752
J. Opt. Soc. Am. A / Vol. 23, No. 7 / July 2006
B̃eff共r, ␻兲 =
冕
K. E. Oughstun and R. A. Albanese
⬁
Beff共r,t兲ei␻tdt
r̃j · k = − i
−⬁
=
储c储
␻
k共␻兲 ⫻ Eeff共r, ␻兲,
共4兲
where k共␻兲 is the wavevector of the plane wave field with
magnitude given by the wave number k共␻兲 = ␻ / c in
vacuum, since the effective local field is essentially a microscopic field. With the temporal Fourier integral representation
r̃j共␻兲 =
冕
the dynamical equation of motion given in Eq. (2) becomes
共␻2 + 2i␦j␻ − ␻j2兲r̃j =
m
qe
=
m
r̃j · Ẽeff =
qe/m
␻ − ␻j2 + 2i␦j␻
2
冋
⫻ 1+
2
␻2
共qe/mc兲2Ẽeff
2
共␻2 − ␻j2 + 2i␦j␻兲2 − 共qe/mc兲2Ẽeff
␻2
qe
冋
aj =
qe/m
␻ − ␻j2 + 2i␦j␻
2
冋
⫻ 1+
关共1 + ir̃j · k兲Ẽeff − i共r̃j · Ẽeff兲k兴,
bj = − i
1 + ir̃j · k
m ␻ −
2
␻j2
+ 2i␦j␻
Ẽeff − i
r̃j · Ẽeff
␻ −
2
␻j2
+ 2i␦j␻
册
k .
共7兲
The electron displacement vector may then be expressed
as a linear combination of the orthogonal pair of vectors k
and Ẽeff as
共8兲
r̃j = ajẼeff + bjk,
where, because of the transversality relation k · Ẽeff = 0,
aj =
r̃j · Ẽeff
2
Ẽeff
2
␻2
共qe/mc兲2Ẽeff
2
共␻2 − ␻j2 + 2i␦j␻兲2 − 共qe/mc兲2Ẽeff
␻2
2
共qe/mc兲2Ẽeff
2
共␻2 − ␻j2 + 2i␦j␻兲2 − 共qe/mc兲2Ẽeff
␻2
,
册
, 共13兲
共14兲
respectively.
The local (or microscopic) induced dipole moment p̃j
⬅ −qer̃j for the jth Lorentz oscillator type is then given by
p̃j共r, ␻兲 = − qe共ajẼeff共r, ␻兲 + bjk兲.
共15兲
If there are Nj Lorentz oscillators per unit volume of the
jth type, then the macroscopic polarization induced in the
medium is given by the summation over all oscillator
types of the spatially averaged locally induced dipole moments as
P̃共r, ␻,Ẽeff兲 =
兺 N 具具p̃ 共r, ␻兲典典
j
j
j
共9兲
,
2
Ẽeff
.
The coefficients aj and bj appearing in Eq. (8) are then
given by
共6兲
r̃j =
册
共12兲
共Ẽeff − ir̃j ⫻ 共k ⫻ Ẽeff兲兲
with formal solution
共11兲
,
共5兲
−⬁
qe
2
共␻2 − ␻j2 + 2i␦j␻兲2 − 共qe/mc兲2Ẽeff
␻2
and substitution of this result into the second relation
gives
⬁
rj共t兲ei␻tdt,
2
共qe/mc兲2Ẽeff
␻2
= 具具Ẽeff共r, ␻兲典典
兺N␣
j j⬜共␻,Ẽeff兲
j
c2
bj =
␻
r̃
2 j
+k
共10兲
· k.
兺 N ␣ 共␻,Ẽ
j j储
eff兲.
共16兲
j
Here
The pair of scalar products appearing in the above expression may be evaluated using the expression given Eq. (7)
as
共0兲
共2兲
␣j⬜共␻,Ẽeff兲 ⬅ ␣j⬜
共␻兲 + ␣j⬜
共␻,Ẽeff兲
=
r̃j · k = − i
r̃j · Ẽeff =
qe
qe
r̃j · Ẽeff
2
m ␻2 − ␻j2 + 2i␦j␻
1 + ir̃j · k
m ␻2 − ␻j2 + 2i␦j␻
k ,
2
Ẽeff
.
Substitution of the second relation into the first then
yields
− q2e /m
␻2 − ␻j2 + 2i␦j␻
冋
⫻ 1+
2
␻2
共qe/mc兲2Ẽeff
2
共␻2 − ␻j2 + 2i␦j␻兲2 − 共qe/mc兲2Ẽeff
␻2
册
共17兲
is defined as the perpendicular component of the atomic
共0兲
polarizability, with ␣j⬜
共␻兲 ⬅ ␣j⬜共␻ , 0兲 = ␣j共␻兲, where ␣j共␻兲
is the classical expression for the atomic polarizability
K. E. Oughstun and R. A. Albanese
Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. A
when magnetic field effects are neglected, and
␣j储共␻,Ẽeff兲 ⬅ i
2
共q3e /m2兲Ẽeff
2
共␻2 − ␻j2 + 2i␦j␻兲2 − 共qe/mc兲2Ẽeff
␻2
共18兲
is defined here as the parallel component of the atomic polarizability, where ␣j储共␻ , 0兲 = 0.
3. CRITICAL FIELD STRENGTH
The atomic polarizability is then seen to be nonlinear in
the local electric field strength when magnetic field effects
共2兲
共␻ , Ẽeff兲 appearing in
are included. The nonlinear term ␣j⬜
Eq. (17) will be negligible in comparison with the linear
共0兲
term ␣j⬜
共␻兲 when the local electric field strength is sufficiently smaller than the critical field strength Ecrit
= Ecrit共␻兲 defined by the condition
冏
2
共qe/mc兲2Ẽcrit
␻2
2
共␻2 − ␻j2 + 2i␦j␻兲2 − 共qe/mc兲2Ẽcrit
␻2
冏
1753
undefined; the estimate given by Eq. (21) yields an approximate value of 7.5⫻ 1011 V / m. Notice that Ecrit → ⬁ at
both of the near-resonance values ␻0± as well as when either ␻ → 0 or ␻ → ⬁. Notice also that since megavolt per
meter electric field strengths are produced by ultrashort
pulsed terawatt laser systems, the minimum critical field
strength for this example is at least five orders of magnitude greater than that which is currently available.
The angular frequency dependence of the relative cor共2兲
共0兲
rection factor ␣j⬜
共␻ , Ẽeff兲 / ␣j⬜
共␻兲 for the perpendicular
atomic polarizability given in Eq. (17) is illustrated in Fig.
3 for the near-infrared resonance line of triply distilled
water for several values of the relative local electric field
intensity about unity. Notice that this correction factor
reaches its peak value at the resonance frequency ␻0
when 兩Ẽeff / 共Ecrit兲min兩 ⬍ 1 and that this peak value bifurcates into a pair of symmetric peaks about this resonance
frequency when 兩Ẽeff / 共Ecrit兲min兩 ⬎ 1, a local minimum now
⬅ 1,
with solution
Ecrit共␻兲 =
mc
共␻2 − ␻j2兲2 + 4␦j2␻2
冑2qe ␻关共␻2 − ␻j2兲2 − 4␦j2␻2兴1/2
.
共19兲
When 兩Ẽeff兩 Ⰶ Ecrit, the linear term in Eq. (17) dominates
共0兲
共␻兲. Notice
the nonlinear term so that ␣j⬜共␻ , Ẽeff兲 ⯝ ␣j⬜
that this critical electric field strength depends on the
value of the applied angular frequency ␻ of the electromagnetic wave field. Furthermore, notice that this critical
field strength in undefined in the angular frequency band
␻ 苸 关␻j− , ␻j+兴, where
␻j± ⬅ 共␻j2 + 2␦j2 ± 2␦j冑␻j2 + ␦j2兲1/2
共20兲
␻j 苸 关␻j− , ␻j+兴.
for each resonance feature j, where
Finally, a
rough estimate of the minimum value for this critical field
strength is seen to be given by
共Ecrit兲min ⬇
mc
冑2qe
␻j =
ប␻j
冑8 ␮ B
共21兲
Fig. 1. Real (solid curve) and imaginary (dashed curve) parts of
the linear atomic polarizability for the near-infrared resonance
line in water.
for each resonance feature, where ␮B ⬅ qeប / 2mc is the
Bohr magneton. The quantity ប␻j is recognized as the energy level state of the harmonically bound electron.
4. NUMERICAL EXAMPLE
Consider a single resonance Lorentz model dielectric 共j
= 0兲 with material parameters representative of the nearinfrared line in triply distilled water at 25 ° C, where ␻0
= 6.19⫻ 1014 r / s and ␦0 = 2.86⫻ 1013 r / s. The angular frequency dependence of the real and imaginary parts of the
共0兲
linear atomic polarizability ␣j⬜
⬅ ␣j共␻兲 for this single resonance line is presented in Fig. 1, and the angular frequency dependence of the critical field strength described
by Eq. (19) is illustrated in Fig. 2, where ␻0− ⯝ 5.91
⫻ 1014 r / s and ␻0+ ⯝ 6.48⫻ 1014 r / s. The minimum critical
electric field strength 共Ecrit兲min ⯝ 1.38⫻ 1011 V / m is seen
to occur at the two points just below and above the frequency band 关␻0− , ␻0+兴 where the critical field strength is
Fig. 2. Angular frequency dependence of the critical local electric field strength (in volts per meter) for the near-infrared resonance line in water.
1754
J. Opt. Soc. Am. A / Vol. 23, No. 7 / July 2006
Fig. 3. Angular frequency dependence of the correction factor
for the perpendicular atomic polarizability for several values of
the relative local electric field strength.
K. E. Oughstun and R. A. Albanese
electric field strength goes to zero and so represents the
classical Lorentz result when magnetic field effects are
neglected. Magnetic field effects are entirely negligible
until the minimum critical field strength is approached
from below. At the critical field strength the real transverse atomic polarizability has just started to bifurcate
into a pair of resonance peaks, one below the linear resonance frequency ␻0 and the other above that characteristic angular frequency. Notice that the downshifted nonlinear resonance is stronger than the upshifted nonlinear
resonance. The downshifted nonlinear resonance peak increases as the local field strength continues to increase
above the minimum critical field strength while the upshifted peak continues to decrease in strength. Similar results are obtained for the imaginary part of the transverse atomic polarizability, illustrated in Fig. 5.
An approximate expression for the locations of these
upshifted and downshifted nonlinear resonance peaks
may be obtained from the appropriate zeros of the denominator in the correction factor appearing in Eq. (17).
The zeros of this expression are given by the roots of the
quadratic equation ␻2 − ␻j2 + 2i␦j␻ = ± 共qe / mc兲Ẽeff␻. Neglect
of the damping term results in the approximate pair of
positive frequency solutions
␻± ⬇
␮BẼeff
ប
再 冋 冉 冊册 冎
±1 + 1 +
ប␻j
␮BẼeff
2
1/2
,
共22兲
where
lim ␻± ⬇ ␻j .
共23兲
˜ →0
E
eff
In the opposite limit when Ẽeff Ⰷ ប␻j / ␮B, the limiting values
Fig. 4. Angular frequency dependence of the real part of the
transverse atomic polarizability for critical and supercritical values of the local electric field strength compared to the linear behavior at zero field strength.
appearing at the resonance frequency ␻0. As this relative
local electric field strength continues to increase above
unity, these two peak values continue to separate as they
also increase in strength. The minimum value at ␻0, however, becomes nearly constant so that the curves cross
each other as the frequency is either increased to the upper peak or decreased to the lower peak from this minimum value point. Finally, since this correction factor is
below unity for all real frequencies when 兩Ẽeff / 共Ecrit兲min兩
⬍ 1, magnetic field effects can be safely neglected in the
Lorentz model of the perpendicular atomic polarizability
when this inequality is well satisfied.
The nonlinear character of the real part of the perpendicular atomic polarizability that is described by Eq. (17)
is presented in Fig. 4 for local electric field strength values increasing above the minimum critical field strength.
The zero field curve in the figure describes the linear
atomic polarizability obtained in the limit as the local
Fig. 5. Angular frequency dependence of the imaginary part of
the transverse atomic polarizability for critical and supercritical
values of the local electric field strength compared with the linear behavior at zero field strength.
K. E. Oughstun and R. A. Albanese
Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. A
1755
resonance about the undamped resonance frequency ␻0
when the local electric field strength is equal to or less
than the minimum critical field strength 共Ecrit兲min given
in Eq. (21). However, as the local field strength exceeds
this minimum critical value, this single resonance peak is
found to split into a pair of resonance structures, one
downshifted and the other uphifted in angular frequency
from the characteristic resonance frequency ␻0 of the Lorentz model dielectric.
5. DISCUSSION
Fig. 6. Angular frequency dependence of the real part of the longitudinal atomic polarizability for critical and subcritical values
of the local electric field strength.
Fig. 7. Angular frequency dependence of the imaginary part of
the longitudinal atomic polarizability for critical and subcritical
values of the local electric field strength.
␻+ ⬇
ប␻j2
2␮BẼeff
␻− ⬇
+
2␮B
ប
ប␻j2
2␮BẼeff
,
Ẽeff ,
The results presented here show the precise manner in
which the magnetic field influences the atomic polarizability in the Lorentz model of resonance polarization. Inclusion of the magnetic field in the Lorentz force relation
results in a microscopic polarization density that contains
both perpendicular and parallel components relative to
the plane wave propagation vector of the local driving
field that are both nonlinear in the local electrical field
strength. This nonlinearity becomes significant when the
local applied electric field strength exceeds a minimum
critical field strength whose value increases linearly with
the resonance frequency. Numerical calculations show
that these nonlinear terms are entirely negligible for effective field strengths that are typically less than
⬃1012 V / m and that they begin to have a significant contribution for field strengths that are typically greater
than ⬃1015 V / m for a highly absorptive material. Fortunately, this critical field strength is at least five orders of
magnitude greater than that which is currently available
in ultrashort pulsed terawatt laser systems. Although
this magnetic field effect is negligible in most practical
situations, it has been shown to be significant in materials exhibiting chirality,9–11 and it may also become important for artificial materials. Nevertheless, seemingly excessive electric field strengths commonly occur in nature;
for example, it is estimated12 that the critical field
strength for the alignment of water molecules for crystallization into polar ice crystals is greater than 109 V / m.
共24兲
共25兲
are obtained.
Similar results are obtained for the frequency dependence of the nonlinear character of the parallel atomic polarizability given in Eq. (18). Although this parallel component of the atomic polarizability vanishes as Ẽeff → 0, it
can exceed the strength of the transverse component of
the atomic polarizability when Ẽeff exceeds the minimum
critical field strength. The frequency dependence of the
real part of the longitudinal (parallel) component is presented in Fig. 6 and the imaginary part in Fig. 7 for two
values of the local electric field strength. Both the real
and the imaginary parts are seen to exhibit a pronounced
Fig. 8. Angular frequency dependence of the relative phasor velocity magnitude of the harmonically bound electron in the Lorentz model.
1756
J. Opt. Soc. Am. A / Vol. 23, No. 7 / July 2006
K. E. Oughstun and R. A. Albanese
Notice that relativistic effects need not be considered in
this extended model when the local field strength is equal
to or below 10% of this minimum critical field strength
since the magnitude of the phasor velocity v ⬅ r̃˙ = −i␻r̃
j
j
j
for the harmonically bound electron, where r̃j is given by
Eq. (8), remains below ⬃0.02c when Ẽeff / 共Emin兲min 艋 0.1
for the example considered here, the maximum value occurring near resonance. However, relativistic effects must
be considered when the minimum critical field strength is
reached and exceeded, especially near resonance, as illustrated in Fig. 8.
The physical origin of the nonlinear term considered
here is due to the diamagnetic effect that appears in the
analysis of the interaction of an electromagnetic field with
a charged particle in the quantum theory of
electrodynamics.13,14 The Hamiltonian for this coupled
system is given by (see Eqs. XIII.71–XIII.72 of Messiah13)
H = H0 −
qe
2mc
H·L+
q2e
8mc2
H
兺
ACKNOWLEDGMENT
The research presented in this paper has been supported
by the U.S. Air Force Office of Scientific Research under
AFOSR grants F49620-01-0306 and USAF 9550-04-10447.
The authors can be reached by e-mail at
[email protected] and [email protected].
REFERENCES
1.
2.
Z
2
taken to be given by the Bohr radius a0 ⬅ ប2 / mq2e ⬇ 5.29
⫻ 10−9 cm, where ប ⬅ h / 2␲ and h is Planck’s constant, the
electric field strength is E ⬇ 5.13⫻ 1011 V / m, in general
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2
rj⬜
,
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