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Constitutive relations and the electromagnetic spectrum in a fluctuating medium
Volume
8, number
OF’TlCS COMMUNICATIONS
2
CONSTITUTIVE
RELATIONS
June 1973
AND THE ELECTROMAGNETIC
IN A FLUCTUATING
SPECTRUM
MEDIUMS
L. MANDEL and E. WOLF Department
of‘ Physics and Astronomy, University Rochester, New York 14627, USA
Received
of Rochester,
16 April 1973
A two-parameter susceptibility function x(t;~) is introduced, to ChdIaCteriZC the local macroscopic properties of a linear isotropic dielectric, whose behavior may change in time. x(t;~) generally is a stochastic variable with respect to its first argument. An expression is derived for the spectral density of the induced polarization in terms of the spcctral densities of the fluctuations of x and of the electric field. In general, this relationship is not a simple convolution. In the special case when the incident electric field is monochromatic, the induced polarization has the usual Brillouintype spectrum.
Recent years have witnessed a great deal of research on molecular scattering, that is largely a consequence of the fact that many scattering experiments that cannot be carried out with light from conventional sources can now be performed with laser beams. The research has already provided valuable new information about thermal fluctuations in gases and liquids, about lattice vibrations in solids, about phase transitions, etc.t. In the analysis of these experiments a time-dependent dielectric response function or susceptibility function characterizing the fluctuating medium is generally employed. However, a response function that depends on only one temporal argument does not adequately describe the macroscopic response of a time-dependent medium, except over a very narrow frequency range of the electromagnetic spectrum, and, therefore, cannot completely account for results of scattering experiments. In particular, one might expect such a response function to prove inadequate when the frequencies of the incident radiation are close to the resonance frequencies of the medium.
In the ‘present note we introduce a two-parameter response function ~(t;r) that characterizes the local macroscopic dielectric response of a linear isotropic medium*. With the help of this function which, in general, is a random function* of its first argument, we derive an expression for the spectral density of the induced polarization in terms of the spectral densities of the fluctuations of the medium and of the electric field. When the electric field is monochromatic, the induced polarization has the usual Brillouin-type spectrum in an exact sense. However, in the general case, the spectral density of the induced polarization is not a simple convolution of the spectral densities of the incident field and of the susceptibility fluctuations. Let us begin by recalling some standard results relating to electromagnetic fields in linear, spatially non-dispersive media, whose macroscopic properties do not depend on time. At each point in the medium the Fourier time transform of the electric field E(t) and of the induced polarization vector P(r) are con+ Generalization of our results to non-isotropic
$ Research supported by the Air Force Office of Scientific Research. t For reviews of some of these researches see, for example, refs. 1l-31.
media and to magnetic media is straightforward. $ Our response function is essentially a stochastic operator of the class studied extensively from the mathematical standpoint by Adomian 141. However, we feel that our treatment in terms of the random process X(t; w) allows somewhat more insight than the use of stochastic Green’s functions.
95
Volume 8, number 2
OPTICS COMMUNICATIONS
netted by the constitutive
relation
P(w) = i(w) E(w) )
(1)
with
h(w)=
s E(t)
eiwr
-cc
dr .
@a>
and
P(w)= /
P(t)eiwrdt,
(Zb)
-_
where i(w) is the susceptibility tensor for the frequency w. For the sake of simplicity we restrict ourselves to an isotropic medium, in which case the susceptibility tensor i(w) may be replaced by a scalar g(w). Taking the Fourier transform of (I) and imposing the requirement of causality, we obtain the following constitutive relation in the time domain: r p(r) = j-
x(t-
t’) E(t)) dt'
(34
dT,
(3b)
-cc
=
s
x(T)E(t--)
0
where
~(7) = (l/In)
s g(w) ePiw7 dw . -cc
(4)
In eq. (3a) x depends on t and i only through the difference t- t’, which is a reflection of the fact that the macroscopic properties of the medium do not depend on time. Since bothp(t) and E(t) are real, ~(7) is necessarily also real, and ~(7) = 0 for T < 0 because of the assumption of causality. Now suppose that the macroscopic response of the medium changes in time (e.g., because of thermal fluctuations), but that the response remains linear. Then, in place of (3b), we postulate the general linear causal relation
.iunc
in which the response function x is labelled by two arguments and is, in most situations, a stationary random function of its first argument. The dependence of x(t;~) on its second argument characterizes the response of the medium to a sufficiently short pulse of electromagnetic radiation. For each value of r, there exists an ensemble of realizations x(t;r), whose PI-operties can, in principle, be obtained from a microscopic treatment of the medium. Similarly, for such a medium the dielectric response function E must also be labelled by two temporal arguments f and ‘T,and the displacement vector D(I) is then related to E(t) via the formula
D(t) - E(t) + 4nP(t)
x(t;T)E(t0
96
7) dr
,
(5)
= j- c(t;T) E(t-
7) dr
,
(5)
0
from which it follows, with the use of eq. (5), that c(t;7)
= S(7) + 4nx(t;7).
(7)
The time scales for the variations of either x(r:~) or e(t;T) with respect to t and 7 are generally quite different. The t-variations, which are associated with thermal fluctuations in most media, are usually confined within a bandwidth of some MHz (often kHz) about zero frequency, whereas the r-variations, associated with optical transitions between atomic states, are centered at frequencies of order 1014-1015 Hz, and may have bandwidths of order 100 MHz. However, for microwave transitions and rapidly fluctuating media the two frequency ranges may approach each other, or even overlap. The Fourier transform y(t;w) of x(t; 7) with respect to its second argumentt T, viz.,
X(t.w)
=s
x(t;T)
eiW7 d7
(8)
0
is another random function of time t. For each value of w there exists an ensemble of realizations of y(t; w) that represents the response of the fluctuating medium to a monochromatic electric field of frequency w. Although a complete expression for our generalized susceptibility can only be obtained from a microscopic
$ We use a circumflex P(t) =I
1973
[as in eqs. (2)] to denote the Fourier r-transform, tilde [as in eq. (S)] to denote the Fourier s-transform, and bar [as in eq. (15b)j to denote theFourier transform with respect to both r and 7.
Volume 8, number 2
OPTICS
treatment of the medium, it is not difficult to see how the generalization arises. Consider first a gas or a liquid whose macroscopic properties are time independent. The susceptibility x(w) at frequency w is then given in terms of the average number of molecules N per unit volume and the mean polarizability O(W) of each molecule by the formula (see for example ref. [5], p. 87) x(w) = N@(W) [ I-
$rNcl(w)]-1
)
[l-
$rN(t)c+J)]-1
.
(10)
The range of validity of this equation can, of course, only be determined from detailed microscopic considerations, but our brief discussion illustrates the origin of the generalized susceptibility. Let us first consider the idealized case when the variations of the medium are strictly deterministic, so that the ensemble reduces to a single realization. Moreover, we first take the electric field to be strictly monochromatic, of frequency ou,
P(w) = + k(w-w
= f [U e-iwOf
+ a*
eiwof]
,
u;w())a
+jT*(--o-w();wu)a*]) (14)
where X(0; wu) = r X(t; wu) e iwf dt -cc =
s _m
dt eiat
s x(t;r) 0
(15a)
eiwcT dr .
(15b’l
Eq. (14) shows that when the electric field is monochromatic, the Fourier spectrum of the induced polarization consists of two shifted contributions from the susceptibility spectrum. Next let us consider the more realistic case when both the electric field and the medium fluctuate and the fluctuations are not deterministic. We shall assume that E(t) and F(t; w) are independent random processes, each of which is stationary at least in the wide sense*. If .we represent the electric field as a Fourier integral (strictly speaking in the sense of the theory of generalized functions), we obtain from (5), with the help of (8), the following expression for P(t): P(t)=(1/2n)
s T(t;w)E(w) -cc
eerw’dw.
(16)
P(t) is now a random process, and it follows from (5) or (16) that, like z(t;w) and E(t), it is also stationary, at least in the wide sense. From (16) we readily deduce that the auto-correlation tensor of P(t) is given by (Pi(t)&(t+
I
1973
used. From (12) it readily follows that the Fourier transform P(w) of P(t) is given by
(9)
which is the Lorentz-Lorenz relation expressed in terms of the susceptibility rather than in terms of the refractive index of the medium, The frequency dependence of the polarizability Q(W) is readily obtained from the Lorentz oscillator model [S, section 2.3.41. Suppose now that the average number density of the molecules undergoes macroscopic fluctuations, e.g., because the medium is traversed by compression waves. Then the average number density N in (9), and hence also the susceptibility, become functions of time. Thus in place of (9) we now have the equation ~(t;w)=N(t)a(w)
June
COMMUNICATIONS
7))
(11)
where a is a constant vector. It then follows from (1 l), (5) and (8), that the induced polarization P(t) is given by P(~)=~[~(t;w~)ae~iw~‘t~(t;~~)a*eiw~f],(12)
The suffices i, k G, k = 1,2,3) label Cartesian components and the sharp brackets denote ensemble aver-
where the relation
* For the definition
Z(r;--w)
= y*(t, 0) )
which is a consequence
(13) of the reality of x (t; r), was
of stationarity in the wide sense see, for example, ref. [6 1, p. 60. The assumed statistical indepcndence of E(t) and y(t, w) is an approximation, which is valid whenever the contribution to the electric field E(t) due to the susceptibility fluctuations is small compared with the remainder of E (t).
97
Volume 8, number 2
OPTICS COMMUNICATIONS
ages. R$‘(w) is the spectral density (power spectrum) tensor of the electric field, which, for a stationary field E(t), satisfies the relation4 (l/271) (+J)
E&J’))
= $;)(c@
(w-G.I’) .
(18)
Let us now introduce the spectral density of ii(t; wO) and the spectral density tensor of P(t), with the help of the Wiener-Khintchine theorem:
k’(~;tio)= J‘ (~(t;aO)~(t+ -cc
7;~~)) eiw7 d7,
June 1973
W(‘)(W) is peaked at frequencies fwo, and is sufficiently narrow compared with the spectral density QZ)(m, o’), in both o and w’, we may approximate ?+‘@)(a) under the integral in eq. (2 1) by a Dirac S-function, and write I4@)(U’) = (I) [6 (WI- Uo) + 6 (w’ + &Jo)] )
(W
where (1) represents the average intensity of the electric field. Eq. (21) then gives the following simple expression for the spectral density of the induced polarization in terms of the spectral density of 7:
(19) 9%(w)
=
s (Pi(t) Pk (t + 7)) eiWT d7 . _m
(20)
P’(w)
= (11274
x (I) [ti~)(w-Wo;Wo)+ On taking the Fourier T-transform of eq. (17) with respect to r, interchanging the order of integrations, and making use of eqs. (19) and (20), we obtain the following expression for the spectral density tensor $$(a):
(21) Eq. (21) shows that, if ?@)(u;cQ’) were independent of the second argument, the spectral density of the induced polarization P(t) would be just the convolution of the spectral densities of E(t) and y(t;w). However, because of the presence of the second argument in MT), which is a consequence of the dispersive properties of the medium, the integral in eq. (21) is somewhat more complicated than a simple convolution, The tensorial features of the spectral density tensors “;5[) and I$% are sometimes unimportant. For example, if the incident light field is fully polarized, as from a laser, then the spectra are adequately represented by scalar functions, and the tensorial indices in eq. (21) can be suppressed. If, in addition, the spectral density 6 According to the Wiener-Khintchine theorem, the spectral density tensor W,$?(w) of the electric field, defined by eq. (18) is, of course, related to the aut@correlation tensor of the electric field by the formula R$)(u)
=_& (Ei(r) Ek(f + 7)) eiw7 dT .
(Cf.. ref. (61, p. 103 and ref. [7].)
98
w(‘ii~(-~-oo;wo)]. (23)
In deriving (23) the condition (13) was used and also the fact that g(f; w) is a wide-sense stationary process. It should however, be noted that most lasers generate light whose bandwidths are of the order of hundres of kHz, or even more, and if this is comparable with _ or greater than - the bandwidth of the fluctuation spectrum of 7, then the approximation (22) is not applicable. This point has been discussed elsewhere [8]. When eq. (23) does apply, the spectral density of the induced polarization P reflects in a simple manner the spectral density of the fluctuations of T(t;wo). In particular if, for a given wo, W(~)(U,U~) has the form of narrow lines centered at frequencies +wl ,(wl< vo), as for example, for acoustic waves in a solid or a liquid, l@)(o) will according to eq. (23) consist of doublets centered at frequencies w. ? w1 and - w. k wl. The well-known Brillouin doublets in the spectrum of the scattered light are a reflection of this feature. In the general case, eq. (21) is the basic relation for the analysis of electromagnetic scattering experiments from fluctuating linear, isotropic media. The equation provides also some insight into the nature of the spectral form of the generalized stiucture function that is sometimes employed in the description of light scattering experiments. (Cf., Komarov and Fisher [9]; see also van Kampen [lo], or Mountain [I I]. This generalized structure function is the electromagnetic analogue of the well-known correlation function introduced by van Hove [ 121 in his theory of neutron scattering.)
Volume
8, number
2
OPTICS
References [l]
[ 21 (31
[4] [5]
June 1973
COMMUNICATIONS
N.C. Ford Jr. and G.B. Benedek, in: Proc. Conf. on Critical Phenomena, eds. M.S. Green and J.V. Sengers (NationaI Bureau of Standards Miscell. Publ. 273, Washington, 1966) p. 150. H.Z. Cummins, in: Quantum optics, ed. R.J. Glauber (Academic Press, New York, 1969) p. 246. H.Z. Cummins and H.L. Swinney, in: Progress in optics, Vol. 8, ed. E. Wolf (North-Holland, Amsterdam, 1970) p. 133. G. Adomian, Rev. Mod. Phys. 35 (1963) 185; J. Math. Phys. 11 (1970) 1069; 12 (1971) 1944, 1948. M. Born and E. Wolf, Principles of optics, 4th Ed. (Pergamon Press, Oxford, 1970).
[6
] W.B. Davenport
171
181 [9]
[ 101 [ill
and W.I. Root, An introduction to the theory of random signals and noise (McGraw-Hill, New York, 1958). R.L. Stratonovick, Topics in the theory of random noise, Vol. 1 (Gordon and Breach, New York, 1963) p. 27-3 1. L. Mandel, Phys. Rev. 181 (1969) 75. L.I. Komarov and I.Z. Fisher, Soviet Phys. JETP 16 (1963) 1358. N.G. van Kampen, in: Quantum optics, ed. R.J. Glauber (Academic Press, New York, 1969) p. 235. R.D. Mountain, J. Res. Natl. Bur. Std. 70A (1966) 207.
1121 L. van Hove, Phys. Rev. 95 (1954)
249.
99
8, number
OF’TlCS COMMUNICATIONS
2
CONSTITUTIVE
RELATIONS
June 1973
AND THE ELECTROMAGNETIC
IN A FLUCTUATING
SPECTRUM
MEDIUMS
L. MANDEL and E. WOLF Department
of‘ Physics and Astronomy, University Rochester, New York 14627, USA
Received
of Rochester,
16 April 1973
A two-parameter susceptibility function x(t;~) is introduced, to ChdIaCteriZC the local macroscopic properties of a linear isotropic dielectric, whose behavior may change in time. x(t;~) generally is a stochastic variable with respect to its first argument. An expression is derived for the spectral density of the induced polarization in terms of the spcctral densities of the fluctuations of x and of the electric field. In general, this relationship is not a simple convolution. In the special case when the incident electric field is monochromatic, the induced polarization has the usual Brillouintype spectrum.
Recent years have witnessed a great deal of research on molecular scattering, that is largely a consequence of the fact that many scattering experiments that cannot be carried out with light from conventional sources can now be performed with laser beams. The research has already provided valuable new information about thermal fluctuations in gases and liquids, about lattice vibrations in solids, about phase transitions, etc.t. In the analysis of these experiments a time-dependent dielectric response function or susceptibility function characterizing the fluctuating medium is generally employed. However, a response function that depends on only one temporal argument does not adequately describe the macroscopic response of a time-dependent medium, except over a very narrow frequency range of the electromagnetic spectrum, and, therefore, cannot completely account for results of scattering experiments. In particular, one might expect such a response function to prove inadequate when the frequencies of the incident radiation are close to the resonance frequencies of the medium.
In the ‘present note we introduce a two-parameter response function ~(t;r) that characterizes the local macroscopic dielectric response of a linear isotropic medium*. With the help of this function which, in general, is a random function* of its first argument, we derive an expression for the spectral density of the induced polarization in terms of the spectral densities of the fluctuations of the medium and of the electric field. When the electric field is monochromatic, the induced polarization has the usual Brillouin-type spectrum in an exact sense. However, in the general case, the spectral density of the induced polarization is not a simple convolution of the spectral densities of the incident field and of the susceptibility fluctuations. Let us begin by recalling some standard results relating to electromagnetic fields in linear, spatially non-dispersive media, whose macroscopic properties do not depend on time. At each point in the medium the Fourier time transform of the electric field E(t) and of the induced polarization vector P(r) are con+ Generalization of our results to non-isotropic
$ Research supported by the Air Force Office of Scientific Research. t For reviews of some of these researches see, for example, refs. 1l-31.
media and to magnetic media is straightforward. $ Our response function is essentially a stochastic operator of the class studied extensively from the mathematical standpoint by Adomian 141. However, we feel that our treatment in terms of the random process X(t; w) allows somewhat more insight than the use of stochastic Green’s functions.
95
Volume 8, number 2
OPTICS COMMUNICATIONS
netted by the constitutive
relation
P(w) = i(w) E(w) )
(1)
with
h(w)=
s E(t)
eiwr
-cc
dr .
@a>
and
P(w)= /
P(t)eiwrdt,
(Zb)
-_
where i(w) is the susceptibility tensor for the frequency w. For the sake of simplicity we restrict ourselves to an isotropic medium, in which case the susceptibility tensor i(w) may be replaced by a scalar g(w). Taking the Fourier transform of (I) and imposing the requirement of causality, we obtain the following constitutive relation in the time domain: r p(r) = j-
x(t-
t’) E(t)) dt'
(34
dT,
(3b)
-cc
=
s
x(T)E(t--)
0
where
~(7) = (l/In)
s g(w) ePiw7 dw . -cc
(4)
In eq. (3a) x depends on t and i only through the difference t- t’, which is a reflection of the fact that the macroscopic properties of the medium do not depend on time. Since bothp(t) and E(t) are real, ~(7) is necessarily also real, and ~(7) = 0 for T < 0 because of the assumption of causality. Now suppose that the macroscopic response of the medium changes in time (e.g., because of thermal fluctuations), but that the response remains linear. Then, in place of (3b), we postulate the general linear causal relation
.iunc
in which the response function x is labelled by two arguments and is, in most situations, a stationary random function of its first argument. The dependence of x(t;~) on its second argument characterizes the response of the medium to a sufficiently short pulse of electromagnetic radiation. For each value of r, there exists an ensemble of realizations x(t;r), whose PI-operties can, in principle, be obtained from a microscopic treatment of the medium. Similarly, for such a medium the dielectric response function E must also be labelled by two temporal arguments f and ‘T,and the displacement vector D(I) is then related to E(t) via the formula
D(t) - E(t) + 4nP(t)
x(t;T)E(t0
96
7) dr
,
(5)
= j- c(t;T) E(t-
7) dr
,
(5)
0
from which it follows, with the use of eq. (5), that c(t;7)
= S(7) + 4nx(t;7).
(7)
The time scales for the variations of either x(r:~) or e(t;T) with respect to t and 7 are generally quite different. The t-variations, which are associated with thermal fluctuations in most media, are usually confined within a bandwidth of some MHz (often kHz) about zero frequency, whereas the r-variations, associated with optical transitions between atomic states, are centered at frequencies of order 1014-1015 Hz, and may have bandwidths of order 100 MHz. However, for microwave transitions and rapidly fluctuating media the two frequency ranges may approach each other, or even overlap. The Fourier transform y(t;w) of x(t; 7) with respect to its second argumentt T, viz.,
X(t.w)
=s
x(t;T)
eiW7 d7
(8)
0
is another random function of time t. For each value of w there exists an ensemble of realizations of y(t; w) that represents the response of the fluctuating medium to a monochromatic electric field of frequency w. Although a complete expression for our generalized susceptibility can only be obtained from a microscopic
$ We use a circumflex P(t) =I
1973
[as in eqs. (2)] to denote the Fourier r-transform, tilde [as in eq. (S)] to denote the Fourier s-transform, and bar [as in eq. (15b)j to denote theFourier transform with respect to both r and 7.
Volume 8, number 2
OPTICS
treatment of the medium, it is not difficult to see how the generalization arises. Consider first a gas or a liquid whose macroscopic properties are time independent. The susceptibility x(w) at frequency w is then given in terms of the average number of molecules N per unit volume and the mean polarizability O(W) of each molecule by the formula (see for example ref. [5], p. 87) x(w) = N@(W) [ I-
$rNcl(w)]-1
)
[l-
$rN(t)c+J)]-1
.
(10)
The range of validity of this equation can, of course, only be determined from detailed microscopic considerations, but our brief discussion illustrates the origin of the generalized susceptibility. Let us first consider the idealized case when the variations of the medium are strictly deterministic, so that the ensemble reduces to a single realization. Moreover, we first take the electric field to be strictly monochromatic, of frequency ou,
P(w) = + k(w-w
= f [U e-iwOf
+ a*
eiwof]
,
u;w())a
+jT*(--o-w();wu)a*]) (14)
where X(0; wu) = r X(t; wu) e iwf dt -cc =
s _m
dt eiat
s x(t;r) 0
(15a)
eiwcT dr .
(15b’l
Eq. (14) shows that when the electric field is monochromatic, the Fourier spectrum of the induced polarization consists of two shifted contributions from the susceptibility spectrum. Next let us consider the more realistic case when both the electric field and the medium fluctuate and the fluctuations are not deterministic. We shall assume that E(t) and F(t; w) are independent random processes, each of which is stationary at least in the wide sense*. If .we represent the electric field as a Fourier integral (strictly speaking in the sense of the theory of generalized functions), we obtain from (5), with the help of (8), the following expression for P(t): P(t)=(1/2n)
s T(t;w)E(w) -cc
eerw’dw.
(16)
P(t) is now a random process, and it follows from (5) or (16) that, like z(t;w) and E(t), it is also stationary, at least in the wide sense. From (16) we readily deduce that the auto-correlation tensor of P(t) is given by (Pi(t)&(t+
I
1973
used. From (12) it readily follows that the Fourier transform P(w) of P(t) is given by
(9)
which is the Lorentz-Lorenz relation expressed in terms of the susceptibility rather than in terms of the refractive index of the medium, The frequency dependence of the polarizability Q(W) is readily obtained from the Lorentz oscillator model [S, section 2.3.41. Suppose now that the average number density of the molecules undergoes macroscopic fluctuations, e.g., because the medium is traversed by compression waves. Then the average number density N in (9), and hence also the susceptibility, become functions of time. Thus in place of (9) we now have the equation ~(t;w)=N(t)a(w)
June
COMMUNICATIONS
7))
(11)
where a is a constant vector. It then follows from (1 l), (5) and (8), that the induced polarization P(t) is given by P(~)=~[~(t;w~)ae~iw~‘t~(t;~~)a*eiw~f],(12)
The suffices i, k G, k = 1,2,3) label Cartesian components and the sharp brackets denote ensemble aver-
where the relation
* For the definition
Z(r;--w)
= y*(t, 0) )
which is a consequence
(13) of the reality of x (t; r), was
of stationarity in the wide sense see, for example, ref. [6 1, p. 60. The assumed statistical indepcndence of E(t) and y(t, w) is an approximation, which is valid whenever the contribution to the electric field E(t) due to the susceptibility fluctuations is small compared with the remainder of E (t).
97
Volume 8, number 2
OPTICS COMMUNICATIONS
ages. R$‘(w) is the spectral density (power spectrum) tensor of the electric field, which, for a stationary field E(t), satisfies the relation4 (l/271) (+J)
E&J’))
= $;)(c@
(w-G.I’) .
(18)
Let us now introduce the spectral density of ii(t; wO) and the spectral density tensor of P(t), with the help of the Wiener-Khintchine theorem:
k’(~;tio)= J‘ (~(t;aO)~(t+ -cc
7;~~)) eiw7 d7,
June 1973
W(‘)(W) is peaked at frequencies fwo, and is sufficiently narrow compared with the spectral density QZ)(m, o’), in both o and w’, we may approximate ?+‘@)(a) under the integral in eq. (2 1) by a Dirac S-function, and write I4@)(U’) = (I) [6 (WI- Uo) + 6 (w’ + &Jo)] )
(W
where (1) represents the average intensity of the electric field. Eq. (21) then gives the following simple expression for the spectral density of the induced polarization in terms of the spectral density of 7:
(19) 9%(w)
=
s (Pi(t) Pk (t + 7)) eiWT d7 . _m
(20)
P’(w)
= (11274
x (I) [ti~)(w-Wo;Wo)+ On taking the Fourier T-transform of eq. (17) with respect to r, interchanging the order of integrations, and making use of eqs. (19) and (20), we obtain the following expression for the spectral density tensor $$(a):
(21) Eq. (21) shows that, if ?@)(u;cQ’) were independent of the second argument, the spectral density of the induced polarization P(t) would be just the convolution of the spectral densities of E(t) and y(t;w). However, because of the presence of the second argument in MT), which is a consequence of the dispersive properties of the medium, the integral in eq. (21) is somewhat more complicated than a simple convolution, The tensorial features of the spectral density tensors “;5[) and I$% are sometimes unimportant. For example, if the incident light field is fully polarized, as from a laser, then the spectra are adequately represented by scalar functions, and the tensorial indices in eq. (21) can be suppressed. If, in addition, the spectral density 6 According to the Wiener-Khintchine theorem, the spectral density tensor W,$?(w) of the electric field, defined by eq. (18) is, of course, related to the aut@correlation tensor of the electric field by the formula R$)(u)
=_& (Ei(r) Ek(f + 7)) eiw7 dT .
(Cf.. ref. (61, p. 103 and ref. [7].)
98
w(‘ii~(-~-oo;wo)]. (23)
In deriving (23) the condition (13) was used and also the fact that g(f; w) is a wide-sense stationary process. It should however, be noted that most lasers generate light whose bandwidths are of the order of hundres of kHz, or even more, and if this is comparable with _ or greater than - the bandwidth of the fluctuation spectrum of 7, then the approximation (22) is not applicable. This point has been discussed elsewhere [8]. When eq. (23) does apply, the spectral density of the induced polarization P reflects in a simple manner the spectral density of the fluctuations of T(t;wo). In particular if, for a given wo, W(~)(U,U~) has the form of narrow lines centered at frequencies +wl ,(wl< vo), as for example, for acoustic waves in a solid or a liquid, l@)(o) will according to eq. (23) consist of doublets centered at frequencies w. ? w1 and - w. k wl. The well-known Brillouin doublets in the spectrum of the scattered light are a reflection of this feature. In the general case, eq. (21) is the basic relation for the analysis of electromagnetic scattering experiments from fluctuating linear, isotropic media. The equation provides also some insight into the nature of the spectral form of the generalized stiucture function that is sometimes employed in the description of light scattering experiments. (Cf., Komarov and Fisher [9]; see also van Kampen [lo], or Mountain [I I]. This generalized structure function is the electromagnetic analogue of the well-known correlation function introduced by van Hove [ 121 in his theory of neutron scattering.)
Volume
8, number
2
OPTICS
References [l]
[ 21 (31
[4] [5]
June 1973
COMMUNICATIONS
N.C. Ford Jr. and G.B. Benedek, in: Proc. Conf. on Critical Phenomena, eds. M.S. Green and J.V. Sengers (NationaI Bureau of Standards Miscell. Publ. 273, Washington, 1966) p. 150. H.Z. Cummins, in: Quantum optics, ed. R.J. Glauber (Academic Press, New York, 1969) p. 246. H.Z. Cummins and H.L. Swinney, in: Progress in optics, Vol. 8, ed. E. Wolf (North-Holland, Amsterdam, 1970) p. 133. G. Adomian, Rev. Mod. Phys. 35 (1963) 185; J. Math. Phys. 11 (1970) 1069; 12 (1971) 1944, 1948. M. Born and E. Wolf, Principles of optics, 4th Ed. (Pergamon Press, Oxford, 1970).
[6
] W.B. Davenport
171
181 [9]
[ 101 [ill
and W.I. Root, An introduction to the theory of random signals and noise (McGraw-Hill, New York, 1958). R.L. Stratonovick, Topics in the theory of random noise, Vol. 1 (Gordon and Breach, New York, 1963) p. 27-3 1. L. Mandel, Phys. Rev. 181 (1969) 75. L.I. Komarov and I.Z. Fisher, Soviet Phys. JETP 16 (1963) 1358. N.G. van Kampen, in: Quantum optics, ed. R.J. Glauber (Academic Press, New York, 1969) p. 235. R.D. Mountain, J. Res. Natl. Bur. Std. 70A (1966) 207.
1121 L. van Hove, Phys. Rev. 95 (1954)
249.
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