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Spectral densities for resonant scattering in a linear fluctuating medium
Volume 10, number 3
OPTICS COMMUNICATIONS
March 1974
SPECTRAL DENSITIES FOR RESONANT SCATTERING IN A L I N E A R F L U C T U A T I N G
MEDIUM
A.V. DURRANT Faculty of Science, The Open University, Milton Keynes, UK Received 13 November 1973 In a recent paper Mandel and Wolf use a two-parameter susceptibility function to derive an expression for the spectral density of the induced polarisation in a linear fluctuating medium. Their treatment is restricted to timestationary fields and media. This paper presents (i) a generalisation of their treatment to include the case of locally stationary fields which is of interest in the study of resonance fluorescence spectroscopy; (ii) a demonstration of the way in which the spectral density of the fluctuations and the atomic spectral response function are combined in the two-parameter description of the susceptibility of an atomic vapour.
1. Introduction Molecular and atomic light-scattering experiments have been used to obtain two different types of information. We have experiments on the resonance fluorescence of atomic vapours, level crossing and double resonance for example, which yield information on the internal electronic structure of the atoms. These experiments are usually carried out without spectral resolution, and the thermal fluctuations of the medium are usually ignored in the analysis. The essential feature is that the scattering occurs in the vicinity of an atomic resonance where the singleatom spectral response function is changing very rapidly. Zeeman shifting of the sub-level resonances then produces profound changes in the intensity and angular distribution of the scattered light; g-values, life-times and oscillator strengths, fine and hyperfine structure constants, have been measured in this way. Reviews of the field may be found in refs. [ 1,2]. In the second type of scattering experiment, to be referred to here as fluctuation scattering, the main interest is in the atomic or molecular structure of the scattering medium as revealed through the spectral characteristics of the scattered light. The essential feature here is that fire thermal (or applied) fluctuations modulate the bulk response of the medium through absorption and emission Doppler effects, and through atomic or molecular interactions. One result of this is the appearance of the fluctuation spectrum as side-bands in the spectrum of the scattered radiation. A wide range of structure problems and fluctuation phenomena have been studied in this way. Refs. [3, 4] give reviews of this work. The experiments are usually carried out in a spectral region remote from a resonance. It is then a good approximation to regard the atomic or molecular spectral response as a constant function over the spectral range of the incident radiation, or equivalently, to regard the atomic or molecular temporal response (impulse response) as a delta function. These two areas of scattering theory have developed more or less independently of one another and from complementary assumptions concerning the response of the medium. A recent paper by Mandel and Wolf [5] has helped to unify the two approaches. They obtained an expression for Ore spectral density of a stationary field of radiation scattered by a stationary fluctuating medium with full account taken of the frequency dependence of the response function. Their paper, which was directed mainly at workers in the field of fluctuation scattering, will also be of interest to those in the field of resonance fluorescence scattering where spectral densities are sometimes of interest. It is the purpose of this paper to continue the interaction between these two areas of scattering theory by (i) showing how the equation (eq. (21), ref. [5]) for the spectral density of the induced polarisation can 262
Volume 10, number 3
OPTICS COMMUNICATIONS
March 1974
be generalised to include the non-stationary fields sometimes used in resonance fluorescence experiments [ 6 - 9 ] ; and (ii) pointing to the case of near-resonant scattering from an atomic vapour where both the atomic response function and the fluctuation spectrum are important.
2. Autocorrelations and spectral densities Following Mandel and Wolf [5] and Adomian [10] we introduce a two-parameter susceptibility tensor X(t; t') in which the first argument is used to describe changes in the macroscopic properties of the medium (due to thermal fuctuations for example), and the second argument is used for the finite response time of the medium. Thus × is a deterministic impulse response with respect to t' and a random function with respect to t. The induced polarisation vector P(t), at a point in the (linear) medium where the electromagnetic wave has electric vector E(t), is then assumed to be given by the convolution
P(t) : e0 / X(t;t')E(t-t') dt',
(1)
where P a n d / f denote the scalar magnitudes, and for simplicity we take X to be the scalar susceptibility of an isotropic medium. From eq. (1) we can write down the autocorrelation of the induced polarisation:
ff/(P)(t,z)-(P(t)P(t+'r)}=eO2 ? dr' f 0
dt" (X(t;t') X[t " + r;t .... )~.
(2)
0
Here we have assumed that the two ensembles E and P are independent (i.e. no multiple scattering). We now introduce ~' = t" t' so I:hat we can write
~(P)(t, T) = e2 / dt' / dr' (X(t; t') X(t + -r; t' + 7')) (E(t- t') E(t- t' + -r- 7')> 0 0 -e 2 / 0
dt'/
dT' ~(x)(t,T;t',7')dA(E)(t
- t',7-r'),
(3)
0
where the symbols if(×) and if(E) have been introduced to denote the autocorrelation functions. In resonance fluorescence experiments it is the intensity of the scattered light, averaged over the temporal fluctuations in E and ×, that is recorded. This is proportional to ~(e)(t,0) (see ref. [9]). Thus, with 7 = 0, eq. (3) is the basic relation for the analysis of resonance fluorescence scattering experiments. An essential feature is the dependence of X on its second parameter (and therefore of if(x) on the primed parameters t', T'), for it is this that contains the interesting details of the internal electronic structure; for example, when the vapour is in a magnetic field B the atomic susceptibility contains terms with factors like exp{-i(co r + 3`B - i F / 2 ) t'}, where cor is the zero-field resonance frequency, 3' the magnetogyric ratio, and F -1 the life-time of the excited state For stationary media ~(x) in eq. (3) is a stationary autocorrelation function of r and a deterministic function of t' and r'. For stationary fields if(E) in eq. (3) is a stationary autocorrelation tunction of (r r'). Under these conditioris eq. (3) is the time-domain equivalent of eq. (21) of Mandel and Wolf [5]. Eq. (3) however is not restricted to stationary fields and media, for the t-dependence of if(x) and the ( t - t')-dependence of if(E) can be used to describe certain kinds of non-stationary behaviour. Light modulation [6, 7], double resonance [2], and pulsed magnetic field [11] experiments, are examples where expressions of the form +q')(t, 0), eq. (3), have been used to describe the non-stationary time dependence of the fluorescence from a vapour. 263
Volume 10, number 3
OPTICS COMMUNICATIONS
March 1974
For stationary fields and media the transformation of eq. (3) into the frequency domain, i.e. into eq. (21) of Mandel and Wolf [5], is an exercise in the use of Fourier transformations and the Wiener-Khintchine theorem along the lines demonstrated in that paper. A generalized form of their eq. (2l) to include some interesting types of non-stationary behaviour can be obtained from eq. (3). Here we shall restrict our attention to cases where the random functions in eq. (3) can be described by locally stationary autocorrelation functions [ 12]. Optical fields can be so described when the intensity is modulated at a rate that is very slow compared with the random intensity fluctuations, as in refs. [6--9] for example (locally stationary susceptibilities will also be included in the treatment but these are of less practical interest). With this restriction the t and r dependence of ~(E) and of ~(X) are mutually independent. This means that the transformation from time r to a frequency co can proceed exactly as in ref. [5]. We now appeal to the symmetry ofeq. (3) with respect to its t and r dependence and assert that the transformation from time t to a frequency ~2 will proceed in a similar manner. The generalised spectral density is therefore qb{P)(co,~)=(e0/2~r)2
fdco'
/d~'~tx)(g2
g2', co
co' ; ~2,' co') ~{£}(co', ~ ' ) .
{4)
The significance of the ~Z-dependence is discussed elsewhere [8, 9]. For stationary fields and media we write 0{£)(co ', S2') = 2n W(£)(co ') 6 (~2'1
and
0(x)(g2, co" ~2', co') = 2n B'{X)(co;co') 6 (fZ).
By carrying out the integrations over g2' and ~ we obtain, essentially eq. (2 l) of ref. [6]:
w(P)(CO)= (e~/2n)
/
W{£)(CO') W(x)(cJ
{5)
co" '~') dco'.
3. Scattering near a resonance For a vapour of non-interacting atoms, we have a simple system for which a microscopic model, based on the Lorentz oscillator, can provide insight into the origin of the two-parameter dependence of If(x). The polarisability of a Lorentz oscillator at rest [9] is (e2/2mcor) (cor - COo - iF/2) -1 which, in the vicinity of the resonance, depends strongly on the incident frequency coo . The thermal fluctuations in the medium are derived from the random motion of the atoms. To include this in the model we must (i) replace c o r by (Jr + vi °'}0/c, the effective resonance frequency of an atom moving with velocity component v i in the direction of incidence; and (ii) introduce a Doppler shift 2v n (coo/C)sm T 0 in the frequency of the induced polarisation as seen by a stationary observer in the direction of the scattering angle 0. Here v n is the atom's velocity component, normal to the bisector of the angle 0, in the scattering plane (see 'also refs. [13, 14]). For lateral scattering (0 ~ 0) from a dilute vapour the quantity of interest is the spectral density of the dipole moment of a single oscillator, summed over the total number of oscillators, N, with account taken of the Doppler shifts. For a gaussian distribution of velocity components, the integral representing this summation is •
2mcor] _ ~f dv
1
cor + vi C°o/C - COO-- iP/2
8(co-coO-2(Vncoo/C)sm(O/2))
exp
\c-&-]
When the integrations over the components o f v are carried out [13, 14] we find that the expression for W(x) to be used in eq. (5) has the form
264
Volume 10, number 3
OPTICS COMMUNICATIONS
,.[-co/2+coO-~r
L S -(ff/53
+ ac
iP
(0/2)
exp-
March 1974
2As~(O/2 )
(6)
where A is a constant independent of 600 and cor. Y, the imaginary part of tile plasma dispersion function [ 15], is a Voigt function centered at co = 2(6o 0 - 60r) rather than co = 0. This leads to a shift in the spectral density of the scattered light away from coO towards the resonance. For near-resonant light, i.e. for light whose spectrum is centered within a few Doppler widths of the resonance, the effect is quite pronounced (see the graph in ref. [14]) and gives rise to the approximate rule that for sideways (0 = 90 °) scattering, the spectral density of fluorescence is always centered at the resonance frequency and has a Doppler width determined only by the vapour. This behaviour contrasts with the simple convolution rule [5] which applies when coo is remote from the resonance. The factor Y is then nearly constant over the significant range of co in eq. (6).
References [1] [2] [3] [4 ] [5] [6] [7] [8] [9] [10] [ 11 ] [12] [13] [14] [15]
B. Budick, Advan. Atom. Molec. Phys. 3 (1967) 73. G.W. Series, in: Quantum optics, ed. S.M. Kay and A. Maitland (Academic Press, London and New York, 1970) p. 395. 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. L. Mandel and E. Wolf, Opt. Commun. 8 (1973) 95. A. Corney and G.W. Series, Proc. Phys. Soc. 83 (1964) 207. O.V. Konstantinov and V.I. Perel', Sov. Phys. JETP 18 (1964) 195. A.V. Durrant, .1. Phys. B: Atom. Molec. Phys. 5 (1972) 133. A.V. Durrant, J. Phys. B: Atom. Molec. Phys. 5 (1972) 1456. G. Adomian, Rev. Mod. Phys. 35 (1963) 185. J.N. Dodd, W.J. Sandle and O.M. Williams, J. Phys. B: Atom. Molec. Phys. 3 (1970) 256. R.A. Silverman, IRE Trans. Inform. Theory 3 (1957) 182. D.G. Hummer, Mon. Not. R. Astron. Soc. 125 (1962) 21. L.G. Henyey, Proc. Nat. Acad. Sci. USA 26 (1940) 50. B.D. Fried and S.D. Conte, The plasma dispersion function (Academic Press, London, 1961).
265
OPTICS COMMUNICATIONS
March 1974
SPECTRAL DENSITIES FOR RESONANT SCATTERING IN A L I N E A R F L U C T U A T I N G
MEDIUM
A.V. DURRANT Faculty of Science, The Open University, Milton Keynes, UK Received 13 November 1973 In a recent paper Mandel and Wolf use a two-parameter susceptibility function to derive an expression for the spectral density of the induced polarisation in a linear fluctuating medium. Their treatment is restricted to timestationary fields and media. This paper presents (i) a generalisation of their treatment to include the case of locally stationary fields which is of interest in the study of resonance fluorescence spectroscopy; (ii) a demonstration of the way in which the spectral density of the fluctuations and the atomic spectral response function are combined in the two-parameter description of the susceptibility of an atomic vapour.
1. Introduction Molecular and atomic light-scattering experiments have been used to obtain two different types of information. We have experiments on the resonance fluorescence of atomic vapours, level crossing and double resonance for example, which yield information on the internal electronic structure of the atoms. These experiments are usually carried out without spectral resolution, and the thermal fluctuations of the medium are usually ignored in the analysis. The essential feature is that the scattering occurs in the vicinity of an atomic resonance where the singleatom spectral response function is changing very rapidly. Zeeman shifting of the sub-level resonances then produces profound changes in the intensity and angular distribution of the scattered light; g-values, life-times and oscillator strengths, fine and hyperfine structure constants, have been measured in this way. Reviews of the field may be found in refs. [ 1,2]. In the second type of scattering experiment, to be referred to here as fluctuation scattering, the main interest is in the atomic or molecular structure of the scattering medium as revealed through the spectral characteristics of the scattered light. The essential feature here is that fire thermal (or applied) fluctuations modulate the bulk response of the medium through absorption and emission Doppler effects, and through atomic or molecular interactions. One result of this is the appearance of the fluctuation spectrum as side-bands in the spectrum of the scattered radiation. A wide range of structure problems and fluctuation phenomena have been studied in this way. Refs. [3, 4] give reviews of this work. The experiments are usually carried out in a spectral region remote from a resonance. It is then a good approximation to regard the atomic or molecular spectral response as a constant function over the spectral range of the incident radiation, or equivalently, to regard the atomic or molecular temporal response (impulse response) as a delta function. These two areas of scattering theory have developed more or less independently of one another and from complementary assumptions concerning the response of the medium. A recent paper by Mandel and Wolf [5] has helped to unify the two approaches. They obtained an expression for Ore spectral density of a stationary field of radiation scattered by a stationary fluctuating medium with full account taken of the frequency dependence of the response function. Their paper, which was directed mainly at workers in the field of fluctuation scattering, will also be of interest to those in the field of resonance fluorescence scattering where spectral densities are sometimes of interest. It is the purpose of this paper to continue the interaction between these two areas of scattering theory by (i) showing how the equation (eq. (21), ref. [5]) for the spectral density of the induced polarisation can 262
Volume 10, number 3
OPTICS COMMUNICATIONS
March 1974
be generalised to include the non-stationary fields sometimes used in resonance fluorescence experiments [ 6 - 9 ] ; and (ii) pointing to the case of near-resonant scattering from an atomic vapour where both the atomic response function and the fluctuation spectrum are important.
2. Autocorrelations and spectral densities Following Mandel and Wolf [5] and Adomian [10] we introduce a two-parameter susceptibility tensor X(t; t') in which the first argument is used to describe changes in the macroscopic properties of the medium (due to thermal fuctuations for example), and the second argument is used for the finite response time of the medium. Thus × is a deterministic impulse response with respect to t' and a random function with respect to t. The induced polarisation vector P(t), at a point in the (linear) medium where the electromagnetic wave has electric vector E(t), is then assumed to be given by the convolution
P(t) : e0 / X(t;t')E(t-t') dt',
(1)
where P a n d / f denote the scalar magnitudes, and for simplicity we take X to be the scalar susceptibility of an isotropic medium. From eq. (1) we can write down the autocorrelation of the induced polarisation:
ff/(P)(t,z)-(P(t)P(t+'r)}=eO2 ? dr' f 0
dt" (X(t;t') X[t " + r;t .... )~
(2)
0
Here we have assumed that the two ensembles E and P are independent (i.e. no multiple scattering). We now introduce ~' = t" t' so I:hat we can write
~(P)(t, T) = e2 / dt' / dr' (X(t; t') X(t + -r; t' + 7')) (E(t- t') E(t- t' + -r- 7')> 0 0 -e 2 / 0
dt'/
dT' ~(x)(t,T;t',7')dA(E)(t
- t',7-r'),
(3)
0
where the symbols if(×) and if(E) have been introduced to denote the autocorrelation functions. In resonance fluorescence experiments it is the intensity of the scattered light, averaged over the temporal fluctuations in E and ×, that is recorded. This is proportional to ~(e)(t,0) (see ref. [9]). Thus, with 7 = 0, eq. (3) is the basic relation for the analysis of resonance fluorescence scattering experiments. An essential feature is the dependence of X on its second parameter (and therefore of if(x) on the primed parameters t', T'), for it is this that contains the interesting details of the internal electronic structure; for example, when the vapour is in a magnetic field B the atomic susceptibility contains terms with factors like exp{-i(co r + 3`B - i F / 2 ) t'}, where cor is the zero-field resonance frequency, 3' the magnetogyric ratio, and F -1 the life-time of the excited state For stationary media ~(x) in eq. (3) is a stationary autocorrelation function of r and a deterministic function of t' and r'. For stationary fields if(E) in eq. (3) is a stationary autocorrelation tunction of (r r'). Under these conditioris eq. (3) is the time-domain equivalent of eq. (21) of Mandel and Wolf [5]. Eq. (3) however is not restricted to stationary fields and media, for the t-dependence of if(x) and the ( t - t')-dependence of if(E) can be used to describe certain kinds of non-stationary behaviour. Light modulation [6, 7], double resonance [2], and pulsed magnetic field [11] experiments, are examples where expressions of the form +q')(t, 0), eq. (3), have been used to describe the non-stationary time dependence of the fluorescence from a vapour. 263
Volume 10, number 3
OPTICS COMMUNICATIONS
March 1974
For stationary fields and media the transformation of eq. (3) into the frequency domain, i.e. into eq. (21) of Mandel and Wolf [5], is an exercise in the use of Fourier transformations and the Wiener-Khintchine theorem along the lines demonstrated in that paper. A generalized form of their eq. (2l) to include some interesting types of non-stationary behaviour can be obtained from eq. (3). Here we shall restrict our attention to cases where the random functions in eq. (3) can be described by locally stationary autocorrelation functions [ 12]. Optical fields can be so described when the intensity is modulated at a rate that is very slow compared with the random intensity fluctuations, as in refs. [6--9] for example (locally stationary susceptibilities will also be included in the treatment but these are of less practical interest). With this restriction the t and r dependence of ~(E) and of ~(X) are mutually independent. This means that the transformation from time r to a frequency co can proceed exactly as in ref. [5]. We now appeal to the symmetry ofeq. (3) with respect to its t and r dependence and assert that the transformation from time t to a frequency ~2 will proceed in a similar manner. The generalised spectral density is therefore qb{P)(co,~)=(e0/2~r)2
fdco'
/d~'~tx)(g2
g2', co
co' ; ~2,' co') ~{£}(co', ~ ' ) .
{4)
The significance of the ~Z-dependence is discussed elsewhere [8, 9]. For stationary fields and media we write 0{£)(co ', S2') = 2n W(£)(co ') 6 (~2'1
and
0(x)(g2, co" ~2', co') = 2n B'{X)(co;co') 6 (fZ).
By carrying out the integrations over g2' and ~ we obtain, essentially eq. (2 l) of ref. [6]:
w(P)(CO)= (e~/2n)
/
W{£)(CO') W(x)(cJ
{5)
co" '~') dco'.
3. Scattering near a resonance For a vapour of non-interacting atoms, we have a simple system for which a microscopic model, based on the Lorentz oscillator, can provide insight into the origin of the two-parameter dependence of If(x). The polarisability of a Lorentz oscillator at rest [9] is (e2/2mcor) (cor - COo - iF/2) -1 which, in the vicinity of the resonance, depends strongly on the incident frequency coo . The thermal fluctuations in the medium are derived from the random motion of the atoms. To include this in the model we must (i) replace c o r by (Jr + vi °'}0/c, the effective resonance frequency of an atom moving with velocity component v i in the direction of incidence; and (ii) introduce a Doppler shift 2v n (coo/C)sm T 0 in the frequency of the induced polarisation as seen by a stationary observer in the direction of the scattering angle 0. Here v n is the atom's velocity component, normal to the bisector of the angle 0, in the scattering plane (see 'also refs. [13, 14]). For lateral scattering (0 ~ 0) from a dilute vapour the quantity of interest is the spectral density of the dipole moment of a single oscillator, summed over the total number of oscillators, N, with account taken of the Doppler shifts. For a gaussian distribution of velocity components, the integral representing this summation is •
2mcor] _ ~f dv
1
cor + vi C°o/C - COO-- iP/2
8(co-coO-2(Vncoo/C)sm(O/2))
exp
\c-&-]
When the integrations over the components o f v are carried out [13, 14] we find that the expression for W(x) to be used in eq. (5) has the form
264
Volume 10, number 3
OPTICS COMMUNICATIONS
,.[-co/2+coO-~r
L S -(ff/53
+ ac
iP
(0/2)
exp-
March 1974
2As~(O/2 )
(6)
where A is a constant independent of 600 and cor. Y, the imaginary part of tile plasma dispersion function [ 15], is a Voigt function centered at co = 2(6o 0 - 60r) rather than co = 0. This leads to a shift in the spectral density of the scattered light away from coO towards the resonance. For near-resonant light, i.e. for light whose spectrum is centered within a few Doppler widths of the resonance, the effect is quite pronounced (see the graph in ref. [14]) and gives rise to the approximate rule that for sideways (0 = 90 °) scattering, the spectral density of fluorescence is always centered at the resonance frequency and has a Doppler width determined only by the vapour. This behaviour contrasts with the simple convolution rule [5] which applies when coo is remote from the resonance. The factor Y is then nearly constant over the significant range of co in eq. (6).
References [1] [2] [3] [4 ] [5] [6] [7] [8] [9] [10] [ 11 ] [12] [13] [14] [15]
B. Budick, Advan. Atom. Molec. Phys. 3 (1967) 73. G.W. Series, in: Quantum optics, ed. S.M. Kay and A. Maitland (Academic Press, London and New York, 1970) p. 395. 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. L. Mandel and E. Wolf, Opt. Commun. 8 (1973) 95. A. Corney and G.W. Series, Proc. Phys. Soc. 83 (1964) 207. O.V. Konstantinov and V.I. Perel', Sov. Phys. JETP 18 (1964) 195. A.V. Durrant, .1. Phys. B: Atom. Molec. Phys. 5 (1972) 133. A.V. Durrant, J. Phys. B: Atom. Molec. Phys. 5 (1972) 1456. G. Adomian, Rev. Mod. Phys. 35 (1963) 185. J.N. Dodd, W.J. Sandle and O.M. Williams, J. Phys. B: Atom. Molec. Phys. 3 (1970) 256. R.A. Silverman, IRE Trans. Inform. Theory 3 (1957) 182. D.G. Hummer, Mon. Not. R. Astron. Soc. 125 (1962) 21. L.G. Henyey, Proc. Nat. Acad. Sci. USA 26 (1940) 50. B.D. Fried and S.D. Conte, The plasma dispersion function (Academic Press, London, 1961).
265