Light scattering in the medium with fluctuating gyrotropy: application to spin noise spectroscopy

0.3 MB·

by  G. G. Kozlov

Light scattering in the medium with fluctuating
gyrotropy: application to spin noise spectroscopy
G. G. Kozlov, I. I. Ryzhov, V. S. Zapasskii
Abstract
7
1 The spin noise signal in the Faraday-rotation-based detection technique can be con-
0 sidered equally correctly either as a manifestation of the spin-flip Raman effect or as
2
a result of light scattering in the medium with fluctuating gyrotropy. In this paper, we

n
presentrigorousdescriptionofthesignalformationprocessuponheterodyningofthefield
a
J scattered due to fluctuating gyrotropy. Along with conventional single-beam experimen-

6 tal arrangement, we consider here a more complicated, but more informative, two-beam
1 configuration that implies the use of an auxiliary light beam passing through the same

scattering volume and delivering additional scattered field to the detector. We show that
]
s the signal in the spin noise spectroscopy arising due to heterodyning of the scattered field
c
i is formed only by the scattered field components whose wave vectors coincide with those
t
p of the probe beam. Therefore, in principle, the detected signal in spin noise spectroscopy
o
can be increased by increasing overlap of the two fields in the momentum space. We also
.
s
show that, in the two-beam geometry, contribution of the auxiliary (tilted) beam to the
c
i detectedsignalisrepresentedbyFouriertransformofthegyrotropyreliefatthedifference
s
y of two wave vectors. This effect can be used to study spin correlations by means of noise
h
spectroscopy.
p
[

1
v Introduction
8
6
2
The spin noise spectroscopy (SNS), first realized in [1], has turned nowadays into a power-
4
0 ful method of studying magnetic resonance and spin dynamics in atomic and semiconductor
.
1 systems (see, e.g., [2, 3, 4] . The most fascinating results of application of the SNS with the
0
greatest progress in sensitivity of the measurements were achieved in physics of semiconductor
7
1 structures, where the novel technique has allowed one not only to considerably move ahead in
v: the magnetic resonance spectroscopy, but also to discover fundamentally new opportunities of
i research. Specifically, it has been established that optical spectroscopy of spin noise (that im-
X
plies measuring wavelength dependence of the spin noise power) makes it possible to decipher
r
a
inner structure of optical transitions [5]. Correlation nature of the SNS allowed one to realize,
onitsbasis, asortofpump-probespectroscopy[6]. Effectivedependenceofthespin-noisesignal
on the light-power density (on the beam cross section) was used to demonstrate the SNS-based
3D tomography [8, 9]. Due to high sensitivity of the SNS, it appeared possible to detect mag-
netic resonance of quasi-free carriers in a single quantum well 20 nm thick [10], to observe the
spin-noise spectrum of a single hole spin in a quantum dot [11], and to realize magnetometry
1
of local magnetic fields (including field of polarized nuclei) in a semiconductor [12, 13]. Due to
these remarkable capabilities of the new technique, it acquired a great popularity during the
last decade.
At the same time, fundamental mechanism underlying the effect of magnetic resonance
in the Faraday rotation noise spectrum remains so far, to a considerable extent, unexplored.
Theoretically, it has been shown in 1983 [14] that this effect is closely related to the spin-flip
Raman scattering, and the detected signal of magnetic resonance is the result of heterodyning
of the scattered light (with shifted frequency), with the local oscillator provided by the probe
laser beam. In this case, the standard experimental geometry we use in the conventional SNS
may appear to be far from optimal. Indeed, we usually collect, on the photodetector, only
the scattered light lying within the solid angle of the probe beam, whereas indicatrix of the
Raman-scattered light may be fairly isotropic. It means that, in the standard experimental
geometry, most part of the scattered light is lost. Therefore, it looks like the detected signal, in
the SNS, can be considerably increased by collecting more efficiently the scattered light. Still,
even if this simple picture is correct, it is not easy to correctly design the experimental setup to
take advantage of the additional scattered field in full measure. First experiments carried out
in this direction [15] and our preliminary analysis of the problem have shown that favourable
solution of this experimental task can be achieved only with allowance for all the factors
affecting the heterodyning process (wave fronts of the reference and scattered waves, shape of
thebeam,volumeofthescatteringmedium,shapeanddimensionsofthephotosensitivesurface,
correlation properties of the gyrotropy, etc. ). Actually, this problem, which we consider to be
fundamental for the SNS method, is rather complicated and needs to be analyzed carefully and
rigorously, with the results of the treatment applicable to real experimental conditions. In our
opinion, computational details of such a treatment and prticularities of used apprpximations
are also highly important.
In this paper, we present such a treatment for a focused Gaussian probe beam propagat-
ing through the medium with fluctuating gyrotropy and analyze in detail mechanism of the
intensity-noise signal formation due to heterodyning of the scattered field on the detector. We
also propose a two-beam experimental arrangement, with the auxiliary light beam tilted with
respect to the probe, that makes it possible to get information about the spatiotemporal cor-
relation function of gyrotropy of the studied system (remind that in conventional SNS only
spatially averaged temporal correlation function is revealed).
Thepaperisorganizedasfollows. InSection1, forcompletenessofthenarrative, wepresent
a brief explanation of what is the Gaussian beam and introduce a model of the polarimetric
detector used in our further analysis. We show here that the detected signal in SNS is con-
tributed only by the scattered field that, in the momentu, space, coincides with that of the
probe. In Section 2, we present basics of the single-scattering theory, apply it to the medium
with gyrotropy randomly modulated in space, and calculate the observed polarimetric signal.
In Sections 3 and 4, we calculate the noise signal observed in the two-beam configuration, when
the auxiliary beam propagating though the medium at some angle to the main probe beam
does not hit the detector and contributes to the signal only by its scattered field. We show
that the spin-noise signal, under these conditions, is proportional to the Fourier component of
the spatial correlation function of gyrotropy at spatial frequency equal to difference between
the two wave vectors. In Section 5, we present calculations for the model of independent para-
magnetic particles (spins) and show that the signal produced by the auxiliary tilted beam is of
2
the same order of magnitude as the one produced by the main probe and, hence, can be easily
detected using the same experimental setup.
1 Detecting polarimetric signal in a confined laser beam
In the simplest version of the light-scattering problem, the probe beam can be taken in the
form of a plane wave. However, in the SNS experiments under consideration, when two light
beams are supposed to be used, with their spatial localization being of crucial importance, this
approximation proves to be inappropriate. So, we will treat Gaussian beams whose electric
fields E (r) are defined by the expression
p
(cid:115)
8W (cosη,0,−sinη) (cid:20) kQ2(X2 +y2)(cid:21)
E (r) = eı(kZ−ωt)kQ exp − (1)
p c (2k +ıQ2Z) 2(2k +ıQ2Z)
where r = (x,y,z), k ≡ ω/c (ω is the optical frequency and c is the speed of light), W – beam
intensity, and
(cid:18)X(cid:19) (cid:18)cosη −sinη(cid:19)(cid:18)x(cid:19)
=
Z sinη cosη z
Field (1) satisfies Maxwell’s equations and represents the beam propagating in the zx plane at
theangleη withrespecttoz axis(η isassumedtobesnall)andpolarizedmostlyinx-direction.
The parameter Q defines the e-level half width 2w of the beam waist by relationship w = 1/Q.
w should be greater than the wavelength λ = 2πc/ω. In our estimations, we accept λ ∼ 1 µm
and w ∼ 30 µm.
In the SNS experiments, we detect small fluctuations of the optical field polarization, and,
therefore, to calculate correctly the SNS signal, we have to specify the model of polarimetric
detector. We suppose the detector to be comprised of two photodiodes PD1 and PD2 (Fig.1)
arranged in two arms of the polarization beamsplitter (BS). The output signal U is obtained
by subtracting photocurrents of the two photodiodes and (to within some unimportant factors)
are given by the expression
ω (cid:90) 2π/ω (cid:90) lx (cid:90) ly (cid:20) (cid:21)
U = dt dx dy Re 2E (x,y,L)−Re 2E (x,y,L) , (2)
x y
2π
0 −lx −ly
where E are the x and y components of the complex input optical field E, 2l are the
x,y x,y
dimensions of sensitive areas of the photodiodes along the x and y directions. We ascribe
physical sense to real part of the complex optical field and, as seen from Eq. (2), the output
signal U represents the difference between intensities of the input optical field in the x and y
polarizations integrated over sensitive areas of the photodiodes and averaged over the optical
period 2π/ω.
In our case, the input optical field E can be presented as a sum of the probe field E
0
(Re E = E ) and the field E (Re E = E ) arising due to scattering of the probe beam
0 0 1 1 1
by the sample with spatially fluctuating gyrotropy. Then, the first-order (with respect to E )
1
contribution u to the polarimetric signal can be written as
1
ω (cid:90) 2π/ω (cid:90) lx (cid:90) ly (cid:20) (cid:21)
u = dt dx dy E (x,y,L)E (x,y,L)−E (x,y,L)E (x,y,L) (3)
1 x0 x1 y0 y1
π
0 −lx −ly
3
This formula shows that the observed signal can be thought of as a result of heterodyning
(mixing) of the unperturbed probe field E with the field of scattering E . Equation (3) also
0 1
shows that , for sufficiently large dimensions of the detector (l (cid:29) λ = 2π/k), polarimetric
x,y
signal u represents projection of the scattered field (in the momentum space) onto the field
1
of the probe beam. This means, in turn, that this signal is controlled by the fraction of the
scattered field whose distribution in space, to a certain extent, reproduces the field of the
probe beam. Specifically, when the probe field represents a plane wave E ∼ eıq0r with the
0
wave vector q , and the scattered field can be presented by a superposition of the plane waves
0
E ∼ (cid:82) dqeıqrS(q), the signal u appears to be proportional to the component of the scattered
1 1
field at the spatial frequency q : u ∼ S(q ).
0 1 0
Let us now calculate the scattered field E .
1
2 Polarimetric signal in the medium with fluctuating
gyrotropy
In this section, we consider scattering of a monochromatic light beam by the medium with
randomly inhomogeneous (spatially fluctuating) gyrotropy. In this case, polarization of the
medium P(r) can be expressed through the electric field E(r) by the expression
P(r) = ı[E(r)G(r)] = ıE(r)×G(r) (4)
where G(r) is the spatially dependent gyration vector. At this stage of our treatment, we
assume the gyration vector to be time-independent. Then, Maxwell’s equations for the elec-
tromagnetic field in the medium can be reduced to the form:
ω
∆E+k2E = −4πk2P−4π grad div P, k ≡ (5)
c
We will search for solution of this equation in the form of series in powers of G(r). The zero
order term E (r) represents the probe beam field which we consider to be known. The first
0
order term E (r) corresponds to the single-scattering approximation which is sufficient for our
1
consideration. This term satisfies the equation
∆E +k2E = −4πık2E (r)×G(r)−4πı grad div E (r)×G(r) (6)
1 1 0 0
SolutionofthisequationcanbeexpressedintermsofGreen’sfunctionΓ(r) = −exp(ıkr)/4πr
of the Helmholtz equation [∆+k2]Γ(r) = δ(r):
(cid:90) exp(ık|r−r(cid:48)|)(cid:20) (cid:21)
E (r) = ı k2E (r(cid:48))×G(r(cid:48))+ grad div E (r(cid:48))×G(r(cid:48)) d3r(cid:48) (7)
1 |r−r(cid:48)| 0 0
Let the sample (we call “sample” the region where G(r) is nonzero) be placed in the vicinity
of the origin of our coordinate system x,y,z. Let the photosensitive surface of the polarimetric
detector be parallel to the xy plane and the detector itself be set at z = L, with L being large
compared with the sample dimensions. Then, as seen from Eq. (7), the scattered field can be
presented as a sum of two contributions:
E (r) = E1(r)+E2(r) (8)
1 1 1
4
ık2 (cid:90)
E1(r) ≡ exp(ık|r−r(cid:48)|)E (r(cid:48))×G(r(cid:48))d3r(cid:48)
1 L 0
ı (cid:90) 1
E2(r) ≡ exp(ık|r−r(cid:48)|) grad div E (r(cid:48))×G(r(cid:48))d3r(cid:48) = grad div E1(r)
1 L 0 k2 1
We will concentrate on calculating the part E1(r) of the scattered field because, in what
1
follows, we will need this field at small scattering angles and, in this case, as it can be directly
checked, only E1(r) is of importance.
1
We take the probe beam in the form of Eq. (1) at η = 0, with the angle φ specifying beam
polarization in the xy plane. Then, the probe field acquires the form
(cid:115)
8W (cosφ,sinφ,0) (cid:20) kQ2(x2 +y2)(cid:21)
E (r) = eı[kz−ωt]kQ exp − , r = (x,y,z) (9)
0 c (2k +ıQ2z) 2(2k +ıQ2z)
Weneedthisfieldintwosubstantiallyseparatedspatialregions: firstly, inEq.(3)atlargevalues
of z ∼ L and, secondly, in Eq.(8) at relatively small values of z within the sample. Calculation
for z ∼ L shows that the field E entering Eq.(3) has the form
0
(cid:115)
(cid:18)E (x,y,L)(cid:19) (cid:18)cosφ(cid:19) 8W k (cid:20) k[x2 +y2](cid:21) (cid:20) k2(x2 +y2)(cid:21)
x0 = sin kL−ωt+ exp − (10)
E (x,y,L) sinφ c QL 2L Q2L2
y0
While deriving these expressions, we assumed that L > z = 4πw2/λ (z is the Rayleigh
c c
length). To calculate the scattered field by Eq. (8), one needs the field (9) at z < z . In this
c
limit, Eq. (9) can be simplified:
(cid:115)
8W (cosφ,sinφ,0) (cid:20) Q2(x2 +y2)(cid:21)
E (r) = eı[kz−ωt]Q exp − , z < z (11)
0 c
c 2 4
Using this relationship, one can calculate the scattered field E1(r) (8) and obtain, for real parts
1
of E and E entering Eq. (3), the following expressions:
x1 y1
(cid:115)
(cid:18)E (cid:19) (cid:18)−sinφ(cid:19) 2W Qk2 (cid:90) (cid:20) Q2(x(cid:48)2 +y(cid:48)2)(cid:21)
x1 = sin[k|r−r(cid:48)|+kz(cid:48)−ωt]exp − G (r(cid:48))d3r(cid:48) (12)
E cosφ c L 4 z
y1
Using Eq. (3) and explicit expressions (10) and (12) for the probe E and scattered E fields,
0 1
we can calculate the polarimetric signal. While averaging the product E E over the optical
x0 x1
period, we come to the integral
ω (cid:90) 2π/ω ω (cid:90) 2π/ω (cid:20) k[x2 +y2](cid:21)
E E dt ∼ sin[k|r−r(cid:48)|+kz(cid:48) −ωt] sin kL−ωt+ dt =
0x 1x
π π 2L
0 0
(cid:20) x2 +y2(cid:21)
= cos k z(cid:48) +|r−r(cid:48)|−L−
2L
The same is obtained for E E . Now, Eq. (3) gives
y0 y1
5
4Wk3sin[2φ] (cid:90) lx (cid:90) ly (cid:20) k2(x2 +y2)(cid:21)
u = − dx dyexp − × (13)
1 cL2 Q2L2
−lx −ly
(cid:90) (cid:20) x2 +y2(cid:21) (cid:20) Q2(x(cid:48)2 +y(cid:48)2)(cid:21)
× cos k z(cid:48) +|r−r(cid:48)|−L− exp − G (r(cid:48))d3r(cid:48),
z
2L 4
with r = (x,y,L) and r(cid:48) = (x(cid:48),y(cid:48),z(cid:48)). The external integration over dxdy runs over the detector
sensitive area, and, therefore, |x|,|y| < l (cid:28) L. We assume that dimensions of the detector
x,y
l exceed the size Lλ/2πw of the probe beam spot at the detector (see Eq. (10)). Then, x
x,y
and y can be estimated as x,y ∼ Lλ/2πw. The internal integration dr(cid:48) runs over the irradiated
volume of the sample. For this reason x(cid:48),y(cid:48) ∼ w and z(cid:48) is of the order of the sample length l .
s
Taking into account that Lλ/2πw,w,l (cid:28) L, we obtain the following expansion for the factor
s
|r−r(cid:48)|:
x2 +y2 x(cid:48)2 +y(cid:48)2 xx(cid:48) +yy(cid:48)
|r−r(cid:48)| ≈ L+ + − −z(cid:48). (14)
2L 2L L
Note that the term ∼ z(cid:48)2 vanishes. Further estimates show that the term (x(cid:48)2+y(cid:48)2)/2L can be
omitted because in our case k(x(cid:48)2 +y(cid:48)2)/2L < π/4 and, finally, we have
x2 +y2 xx(cid:48) +yy(cid:48)
|r−r(cid:48)| ≈ L+ −z(cid:48) − (15)
2L L
Using this formula, we can evaluate the product of the cosine functions in (13) as
(cid:20) x2 +y2(cid:21) (cid:20)xx(cid:48) +yy(cid:48)(cid:21)
cos k z(cid:48) +|r−r(cid:48)|−L− = cosk (16)
2L L
As was mentioned above, the detector dimensions are assumed to be greater than the size of
the probe beam spot: l > Lλ/2πw. This allows one to extend integration over the detector
x,y
surface in (13) to infinity: |l | → ∞ and to calculate all integrals using the formula
x,y
(cid:90) (cid:114)π (cid:18) β2(cid:19)
dxexp[−αx2 +ıβx] = exp − . (17)
α 4α
For example, the integral with cosine function in Eq.(16) (we denote it I ) can be calculated
1
as follows:
(cid:90) ∞ (cid:90) ∞ (cid:20) k2(x2 +y2)(cid:21) (cid:20)xx(cid:48) +yy(cid:48)(cid:21)
I = dx dy exp − cosk = (18)
1 Q2L2 L
−∞ −∞
(cid:90) ∞ (cid:90) ∞ (cid:20) k2(x2 +y2) xx(cid:48) +yy(cid:48)(cid:21)
= Re dx dy exp − +ık =
Q2L2 L
−∞ −∞
(cid:90) ∞ (cid:20) k2x2 xx(cid:48)(cid:21) (cid:90) ∞ (cid:20) k2y2 yy(cid:48)(cid:21)
= Re dxexp − +ık dy exp − +ık =
Q2L2 L Q2L2 L
−∞ −∞
πQ2L2 (cid:18) [x(cid:48)2 +y(cid:48)2]Q2(cid:19)
= exp −
k2 4
Substituting (18) into (13), we obtain the following expression for the polarimetric signal:
6
Figure 1: Detecting the noise signal produced by the auxiliary beam.
4WkπQ2sin[2φ] (cid:90) (cid:20) Q2(x(cid:48)2 +y(cid:48)2)(cid:21)
u = − exp − G (r(cid:48))d3r(cid:48) (19)
1 z
c 2
V
Remind that this formula is valid if the sample length l is smaller than the Rayleigh
s
length, l < z (see definition of the Rayleigh length after Eq. (10)) and the probe beam spot
s c
is smaller than the detector photosensitive area, l (cid:29) Lλ/2πw. It is seen from Eq. (19) that
x,y
the polarimetric signal is, in fact, proportional to z-component of the gyration averaged over
irradiated volume of the sample, as is usually implied intuitively.
Equation (19) allows one to obtain the expression for the magnetization noise power spec-
trum observed in the SNS. In this case, G(r) is proportional to instantaneous spontaneous
magnetization of the sample randomly fluctuating both in space, and in time. If characteristic
frequencies of this field are much lower than the optical frequency ω, one can use Eq. (19) for
calculating the random polarimetric signal by substituting G(r) → G(r,t). The noise power
spectrumN(ν)isdefinedasFouriertransformofcorrelationfunctionofthepolarimetricsignal.
UsingEq. (19), thenoisepowerspectrumN(ν)canbeexpressedintermsofthespatiotemporal
correlation function of the gyrotropy G(r,t):
(cid:90) 16W2k2π2Q4sin2[2φ]
N(ν) = dt(cid:104)u (t)u (0)(cid:105)eıνt = × (20)
1 1 c2
(cid:90) (cid:90) (cid:90) (cid:20) Q2(x(cid:48)2 +y(cid:48)2 +x2 +y2)(cid:21)
× dteıνt d3r d3r(cid:48)exp − (cid:104)G (r(cid:48),0)G (r,t)(cid:105)
z z
2
V V
Tocalculatethecorrelationfunction(cid:104)G (r(cid:48),0)G (r,t)(cid:105)enteringEq. (20), oneshouldspecify
z z
a particular model of the gyratropic medium. The example of such a model (the model of
independent paramagnetic atoms with fluctuating magnetization) will be described in Section
5. In the next section, we will calculate the plarimetric signal produced by an auxiliary tilted
beam that produces a scattered field but does not irradiate the detector (see Fig. 1).
3 Detecting scattered field of a tilted beam
Let the sample be illuminated by an auxiliary light beam (AB) propagating at the angle Θ with
respect to the main probe beam (Fig. (1)). Note that AB does not hit the detector, but the
7
scattered field of this beam may provide additional contribution to the detected polarimetric
signal, and our goal now is to calculate value of this contribution.
The calculation can be performed in the same way as in the previous section with the
following changes. The scattered field is calculated using Eq. (8) with the field E (r) replaced
0
by Et(r), where Et(r) represents the field of the auxiliary (tilted) beam. The field Et(r) can
0 0 0
be obtained by rotating E (r) by the angle Θ around the axis (cosφ,sinφ,0) parallel to the
0
direction of polarization of the probe beam 1:
Et(r) = ME (Mr). (21)
0 0
Here, the matrix M is defined as
 cosΘsin2φ+cos2φ [1−cosΘ]sinφcosφ −sinφsinΘ
M = R(−φ)H(Θ)R(φ) = [1−cosΘ]sinφcosφ cosΘcos2φ+sin2φ cosφsinΘ  =
 
sinΘsinφ −sinΘcosφ cosΘ
(22)
1− 1Θ2sin2φ 1Θ2sinφcosφ −Θsinφ
2 2
=  1Θ2sinφcosφ 1− 1Θ2cos2φ Θcosφ +O(Θ3)
 2 2 
Θsinφ −Θcosφ 1− 1Θ2
2
Therefore, the field Et(r) is defined by the expression
0
(cid:115)
2W (cid:20) (cid:21) (cid:20) Q2[X2(r)+Y2(r)](cid:21)
Et(r) = Q t(cosφ,sinφ,0) expı kZ(r)−ωt exp − (23)
0 c 4
where
X(r)  cosΘsin2φ+cos2φ [1−cosΘ]sinφcosφ −sinφsinΘx δx
Y(r) ≡ [1−cosΘ]sinφcosφ cosΘcos2φ+sin2φ cosφsinΘ y+δy
      
Z(r) sinΘsinφ −sinΘcosφ cosΘ z δz
(24)
with r = (x,y,z). We denote by W intensity of the AB and take into account its possible
t
spatial shift (δx,δy,δz). Substituting Et(r(cid:48)) (23) into Eq. (8) instead of E (r(cid:48)), one can obtain
0 0
the following expression for the scattered field produced by AB:
(cid:115)
(cid:18)Et (cid:19) (cid:18)−sinφ(cid:19) 2W Qk2 (cid:90) (cid:20) Q2(X(cid:48)2 +Y(cid:48)2)(cid:21)
1x = t sin[k|r−r(cid:48)|+kZ(cid:48) −ωt]exp − G (r(cid:48))d3r(cid:48)
Et cosφ c L 4 z
1y
(25)
where X(cid:48) = X(r(cid:48)), Y(cid:48) = Y(r(cid:48)) and Z(cid:48) = Z(r(cid:48)), with the functions X(r(cid:48)), Y(r(cid:48)), Z(r(cid:48)) defined by
Eq. (24) with substitution x,y,z → x(cid:48),y(cid:48),z(cid:48). This formula has the same sense as Eq. (12); for
clarity we supply components of the scattered field by superscript t. Taking into account this
replacements, one can get the relationship for polarimetric signal produced by the AB (instead
of Eq. (13))
1Thus, polarizations of the tilted and the probe beams are the same
8
√
ut = −4 WWtk3sin[2φ] (cid:90) lx dx(cid:90) ly dyexp(cid:20)− k2(x2 +y2)(cid:21)× (26)
1 cL2 Q2L2
−lx −ly
(cid:90) (cid:20) x2 +y2(cid:21) (cid:20) Q2(X(cid:48)2 +Y(cid:48)2)(cid:21)
× cos k |r−r(cid:48)|+Z(cid:48) −L− exp − G (r(cid:48))d3r(cid:48)
z
2L 4
Calculation of intergrals can be made as in the previous section, and the final result for the
polarimetric signal produced by the AB is:
√
4 WW kπQ2sin[2φ] (cid:90) (cid:20) Q2(x(cid:48)2 +y(cid:48)2 +X(cid:48)2 +Y(cid:48)2)(cid:21)
ut = − t cosk[z(cid:48)−Z(cid:48)]exp − G (r(cid:48))d3r(cid:48) (27)
1 c 4 z
V
where r(cid:48) = (x(cid:48),y(cid:48),z(cid:48))
X(cid:48)  cosΘsin2φ+cos2φ [1−cosΘ]sinφcosφ −sinφsinΘx(cid:48) δx
Y(cid:48)  = [1−cosΘ]sinφcosφ cosΘcos2φ+sin2φ cosφsinΘ y(cid:48)+δy
      
Z(cid:48) sinΘsinφ −sinΘcosφ cosΘ z(cid:48) δz
(28)
One can see that ut is proportional to overlap of the two beams and vanishes at large shifts
1
δx,δy,δz. The trigonometric factor cosk[z(cid:48)−Z(cid:48)], in fact, singles out harmonic of the gyrotropy
with the spatial frequency equal to difference between the wave vectors of the two beams. Total
signal in the presence of two beams is the sum of (19) and (27): u + ut. Remind that the
1 1
angle Θ should not be too large; otherwise, one should take into account the component E2(r)
1
in Eq. (8).
4 Noise signal in the two-beam configuration
The noise signal produced by the two beams in the configuration of Fig. 1 is calculated as
Fourier transform of correlation function of the total polarimetric signal u = u + ut. It
1 1
consists of 3 terms:
(cid:90) (cid:90) (cid:20) (cid:21)
N (ν) = dteıνt(cid:104)u(0)u(t)(cid:105) = dteıνt (cid:104)u (0)u (t)(cid:105)+2(cid:104)u (0)ut(t)(cid:105)+(cid:104)ut(0)ut(t)(cid:105) (29)
t 1 1 1 1 1 1
Using Eqs. (19) and (27), one can write the expressions for each of them. The first term has
been already calculated and is given by Eq. (20). For the correlator entering the last term, we
have
16WW k2π2Q4sin2[2φ] (cid:90) (cid:90)
(cid:104)ut(0)ut(t)(cid:105) = t d3r d3r(cid:48)cosk[z −Z]cosk[z(cid:48) −Z(cid:48)]× (30)
1 1 c2
V V
(cid:20) Q2(X2 +Y2 +x2 +y2 +X(cid:48)2 +Y(cid:48)2 +x(cid:48)2 +y(cid:48)2)(cid:21)
×exp − ×(cid:104)G (r(cid:48),0)G (r,t)(cid:105),
z z
4
where x,y,z → r and X,Y,Z are defined by Eq. (28)
X  cosΘsin2φ+cos2φ [1−cosΘ]sinφcosφ −sinφsinΘx δx
Y  = [1−cosΘ]sinφcosφ cosΘcos2φ+sin2φ cosφsinΘ y+δy (31)
      
Z sinΘsinφ −sinΘcosφ cosΘ z δz
9
X(cid:48),Y(cid:48),Z(cid:48) are similar functions of x(cid:48),y(cid:48),z(cid:48) → r(cid:48).
Finally, the cross correlator (cid:104)ut(0)u (t)(cid:105) can be written as
1 1
√
16W WW k2π2Q4sin2[2φ] (cid:90) (cid:90)
(cid:104)ut(0)u (t)(cid:105) = t d3r d3r(cid:48)cosk[z −Z]× (32)
1 1 c2
V V
(cid:20) Q2(X2 +Y2 +x2 +y2) Q2(x(cid:48)2 +y(cid:48)2)(cid:21)
×exp − − ×(cid:104)G (r(cid:48),0)G (r,t)(cid:105)
z z
4 2
Consider now physical sense of different factors entering Eqs.(20), (30), and (32).
Exponential factor reduces the region of integration down to the region of overlapping
of the two beams. If Θ is not too large and l Θ < w, this region is close to “the beam volume
s
within the sample”. In this case, the exponential factor can be calculated at X = x,Y = y,Z =
z,X(cid:48) = x(cid:48),Y(cid:48) = y(cid:48),Z(cid:48) = z(cid:48). Note that it is rather difficult to satisfy the condition l Θ < w in a
s
real experiment. For this reason, the overlapping factor may considerably reduce contribution
of the AB to the polarimetric signal.
Trigonometric factor at small angles Θ is controlled by the difference between wave
vectors of the two beams because the cosine argument can be evaluated as z −Z = [cosφy −
sinφx]Θ.
Correlation function (cid:104)G (r(cid:48),0)G (r,t)(cid:105) is determined by particular model of the gy-
z z
rotropic medium. For homogeneous media, it depends on the difference r−r(cid:48) of the spa-
tial arguments. For the model of independent spins, described below (cid:104)G (r(cid:48),0)G (r,t)(cid:105) ∼
z z
δ(r−r(cid:48))e−|t|/τ cosω t
0
Thus, the integrals entering Eqs.(20), (30), and (32) can be calculated for any particular
model of the gyrotropic medium. In the next section, we will present calculations for the model
of independent paramagnetic particles (spins). Still, the following general remark should be
made. Let the beam waist 4w and the sample length l be much greater than the gyrotropy
s
correlation radius R and spatial period 2π/kΘ related to the difference of wave vectors of
c
the two beams: 4w,l (cid:29) R ,2π/kΘ. Then, one can substitute variables in the integrals
s c
entering Eqs.(20), (30), and (32) in the following way: r,r(cid:48) → R ≡ r−r(cid:48),R(cid:48) ≡ r+r(cid:48) and
take advantage of the fact that the correlator (cid:104)G (r(cid:48),0)G (r,t)(cid:105) depends on difference of its
z z
arguments:
(cid:104)G (r(cid:48),0)G (r,t)(cid:105) ≡ K(r−r(cid:48),t) (33)
z z
Then, the integral over R ≡ r−r(cid:48) in Eq. (20) can be estimated as the average of K(R,t)
over irradiated volume of the sample V . The integration over R(cid:48) ≡ r+r(cid:48) gives this volume
b
itself, and we obtain
16W2k2π2Q4sin2[2φ] (cid:90) (cid:90) (cid:20) Q2(x(cid:48)2 +y(cid:48)2 +x2 +y2)(cid:21)
N(ν) = dteıνt drdr(cid:48)exp − K(r−r(cid:48),t) ∼
c2 2
V
(34)
W2l sin2[2φ] (cid:90) (cid:90)
∼ s dteıνt dRK(R,t).
S
V
b
Here, we denote the cross section area of the beam by S ≡ 4πw2 and take into account that
w = 1/Q and that irradiated volume of the sample is V = Sl , where l is the sample length.
b s s
10