Experiments on interaction of high-power laser radiation with matter have revealed ... We present in the present paper a theoretical foundation for the use of the Faraday ...... 4.2.2), and also the average value of the magnetic field Bl(Y) (see Sec. .... on Laser Fusion Program, Institute of Laser Engineering, Okasa, VII-3, 162.
Journal of Soviet Laser Research 11(1): pp.1-32 (1990).
FARADAY-ROTATION METHOD FOR MAGNETIC-FIELD DIAGNOSTICS IN A LASER PLASMA
T. Pisarczyk, A. A. Rupasov, G. S. Sarkisov, and A. S. Shikanov
The possibility of using the Faraday effect for the diagnostics of magnetic fields in a dense plasma is theoretically demonstrate& A procedure for measuring the plane of polariztion of the probing radiation is examined in detail, with account taken of the wave polarization and of the presence of plasma self-luminosity. A procedure for determining the spatial distribution of the magnetic field is describecL A new three-channel polarointerferometer scheme is propose~ The idea behind the method is illustrated with reconstruction of the magnetic fields in a laser plasma as an example. 1. INTRODUCTION Experiments on interaction of high-power laser radiation with matter have revealed the presence of megagauss spontaneous magnetic fields (SMF) produced in dense layers of the laser plasma (LP) [1-20]. The study of SMF is of great interest in connection with laser fusion, since megagauss magnetic fields can alter substantially the physical picture of the ablation of spherical targets. Since SMF are capable of lowering the heat flow from the radiation-absorption region in the interior of the target, which raises the temperature and pressure in the corona and lowers the evaporation rate, hence adversely affecting the symmetry of the thermal heating [21, 22], SMF can promote absorption of the radiation and its reradiation from the plasma [23], the development of Rayleigh--Taylor instability [24], the target heating by fast electrons [24, 25], the increase of the bremsstrahlung [25], and also the lowering the heat flux from D--T fuel to the target shell and the increase of the effective mean free path of the a particles in a D--T plasma [27]. Several methods of generating SMF in LP have been proposed by now. Foremost are the photo-emf mechanism due to inhomogeneity of the heating beam over its cross section [28], and mechanisms connected with resonance absorption [29], evolution of parameteric [30] and magnetic--thermal instabilities [31], and the high level of Langmuir noise in a turbulent plasma [32]. A rather detailed analysis of the various SMF generation mechanisms is given in [33]. Specific field-generation mechanisms require a detailed analysis of the LP expansion conditions. In the case of a plane target, thermoelectronic generation of SMF is governed by the very geometry of the plasma formation. In the case of an ideal spherical geometry there are no SMF. In real experiments, however, there are always deviations from spherical symmetry of both the target and of the irradiation geometry. Hydrodynamic instability can lead to an appreciable growth of these initially small perturbations and, owing to the ensuing plasma inhomogeneity, to the appearance of crossed temperature and pressure gradients, and therefore also to generation of MF. When the latter reach the critical surface, they can initiate magnetothermal instability in the region n e < nc, where v T and xTne are parallel. In this region, the initial MF leads to the onset of a crossed heat flow and to rotation of v T relative to Vne, causing an appreciable enhancement of the MF. Many investigations have been marly by now of SMF generation in an LP. It will be shown in Sec. 2 the the most instructive research method is based on the magnetooptic Faraday effect, which was successfully used in several of the world's laboratories to investigate SMF and LP, as well as MF in electric-discharge plasma, viz., pinches, plasma loci, and others. Notwithstanding the basic simplicity of the method, it is quite subtle and requires a detailed methodological implementation, without which a correct performance of the experiments is possible. This question, however, has not received its due treatment in the literature. We present in the present paper a theoretical foundation for the use of the Faraday effect to measure magnetic fields in a dense plasma (Sec. 3). We analyze the methodology for important tasks of polarimetric measurements, such as the choice of the optimal uncrossing angles of the polarizers, allowance for the influence of the plasma self-luminosity, Laser-Plasma Laboratory, Lebdev Physics Institute. Translated from Preprint No. 135 of the Lebedev Institute of Physics, Academy of Sciences of the USSR, Moscow, 1989. 0270-2010/90/1101-0001512.50
9
Plenum Publishing Corporation
1
t7
b
Q.§ I c
d
e
Fig. 1
influence and allowance for depolarization of the probing radiation in the plasma, and analysis of polarimetry-measurement errors (See. 4.1). We describe the procedure of reconstructing the spatial distribution of the MF, based both on the use of the mean MF concept and on a solution of the Abel equation (See. 4.2). The three-channel polar-interferometer system, used by the authors in a number of laser facilities and optimized on the basis of the described methodological approach is described. The methodological treatment is illustrated using as an example measurements of magnetic fields in a laser plasma produced by single-beam irradiation of a plane target (See. 5). 2. EXPERIMENTAL METHODS OF INVESTIGATING SPONTANEOUS MAGNETIC FIELDS IN A LASER PLASMA Recording and investigating SMF in LP is a complicated experimental task. The main reasons are the small size (-100 ffm) and high electron densities (~1021 cm -3 of the plasma regions in which the SMF are generated. Another important factor is the short lifetime ( - 1 nsec) of the SMF. Nevertheless, a number of methods for the diagnostics of SMF in LP have by now been developed. We describe here briefly these methods and analyze their capabilities.
1) Magnetic Probes The most widely used method of measuring MF in a plasma is based on the use of magnetic probes (see Fig. la). Such a probe is a cylindrical coil of small diameter (~1 mm) directly placed in the plasma in a manner that does not disturb strongly the plasma parameters. A change in the magnetic flux @ passing through the coil cross section leads to the appearance of a potential difference d~ z u=-~-=-,,,t~
(2.1)
where S is the coil cross-sectional area, N the number of turns, and B• the component of the magnetic-field induction along the normal to the plane of the loop.
a~
z
Fig. 2
P,
P
Pit !
T Fig. 3
Magnetic probes can be used to measure the time dependence of magnetic fields in a plasma local region bounded by the loop size, with a time resolution determined by the coil inductance. This method is widely used to investigate extended plasma objects of low density, when the use of a large number of sensors can yield the spatiotemporal structure of the magnetic field. The small size of the LP makes it impossible to measure the SMF with probes directly in the region of the plasma focus. The magnetic probe is usualy placed several millimeters away from the focal spot. It is asumed that the observed fields are directly connected with the SMF generated in the focal region, and are due to their diffusion over the residual-gas plasma and to the convection in the expanding LP. Extrapolation of the recorded fields by 2-3 orders into the region of the critical density of a LP is, however, quite problematic. These substantial restrictions notwithstanding, the first registration of the magnetic moment of a spark produced by air breakdown in the focus of a ruby laser [1] was carried out with a magnetic probe. A number of studies [2-5] were devoted to the use of these procedures for the investigation of SMF in LP of various solid targets. 2) C u r r e n t P r o b e s
The SMF were measured in [6, 7] by the current-probe method, which yielded the time-resolved current distribution over the surface of a target, from which the magnetic-field distribution in the plasma was obtained by using the Maxwell equations. The experimental setup is shown in Fig. lb. One end of a thin copper wire is secured to the rear surface of the target, and the other end is led through a small opening to the front surface and is insulated from the target by a dielectric. At the instant of action of the laser radiation, a current pulse passed through the probe and induced in a secondary circuit an emf proportional to the derivative of the current with respect to time: _ dI
(2.2)
where k is a proportionality coefficient and I is the current. The temporal resolution of a probe + oscilloscope system was - 4 nsec (at a laser pulse duration --30 nsec). Varying the distance r from shot to shot gave the two-dimensional distribution of the current over the target surface. The current was averaged over the area of the end face of the probe (0.3 mm in diameter). Note that this method makes it possible to determine only that part of the magnetic field in the LP which is due to the current lines that are closed on the target. Yet the main contribution to the generated SMF is due to the motion of the charges over trajectories closed inside the plasma.
t
T
I
F Fig. 4
3) Magnetic Tape An original method of finding the SMF distribution by using ordinary sound-recording magnetic tape was proposed and implemented in [8, 9] (see Fig. lc). Prior to the shot the magnetic domains are randomly oriented in the tape. An external magnetic field causes some of the domains in the tape to assume a definite orientation, and their number is proportional to the field strength. It should be noted that this proportionality is preserved up to an external field B 0. If the field exceeds B0, the proportionality is violated, since all the domains are already in the proper orientation and the tape magnetization has reached saturation. In the experiment we used a tape containing 7-Fe20 3 particles (average particle dimension -0.1/zm, length ~0.5/~m, in which case a spatial resolution ~ 2 / z m is ensured). After the shot, regions with remanent magnetization are produced on the tape near the focal spot. To visualize the magnetic-field distribution the film was immersed in a liquid containing Fe30 4 particles of average diameter - 1 0 0 / ~ After evaporation of the liquid, the Fe304 particles remained on the tape and their number was distributed over the surface in proportion to the tape magnetization. It should be noted that this original method yielded information on the two-dimensional structure of the magnetic-field distribution in a tape region directly in contact with the plasma. This method yields only, however a lower-bound estimate of the maximum magnetic field during the entire plasma lifetime. 4) Zeeman Effect The SMF in an LP was measured in [10-11] by using the Zeeman effect. The plasma luminescence was measured in a direction transverse to the magnetic-field lines (see Fig. ld). The observed spectral line was split in this case into 3 components: ~, tr+, tr-. The line chosen for the splitting was ls2s3Sl--ls2p3Pi,2, 0 of the heliumlike ion C4+ which produces the triplet 2270.9/~ (J = 2), 2277.9 A, (J = 1), and 2277.3 A (J = 0). This triplet was chosen for the following three reasons: 1) relatively low sensitivity to Stark broadening; 2) possibility of using standard optical equipment; 3) emission of lines from the hot and dense region of the plasma; 4) simple interpretation of the Zeeman splitting in the region of medium magnetic fields (~100 kG). The ~ and a components were separated in [19] by a Wollaston prism. Each of the components was then directed to a monochromator. The monochromator output signals were fed to semiconductor detectors which permitted observation of the amplitude--frequency charcteristics of the received signals. By varying from shot to shot the position of the exit slit of the monochromator (for good reproducibility of the laser flashes), information was obtained on the evolution of the entire investigated section of the spectrum. Thus, measurement of the Zeeman splitting makes it possible to determine, with high time resolution, the average magnetic field along the observation line. It should be noted that this procedure is complicated and laborious, both in the performance of the experiment and in the interpretation of the results, since measurement of Zeeman splitting in an LP calls for taking into account various mechanisms that lead to displacement and broadening of the lines. Foremost are the broadening due to the thermal Doppler effect, the Doppler line shift due to the directional motion of the plasma, and the Stark and apparatus line broadenings.
4
5) Faraday Effect In [12-20] the SMF in an LP was determined by using the Faraday effect. Since actual implementations of Faraday measurements in an LP differ strongly from one another, we shall not describe any of them, but dwell only on the basic principles of this diagnostic method. In the Faraday magnetooptic effect the polarization plane of a probing electromagnetic wave is rotated as the wave propagates in the plasma along the magnetic-field force lines (see Fig. le). The rotation angle is proportional to the integral, taken along the path L, of the product of the electron density n e by the longitudinal component B I (component of the probing beam along the vector k) of the magnetic induction vector: 5
~;BllrtedL'
(2.3)
o
Interferometric measurements of the plasma are carried out in parallel with the polarimetry. The phase shift of the probing wave is proportional in this case to the integral of the electron velocity along the path: L
ai.
(2.4)
o
The ratio of these integrals yields the average component B'I of the magnetic field induction along the observation line. Thus, simultaneous polarimetric and interferometric measurements in an LP yields, with one flash, the two-dimensional distribution of the averaged magnetic field in the very interior of the plasma flare. If, however, the LP has a clearly pronounced axial symmetry (i.e., n e = ne(r), B = B ( r ) ) , it is possible to reconstruct mathematically the three-dimensional distribution of the mangetic induction vector B(r). Undisputed advantages of this procedure are the high spatial ( - 1 /~m) and temporal (--0.1 nsec) resolutions. The spatial resolution is governed by the quality of the optical system and by the "blurring" of the plasma image during the probing time. The temporal resolution is governed by the duration of the probing laser pulse or by the operating speed of the time-analyzing apparatus. Shortcomings include the limited (by refraction) depth of penetration of the probing beam into the region of high electron densities of the LP. This method can therefore be used to investigate only relatively the low-density coronal part of the plasma flare with an electric density not exceeding -1/100 of the critical value for the probing radiation. It can be seen from the foregoing analysis of the SMF investigation methods that the preferred diagnostic procedure is based on the use of the Faraday effect, since is the only method that yields, in one flash, the three-dimensional distribution of the magnetic field B(r) in the very interior of the plasma flare, with high spatial and temporal resolution. 3. THEORETICAL FOUNDATIONS OF POLARIMETRIC MEASUREMENTS IN A PLASMA We present, on the basis of the literature data [34-39], a systematic exposition of the variation of the polarization state of an electromagnetic wave (EMW) propagating in a homogeneous magnetoactive plasma (and also in an inhomogeneous magnetoactive plasma in the geometric-optics approximation). To this end we need the following quantities: 1) the plasma electron frequency ~op = ( 4 r m e e a / a e )
t / 2 =5, 64" ! 0 " ~
[soc- 1]
(3.1)
2) the electron cyclotron frequency coB:eB/~eC=t,76.107B
[see -1]
(3.2)
3) three dimensionless parameters u=;~ l~o ~ ; p
u=co~ /to ~ ;
s=v/co
(3.3)
where B is the magnetic-field induction (in G), co, the cyclic frequency of the EMW, and v, the effective collision (usually electron--pion) frequency. Let a high-frequency EMW of frequency co > > f2B (Q~ is the ion-cyclotron frequency) propagate in a homogeneous magnetoactive plasma. The angle between the wave vector k and the magnetic-field induction vector B is equal to a (see Fig. 2). The influence of the ions cannotbe neglected for such waves. The solution of the wave equation for a plane harmonic wave of type Exy = Eoxye+--i~ z leads to a dispersion equation whose solution is of the form
5
(r~-~%~
2 = I-
2,JC!-v-Ls9 2CI -~s3C~-v-Ls)-'~..cLr~cx+-(uzSLrt~+4U(~,-~-LS)ZCc~s2~] ~ / ~ "
'
(3.4)
The upper sign in (3.4) yields the quantity (n -- iZ)22 and corresponds to the "ordinary" (o) wave, while the lower gives (n iX)]2 and corresponds to the "extraordinary" (e) wave. In the absence of absorption, Eq. (3.4) takes the simpler form 2~J( I -~)
=~ -
r~ 2
(3.5)
2 ( ! _ ~ ) _ u S ~ r ~ z c x +_ (,~zStrt4 c~+4zzCi _ ~ ) ~ C o s ~ c x j ~ /= "
.2
From the solution of the wave equation we can obtain the polarization coefficients of "o" and "e" waves [34]: K = ET~ .2 _ _ t 2 ~r t .z cz:" ,2 uSLrtza 7- (u25Lrt4a*4~zfl-u-~s)2CcsZcxl:/2 9
(3.6)
In the absence of absorption, Eq. (3.6) becomes K
=- t t ,2
vStr~a
2 E f f ' ( I -•9. r u fu~Sf.rt4-a+4ufI_~)zCosZa],/2
(3.7) "
We see that in the general case both types of wave are elliptically polarized, with the "o" and "e" waves rotating counterclockwise and clockwise, respectively. In normal waves the electric vector E 0 oscillates in the X O Y plane (see Fig. 2), and the ellipse axes are parallel to the axes X and Y. It is easily seen that the "o" and "e" waves are "orthogonal" [34]: K .~=1
(3.8)
IK1,21 is the ratio of the ellipse axes. We consider two particular cases. For an EMW propagating along the B force lines in the plasma, i.e., for a = 0, we obtain from (3.7), for normal waves: K 1 = --i, K 2 = +i, i.e., the normal waves are circularly polarized. For an EMW propagating across the B force lines in the plasma we obtain from (3.7) for normal waves K 1 = 0, K 2 = --i" ~, i.e. the normal waves are linearly polarized. In all the remaining cases, i.e., for 0 < a 90~ the normal waves are ellipticaUy polarized. Thus, when an E M W of arbitrary polarization is incident on a plasma, "o" and "e" waves propagate in the plasma and their amplitudes, phases, and ellipticities are determined by the properties of the medium and by the parameters of the incident wave. In the linear approximation, normal waves propagate independent of each other, since they are exact solutions of the wave equation. Consequently, the electric field of the wave can be represented by the superposition E o = E 1 + E2, where E I and E 2 are the electric field vectors of the "e" and "o" waves, respectively. Since the normal waves propagate in the plasma with different phase velocities, their phase difference at the exit from the plasma will differ from that at the entrance. The difference between the phase velocities of the normal waves leads thus to a change of the polarization of the EMW as a result of its propagation through the magnetoactive medium. We examine now how the dependence of the E M W polarization state on the parameters of the normal waves. In the coordinate frame shown in Fig. 2, we can represent the real projections of the electric vectors E x and Ey for the "o" and "e" waves in the form: for the "e" wave - E1 x =a" St rt(~0t.3 (3.9) El y=cLp" CosCto~.)
for the "o" wave E2. =0 p. S t nC~0t -~.~
(3.10)
E2y=-b" CosCtoL - ~ )
where a and b are the amplitudes of the normal waves, ~o is the phase difference between the normal waves, and p = --i "K 1 is the degree of ellipticity of the normal waves. It is easy to show that p E [--1, 0]. Summing Eqs. (3.9) and (3.10) we get an expression for the resultant wave: Ex=E: x +Ezx =C .Sf.rt(~ot-(~x..3 Ey=E~ y+E2y=D . S(.rtCr
(3.11)
'd C
Fig. 5
where Ca =cza +Cbp32+2ebp "Co5~ ;
~=Cap3a+tz-2abp. Cose ;
u+Op.Coso
b.Scn#
We see that (3.11) is the equation of an ellipse in parametric form. It can be shown that the degree of elliptici~ k o f t h e resultant wave (i.e.,the ratio of t h e m ~ o r a x i s to t h e m ! n o r ) i s given by [35]
C! +6aDCI+p2)
(3.12)
The angle ~0 betweeen the ellipse axis and the OX axis can be written in the form
tb,.~=+_.2
C! +paDG. 5~rup Ct-p~ )C S~-I )+4pG.cosa
(3.13)
where G = a/b. The phase difference A~o between normal waves after passage of an EMW through a plasma of thickness L is given by [34] L /~o=-~JCr~a-r~ t )dL
(3.14)
O
where 2 is the wavelength of the electromagnetic radiation. Two limiting cases are well known: 1) the Faraday effect (a = 0) and 2) the Cotton--Mouton effect (a = 90~ Let us show that both eases can be simply obtained from the general equations (3.12)-(3.14). In the case of the Faraday effect the polarization coefficient of the normal waves is K1,2 = +_i, i.e., as indicated above, normal waves have left-hand anti right-hand circular polarizations, with p = --1. We introduce, for convenience, an additional angle 7 such that tan ? = G. For arbitrary 7 Eqs. (3.12) and (3.13) acquire the much simpler form: /~=~~C,-r/4-~,)
(3.15)
~---I"O 9
(3.16)
Thus, for the general form of an elliptically polarized EMW, the Faraday effect leads to rotation of the entire polarization ellipse. There is no wave deopolarization in this case. If the E M W is plane-polarized, the Faraday effect leads simply to rotation of the polarization plane. If electron--ion collisions can not be neglected (s ~ 0) we have a change in the ellipticity of the wave. This phenomenon is due to the anistropy of the damping of the normal waves in the plasma [36]. From the foregoing equations we easily obtain an expression for the polarization-plane rotation angle of a linear EMW. Using (3.14) and (3.16) we express the change of the rotation angle in the form
L Alp=~I("7"tz-r~t ) d ~ 9
(3.17)
0
The difference n 2 -- n 1 is determined from (3.5). F o r a = 0 we have rl"
=I -
v
L ,2.
(3.18)
I ~ ~TE- "
When the conditions u, v < < 1 are met (this is realized in practically all the experiments on LP probing in the optical band) we obtain rt
We can express the difference n 2
=~-
73 2(" 1 _+~CY') ~
(3.19)
-- n I in the form TL - , q . 2. !.
'j LgU-' _ I -'O.
mz.~z~ 9 3
(3.20)
Substituting (3.20) in (3.17) we get I.,
I..
~w_rr r ~ . ~ s ~ L _ -Z,I0 ~
e"k2. Irt BdL 9 2rta2.c" 0
(3.21)
Calculating the value of the coefficient, we obtain the final form of the equation for the Faraday rotation of the E M W polarization plane in the plasma: t. A~=2.52.iO-t'k2IrtBdL
[rad]
(3.22)
0
where L and ;t are in cm, n e in cm -3, and B in Gauss. In the case of the C o t t o n - M o u t o n effect, the polarization coefficients of normal waves are K 1 = 0 and K z = --i" 0% i.e., as already noted, normal waves are linearly polarized and p = 0. For arbitrary p = 0, Eqs. (3.12) and (3.13) take the form (3.23)
&g2~p=t ~2T" 5 (.n~o,
(3.24)
I.~t us analyze on the basis of these equations the main distinctive features of the change of polarization state of an EMW. A distinction can be made between three different cases: 1) y < ~r/4, 2) 7 > at/4, 3) y = ~z/4. Note that the quantity y does not vary in the Cotton--Mouton effect and depends only on the initial ellipticity of the E M W and on the angle between the major axis of the ellipse and the O Y axis: Cos ~ 2 y : g o s z 2~" Cos ~ ~'2~rc L~C:~..)),
(3.25)
If a plane-parallel E M W is incident on a plasma at an angle ~p < at/4 the degree of ellipticity of the E M W changes, with increase of the phase shift, periodically from k = 0 at A~o = 0 to k = tan y at AT = at/2 with a period at. The inclination of the major ellipse also changes from g, at A~o = 0 to ~ at A~o = ~r with a period 2at. Thus, the polarization ellipse "pulsates" relative to the O Y axis in an angle range +_q,; furthermore, k becomes minimal when I~1 reaches a maximum and maximal when ~0 -- 0. If a plane-polarized wave is incident on a plasma at an angle ~p > ~r/4, a similar "pulsation" of the polarization state relative to the O X axis takes place betwen the angles ~p and ar -- % In this case k reaches a maximum at g, = zc/2. A t g, = at/4, there is no systematic rotation of the polarization plane. With increase of the phase shift, the degree of ellipticity of the wave increases, and at ~, = ~r the polarization becomes circular. With further increase of phase, the ellipticity of the wave decreases, but in this case g, = ---st/4. Thus, a jumplike change of the inclination of the major polarization axis of the ellipse takes place at g, = at/4. F o r ~p = 0 and g, = Jr/2, no change takes place in the E M W polarization state. We obtain now an expression for the phase shift in the Cotton--Mouton effect. simpler form
At a = 90 ~ Eq. (3.6) takes the
rt2
=f_ . ,2
2,J(:-v3 2C1 - u . ) - u z u
(3.26)
Putting u, v
where v0 and T are the plasma velocity and temperature, Ln, Lv, L T are the characteristic scales of the density, velocity, and temperature, n is the electron density, n c is the critical density of the electrons, and ~, is the angle betwen the vector E of the EMW and electron-density gradient. Since the rotation mechanisms are additive, the resultant rotation ~ of the EMW polarization plane is the sum of the components listed above: ~)=~B+~n +~v+~T 9
10
(3.40)
Note that a contribution to the rotation angle is made by gradient components perpendicular to the EMW propagation direction. For the typical LP plasma parameters (n/n c > Vo/C, L n ~ L v, y / L > 1) it can be shown that the following inequalities are met: ~B>~n)~v)~Z
9
(3.41)
Thus, the condition under which the Faraday-rotation method can be used for SMF diagnostics is formulated as follows [39]: (oBLn > Y
c
an
n.
(3.42)
For typical LP parameters L n - 3" 10 -3 cm, y - 5" 10 - z cm, n e - 102~ cm -3, n c ~ 4 " 1021 cm -3 (for 2 = 534 nm), the inequality is satisfied for B > 30 kG. In the case of steep gradients ( L n < 3" 10 -3 cm) and weak fields, however, the two effects can be comparable. 4. PROCEDURE FOR MEASURING MAGNETIC FIELDS IN A DENSE PLASMA USING THE FARADAY EFFECT Magnetic-field measurements comprise three principal stages: 1) polarimetric measurements; 2) interferometric measurements; 3) determination of the spatial distribution of the MF. We shall not consider in this paper interferometric measurements of the electron density in a plasma, since this topic is treated in detail in many papers (e.g., [41]). The first and third stages have not been sufficiently well covered in the literature, and we shall therefore pay particular attention to them. 4.1. Procedure for Polarimetric Measurements 4.1.1. B a s i c Premises.
T h e basic setup for the measurement of SMF in LP by the Faraday-rotation method is shown in
Fig. 3. The principal element of the setup is a system of two crossed polarizers (P1, P2), between which is located the investigated plasma (P). The polarizers used are two Glan prisms made of Iceland spar. The plasma is probed by pulsed laser radiation synchronized with the heating beam. After passing through the plasma, the rotated probing beam is incident on the analyzer (P2) where it is split into two parts. The part corresponding to the projection of the electric vector E 0 on the analyzer-transmission direction lands in the "Faraday" channel (F-channel), while the part corresponding to the E 0 projection on an axis perpendicular to the transmission direction (i.e., the radiation fraction reflected from the inner canted face of the Glan prism) lands in the "tenebral" (shadow) channel (T-channel). Part of the T-channel radiation is used to obtain the interference pattern of the plasma (I channel). The image is recorded on photographic film. Let us examine in detail the procedural features of measuring the angle of rotation of the polarization plane of the probing radiation. Let the electric vector E 0 of a plane polarized EMW undergo rotation through an angle ~o after passing through the plasma. The signals in the F and T channels (for fully crossed polarizers) will then be E 0 sin ~, and E 0 cos ~o respectively (see Fig. 4). The intensities in both channels will be connected with the intensity J0 of the probing wave by the relation JF:Jo ' ( S t rtZ g+ l~Q ' Coa~ 9 )
JT:go 9(Co5 2 ~ + ~ o
" 5 La 2 r
(4.1)
where k 0 is the polarimeter contrast (k 0 = JF/JT at ~o = 0). To simplify the mathematics, we regard the polarimeter contrast as the residual ellipticity of the probing wave after its passage through the first polarizer. The light flux J(t) darkens the photographic film in proportion to the energy of the photons incident per square centimeter of the corresponding layer during the exposure time t 0, i.e., the exposure H is given by to H:~J( t.)dt (4.2) o
The difference between the photographic densities at two points of the film is connected with the corresponding exposures by the equation O~ -Da : y L~C-h ~Ha.)
(4.3)
where y is the contrast factor of the film, while D 1 and D 2 ase the fiim photographic densities at the points 1 and 2.
11
/)F
,0
__~k
J
9
9
a,
oI
,r---
,If:
r
~
.%
Fig. 7
If the intensity ratio Jl(t)/J2(O is constant in the entire interval t E [0; to] (this holds true if t o < ~, where T is the characteristic time of the change of the parameters n e and B of the plasma), then Eq. (4.3) can be written in the form
DI -D~=7L~CJI/J2.)
(4.4)
Using (4.1) and (4.4) it is easy to obtain an expression for the rotation angle to in terms of the photographic densities at the corresponding points of the Faraday (DF) and shadow (DT) channels
t~.!p=, kr/~F.
IodDF-l~)/r-.ho
(4.5)
- ~ T / ~ F ~o 9I 0(DF-DT)/Z
where kick F is the ratio of the transmission coefficients of the filters of the T and F channels. In most experiments, one introduces betweeen the transmission axes of the polarizer and analyzer an initial uncrossing o o ~o0, the need for which will be explained below. The ratio kT]k F can be expresed in terms of the densities D F and D T at the mutually corresponding points of the F and T channels (corresponding to the plasma regions where the polarization plane has not been rotated) in the following manner: kT
;~'F --[
ko + t.~ , ~ o
"
! +.~ot,~;zpo
r ..... ] "I0"Jr-uF'/T "
(4.6)
Equation (4.5) can then be written in the final form t. ~2 (" VpO +~.~_-- , I - '~s
I -I. ~o
where we have introduced the notation
(4.7)
~_,-r~,F-uFv-, ~" . , o . r u"" , - u ~- jX ,
I=JF/JT=[ ~o+~a~o
I+,%"ikfpo
]i
O~D,/y .
Equation (4.7) was obtained under the following asumptions: 1) the plasma self-luminosity can be neglected; 2) there is no depolarization of the probing beam. The influence of these factors on the determined value of the rotation angle will be analyzed below. We must emphasize the following feature of Eq. (4.7). The difference between zXDF = D F -- D F and zero can be due to four factors: 1) rotation of the polarization plane; 2) inhomogeneitiy of the intensity J0 over the cross section of the probing beam; 3) refraction of the probing beam in the plasma; 4) absorption of the probing beam in the plasma. On the other hand, the deviation of ADT = D T -- DT2 from zero can be the result of only the three factors 2, 3, and 4. Therefore, introduction of the T channel into the polarimeter makes it possible to exclude the influence of factors 2-4 on the determined value of the rotation angle. For a more detailed analysis of Eq. (4.7), we show schematically in Fig. 5 the different regions of a LP for plane (a) and spherical (b) targets. Region A corresponds to the plasma flare region, which is transparent to the probing radiation; B is the region where refraction and absorption take place; C is the region opaque to the probing radiation (where n e > 0.01" nc, with n c the critical density of the electrons for the probing wave).
12
4
/
----
3,
t, - ~ a -r162-r162 -03 -O,a -Od ,
1
o.2 o,~ ~),r 0.,-o.~ -i
,-s
.-6 Fig. 8
A typical form of the dependence of the film density in the T channel on X a t fixed Y, i.e., DT(X), is shown in Fig. 6, where the following notation is used: D o the film-fogging density; D 1 is the density corresponding to the start of the linear section of the film characteristic curve; D 2 is the density corresponding to the end of the linear section of the film characteristic; R is the radius of the opacity region of the LP in the given cross section Y. The numbers denote the following plasma regions: 1-2 and 5-6 the transparency region; 2-3 and 4-5 the refraction and absorption region, and 3-4 the opacity region. A typical plot of the film density in the F channel as a function of X at a fixed cross section Y, i.e., DF(x), is shown in Fig. 7. A distinction is made betwen two different cases: ~o0 ~ 0 and b) ~o0 = 0. The notation in the figures is the following: 1-2 and 5-6 mark the region where there are no rotations; 2-3 is the region of positive rotations; 4-5 is the region of negative rotations; 3-4 is the plasma opacity region. By positive rotation we mean rotation of the polarization plane towards the analyzer transmission axis, and by negative we mean rotation in the opposite direction. We see that at 5o0 = 0 it is impossible to distinguish the sign of the polarization-plane rotation, inasmuch as in this case any rotation increases the film density. In the case ~o ;~ 0 positive rotation of the polarization plane increases the film density, while negative rotation decreases it. Figure 8 shows a family of ~o(AD) curves for 7 = 1.5 (at different uncrossing angles ~o0), obtained from Eq. (4.7). As AD is increased the absolute value of ~o(AD) increases. It can be seen that the plots are not symmetrical relative to the rotation direction. The only exceptions are the curves ~o0 = 45 ~ and ?0 = 0. In the first case ~o(AD) is linear, and in the second case the rotation direction cannot be identified at all and only the modulus of the rotation angle can be determined. Plots of a(AD) for ~o0 = 5 ~ at different film contrast factors ), are shown in Fig. 9. A particularly strong difference between the curves is observed when the positive rotation angle ? is increased. The character of the plots attests to the need for as acurate a determination of ~, as possible, for otherwise the value of ~o will be determined with a large error, with the error in the determination of the positive rotations larger than for the negative ones. Figure 10 shows plots of the sensitivity d(AD)/d~o of the polarimeter (for a rotation through +1 ~ against the uncrossing angle ~o0 for different values of ~, of the film. It is seen from this figure that variation of the uncrossing makes possible variation of the sensitivity -- by a factor of - 5 0 . The strongest sensitivity change (by about 10 times) takes place when 5o0 is changed from 0 to 5 ~ When the uncrossing is changed from 10~ to 45 ~ the sensitivity of the polarimeter changes little (by a factor of --3).
4.1.2. Choice of Optimal Uncrossing Angle. For a correct measurement of the rotation of the polarization plane it is necessary that the phtographic densities in the F and T channels lie on the linear section on the film characteristic, i.e., DF, D T ~ [D 1, D2]. The limits D 1 and D 2 of the dynamic range of the film call for a corresponding choice of the polarimeter sensitivity, i.e., for introduction of a definite uncrossing +__~Oma x. The choice of the optimal value of ?0 is one of the conditions 13
3
2 t.
-t
Fig. 9
O.4
~r a,6
~3
O,a o,t
O Fig. 10
for a correct measurement of the rotation angle of the polarization plane. On the one hand, for the function to(AD) to be single-valued it is necessary that too be not less than tOmax. On the other hand, increasing to0 above its optimum value lowers the sensitivity of the system and increases thus the angle-measurement error. The angle ~o0 should be such that the density due to the rotation +tOmax be equal to D2, and the density due to the rotation "--~Omax be equal to D 1. It is necessary here also to choose the proper value of D~. Simultaneous solution of Eq. (4.7) for the angles to = +tOmax and to = --3Omaxleads to a biquadratic polynomial whose solution yields the optimal uncrossing angle to0:
:-5('!~)/SCy)->."- v .
(4.28)
Calculating f(r) for different ~" (Yi) we obtain a set o f ~ (y) curves, which determines the range of variation of the function'f(r). Note that this approach to the determinatin of the error of fly) can be used in any method of numerically solving the Abel equation. We now demonstrate, using trial functions, the difference betwen the above methods. We use for B(r) and ne(r ) the expressions BCr)=JrC~ - r ) rt ( r ) = I - r z 9 (4.29)
27
Since
B(r)
and
ne(r)
are specified analytically, we can obtain from (4.7) expressions for S•(y) and
5~=(e'3-z/e~2)cl-Y23'/~*(2/2y'-eYz')~r~("
~ .(~._,a) x/a, ~ J
Sn(y): (4.30)
5 n = 2 / 3 " ( I - ~ ) */a 9 Figure 25 shows the percentage deviations of the determined function B'(r) from its analytic form (4.29). Curves 1 and 2 were determined by the Van Voorhis and the Gegenbauer-polynomials methods, respectively. It can be seen that both methods reproduce the specified MF profile with high accuracy (1-3%) in practically the entire range of r (except for the peripheral region, where the error of B(r) increases steeply in practically all methods used to determine f(r)). The Van Voorhis method describes the behavior of B(r) better. Great interest attaches to the problem of the influence of the measurement errors of SB0,) and Sn(Y) on the accuracy with which the MF is determined. To this end a so-called "Gauss noise" with a definite variance is superimposed on the analytic SB(y) and Sn(Y) (i.e., it is assumed that the error distribution in each measurement takes a Gaussian form). For SB(Y) the variance is taken to be ASn = 0.02 ( ~ 5 % of the maximum rotation angle), and for Sn(Y): ASn = 0.06 ( - 1 0 % of the maximum band shift). The error in the determination of the MF was found by a statistical method (see above). Figure 26 shows typical plots indicative of the error of B(r) for two error distributions (Fig. 26a -- the Gegenbauer-polynomial method, Fig. 26b -- the Van Voorhis method). It can be seen that determination of the MF distribution by the Gegenbauer-polynomial method is preferable both in the sense of the absolute values of the errors and in e sense of the stability of the procedure to "noises" of the input functions SB(Y) and Sn(Y). It should be noted that the instability of the Van Voorhis procedure to "noise" can be substantially reduced by preliminary smoothing of the experimental points, decreasing thereby the influence of random spikes of SBfY) and Sn(Y) on the determined values of the MF. A more complete comparative analysis of the various methods of solving the Abel equation, using a large class of trial functions, is carried out in [45]. 5. THREE-CHANNEL POLAROINTERFEROMETER SYSTEM In most studies reported to date [12-20] the polarization-plane rotation angle was measured by the single-channel procedure. A two-channel polarimeter system was developed in [19]. To determine the degree of ellipticity of the probing wave, three- and four-channel polarimeters were used in [13] and [17]. The weak spot of [12, 13, 17] is the absence of interferometric measurements. In addition, a common shortcoming of the studies cited above is the presence, between the polarizer and the analyzer, of various optical elements that can depolarize the probing radiation to one degree or another, and lower thereby the accuracy with which the rotation angles are measured. We propose here a new three-channel polarointerferometer system that combines simultaneously the Faraday, "tenebral" (shadow), and interferometric recording channels. The possibility of instrumental polarization of the probing radiation is completely excluded in this system. The optical diagram of the polarointerferometer is shown in Fig. 27. A plane-polarized probing beam shaped by an input polarizer (1) passes through the plasma (2) and is incident on an analyzing Iceland-spar wedge (4) whose axis makes some small uncrossing angle 9'o with the transmission direction of the input polarizer (1). The probing wave is resolved in analyzer (4) into ordinary "o" and an extraordinary ~e' components with mutually perpendicular polarizations. The ~o" wave corresponds to the Faraday channel (the polarization makes an angle 90~ -- 9"o with the initial one) and the "e" wave corresponds to the shadow channel (its polarization makes an angle 9"o with the initial one). Angular separation of the ~o~ and 'e" waves takes place at the exit from the wedge (4). The lens (5) projects the image of the plasma (2) on the recording planes (8) and (9). Two focal spots of the "o" and "e" waves, with a distance A between them, are formed then in the focal plane. The orientation of the polarizer (7) is chosen such that its transmission direction makes an angle 90~ with the transmission angle of the input polarizer (1). The polarizer (7) separates components of equal intensity from the "o" and 'e" parts of the beam. The intensities of these components can be represented in the form J0 = Je = J" sin2 9"o" c~ 9"0, where J is the intensity of the light prior to entry into the analyzer (4). In the region where the beams overlap in the registration plane, an interference pattern is produced. It is necessary in this case that the probing beam cross section exceed the plasma flare and be shifted sideways from the latter. This is accomplished by introducing the knife-edge (3) or by suitable choice of the target construction. All this produces simultaneously in the registration plane (8) an interference pattern (in the region of overlap of the perturbed part of one beam and the unperturbed part of the other) and a Faradaygram (where there is no beam overlap, see Fig. 27).
28
The shadow image of the plasma is produced in the plane (9) upon reflection of the "e" wave from the inner face of the Glan prism (7). Its intensity is Je' = J" c~ ~'0" The image formed by reflection of the "o" wave has the much lower intensity J0' = J" sin4 g0. To vary the widths AS of the interference fringes, a second wedge of Iceland spar (6) is placed between the lens (5) and the polarizer (7). The optical axis of this wedge is parallel to the optical axis of the analyzing wedge (4), so that the latter refracts the beams differently without varying the beam polarization. Variation of the distance between the wedge (6) and the lens (5) makes it possible to vary A and thus regulate the widths AS of the interference fringes. The beam overlap S in the registration plane (8) remains practically unchanged in this case. A major advantage of this system compared with the one described in [18] is the absence of any depolarizing elements between the plasma and the analyzer. This permits, in particular, replacement of the lens by complex high-grade objectives even though they can produce substantial depolarization. The fact that the Glan prism has a working angular aperture < 8 ~ certain limitations are imposed on the value of/~ (the angle between the "o" and "e" waves past the wedge (6)), on the aperture D of the probing beam, and on the focal length F of the lens. For normal operation of the polarointerferometer it is necessary to check beforehand the condition f3+2arc L8C D/2F)
