Efficient polarization analyzer for Lymana radiation

Stanislaw Chwirot, Sile Nic Chormaic, Darek Dziczek, and James Slevin

A new efficient polarization analyzer for Lyman,, radiation is described and compared with previouslyreported instruments.

Key words: Vacuum ultraviolet polarizers, Lyman.

IntroductionAtomic hydrogen occupies a central position in thehistory of 20th-century physics. During the firsthalf of this century, experimental investigations ofthe spectrum of hydrogen provided the basis for thedevelopment of modern theories of atomic structure.With the relatively recent advent of Doppler-freelaser spectroscopy and the availability of lasers capa-ble of spanning excitation to the first excited state ofhydrogen by means of two-photon absorption, thelast 15 years have witnessed a worldwide renaissancein atomic hydrogen spectroscopy, yielding new mea-surements of fundamental physical constants withunprecedented accuracy.' It is thus with some justi-fication that atomic hydrogen, the simplest and mostabundant element in the universe, is often describedas the Rosetta stone of modern physics.

Despite the well-known difficulties of polarizationmeasurements in the vacuum ultraviolet (VUV), therehas been great interest in the polarization of theLymana (La) line, which is the first resonance lineof atomic hydrogen. In general, measurements ofatomic-line polarizations provide detailed informa-tion on the mechanisms responsible for populatingexcited states; they are of interest in astronomicaland astrophysical applications as well as in laboratoryplasma and collision experiments.

Several experimental arrangements for polariza-tion observations of the La line that use differentoptical components have been reported in the litera-ture, each attempting to combine the efficiency of the

S. Chwirot and D. Dziczek are with the Institute of Physics,Nicholas Copernicus University, Grudziadzka 5-7, PI. 87100 To-run, Poland. S. N. Chormaic and J. Slevin are with the Depart-ment of Experimental Physics, St. Patrick's College, Maynooth,County Kildare, Ireland.

Received 20 November 1991.0003-6935/93/091583-07$05.00/0.© 1993 Optical Society of America.

analyzer with the accuracy of the measurement.To our knowledge we report a new, efficient polarizerfor La radiation; we compare it with previously re-ported techniques. We believe our approach repre-sents the best choice of optical elements currentlyavailable for the measurement of both the linear andcircular polarization of this important spectral line.

Optical ElementsBecause La radiation is relatively well transmitted bylithium fluoride (- 60% transmission for a thicknessof 2 mm at 120 nm with a cutoff of 105 nm2), thismaterial is the natural first choice for the componentsof the polarizing optics. However, feasibility studiescarried out by several authors3-5 yielded discouragingresults and highlighted the difficulties in obtainingstable optical elements fabricated from LiF, primarilybecause of the inherent instability of this materialthat results from its hygroscopic properties.

Because of the relatively large reflection coeffi-cients of several materials for radiation in the VUV, apolarizer based on the well-known phenomenon ofpolarization by reflection was suggested and testedfor both linear6 7 and circular8 polarization measure-ments. McIlrath8 investigated reflection of the Laline from several different materials and constructedthe first circular polarizer for this radiation by using areflecting surface as both a retarder and a linearpolarizer. The original construction by McIlrathemployed an A12 03 mirror as the linear polarizer andan aluminum mirror as the retarder. However, bothof these materials are well known for their long-terminstability and the poor reproducibility of their opti-cal constants. Because of this knowledge, later ver-sions of the basic McIlrath design9"10 used gold mir-rors for both purposes.

The most recent development in the field of Lapolarization measurement involves the use of high-transmitting, zero-order X/4 plates, which are fabri-cated from magnesium fluoride and are now available

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1583

commercially. With its high transmission [ 70%(Ref. 11)] and temporal stability, MgF 2 is a particu-larly suitable choice of material. Two instrumentshave been reported that use such a MgF2 retardercombined with a flat LiF mirror 2"13 or a pile of fourMgF2 plates" as a linear polarization analyzer.

The polarizer we report on here employs a X/4 platefabricated from MgF2 and purchased commercially,followed by a quartz (fused-silica) mirror that worksas a linear polarizer. Silica was chosen as a reflectorafter an exhaustive search of the literature for amaterial combining stability with high reflectanceand polarization efficiency.' 4

Polarization AnalyzerFigure 1 is a schematic representation of the May-nooth polarizer. The incident La radiation passesthrough the X/4 plate and then, after reflection fromthe quartz mirror, is incident on a channeltronsensitized to La radiation by a CsI-based coatingdeveloped in this laboratory with a quantum effi-ciency estimated at 15%. The X/4 plate for the Laline is a zero-order plate (14.5 mm diameter) com-posed of two MgF2 plates of opposite orientation witha total thickness of 0.8 mm.'5

The linear polarization analyzer is fabricated fromsilica polished to a flatness of X/50, i.e., much betterthan the La photon wavelength. The complex refrac-tive index represented by the constants n = 2.240 andk = 0.715 used for calculating the optical propertiesof quartz were taken from tables presented by Palik14

and yielded a polarizing angle of 670 at 121.6 nm.The mirror was set for an incidence angle of 700,slightly larger than the polarizing angle, in order toincrease the total reflectance by 3%. The reflectoris rotated by a stepper motor operated inside thevacuum tank. The position of the mirror is con-stantly checked during operation by reference to afixed point, whose orientation with respect to the fastaxis of the X/4 plate is accurate to better than 10.

Calculations performed for the silica mirror for thereflection of the La line at Oi = 70° yielded theoreticalestimates of the polarization efficiency P = 92% andof the total reflectance R = 29%. Because silica doesnot react with water and its high absorption of Laradiation means that multiple reflections do notaffect its operation, we can expect that the experimen-tal parameters achievable in practice should not differsignificantly from the theoretical values. This as-

MgF2RETARDER

sumption was supported by the application of thesame optical constants to the data obtained by otherauthors for Oi = 450 (Ref. 6): the reported measure-ments yielded P = 53% and R = 20% whereas ourestimates for this angle were P = 53.6% and R =19.6%.

The Bielefeld group 2 reported a polarization effi-ciency or polarisance P = 65% for a curved Suprasil(ultraviolet variant of a fused-silica) mirror of 120mm diameter set at Oi = 600. Although the theoreti-cal value for a flat-silica reflector operated at the sameangle is P = 87%, the difference can in this case beexplained by the curvature and large diameter of theSuprasil mirror that resulted in a significant spreadof the angles of incidence, and also by differencesbetween the optical constants of pure silica andSuprasil.

In spite of the general confidence in the quality ofthe optical data for quartz and the reliability of theresulting theoretical estimation of polarisance, theprecision , the determination of P is so critical forthe accu; acy of the polarization data that we decidedto tes the performance of our reflector directly bymeasuring the apparent linear polarization of the L,.beam reflected from an identical reflector located infront of the analyzer working without the retardingplate (see Fig. 2 and Nbrrenberg's polariscope geome-try 6 ). The source of the La radiation was a beam ofhydrogen atoms cross-fired with an electron beam ofincident energy 100 eV. According to the publisheddata, the linear polarization degree is 6% for the Lafluorescence of the atoms detected at right angles toboth beams.7 Thus, because the axial symmetry ofthe excitation process excludes the possibility of anycircular polarization, the radiation can be consideredto be almost completely unpolarized and suitable fortesting the polarizer.

Figure 3 illustrates the data obtained by sweepingthe analyzer in the angular range -90° to +90around its main axis. The datum points were fittedwith a function of the form y = A cos2 f3 + B, which ischaracteristic of crossed polarizers, by using a routinenonlinear least-squares-fitted program. The fit yieldsvalues for the constants A and B.

It can readily be shown (see Appendix A) that for acompletely unpolarized light source the fitted param-

QUARTZMIRROR 70,

COLIMTO ~ QUARTZ STEPPERMIRROR MOTOR

L RADIATION L,,RADIATION

Fig. 2. Schematic representation of the geometry of the polari-sance measurement.

1584 APPLIED OPTICS / Vol. 32, No. 9 / 20 March 1993

" QUARTZ STEPPER| _ MIRROR MOTOR

Fig. 1. Schematic representation of the Maynooth analyzer.

-9df -60, -30o O 300 600 90°

Angle (I between the incidence planes of the mirrors

Fig. 3. Intensity of unpolarized L,,: radiation after reflection fromtwo parallel quartz mirrors aligned for an angle of incidence of 700registered as a function of the angle between the polarizationdirections of the two mirrors. The solid curve represents theleast-squares fit of the function y = A cos2 p + to the experimen-tal data.

eters are related to the polarisance of the two-mirrorsystem by the expression

A _ R 2 + R2 - 2RRII= A + 2B) - R, 2 + R2 + 2RRI,

_ (R± - R)'

(R + R)2

The polarisance P is defined as the ratio (I, - IIIV(I, + III) for the intensities I, and III measured at twoorthogonal positions of the detector. For a singlemirror it is given by17

P = Q. (2)

The L, radiation emitted by our source had, however,a residual linear polarization characterized by thenormalized Stokes parameter s = -6% (Ref. 7).Using the Muller matrix formalism (see Appendix A),we found in such a case that the reflectances of thesingle mirror for light polarized parallel and perpen-dicular to the plane of incidence are given by

R ='MA + B + [(A + B)2 - (1 - S12)B2]1/2(1- sl)

B11 2R= (4)

The polarisance of the single mirror can then be!found by substituting these values into the formuladefining the polarisance:

-R1 I ()1 11 (5)RF + RI,

For s = 0 the expressions of Eqs. (2) and (5) are

identical. However, the effect on P of including thefinite polarization of the light in. this way is verysmall, changing the resulting polarisance by less than1%.

With the above procedures the data of Fig. 3 yieldeda value for the polarisance of the silica mirror givenby'P = 85.5% 1%. The polarisance calculatedfrom the optical constants is P = 92%; becauseestimates of the errors in these data are not quoted, itis not possible to estimate the accuracy of this value.The measured value is, however, significantly lessthan expected; it is likely to lie well outside thetheoretical limits, even when a generous allowance ismade for these errors. The most likely cause of thisdisparity does not lie with the errors in the publishedconstants, but rather relates to the particular environ-ment in which the current polarizer was used. Thequartz reflector used in the current experiment hadbeen in continuous use in a relatively poor vacuum ofapproximately 6 x 10-7 Torr for a period of over 1year. Thus it is possible that a surface film of pumpoil developed, which for even a thin layer would alterthe reflecting properties of the quartz. This possibil-ity highlights the importance of a periodic calibrationof the polarizer, especially in applications wherecontamination could be a factor. It is clear that theoptical constants should be used only with consider-able caution for the determination of the actualpolarisance of a particular device.

As an additional test of the analyzer, we have alsoperformed a measurement of the circular polarizationdegree of the La fluorescence resulting from the decayof hydrogen atoms excited by the impact of 54.4-eVelectrons scattered at = 10°. A value of P3 =

-0.335 ± 0.050 was obtained. This value is in goodagreement with the only published value for thisparameter,1 0 which is P3 = -0.444 ± 0.140.

General Considerations

The use of Stokes parameters to characterize theproperties of light is well known (see, e.g., Ref. 16),and the transmission and reflection properties ofoptical devices can be most conveniently described interms of these parameters. In particular the inten-sity of the light reaching the detector through theretarder-polarizer combination is given by the well-known expressions

I(13, ) = RT[So - P(S1 cos 2 + S2 cos 8 sin 2

+ S3 sin sin 2)], (6)

where R and T are the total reflectance and externaltransmittance of the two optical elements, Si are theStokes parameters, i is the angle that the main axisof the reflector plates makes with an axis fixed in theX/4 plate, is the phase shift introduced by theretarder, and P is the polarisance of the mirrordefined as the polarization degree of the light result-ing from the reflection of a completely unpolarizedincident beam.

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1585

Equation (6) suggests many different methods ofmeasuring the Stokes parameters of interest. How-ever, from a point of view of minimizing the errorsrelated to the experimental uncertainties, an opti-mum set of the six measuring angles below is conven-tionally chosen, yielding

S = I(O, 0) + I(90, 0),

S1 = I(O, 0) - I(90, 0),

S 2 = I(45, 0) - I(135, 0),

S3 = I(45, 90) - I(135, 90). (7)

This procedure formed the basis for the design andconstruction of the current analyzer.

The seemingly straightforward problem of a deter-mination of the Stokes parameters becomes morecomplicated in practice, primarily because of thesensitivity of the experimental data to some of thepolarizer's parameters. To see how this sensitivityoccurs and also to facilitate a comparison of differentanalyzers developed for the La radiation, we find itessential to consider in some detail all of the possiblesources of the experimental errors involved in suchmeasurements.

First consider the situation in which the errors in 13and are negligibly small. Then, for any of thecombinations of the angles used in Eq. (7), the flux ofthe light reaching the detector during a time period ris given by

I+ = RTSor[l Psi], (8)

and the normalized Stokes parameter can be calcu-lated as

Si = S (9)

It can easily be shown that the relative error of themeasured value of si is given by

Ls -AI, + I+ AL AlP-~ = 2- + 2 - (10)

Si 12(I+ + I-)(I+ - I) P

where A indicates the experimental error assigned toa given quantity.

The single-photon-counting technique in in fact theonly one suitable for a detection of weak fluxes of theLa radiation, and thus the estimate of Al is given by

AI = VTt. (11)

Substituting Eq. (11) into Eq. (10) results in

As, = 1 Fi + AP (12)Si 2(RTS0T)1/2 P

where

= (1 - X3)(1 + Xi ) 1/2

Fi = ~Xi2[(1 + x)(1 - 2)] 1/2

and 0 < xi =Psi < 1.An interesting feature of Eq. (12) is that it contains

an additive term related entirely to the imperfection(P < 1) of the linear polarization analyzer. In otherwords, the accuracy AP of the determination of itspolarisance P sets an asymptotic minimum value ofthe error of any measurement of any Stokes parame-ter that cannot be reduced by the experimentalprocedure. This fact, although quite obvious, wasnot explicitly discussed in the earlier papers on thissubject.

It follows from the above analysis that special careshould be taken to determine P as precisely aspossible. It is not possible to decide in a general waywhether it is better to measure it directly or todetermine it on the basis of tabulated optical con-stants. This problem has to be analyzed carefully,taking into account the reliability of both the experi-mental setup devised for such a test and the tabulatedvalues of the real n and imaginary k components ofthe refractive index usually presented to the thirddecimal place in the literature but often withoutestimates of the experimental errors. Periodic mea-surements of P are in any case essential for unstablematerials such as LiF and perhaps also for piles ofplate polarizers, which are sensitive to the formationof surface films.1 7 In contrast, for stable materialswith well-known optical constants determined inspecialized laboratories, it may be better to rely on theoptical data than to attempt measurements with lightsources that may not provide unpolarized light andthat may also be subject to other systematic errors.

The second term on the right-hand side of Eq. (12)is composed of two factors. The first, proportionalto [RTSoT]-/ 2 , shows that because the total intensityof the detected radiation usually depends more on thesystem under investigation than on the investigator,the time X required to achieve a particular level ofaccuracy of the measurement can be reduced only byusing elements of the highest possible reflectances ortransmittances.

The meaning of the second factor Fi is not immedi-ately transparent. However, looking at a plot of Fiversus xi = Psi (Fig. 4), one realizes immediately howimportant it is to use a linear polarization analyzer ofthe highest possible polarisance P. This use is espe-cially true in the case of measurements of Stokesparameters in which the values are small, e.g., < 0.2.This is a common experience in many modern experi-ments, where there is normally a strong depolariza-tion caused by either internal atomic fields or inter-atomic interactions.

In summary, then, the important parameters forthe performance of any polarization analyzer are R,T, P, and AP. Special care has to be taken to ensurea precise determination of the polarisance and its

1586 APPLIED OPTICS / Vol. 32, No. 9 / 20 March 1993

Xi=Psi

Fig. 4. Graphical illustration of the function F, showing thesignificance of the polarisance P for a reduction of the relative errorof a determination of the Stokes parameters. Note a rapidincrease of Fj in the region of small values of the argument xi = Psi.

long-term stability and to ensure that the value ofthis parameter is a maximum.

The experimental errors related to the alignment ofthe fast axis of the retarder and polarizer angle [ aswell as errors in the phase shift 8 are typical problemsof polarization measurements, so we shall not discussthem in detail here. The only remark we make isthat whatever experimental procedure is used, anydata on the linear polarization obtained with the useof a device containing a retarder will always carry anunavoidable error related to those two angles.

Comparison with Other PolarizersThe current analyzer is similar to that of the Bielefeldgroup.12"13 The essential difference between the twodevices relates to the different materials used for theanalysis of the linear polarization. We employed asilica mirror acting at Oi = 70°, whereas a LiF mirrorwith 0i = 60° was used in the Bielefeld analyzer. Thereflecting properties of a clean LiF surface make it anexcellent material for a polarizing mirror, and be-cause the imaginary component of its refractive indexk is extremely small for the L line, k - 10-6,14Samson2 suggested that it could provide a polarisanceof 100% at this wavelength. However, it was alsorealized that the hygroscopic properties of LiF wouldalmost certainly lead to a reduction of the polarisancein applications conducted over long time periods, andfurthermore that the theoretical polarisance waseven more significantly reduced in practice by reflec-tions from the back surface of the LiF mirror, result-ing in multiple reflections inside it.7 The authors7concluded that the apparent polarisance of their LiF

reflector was only 94%; a more recent estimate bythe Bielefeld group was 90%.12 Because both groupsused a similar technique to measure the apparentpolarisance, the observed difference once more pointsto the problem of the temporal instability of theoptical properties of LiF. The theoretical value ofthe total reflectance at the polarizing angle Oi = 600for L. radiation calculated by assuming n = 1.624 andk = 7.7 x 10-7 (Ref. 14) isR = 11%.

Although both the Bielefeld and Maynooth instru-ments combine a transmitting retarder with a reflect-ing linear polarization analyzer, the Windsor construc-tion9 employs two Au mirrors in a double-reflectionconfiguration. The MUnster analyser11 in contrast,exclusively relies on transmitting optics. In ouropinion the Windsor approach, which is the onlypossible one for the spectral region below the LiFcutoff, does not ensure the optimum performance forthe analysis of the La radiation. Assuming the opti-cal constants of Au to be n = 1.249 and k = 0.979(Ref. 14) and angle of incidence to be 57.5° (Ref. 9),one finds that the external transmittance (i.e., thetotal reflectance) of the two combined mirrors systemis only 6% for the L,, line and that the polarisance ofa Au reflector working as a linear polarization ana-lyzer is P = 68%. Moreover, the Windsor instru-ment requires a determination of the linear polariza-tion fraction of the incident radiation as a necessaryfirst step before any measurement of its circularpolarization can be made. Hence the accuracy of thelatter strongly depends on the errors associated withthe values of the Stokes parameters S, and S2.9

According to Winter and Ortjohann," the totalexternal transmittance of the MUnster instrument ishigher at T = 7%, and the polarisance of the pile ofMgF2 plates as a linear polarizer isP = 85%.'1 It canbe seen that although these values compare favorablywith those for the Windsor analyzer, they are stilllower than the appropriate values characterizingboth the Bielefeld and Maynooth instruments. It isalso worth mentioning that the optical data for MgF2taken from Samson2 (n = 1.30, k = 0.45) yielded avalue of P = 39% for the pile of plates described by theMUnster group,'1 i.e., a polarisance much differentfrom that measured by the authors. A much betterestimate of P = 78% was, however, obtained by usingn = 1.660 (a linear interpolation of the data presentedby the Oriel Corporation 8 ). These two facts seem tosuggest that Samson's data for MgF2 are in error andshould not be used in any future calculations.

Both the Windsor and the MUnster analyzers in-volve a retarder as an active element for the measure-ments of the linear polarization of the radiationunder investigation and thus provide data that carryan unnecessary error related to the alignment and tothe phase shift. This source of error is avoided inthe other two analyzers. Although theoretically theMUnster device could also achieve an analysis of thelinear polarization by rotating the pile of platespolarizer, this would mechanically be a more compli-

20 March 1993 / Vol. 32, No. 9 / APPLIED OPTICS 1587

Table 1. Important Characteristics of Different Analyzers

Total PolarisanceExternal of Linear

Transmittance Polarization Long-Term StabilityHeading (%) Analyzer (%) of Polarisance Acceptance Angle

Double reflector (gold) = 6 68 May be reduced by up to 20% by forma- Reduced by dependence of the phasetion of surface films'9 shift on the angle of incidence

Pile of plates with X/4 7 85 May be reduced by formation of surface Reduced by the geometry of the pile ofplate films17 plates polarizer

LiF reflector with X/4 < 8 90 May be reduced due to hygroscopic Limited only by the width of the Brew-plate properties2 '3 ster minima of reflectances

Quartz reflectort with X/4 20 92 May be reduced by up to 8% by forma- Limited only by the width of the Brew-plate tion of surface films ster minima of reflectances

cated task; probably for this reason the authorsdecided to rotate the retarder instead.

With the exception of the Minster analyzer allother reflectors discussed here can also be used forlinear polarization measurements at wavelengthsmuch shorter than La. Down to - 70.0 nm, the silicamirror ensures both a higher polarisance and ahigher total reflectance than the Au reflector. Thelatter shows a higher reflectance in the importantregion of the helium resonance lines, but its polari-sance is still lower than that of silica. Calculationsperformed for 59.0 nm that used tabulated opticalconstants showed that the best candidate for a linearpolarization analyzer at this wavelength could be LiF(R 17%, P 72% at Oi = 55°), whereas Au wascharacterized by R 21%, P 66% at Oi = 550 andsilica by R 12%, P 68% at Oi = 45°. Comparingthese numbers, we find it clear that the silica mirroris also an excellent linear polarization analyzer in thisregion of the spectrum, ensuring good temporal stabil-ity and a good polarisance.

ConclusionWe have described and evaluated a new VUV ana-lyzer, which is specifically designed for measuring thecircular polarization of the important La line ofatomic hydrogen. The analyzer, based on a commer-cially available /4 plate and quartz reflector, hasexcellent polarizability and a relatively high reflec-tance. It is a compact device that is easy to manufac-ture, relatively inexpensive, and that uses materialswith excellent long-term stability. Furthermore, theanalyzer, with its X/4 plate removed, performs well asa linear polarization analyzer down to wavelengths of

60.0 nm where the He resonance lines lie, compar-ing favorably in this spectral region with the Aureflectors of the Windsor polarizer.

To facilitate a comparison between the differentapproaches to polarization analysis in the VUV, wesummarize in Table 1 the important characteristicsof all the different analyzers reported in the literature.It is clear from this table that there is a widespectrum of performance, and great care must beexercised in the particular choice of the analyzer

configuration to be used if accurate and reliablepolarization data are to be obtained.

Appendix AThe most typical approach to a determination of thepolarisance of a reflecting linear polarizer is to per-form a measurement by using a system composed oftwo identical reflectors aligned in the so-called Nbrren-berg's polariscope geometry16 and to measure theintensity of radiation transmitted by such a system asa function of the angle 3 between the incidence planesof the mirrors.

Let M be a Miller matrix of a single reflector 9:

(R1 + R) -(R, - R)

. I -(Rj - R) (R, + R)M= K ° 0

O 0

0 0

0 0

21(,R1 0

0 2 R±R

(Al)

where R, and R are the reflectances for componentsof the incident light polarized perpendicular andparallel to the plane of incidence. The appropriaterotation matrix R(13) is given by9

I1 o o o

O cos 2P sin 2,P 0

O() -sin 2P cos 2,P 0

O O 0 1

(A2)

The Stokes vectors SW and S(d) of the light incidenton and transmitted by the system are then related bythe operator equation

S(d) = MR(I)MS(,) (A3)

If the incident light is completely unpolarized (anideal case but extremely difficult to achieve in prac-tice), its Stokes vector contains only one nonvanish-ing component S(). The intensity of the light trans-

1588 APPLIED OPTICS / Vol. 32, No. 9 / 20 March 1993

mitted by the system under consideration is then

Id (R, - RII) 2cos2 13 + 2RRII. (A4)

Comparing relation (A4) with the function

y = A cos2 3 + B (A5)

one immediately finds that the instrumental polariza-tion of the system defined as

(Im- -. Iin) (6(Im. +mm) (A6)

can be calculated as

A

Q (A +2B) (A7)

The situation becomes more complicated if theincident light contains a residual linearly polarizedcomponent, characterized (say) by a normalized Stokesparameter s1 = S1/So. Using equation (3), one againfinds that in such a case the intensity of the detectedlight is given by

Id [(R 1 - R11)2

- s(R 2

- R 2)]cos2

+ 2RRII,

(A8)

and that the fitted parameters A and B of the function(A5) are related to R, and RI, by

A = (R, - R11)2

- sl(R 2 - R 2),

B = 2RR 1. (A9)

In this more general case it is easier to solve thesystem of Eq. (A9) than to try to relate the measuredinstrumental polarization Q directly to the polari-sance of the single mirror. The system of Eq. (A9)leads to an algebraic equation of the fourth order.The solution of the equation yields the followingexpressions for the reflectances:

{A + B + [(A + B)2 - ( - s 2 )B2]1/ 2l 1/2R, = 2(1 - s) J'(A1O)

BRI2 = - (A1)

and the polarisance of the single mirror can be easilycalculated by substituting these expressions into Eq.(5).

We are grateful to N. Murphy and W. Lanigan for

their technical advice and support. This researchwas supported by EOLAS, the Irish Science andTechnology Agency, and by grant 0169c of the Euro-pean Community Science program. S. C. acknowl-edges the hospitality of St. Patrick's College, May-nooth and the financial assistance from the the PolishNational Research Council under grant KBN 2-0351-91-01.

References1. A. I. Ferguson and J. M. Tolchard, "Laser spectroscopy of

atomic hydrogen," Contempt. Phys. 28, 383-405 (1987).2. J. A. R. Samson, Techniques of VUV Spectroscopy (Wiley, New

York, 1967), Chap. 6, p. 180; Tab. 9.3, p. 312.3. E. G. Schneider, "Optical properties of lithium fluoride in the

extreme ultraviolet," Phys. Rev. 49, 341-345 (1936).4. W. C. Walker, "Factors affecting the transmittance of lithium

fluoride in the vacuum ultraviolet," Appl. Opt. 3, 1457-1461(1964).

5. J. Davis, "Pile-of-plates polarizer for vacuum ultraviolet," J.Opt. Soc. Am. 56, 837-841 (1966).

6. K. Rabinovitch, L. R. Canfield, and R. P. Madden, "A methodfor measuring polarization in the vacuum ultraviolet," Appl.Opt. 4, 1005-1009 (1965).

7. W. R. Ott, W. E. Kaupilla, and W. L. Fite, "Polarization ofLyman-alpha radiation produced in collisions of electrons andhydrogen atoms," Phys. Rev. Lett. 19, 1361-1363 (1967).

8. T. J. McIlrath, "Circular polarizer for Lyman-alpha flux," J.Opt. Soc. Am. 58, 506-510 (1968).

9. W. B. Westerveld, K. Becker, P. W. Zetner, J. J. Corr, and J. W.McConkey, "Production and measurement of circular polariza-tion in the VUV," Appl. Opt. 24, 2256-2262 (1985).

10. J. F. Williams, "Polarisation and angular correlations ofexcited P states of helium and hydrogen," Aust. J. Phys. 39,621-632 (1986).

11. H. Winter and H. W. Ortjohann, "High transmission polariza-tion analyzer for Lyman-a radiation," Rev. Sci. Instrum. 58,359-362 (1987).

12. R. Hippler, M. Faust, R. Wolf, H. Kleinpoppen, and H. 0. Lutz,"Polarization studies of H(2p) charge-exchange excitation:H-Ar collisions," Phys. Rev. A 31, 1399-1404 (1985).

13. R. Hippler, M. Faust, R. Wolf, H. Kleinpoppen, and H. 0. Lutz,"Polarization studies of H(2p) excitation: H-He collisions,"Phys. Rev. A 36, 4644-4651 (1987).

14. E. D. Palik, Handbook of Optical Constants of Solids (Academ-ic, Orlando, Fla., 1985), table X, p. 753.

15. Fa. Halle Nachfl., Berlin, Germany.16. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,

1975), Chap. 1, p. 46.17. D. Clarke and J. F. Grainger, Polarized Light and Optical

Measurements (Pergamon, Oxford, 1971), Chap. 3, p. 91.18. "Optics and filters," in The Oriel Catalogue (Oriel Corpora-

tion, Stratford, Conn., 1990), Vol. III.19. J. J. Corr, P. J. M. van der Burgt, P. Plessis, M. Khakoo, P.

Hammond, and J. W. McConkey, "Coherence parameter mea-surements for electrons scattering off heavy noble gas targets,"J. Phys. B 24, 1069-1085 (1991).

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