Coherency matrix polarization measurements: applicationto magnetooptic garnet films
Herbert Engstrom
A method of completely determining the polarization of light is described. An algorithm that allows a fastnumerical evaluation of the coherency matrix based on the results of four nonequivalent polarizationmeasurements is presented. The method has been applied to determine the degree of polarization, ellipticity,and orientation of the major axis of light transmitted through a magnetooptic thin film on a garnet substrateas a function of magnetic field. Representative results are given. Key words: Coherency matrix, polariza-tion matrix, Mueller matrix, Jones matrix, Stokes parameters, magnetooptic, Faraday rotation, garnet, thinfilm.
I. IntroductionMaterials exhibiting magnetooptic effects-Kerr
and Faraday rotation-are finding increasing applica-tion in a wide variety of areas.' Among these areoptical data storage, 2 magnetic field sensors,3 and opti-cal isolators.4 In many of these materials the magne-tooptic effects are quite small, and their characteriza-tion requires careful calibration, measurement, andanalytical procedures.
Two approaches are in common use for analyzingpolarization data. The first uses Jones matrices, 5-7 inwhich the two-component complex electric field vectorof the light,
E = ('), (1)
is transformed by each polarizing element in the opti-cal system. The transformations are represented by 2X 2 complex Jones matrices, which account both forattenuation and relative phase shift of the compo-nents. The transformation of the polarization due tothe entire optical system can be expressed as a singleJones matrix, which is simply the product of all theJones matrices of the individual elements. The Jonesmatrix formalism has the virtues of simplicity and
When this work was done the author was with Verbatim Corpora-tion, 435 Indio Way, Sunnyvale, California 94086. The author isnow with Tencor Instruments, 2400 Charleston Rd., Mountain View,CA 94043.
intuitiveness, but it has the disadvantage that it isappropriate only for systems involving 100% polarizedlight.
The second approach, that of the Mueller matri-ces,5 8 describes the light in terms of the four Stokesparameters, given by
SO = (EJ 2) + (IEI2),
S1 = (IE"2) - (IEI2),(2)
S2 = (E.E>) + (E;E,),
S3 =-WExE, - (E;EY))
which completely describe the irradiance and polariza-tion of the light. Polarizing elements of the opticalsystem are represented by 4 X 4 Mueller matrices,which multiply a four-element Stokes vector whosecomponents are the four Stokes parameters. Thisapproach allows the treatment of partially polarizedlight, and in fact the degree of polarization is given by
P = (S2 + S2 + S2)1 2/S0.-\1 2 TA3J ~ (3)
The Mueller matrix approach has another minor ad-vantage in that the Stokes parameters are all real, andtherefore the Mueller matrices are also real. The dis-advantage of the Mueller matrix approach is that thealgebra involved in determining the Mueller matrix ofanything but the simplest otical system is long andtedious.
A formalism that combines the simplicity of theJones matrices with the ability to handle partiallypolarized light is that of the coherency matrix.910
Kim et al."1 showed that this approach can be appliedto the polarization of light transmitted through a ran-dom medium. They demonstrated that the Mueller
matrix for such a system corresponds to a statisticalensemble of Jones matrices. Their work was theoreti-cal, and they did not make direct application of theformalism to the interpretation of experimental mea-surements.
I show here how the coherency matrix formalism canbe applied easily to the interpretation of experimentalpolarization measurements. In Sec. II an algorithmfor determining the elements of the coherency matrixis derived, and formulas for the degree of polarization,the ellipticity of the polarized component, and theorientation of the major axis of the polarization ellipseare given. Finally, in Sec. III the utility of the methodis demonstrated by its application to the interpreta-tion of experimental data from light transmittedthrough a magnetooptic film garnet grown on a garnetsubstrate by liquid phase epitaxy.12
II. Determining the Coherency Matrix and Polarization
A. TheoryThe coherency matrix is given in terms of the electric
field E of the light by9' 10
Assuming perfect components (no reflection losses,exact quarterwave retardation, etc.) it is easy to deter-mine the Jones matrices associated with the above fourmeasurements:
L, =( I ),/ 0\
L2 =( 0 )(O
(9)
L3 = 2( 1),
21 1 i
L4 =(V )
The equations expressed by Eq. (8) have the solution
Ii = Y1,
i2 Y3- 2(Y + Y2),
3 = Y4 -/2(Yl + Y2),
14 = Y2-
(10)
J = (EEf = ((ExEy)(EXEy h
(EYE>* /
J is Hermitian, and it will be helpful to write it in theform
j il 2 +iia) (5)
i2 - hj 4G 5
where ji... ,14 are real.As in the case of the Stokes parameters, the coheren-
cy matrix has four independent parameters, and itrequires at least four nonequivalent measurements tofind them all. Traditionally, these four measure-ments are taken to be measurements of the irradianceof light transmitted through the following devices: (1)a polarization analyzer oriented with its axis of maxi-mum transmission at 00 (i.e., parallel to the x-axis); (2)the analyzer at 900; (3) the analyzer at 450; and (4) aquarterwave retarder with its slow axis at 90° followedby the analyzer oriented at 45°.
For the above measurements it is not difficult tosolve for the coherency matrix elements explicitly.The details are given in Goodman,9 whose notation Ifollow. The procedure is to represent each of thepolarization devices by its associated Jones matrix, Li,with i = 1, . . . ,4. The electric field transmittedthrough device i is
Ei = LiE. (6)
The coherency matrix J for the light transmittedthrough device i is found by substituting Eq. (6) intoEq. (4):
From a purely practical standpoint solving for Jinvolves some tedious algebra if either a different set ofmeasurements is made, or, more commonly, the polar-ization elements cannot be represented by the simplematrices of Eqs. (9). The latter situation often occurswhen the polarization elements do not perfectly con-form to the ideal extinction ratio, orientation angles, orretardation. For example, the retardation plate mayretard by exactly one quarterwave at a particularwavelength but not at others of interest. Also, in thegeneral case it is useful to incorporate reflection lossesof each element directly into its Jones matrix. In theseand similar conditions the evaluation of the four equa-tions represented by Eq. (8) becomes time-consumingand prone to error. It is useful, therefore, to solve for Jnumerically.
B. Numerical Evaluation of JTo evaluate J numerically we take advantage of the
fact that Eq. (8) represents equations that are linear inthe elements of J. Thus, if Eq. (8) is completelyexpanded, it may be written as follows:
Yi = I aik-k=1
In matrix form Eq. (11) becomes
Y = AJX,
(11)
(12)
where I have now defined the column vectors
Y = AY3,
\4
iX= LiJLt.
Irradiance yi of the transmitted light is given by
and where A is yet to be determined.Let us now define four new basis vectors, J(c) whose
elements are given by
(J(C)), = k. (15)
Just as the column vector of Eq. (14) corresponds tothe coherency matrix of Eq. (5), the basis vectors of Eq.(15) have corresponding coherency matrices given by
1 (O 0)
So= l + 14,
SI = j1l-4,(21)
S2 = 212,
S3 =2j3.
The irradiance is S0 and the polarization is given byEq. (3). The algebra involved in finding the ellipticityand major axis orientation is lengthy but straightfor-ward. The details are given in Born and Wolf,13 whodefine an auxiliary angle x that is found from
sin2x = S3 /So. (22)
Then the ratio of length b of the minor axis of theellipse to that of a of the major axis is given by
b/a = F tanx, (23)
where x > 0 corresponds to right-hand elliptical polar-ization. Orientation angle V/ of the major axis with
(16) respect to the x-axis is given by
tan2& = S2/SO. (24)
J4 =( 0 ).J4=(O I)
Now if we substitute each of the four basis coherencymatrices given by Eqs. (16) into Eq. (8) for the irra-diances for each of the four Jones matrices of theoptical system, we find sixteen values of y:
Yi = Tr(LiJkLt). (17)
As we did for Eq. (11), Eq. (17) may be written out as
4 4
Yik alJici = l ailek, = ai,. (18)1=1 1=1
Comparing Eqs. (17) and (18) shows
aik = Tr(LiJkLt). (19)
Having determined A, we find the coherency columnvector from Eq. (12):
J(c) = A-'Y. (20)
D. Recapitulation of the AlgorithmHere is a brief summary of the method of determin-
ing the coherency matrix and the geometric parame-ters that describe the polarization:
(1) Set up four separate configurations of polariza-tion devices characterized by Jones matricesL 1, . . . L4. The devices must include at least one re-tardation element, and the four Jones matrices mustbe linearly independent.
(2) Measure the irradiance of the light transmittedthrough each of these devices and express the result asthe column vector given by Eq. (13).
(3) Define the basis coherency matrices Jk for k =1, . . . ,4 given by Eq. (16).
(4) Construct the 4 X 4 matrix A whose elements are
(25)aik = Tr(LiJkLt).
(5) Evaluate the vector J(C) given by
J(c) = A-1Y
ilii
=i3
The coherency matrix is then given by Eq. (5).
C. Geometric InterpretationIn polarization measurements, and particularly
those relating to magnetooptic materials, the impor-tant quantities are the ellipticity and the rotation ofthe plane of polarization induced by the material. Inaddition, it is always useful to check for absorption andreflection losses by measuring the total irradiance andto monitor the degree of polarization to see if depolar-ization occurs. All the above parameters can be calcu-lated from the coherency matrix.
To find the degree of polarization and the ellipticityand orientation of the major axis of the polarizationellipse, it is useful to write the Stokes parameters interms of the elements of the coherency matrix. Com-paring Eqs. (2), (4), and (5) shows that
(26)
(6) Calculate the Stokes parameters given by Eq.(21).
(7) Evaluate P, b/a, and 6, using Eqs.(24).
(3) and (22)-
Ill. Application to Magnetooptic Garnet FilmsI applied the above analytical technique to deter-
mine accurately the Faraday rotation of light trans-mitted through two magnetooptic films grown by liq-uid phase epitaxy onto garnet substrates. Thecomposition of the films was14 (Bi1.08TmO.07Gd0.95-Yo.go)(Fe3.92Gao. 76Yo.3oTmo.02)012 . That of the garnet
REF. LOCK-INHINPUTFig. 1. Apparatus used in measuring the polarization of light. Thelaser diode emits at 780 nm, and the retarder is a quarterwave plateat this wavelength. For three measurements the retarder is re-moved and the second Glan prism analyzer is oriented with itstransmission axis at 00, 450, and 90°. For the fourth measurement
the retarder is inserted and the Glan prism is oriented at 45°.
substrate was (Gd2 .6 8Cao.3 2) (Ga4 .02Mgo.33Zro.65 )012.The approximate thicknesses of the thin film layerswere 3 and 30 Am, respectively.
Figure 1 shows the apparatus. The light source wasa GaAs laser diode emitting at 780 nm. A lens andaperture provided a well-collimated beam of -2-mmdiameter. The irradiance was measured using a com-bination mechanical chopper, PIN photodiode, andlock-in detector. Following the chopper a Glan-Thompson prism with its transmission axis verticalensured that the light was 100% linearly polarized.The sample under study was mounted in a coil whosemagnetic field was 16 kA/m (200 Oe) for a dc current of200 mA, the maximum current used. The analyzerwas a second Glan-Thompson prism, which was setwith its transmission axis at 00 (horizontally), at 900,and at 45°. For the fourth measurement the quarter-wave retarder shown in the diagram was inserted intothe light path in front of the analyzer, which was set formaximum transmission at 450 for that measurement.
To illustrate the calculations involved let us consid-er the experimental results for the 30-Am sample in afield of 16 kA/m. The signals for the four measure-ments were (1) analyzer at 00: 3.655 mV; (2) analyzerat 90°: 25.31 mV; (3) analyzer at 45°: 7.08 mV; and(4) X/4 retarder plus analyzer at 450: 14.16 mV.These measurements yielded values of i.... ,j4 in Eq.(26) of 3.85, 26.74, -7.83, and 0.61, respectively.Equations (23), (3), and (22)-(24) gave the values P =90.6%, b/a = 0.022, and V = 107.20. The value of &corresponds to a Faraday rotation of 17.20.
The Faraday rotation for the two samples is shownin Fig. 2 where the triangles correspond to the 3 -Amfilm sample and the circles to that of the 30-jim film.The straight lines are the best least-squares fits andhave slopes of 0.203 4 0.005 and 1.075 + 0.0030 /(kA/m), respectively.
MAGNETIC FIELD (kA/m)Fig. 2. Faraday rotation of the plane of polarization of the light.Triangles represent data from the Bi:YIG sample having 3-gm thickLPE surface films; circles correspond to the sample with 30-gm
films.
100 _
0
-
N
F-i
0
n
95 -
90 _
85 H
800 -10 0 10
MAGNETIC FIELD
20
(kA/m)Fig. 3. Degree of polarization of light transmitted through the
samples having 3-gm films (triangles) and 30-gm films (circles).
The polarization of the transmitted light is shown inFig. 3. The results indicated that the light transmit-ted by the 3-Am sample remained exactly 100% polar-ized at all values of the field. The optically denser 30-,m sample, by contrast, had a minimum polarizationof 84.3% in zero H-field and values of -90.7% at h16kA/m. Examination under a polarizing microscopeshowed that the domains of the 3-rim sample were -10,m wide; that is, the domains were substantially widerthan the film was thick. By contrast the domains ofthe 30-Mm film were found to be -20 m wide, thinnerthan the films. As the H-field increases, so does thewidth of the domains. I suspect that this domaingrowth may be related to the decrease in the depolar-ization from the sample.
Only a very slight ellipticity (b/a 0.02 for eachsample) was found. This value is small enough to be
attributable to experimental error; analysis indicatedthat an error of 1% in the value of the efficiency of theretarder combined with 1% in the signal could accountfor the measured value of ellipticity.
IV. Concluding RemarksThe coherency matrix approach to polarization
measurements has proved to be simple, accurate, andfast. Matrix A-' in Eq. (26) need be evaluated onlyonce, so that a single matrix multiplication yields thecomponents of the coherency matrix. The speed ofthe calculation is such that polarization measurementscould be made in real time. For example, the additionof three 50% nonpolarizing beamsplitters, three detec-tors, and three analyzers to the apparatus of Fig. 1would permit all four measurements to be made simul-taneously.
I would like to thank John B. Ings of the LittonIndustries Airtron Division for providing the samplesused in this study. I would also like to thank AlanMarchant of Verbatim for his support and help withthe microscope measurements, and James Colvin andNeil Wilber, who made the polarization measure-ments.
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