OPTICS LETTERS / Vol. 14, No. 10 / May 15,1989. Influence of the input state of polarization on interferometric and polarimetric measurements of birefringence.
OPTICS LETTERS / Vol. 14, No. 10 / May 15,1989
476
Influence of the input state of polarization on interferometric
and polarimetric measurements of birefringence T. Tambosso and S. Donati Department of Electronics, University of Pavia,27100Pavia, Italy Received June 10, 1988; accepted February 24, 1989 Using a general model to schematize polarimetric and interferometric
measurements,
we derive the dependence of
the output signal on the input state of polarization. We show that in any birefringent medium, in addition to the eigenvectors that maximize the interferometric phase signal, there exist dual vectors that maximize the amplitude of the polarimetric signal. The dual vectors lie on the maximum circle normal to the eigenvector axis.
The evolution of the state of polarization (SOP) is an important issue for coherent communications and optical fiber sensors. In interferometric sensors and coherent receivers, fluctuations in the output SOP induce the so-called polarization fading of the photodetected signal,'-
4
for which polarization
controllers 2'4
and polarization diversity have been devised. In polarimetric fiber sensors5 fluctuations of the SOP directly reflect themselves as a measurement error whose effect has not yet been analyzed in detail.
Also,
it is well known that interferometric and polarimetric measurements are complementary because they involve the phase and the amplitude, respectively, of the signal.
As shown below, for those input SOP's maxi-
mizing the information in one measurement, the other measurement carries no information. The aim of this Letter is to generalize the analysis of interferometric and polarimetric measurements in birefringent media so as to find which SOP is best suited to perform the measurement and which error is eventually generated. First, let us model the interferometric measurement as in Fig. 1(A), where the input SOP is described
by the Jones vector Va. After propagation the output vector Vu is combined onto the photodetector with the input delayed by an arbitrary phase 0o. The current signal at the photodetector output is then
With Al and A 2 normalized (IAI = = 0, we have IR112 +
IA21 =
1) and A, - A2
2 IR21 = 1.
(4)
Now let us write the output vector Vuas Vu = J *Vi,
(5)
where J is the Jones matrix of the medium. By inserting Eq. (3) into Eq. (5), since J *Al = XAj and J *A2 = X2 A2 for the eigenvectors and X, = X2* = exp(iAz) forthe eigenvalues XI and X2 of a Jones matrix J of a
nondissipative medium (with the determinant equal to unity), we get easily for Vu Vu = RAj exp(iAz) + R2A2 exp(-iAz),
(6)
where z is the path length and A is the birefringence coefficient of the medium. [Specifically, A = f1/2 for a linear birefringence, A = 03/2 for a circular birefringence, A = (1/2)(f12 + 032)1/2 for a coexisting linear and
circular birefringence, etc.] Combining Eqs. (2), (3), and (6), one has P = [cos Az + i(1R212
-IR,1
2
)sin Az]exp(i
0o),
and the interferometric signal therefore reads Re P = cos Az cos 00 - (IR 2 12
-
IR,12 )sin Az sin 0 0 . (8)
i ol vi exp(ifo) + VUI2 = IV,12 +
(7)
2
lVu1 + 2 Re[Vi exp(iko)Vu*].
(1)
The interferometric signal, contained in the last term of relation (1), is the real part of the scalar product P between the input and output SOP's, P = Vi exp(iko) - Vu*
e
(A) (A)
Propagation 1
Mediumei o
i
Vieo
(2)
Now we explicitly define Vi in terms of the eigenvec-
tors (or polarization modes) of the medium Al and A2 as Vi = R Aj + R2 A 2 ,
(3)
where the coefficients RI and R2 are complex numbers (given by the scalar product of Vi with Al and A2 ). 0146-9592/89/100476-03$2.00/0
(B)
Fig. 1.
Models of (A) interferometric
measurements. BS, beam splitter. © 1989 Optical Society of America
and (B) polarimetric
May 15,1989 / Vol. 14, No. 10 / OPTICS LETTERS
477
To discuss the dependence of the interferometric signal from the input SOP, we first note that if one of the eigenvectors of the medium is chosen as the input (R 1 = 1 and R 2
=
0 or vice versa) the result is
Re P = cos(0 0 t Az), as expected for a measurement
(9)
of a phase shift LAz
with an added reference phase po. For all other input SOP's (R1 and R 2 different from zero) the signal is smaller than the above value, and in the case of IR11 = IR 21 it becomes
Re P = cos Az cos ko,
(10)
i.e., it is modulated in amplitude by the bias term cos 0 0. This is the extreme case of maximum fading of polarization. 1' 4 Alternatively, Eqs. (9) and (10) can be
interpreted in terms of phase- and amplitude-modu-
Fig. 2. Representation
of eigenvectors Al and A 2 and dual
vectors (circle D).
lated signals. This is easily seen by letting /o = wt, so
that the above results are also extended to the case of a heterodyne detection scheme. If an eigenvector Al or A2 is used as the input, the beating (or interferometric) signal given by Eq. (9) contains the birefringence information Az as a phase-modulated term of the car-
rier wt, and there is no information in amplitude. The reverse is true if the SOP's with !R11 = 1R 2 1 are used as input: the information on Az is contained in the amplitude-modulation
term cos Az of the carrier cos wt,
and there is no information in phase. We call these SOP's, obtained by equal weighting (IR11 = IR2 1) of the eigenvectors Al and A 2 , the dual vectors of the eigenvectors. On the Poincar6 sphere the dual vectors lie on the maximum circle normal to the axis connecting the eigenvectors. is described by a rotation 6-8
Since the propagation
of the Poincar6 sphere around the eigenvector axis, an input dual vector will change its SOP on propagation but still be a dual vector [this can also be seen by
comparing Eqs. (3) and (6)]. The intersections of the dual-vector circle and of the Poincar6 sphere equatorial circle give the two linearly polarized dual vectors of the medium. If 2 bA and 2 TA are the coordinates
(inclination and ellipticity) of the eigenvectors, those of the two linear dual vectors are 24)= i7r/2 + 2 4 A and 2-I' = 0. For a medium presenting a distributed linear
(All)and circular (0i,)birefringence, in which the rotation velocity7' 8 of the Poincar6 sphere is Q = Q1 + Qc with 12I =
(f1
2
+ 132)l/2,
Fig. 3. Geometrical interpretation of the scalar product P and of weights R 1 and R 2 -
we have IPI = 1 and 0 = ±Az + 0o, while for the dual vectors (IR11= IR21 = 1/42) we have IPI= CosAz and k =
'00. On the Poincar6 sphere the modulus JP1of the scalar product has a simple geometrical interpretation related to the angle ,Abetween the input and output SOP vectors (Fig. 3). It can be shown9 that 1PI = vi
we show in Fig. 2 the eigenvec-
vv>*I = cosM/2.
(14)
D).
The projection of Vi on the eigenstates axis intercepts two segments AXCand A2 C that are proportional to
2.
nected to jR1 12and IR2 12through Eq. (13) but has no direct geometrical interpretation on the Poincar6
tors Al and A2 and the locus of the dual vectors (circle Here the angular coordinates 6' 7 of mode A2 are 2=A 00 (inclination) and 2 'IfA = 2 arctan a (ellipticity), where a = (0,/2)/(G1/2+ A) and A = (012 + 032)1/2/
Further insight can be obtained by expressing the scalar product P [Eq. (2)] in the form P = IPAexp(iW),
(11)
so that from Eq. (7) one finds that 2 JP1 =
=
1-
arctan[(IR
2 2 2 41R 1 11R 2 1 sin 2 12 -IR,
2
1
Az,
(12)
)tan Az] + 1O.
(13)
For the eigenvectors (R1 = 0 and R2
=
1 or vice versa)
IR112 and
IR212 (Ref. 10) (Fig. 3).
The phase
0 is con-
sphere, because o/z is in general dependent on z.
Both the modulus and the phase of the scalar product P depend in general on the phase shift variation Az [Eqs. (12) and (13)]. These two quantities have the meaning of the information content carried by the amplitude and phase, respectively. We may maximize the information contained in the phase by requiring that the modulus is independent of Az or, equiva-
lently, that alPl/aAz = O.
(15)
478
OPTICS LETTERS / Vol. 14, No. 10 / May 15, 1989
By inserting Eq. (12) into Eq. (15) we obtain as a condition 1R just by the 11 1R 21 = 0, which is satisfied eigenvectors; conversely, if we maximize the information contained in the amplitude by letting aolqAz = 0,
(16)
IR11 =
IR2 1, which is the one
we obtain the condition
defining the dual vectors. Last, we note that the above analysis is readily extended to the case of birefringence in both arms of the interferometer.
For this case we can go back 4 to that
of Fig. 1(A)by considering J *Jr-l as the Jones matrix of the equivalent medium, where JV' is the inverse of the Jones matrix of the reference-arm birefringence. The general scheme of a polarimetric measurement can be modeled as in Fig. 1(B). Here Vi and Vu are the
Jones vectors at the input and output of the birefringent medium, Vu may be modified in Vu' by means of a
suitable rotator and retarder combination, and V>' is analyzed in its components Vu/' and VJy'with respect to axes S1 and S2.
Conceptually the polarimetric signal is maximized by applying a rotation and a retardance such that the input vector Vi is brought to coincide with the Si axis and the output vector Vu' is brought to lie on the equator of the Poincare sphere (S3 = 0). In view of the equivalence theorem,6 this is always possible and physically realizable. Since any rotation of the Poincare sphere leaves the distances unchanged, 6' 8 we have
(17)
VUSi = VuVi =,
where , is obtained by the differential equation of polarization evolution, 6' 8 i.e.,
(18)
and the angle QV can be related to longitudes IQ and 4 'v and latitudes Tg and Tv through spherical trigonometry as 2
(xI' -ITV).
(19)
Limiting ourselves to the case of constant Q,so that V rotates around Q with a constant angle QV, and in general letting
11Q = 2A,8 we have A = 2Az sin QVi.
(20)
Since the output Vu' has coordinates D = , and T = 0, it can be written as the Jones vector, V
[cos u/2/ =cos
[sin
and TI's,as A, = A[1
-
COS2 2(4Q9-
2 1,)COS 2(
A'z
(21)
c2/2 sin ci[A'z
where the birefringence coefficient A' is defined ex-
-
i)]1/2.
(22)
As special cases of Eqs. (21) and (22), we consider the eigenvectors for which Vi 11 .,A' = 0, and no
polarimetric signal is developed, and the dual vectors for which Vi I Q,A' = A, and the output vector has the components cos Az and sin Az.
Another useful expression can be obtained in terms of the weight coefficients R, and R 2 by writing Vi as in
Eq. (3) and Vu as in Eq. (6). From the projection of the input vector on the eigenvector axis Q the two A1C = 21R212 and A 2 C = 21R112 are found, which allow us to evaluate sin Q V as (1 - cos2 QV)1/2= 2 1 2 [1 - (1 - 1AC) ] / = 21R 1 I IR21. Therefore we can segments
rewrite Eq. (22) in the form (23)
A' = 2AIR 1 1 IR2 1-
It is interesting to note that, with the rotator and retarder compensation, the output signal Vu' in a po-
larimetric measurement can always be brought to the form of cos and sin components of A'z. The reduction factor A'/A = 2IR 21 IR 11is independent of the argument of R 2 and R, or, equivalently, from the longitude of V
with respect to Q;A'/A is unity for the dual vectors and decreases as it moves on the Poincar6 sphere toward the eigenvectors, for which it is zero.
In conclusion, we have derived expressions regarding interferometric and polarimetric measurements in terms of input SOP's. We have shown that the dual vectors maximize the polarimetric amplitude and give a zero output for the interferometric
du = 1QX Vldz = Q sin QV dz,
cos QV = cos 2 ( 4 Q-bv)cos
plicitly, in terms of input vector angular coordinates cib
output, as op-
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tum Electron. QE-18, 626 (1982). 6. P. S. Theocaris and E. E. Gdoutos, Matrix Theory of Photoelasticity (Springer-Verlag, New York, 1979).
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