690
J. Opt.
Soc. Am.
B/Vol.
5, No.
3/March
1988
Theory of polarization and spatial information recovery by
modal dispersal and phase conjugation
Yasuo Tomita, Ram
Yahalom, and Amnon Yariv
Department of Applied Physics, 128-95, California Institute of Technology, Pasadena, California 91125
Received July 14, 1987; accepted November 16, 1987
A general theory of polarization and spatial information recovery by modal dispersal and phase conjugation is
presented by means of a coherency matrix formalism. The theory is applied to a system that consists of a
multimode modal-scrambling fiber terminated by a conventional phase-conjugate mirror that reflects only one
polarization component. The degree of polarization and the signal-to-noise ratio of the reconstructed field are
discussed
as a function of input-beam launching conditions. Some
experimental results are also shown for
comparison with the theory.
1.
INTRODUCTION
It is well known that when a polarized laser beam is launched
into a multimode modal-scrambling fiber, the initially cou-
pled fiber modes are scrambled among all the fiber modes,
including those of orthogonal polarization by strong inter-
modal coupling, and consequently the output beam from the
fiber shows speckled spatial structures and depolarization.'
These types of beam aberration, i.e., wave-front distortion
and polarization scrambling, can be corrected if the aberrat-
ed fields including both orthogonal polarizations are phase
conjugated.
2
-8 However, it
was found recently that
even
when only one polarization component of the forward-trav-
eling beam from the fiber was reflected by an ordinary
phase-conjugate mirror (PCM) and fed back into the fiber,
the resultant beam, emerging from the input plane of the
fiber, could be a phase-conjugate replica of the original input
beam including its original polarization state, i.e., this would
be true phase conjugation of vector wave fronts.
9
Since this first observation in such a fiber-coupled phase-
conjugate mirror (FCPCM), a theoretical model'
0
was pro-
posed, and a number of new applications, including correc-
tion of nonreciprocal distortions," correction of lossy ampli-
tude distortions,1
2
temporal data channeling between
beams,'
3
and all-optical beam thresholding,1
4
have been re-
ported. In addition, the fidelity of the true phase conjuga-
tion using the FCPCM was also studied,
5
and it was found
that the fidelity was strongly dependent on launching condi-
tions of an input beam [i.e., input-beam numerical apertures
(N.A.'s)]: if an input-beam N.A. is close to the fiber's N.A.,
then the reflected beam within such an input-beam N.A.
carries nearly the same amount of the noise power as that of
the true phase-conjugate beam, resulting in the degradation
of polarization and spatial information recovery. Although
this input-beam N.A. dependence was explained by a phe-
nomenological model'
5
and its asymptotic case (i.e., a case of
a large N.A. input) was also reported,1
6
a detailed analysis
has not been given so far.
In this paper we present a general theoretical description
of polarization and spatial information recovery using the
FCPCM by means of a coherency matrix formalism. The
treatment involves the analysis of the polarization state of
the field transmitted through the fiber and the dependence
of the polarization recovery and the signal-to-noise ratio
(SNR) of the phase-conjugate field on the input-beam N.A.
The effects of the modal-scrambling property of the fiber
and the fidelity of phase conjugation by the PCM on the
polarization recovery are also discussed. The physical pro-
cesses considered in the present analysis are twofold: (1) a
(time-reversed) phase-conjugation process, which is deter-
ministic in nature; and (2) a scattering process in the fiber,
which results from partial phase conjugation of the mode-
scrambled field and is seemingly
completely random (or
stochastic) but is in fact constrained by the unitarity condi-
tion of the scattering matrix (i.e., the energy-conservation
condition). In this case we take the coupling strength in this
scattering process to be essentially the same among all the
fiber guided modes but its relative phases to be random
under the constraint of the unitarity condition. In the anal-
ysis of the polarization properties of the phase-conjugate
field, unlike in the treatment of the Jones calculus for ran-
dom media,1
7
we do not resort to a statistical ensemble
average over the coherency matrix elements but use the
modal averaging'
0
over phase-mismatched fields in this
phase-conjugation process, which may be analogous to the
phase-matching condition in the coupled-mode theory.18
The main reason for our approach is that one usually treats
only a single fiber in phase-conjugation experiments. In the
SNR treatment, however, we simplify the analysis by using
an a priori knowledge of the statistical properties of a
(speckle) noise field instead of considering statistical prop-
erties of the scattering matrix and its relation to the proper-
ties of the noise field. This is done by assuming a probabili-
ty-density function of the outcoupled noise field in the free
space, and the SNR can then be obtained from the root-
mean-square (rms) value of (statistical) intensity fluctua-
tions of the noise field.
With the above treatment the present analysis enables us
to evaluate theoretically the fidelity of polarization and spa-
tial information recovery by using the FCPCM and to give a
0740-3224/88/030690-11$02.00 Ā© 1988 Optical Society of America
Tomita et al.
Vol. 5, No. 3/March 1988/J. Opt. Soc. Am. B 691
criterion for the limitation of the use of the FCPCM. Some
experimental results are also shown for comparison with the
theory.
2. BASIC FORMULATION USING
SCATTERING MATRICES
Figure 1 shows a schematic of the FCPCM.
An image-
bearing incident field E) is launched into a multimode
modal-scrambling fiber, which is assumed to be linear with
negligible loss. Because of the strong intermodal coupling
in the fiber, the input power initially coupled into any one
fiber guided mode is distributed essentially uniformly
among all the other spatial and polarization modes during
propagation, and the outcoupled beam E(2) from the fiber
exhibits speckled spatial structures and nearly complete de-
polarization. The PCM, e.g., a self-pumped PCM,1
9
is
placed after a polarizer (set to the x direction) and phase
conjugates only the x component of the field E(2). The
phase-conjugate field E(3) retraces the original path and is
launched into the output side of the fiber. After the propa-
gation and the strong intermodal coupling in the fiber, the
left-traveling field forms the output field E(
4
) at the input
end of the fiber.
By using the same notation as in Ref. 10, the input field
E(1) is expressed, in terms of the fiber guided modes, as
N
E(1)= [a(')ex. + ae,.]
n=1
[A'(l)](
)
where N is the total number of the fiber guided modes in one
polarization; exn is the nth transverse fiber guided mode,
which is predominantly x polarized; eyn is the nth y-polar-
ized mode; and A(') and A(') are column vectors of rank N
whose elements are the complex amplitudes an and a,
respectively. Note that we neglect the coupling into other
possible fiber modes (e.g., leaky and radiation modes) for
simplicity of the analysis.
The output field E(4)
is expressed as
E(4) = rM'CM*[E(l)]*, (2)
where r is the PCM amplitude reflectivity, M is the scatter-
ing matrix of the
fiber in the forward direction given by
Lmyx
Myy_
in which Mij (i, j = x, y) are N X Nsubmatrices, and M' is the
scattering matrix in the backward direction. In addition,
the matrix C, representing the removal of the y polarization
by the polarizer, is given by
C =[I ]
(4)
where I is an N X N unit matrix. We note that a mode-
independent (scalar) reflectivity ofthe PCM is assumed in
Eq. (2). If a mode-dependent reflectivity is taken into ac-
count, r should be replaced by a 2N X 2N matrix. We
discuss this effect in Section 4. In what follows we examine
Multimode Fiber
Polarizer PCM
E (2) E 2)
-->3 ) Z Ht(1 D
x
, KZ
ye z
Fig. 1. Schematic of the FCPCM for polarization and spatial infor-
mation recovery. The (polarization- and modal-scrambling) multi-
mode fiber is assumed to be linear with negligible loss.
the properties of the scattering matrices and express the
fields E(2) and E(
4
) in terms of the
scattering matrix ele-
ments.
Because of the conservation of the energy in a lossless
linear fiber, we require the following unitarity condition
10
:
MtM =[I 0], (5)
where t denotes the Hermite transpose operation. By using
Eq. (3), Eq. (5) can be translated into the following sum
rules:
(Mxx)ik(Mxx),k' + (Myx)ik(Myx)k'= 0
(Myy)ik(Myy)k' + (Mxy)ik(Mxy)lk' = 0
(Mxy)ik(M..)i'k'+ (MYY)ik(Myx)i k'= 0
(MXX)ik(Mxy)iko + (Myx)ik(Myy)k' = 0,
(6a)
(6b)
(6c)
(6d)
and
(MXX)ki(MXX)ki + (My)ki(Mxy)ki = kk
(Myy)ki(Myy)k'i + (Myx)ki(Myx)k'i = 0
(My.)ki(M.dk'i + (MYY)ki(Mxy)ki = 0,
(MXX)ki(Myx)k'i + (Mxy)ki(Myy)*'i = 0,
(7a)
(7b)
(7c)
(7d)
where
henceforth summation over repeated indices is under-
stood.
For the time-reversal symmetry of
any fields in a lossless
linear fiber we also require that
9
"10
M'M* = ]0.
From Eqs. (5) and (8) we obtain
M = Mt,
(8)
(9)
where t denotes the transpose operation. By using the sub-
matrices given in Eq. (3), Eq. (9) can be rewritten as
(MXX)ij =(MXx)ji'
(Myy)ij = (Myy)ji,
(M'y)j
= (My)jL.
(M ) = (M )i
(lOa)
(lOb)
(lOc)
(lOd)
Here we note that the elements of the scattering matrices are
interrelated by the constraint given by Eqs. (6), (7), and (10).
Tomita et al.
692 J. Opt. Soc. Am. B/Vol. 5, No. 3/March 1988
3. SPATIAL AND POLARIZATION
PROPERTIES OF THE FIELD E(
2
)
With the relation E(2) = ME(), we express the correlations
between the 2N modes of the
field E(
2
) by means of the
following 2N X 2N Hermitian coherency matrix:
L(2)- (E(2)E(2)t)
[L(
2
) L(2)1
- = x~~~~x xy b (1
M(E(1)E(1) )M I = L(2)t L(2)]
LXY YY
where (. . .) denotes the time average
and LM (i, j x, y) are
N X N matrices given by
L(') = MXXL'x)Mx + MXXL)MtY
+ MXYL)
t
MX + MXYLyJMxY, (12a)
L(2) = M XL()Mt
+ MYXLJ)MtY
+ LYYI4?M + M (12b)
L(') = M QL~)At + M L(l)AMt
+ MXYLI)
t
My + MXYL(I)MtY,
(12c)
in which L() (EME(1)t) denotes the correlations between
the 2N modes of the input field EM
1
). We note that the effect
of a possible decrease of the temporal coherence of the light
source at the output, which is due to the modal dispersion in
the fiber,
20
is not taken into account in the present analysis.
We now introduce the following modified 2 X 2 coherency
matrix:
j()-Ej1) j
2
)
c2)[ Xj(
2
)* j
2
) I
xy jyy
Here each element J(j2) (i, j = x, y) is given by
N N
j(j2) = J [L keike*jldxdy
k=1 1=1
N
= (const.) X [L,()]k
k=1
= (const.) X Tr[L(?)].
(13a)
(L1j,= (M4 )kI
2
[L
yy = {yx l Xx)kk + (Myx)ik(Myx)ik'(Lx kk'
(k= M kk'
[Lx)i =Y (Mxx~k(MyxAik'[Lxx kk'-
(14b)
(14c)
As was mentioned in Section 1, we assume that, because of
the strong intermodal coupling in the fiber, the amplitudes
of the matrix elements Mij, i.e., the coupling strength be-
tween modes, are essentially the same (or symmetrically and
widely distributed with respect to the diagonal elements
Mii), while their relative phases are distributed essentially
uniformly over the -7r - +7r interval (henceforth we refer to
this as the random coupling approximation; see Appendix
A). Then we see from Eqs. (14) that the input power initial-
ly coupled into any one fiber guided mode is redistributed
among all the other fiber guided modes, including those of
the orthogonal y polarization during propagation. In addi-
tion, the out-coupled different spatial modes possessing ran-
dom phases interfere with one another at any point, result-
ing in the speckled spatial structures in the free space.
The polarization state of the field E(
2
) can be obtained by
using J(
2
). From Eqs. (13) and (14) we have
JX
2
X = akk[Lv.)]kk + akk[Lxx] kk'
(k k')
J(y
2
y) = (6kk - akk)[Lxx)]kk
- akk'[Lx(x] kk'
(kk)
J
2
) = bkk[LT] kk
(15a)
(15b)
(15c)
where we used the sum rules given in Eqs. (6) and introduced
the following parameters:
(16a)
and
bkkE-(Mxx)ik(Myx)ik.
(13b)
(16b)
Here we note that the terms akk, and bkk' in Eqs. (16) are
(k 6k')
much smaller than akk because of the modal averaging in the
random-coupling approximation. The Stokes parameters
(So, S
1
, S
2
, S
3
) and the degree of polarization [(
2
)] (Ref. 22) of
the field E(
2
) are then given by
In Eq. (13b) Tr denotes a trace of a matrix, and we have used
the orthogonality of the fiber modes,
21
i.e., ff, eime*jndxdy =
(constant) X amn (i, j = x, y; m, n = 1,... , N), where
denotes the whole fiber cross section and a circular fiber is
assumed, so that we can neglect the contributions of the off-
diagonal elements of LX2), L2) , and L(2) to J(2) on
the detec-
tion of the field E02) over o-. We note that, unlike in the usual
definition of the coherency matrix,
22
the elements of J(
2
)
have the dimensionality of power [hereafter, however, we
shall omit the constant in Eq. (13b) for brevity].
For the sake of simplicity we consider the x-polarized
input here. Then, with L(') = L(') = 0, we can write the
diagonal elements in Eqs. (12) as follows:
[Lxx)] = (Mxx)ik(Mxx)ik'[Lx(x]kk'
|(Mxx)ikl'[L()]kk + (Mxx)ik(Mxx)ik'[L(1}]
lk'
(k#k')
N
so JMx + J = [L(1)]kh
k=l
-J Jx2x)- J () = 2ak[L(1)]k + 2akk'[L'xx)] kk'
-SO
Y ~ ~~~~~~~~(k Sk'
S2-2y) + y) = Rtk'[L(1)1kk'),
i[J-(2x) -Jxy)] = 2 Imjbkk,[Lxx]kk'1
and
p(
2
)
(S 2 + s + )1/2
So
(17a)
(17b)
(17c)
(17d)
(18)
By using the random-coupling approximation, i.e., akk 0.5
for any k, p(
2
) can be reduced to
p(
2
) - (
2
+ iU12)l/
2
!
(19)
(14a) where
Tomita et al.
akk'ī (Mxx)ik(Mxx)i*k'
Vol. 5, No. 3/March
1988/J. Opt.
Soc. Am. B
693
Table 1. Experimental Data of the Stokes
Parameters
and the Degree of
Polarization of the
Field E(
2
)a
Input-Beam
N.A. Si/so S2/SO
s3/so p(
2
)
0.02
0.003 0.008 -0.016 0.018
0.11
0.033 0.023
-0.028 0.049
0.25
0.004 0.041 -0.021 0.046
In the experiment a multimode (N.A.fiber = 0.29; 5 m long)
graded-index
fiber and a linearly
polarized input (X = 5145 A) were
used.
2akk4[Lxx]
kk'
q
(k k')
(20a)
so
and
2bkk'[L(x)]kk'
U =.
- .
(20b)
sO
We thus find
that q and u, which are expressed by akh'
(k k)
and bkk', respectively, are responsible for the
residual polar-
ization of the field E(
2
). Table 1
shows the experimental
data of
the Stokes parameters and the degree of polarization
of the field E(
2
) for the different values of input-beam
N.A.'s.
It is seen that sl,
s2, and S3 are much smaller than so; there-
fore
the degree of polarization p(
2
) is much smaller
than
unity, i.e., Iql and lul are much
smaller than unity, and conse-
quently the field E(
2
) is almost
completely depolarized, inde-
pendently of the input-beam N.A.'s. These data
clarify the
validity
of the random-coupling approximation and the
as-
sumption of the
modal averaging by which the cross terms
akk, and bkk'
are much smaller than unity. We note that
(k k')
the parameter
q will play an important role in the fidelity
of
the reconstruction of the
original information, as we discuss
in Section 4.
=j
{(kxx)ki(A4xx)kj
(I =d
I)
D 0i - lĀ° (i = I)
= -(Myy)ki(Myy)kj
(i i D
Qi= (Mxx)ki(Mxy)k;.
Then the field E(
4
) given by Eq. (21)
becomes
E(4) = l/
2
r[E(l)]* + V,
(23a)
(23b)
(23c)
(24)
where V = rS
2
[E(O)1*. The first term
on the right-hand
side
of Eq.
(24) corresponds to the true phase-conjugate replica
of
the input field
E(l, while the
second term
corresponds to
the
noise-possessing
random phases
in the field
E(
4
).
Figure 2 shows a diagrammatic
explanation of the forma-
tion of the
field E(
4
). The
ith fiber guided
mode of the x
polarization that is excited initially at the input
plane of the
fiber is coupled into all the fiber guided modes
at the output
plane
in the forward direction. After the elimination of
the
y-polarized component
and phase conjugation of the x-po-
larized component,
each mode at the output plane is, again,
coupled into all the fiber
guided modes at the input plane in
the backward
direction. In Fig. 2(a)
the (time-reversed)
paths in the backward direction are
deterministic and are
exactly the same as those in the forward direction,
resulting
in a
constructive coherent superposition of the scattered
fields at each
mode at the input plane. Because of the
constructive interference
this true phase-conjugate field,
corresponding
to the term l/
2
r[E(l]* in
Eq. (24), has almost
one half of the total reflected power.
[Note the factor 1/2 in
Eq. (24) and remember that almost one half of
the power of
the field E(2) is eliminated
by the polarizer
and the remain-
input
plane
output plane
4. SPATIAL AND
POLARIZATION
PROPERTIES
OF THE FIELD E(
4
)
In Section
3 we showed that the field E(
2
) suffers spatial
distortions and nearly complete
depolarization because of
the strong intermodal coupling
in the fiber. In this section
we show that such a distorted and depolarized
field can be
corrected,
under certain conditions, even when only
one
polarization component of the field E(
2
) is phase
conjugated.
First we rewrite Eq. (2) as
E(4) = rS[E(i)]*,
(21)
where the scattering matrix S in the
round-trip propagation
is given by S = M'CM*. Here we again use
the random-
coupling
approximation, i.e., Yi 1 (Mxx) ik12, ZN
1
(Mx)ikI2
0.5. Then S = S
1
+ S
2
, where
2s
[0
]
and
S2 = [Qt D,
in which D, D', and Q are N X N
submatrices given by
(22a)
x-polarized
modes
y-polarized
modes
(a)
ii
I -- ~~~~~~
I - I -. -
I
- - --
n
I I(b
(22b)
Fig. 2. Diagrammatic
description
of the formation
of the field
E(
4)
:
(a) deterministic
phase-conjugate paths that result in true phase
conjugation of the input field E); (b) randomly
scattered phase-
conjugate
paths that
result in the
noise.
Tomita et al.
II
k9
i
I
I
694
J. Opt.
Soc.
Am. B/Vol.
5,
No. 3/March
1988
der is reflected
by the PCM.]
On the other
hand, in Fig.
2(b)
the remainder
of the
paths in the backward
direction
are
random and
different
from those
in the forward
direction,
and
therefore
because of
the random
interference
at each
mode
at the input
plane these field
components form
the
noise
V given
in Eq. (24).
We will
see below
that the
total
power
of this
noise is nearly
the same
as those of
the true
phase-conjugate
field,
but
it is distributed
essentially
uni-
formly among
all the fiber
guided modes,
independently
of
the input-beam
N.A.'s.
For this reason
the noise power
per
mode
is much
smaller than
that
of the true
phase-conjugate
field,
provided
that the
input field
initially
excites
only a
small fraction of the
fiber guided modes
(i.e., a small input-
beam
N.A. is used) and
that the detection
is made within
such a small
input-beam N.A.
In this case
we can actually
neglect
such noise
contributions
designated
by V
in Eq. (24),
and the field E(4)
can be the true phase-conjugate
replica of
the input field El.
The correlations
between the 2N modes
of the field E(
4
)
can also be expressed
by means
of the following
2N X
2N
Hermitian coherency
matrix:
L (4)- (E (4)E4
M)
= Irl2SL(l)*St
= IrIĀ¶1
4
L(l)*
+ S
2
L(I)*S
2
t + 1/2[S
2
L(l)*
+ L(l)*S
2
t]1.
(25)
In the right-hand
side of the
last equation above,
the first
term corresponds
to a time-reversed polarization state
of the
input field
El, while the rest
of the terms correspond
to the
noise.
Since
the expression
given by
Eq. (25) is
linear in L(l),
it is sufficient
to consider
the case
of the x-polarized
inci-
dence for simplicity. In this
case we can express the noise
terms in Eq.
(25), in terms of the
submatrices D and
Q given
by Eqs. (23), as
S
2
L(l)*S
2
t = [DL)Dt
DLxx) 1
QtL(1)*Q,
DL('*
+ L('*Dt
[
2
L(l)*
+ L(l)*S
2
t]
-L Qf~
2
2
tI)
L()*Q]
O j
Each diagonal
element in
Eq. (26a) can be
rewritten as
[DL(')*Dt]ii
= ID
l
2
[L(lJ
1
% +
DD*,[L(1)I*
1u, (27a)
[QtL(l)*Q]ii IQ I
2
[L(1x]%+ Q*iQ
1
,[L()]*
Ij, (27b)
(1i
1')
[DL(')*Q]ii
= DiQlti[L(1)1*
(27c)
where the
summation over and
1' is understood. In the
above expressions Eq. (27a) corresponds to the
noise power
of the x-polarized ith fiber
guided mode of the field E
This consists of
the interference between the other initial
modes that are finally
coupled into the x-polarized
ith fiber
guided mode through different
scattering paths after the
round-trip
propagation. Likewise Eq.
(27b) corresponds to
the y-polarized noise power of the
ith fiber guided mode.
Each diagonal
element in Eq. (26b) can also be rewritten
as
[DL(')* + L(1)*Dt]ii = 2 ReJDiJL(1)]*J,
[L(x)*Q] = QuJ [L()]
1
(28a)
(28b)
where the summation
over I is again understood.
Equation
(28a)
corresponds
to the interference
between the
true
phase-conjugate
field and
the noise field
at the
ith fiber
guided
mode of the x
polarization. This
term is related to
the
residual polarization
of the field E(
2
).
Note that Da = ail
for i 0 1 and therefore
the total power
of this noise contribu-
tion, Y ,[DL')*
+ L(Z)*Dtlii/2,
is equal
to qso/2;
see Eqs.
(16a),
(20a), and (23a).}
We also note
that this noise
is
distributed
only inside the input-beam
modal distribution
[Q)]ii, i.e.,
this noise is
x polarized.
In
order to
estimate the
ratio of
this noise
power to
the
true phase-conjugate
beam
power per
mode, we consider
the
simple form
of the scattering
matrix elements,
Mij =
1N
exp i(k0).1
0
Then,
by inserting this form into
Eqs. (27), we
immediately
see that
[DL(1)*Dtii and
[QtL2*)*Q]ii
are of the
order of
so/N independently of the
mode number i, where the
total
input power
s
0
is given
by Eq. (17a).
It is therefore
seen that the
x- and y-polarized
noise powers
given by
Eqs.
(27) do not
differ from
each other significantly
at any
ith
mode, so that the noise
power of the field E(
4
) is almost
essentially uniformly
distributed among all the fiber
guided
modes, independently
of the input-beam
N.A., i.e., of the
distribution
of [L(1)]ii.
We will see
below that this
noise is
nearly
completely
depolarized. The
ratio of the noise
power
to the true
phase-conjugate
beam power
per mode is
of the
order of Mo/N (Mo
is the number of the fiber guided
modes
that are excited initially).
This ratio can be negligibly
small
when MO/N
<< 1, justifying the fact that the
noise V in Eq.
(24)
can be neglected
for small N.A.
inputs. In addition,
the
noise given by Eq. (28a) is of the
order of IqisO/Mo, where q is
given by Eq. (20a). The
ratio of this noise power to the true
phase-conjugate
beam power
per mode is of the
order of qi,
which
is also much smaller
than unity and
is independent of
the input-beam
N.A.'s. Although this noise is x polarized,
its spatial structure is distorted
because of the random
phases. Therefore
we refer to this as
the polarized noise.
We are now in
a position to evaluate quantitatively the
polarization
recovery of the input field E(
1
M as a function
of
input-beam N.A.'s.
Suppose that the
whole power of the
field E(
4
) is detected. Then, according
to the definition of
the coherency
matrix [see Eqs. (13)], the polarization state
of
the field
E(
4
) is expressed by means of the following 2
X 2
coherency matrix of
the field E(
4
):
4) =J(x4x) X4
jl
4
)* 1j4)l
L
j yyJ
= /
4
1IrI
2
l+ Jn$oise,
where
(30a)
and
,j(
4
) = fTr[L ) - /irl2so Tr[Lx(y)
noise
Tr[L 4)] Tr[L4)]J
(30b)
(29)
Tomita
et al.
,1(1) = 0
0-