A new expansion for the Jones matrix of a transmission medium is used to describe high-order polarisation dispersion. Each term in the expansion is characterised by a pair of principal states and the corresponding dispersion parameters. With these descriptors, a new expression for pulse deformation is derived and confirmed by simulation

Eyal, A.

Marshall, W. K.

Tur, M.

Yariv, A.

English

Caltech Authors - Main

IC, and then to a PC for display. Codes were reliably read for 
image distances up to 120 mm, and work is continuing on enhanc- 
ing the scan length and working distance. 
Acknowledgment: The support of EPSRC, decoder construction by 
C.-L. Chew, and precision machining by P. Jones are gratefully 
acknowledged. 
0 IEE 1999 
Electronics Letters Online No: 19991139 
D 01: 10. I049/el: I9991 I39 
D.A. Roberts, R.R.A. Syms, A.S. Holmes and E.M. Yeatman 
(Optical and Semiconductor Devices Section, Department of Electrical 
and Electronic Engineering, Imperial College, Exhibition Road, London 
S W7 2BT, United Kingdom) 
19 July 1999 
References 
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York, 1991) 
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interference tomograph and scanning laser opthalmoscope’, 
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scanning confocal optical microscope’, J. Microelectromech. Syst., 
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deflection’, Appl. Opt., 1978, 17, pp. 32943297 
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with piezoelectric element’. US Patent 5,521,367, 28th May 1996 
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IEEE Photonics Technol. Lett., 1996, 8, pp. 95-97 
Representation of second-order polarisation 
mode dispersion 
A. Eyal, W.K. Marshall, M. Tur and  A. Yariv 
A new expansion for the Jones matrix of a transmission medium 
is used to describe high-order polarisation dispersion. Each term 
in the expansion is characterised by a pair of principal states and 
the corresponding dispersion parameters. With these descriptors, 
a new expression for pulse deformation is derived and codirmed 
by simulation. 
Introduction: Polarisation mode dispersion (PMD) in singlemode 
fibres has been extensively studied in recent years. The first-order 
effects of PMD are conveniently described in terms of the princi- 
pal states of polarisation (PSPs) and their differential group delay 
(DGD) [l]. Second-order effects of PMD arise from the variation 
of these descriptors with optical frequency [2, 31. Like chromatic 
dispersion, the effect of second-order PMD becomes increasingly 
important as the transmission bandwidth increases. It is thus 
important to have a simple method for representing and character- 
ising the effects of second-order PMD. Until now, the analysis of 
second-order PMD was based on expanding the Jones matrix of 
the transmission medium near the carrier frequency using a Taylor 
series or by considering the variations of the PMD-vector on the 
Poincark sphere. In this Letter we propose a new description of 
PMD which is based on an exponential expansion of the Jones 
matrix of the transmission medium. According to this description, 
different orders of PMD are treated separately as different subsys- 
tems and the overall behaviour of the transmission medium is 
obtained from serially concatenating these systems. Generalising 
Poole and Wagner’s phenomenological approach [ 11, each subsys- 
tem is described in terms of a pair of principal states and the cor- 
responding dispersion parameters. With the use of this simple 
model a new expression for pulse deformation due to second-order 
PMD is derived. The validity of the model is con t i i ed  using an 
’exact’ numerical simulation. 
PMD representation: Let the field at the input of a linear transmis- 
sion medium be described by: ztn = AeLo‘Ein where gin is a unit 
Jones vector representing the input state of polarisation, w is the 
optical frequency and A is the complex amplitude of the field. 
Using Jones’ formulation, the output field is given by: z,,, = 
T(w)Z,, where T(w) is a frequency-dependent 2 x 2 complex 
matrix [4]. In realistic optical transmission media T(w) often pos- 
sesses a complex frequency dependence. Therefore, to investigate 
the dispersive properties of the medium, it is useful to expand 
T(w) near the carrier frequency o, in powers of o - o,. In conven- 
tional analysis of pulse propagation in dispersive, isotropic media 
it is customary to represent the frequency dependent phase 
acquired by the propagating wave by its Taylor expansion near 
00, rather then expanding the field itself. To generalise this to the 
non-isotropic case, we propose the following expansion of the 
medium Jones matrix: 
where C2 << o,, To T(o,) and the parameters denoted by N, (k = 
1 ,  2, 3, ...) are 2 x 2 complex matrices which can be found by suc- 
cessively differentiating eqn. 1 with respect to Q and substituting 
SZ = 0. The first two terms in the expansion are given by 
Here the primes denote differentiation with respect to frequency. 
It can be easily seen that as long as T(o) is non-singular (i.e. the 
system does not contain an ideal polariser) and differentiable M 
times, then Nk will be well defined for k = 1, 2, 3, ..., M. We 
denote the eigenvectors and eigenvalues of Nk as &,* and a, 
respectively. The eigenvectors of NI, el*, were shown to be the 
input PSPs of the medium and the imaginary part of (al+ -al-) is 
the DGD associated with them [5]. We now utilise the well-known 
theorem that if A is a non-singular matrix with eigenvalues a,, 
then eA and A have the same eigenvectors, and the eigenvalues of 
& are eak. Therefore 
T(UO + 0) = 
To lP~Qi(fl)P;’] LP2Q2(fl)P;1J LP3Q3(fl)P311 . . . 
(3) 
where 
k = 1 , 2 , 3 ,  ... (4) 
The main advantages of this representation is that it enables us to 
consider the effects of the different orders of PMD separately and 
that high-order terms have the same form as the first-order term. 
Each order of PMD is represented as a separate optical system 
with eigenstates eh and eigenvalues a,. We refer to the parame- 
ter denoted by as the input principal states of polarisation of 
the kth order and to the difference between the imaginary parts of 
a,+ and a,- as the differential group-delay dispersion (DGDD) of 
the kth order. 
First-order PMD: eqn. 3 can be used to study the propagation of 
short pulses in a polarisation dispersive transmission medium. Let 
A(t) and A,@) describe the amplitude of an optical field at the 
input of the transmission medium and the corresponding Fourier 
transform, respectively. We consider first the case where only the 
zero and first-order terms in eqn. 3 are si&icant. The output 
field in this case is given by 
&,,t(t) E O3 Af(fl)ei(”o+”)tT o Pi &I (0) PF1 gindfl 
(5) 
2n L 
In the absence of polarisation dependent loss or gain @‘DUG) 
T(w) is proportional to a unitary matrix. In this case a,* are 
purely imaginary and integration of eqn. 5 yields 
where clf are defined by [cl+ c1-IT = PI-l&,, alfl = Im{al,} and 
E,,,(t) is given by: 
1658 ELECTRONICS LETTERS 16th September 7999 Vol. 35 No. I9  
Identifying TOPI as a matrix in which the columns are the output 
PSPs we obtain the well-known result that the output field due to 
fxst-order PMD in the basis of the output PSPs is composed of 
two differently delayed replicas of the input field with weights 
given by the projections of the input SOP on the input PSPs. In 
the presence of PDUG the PSPs are no longer orthogonal and 
experience differential losdgain-dispersion. While this necessitates 
minor modifications to eqn. 6, they will not be considered here 
for brevity. 
Second-order PMD - pulse deformation: The output field in the 
presence of second-order PMD is given by 
SOut(t) 2?T Iw { Af(R)ei(Wo+")tTo[P 1Qi (fl)P,l] 
x [P~?Q~(fl)€';~]2~~}dfl (8 )  
Introducing the Fourier transform pairs 
(9) 
we obtain 
where c2* are defined by [c2+ c2_IT = P2-'&,. Finally we use eqn. 6 
to obtain 
where the k,, are the elements of the matrix P,-'P2 and the linear 
that the output field in the basis of the (first-order) output PSPs is 
composed of four filtered versions of the input pulse. Two of them 
correspond to the slow P,SP and haye the same group delay a]+, 
but different envelopes, A + ( . )  and A_(.) .  The other two have the 
same two envelopes A + ( . )  and A - ( )  but are delayed by ai-. The 
weights of the four different components are determined by the 
projections of the input SOP on the second-order PSPs and the 
projections of the second-order PSPs on the first order PSPs. 
losdgain polarisation dispersion is neglected. This result shows 
ai 0.16- 
Z 0.14- 
3 
Q 
E 0.12- 
._ % 0.10- 
7J 
- 
0.08- ' 0.06- 
0.04- 
0.02 - 
-500 
O L  
time, ps 
a 
time, ps 
b m  
Fig. 1 Comparison between direct calculation of system response and 
model presented 
a Input pulse and output pulses in basis of first-order PSPs 
b Input pulse and pulses at output of compensated medium on basis 
of second-order PSPs 
- - _ -  input pulse 
. . . . . . . . . . . modelled output pulse 
simulation output pulse 
ELECTRONICS LETTERS 16th September 1999 Vo 
Simulation: To check the validity of the model a computer simula- 
tion was used. The simulation was composed of the following 
stages: first the Jones matrix, T(oi), of a cascade of 1000 randomly 
oriented linearly-birefringent fibre segments was calculated at 200 
consecutive frequencies. Next eqns. 2 and 4 were used to evaluate 
the matrices NI, N2 and N3 as well as the corresponding principal 
polarisation states, the DGD and the DGDD at the centre fre- 
quency %. To be able to consider the second-order effect of PMD 
separately, we compensated for the first-order effects by left-multi- 
plying T(o,) with dml-wO)NITO-l. We then calculated the time 
response of the compensated system to a Gaussian input. The 
pulsewidth, 118ps FWHM, was chosen such that third-order 
effects would be negligible compared to those of the second-order. 
Results: Fig. 1 shows a comparison between the result of the 
direct calculation of the system response and that according to our 
model using eqns. 9 - 11. The response of the medium in the basis 
of the first-order PSPs is plotted on the left side. The two broad- 
ened pulses correspond to the fast and slow PSPs and are accord- 
ingly delayed differently with a DGD of 26ps. In addition the 
pulses are non-Gaussian. The response of the compensated 
medium in the basis of the second-order PSPs is plotted on the 
right side of Fig. 1. It can be seen that the pulses are now equally 
delayed. In accordance with our model they retain their Gaussian 
shape but they are broadened differently with a DGDD of 
2 9 0 0 ~ s ~ .  
Conclusion: We have developed a new representation for high- 
order PMD. According to this representation the effect of each 
order of PMD is described by a pair of principal states and the 
corresponding dispersion parameters. The representation has been 
used to analyse the deformation of short pulses due to second- 
order PMD. 
0 IEE 1999 
Electronics Letters Online No: 19991 136 
DOI: IO. I049/el:I9991 I36 
A. Eyal, W.K. Marshall and A. Yariv (Department of Applied Physics, 
California Institute of Technology, Mail Stop 128-95. Pasadena, CA 
91125, USA) 
E-mail: avishay@its.caltech.edu 
M. Tur (Department of Interdisciplinary Studies, Faculty of 
Engineering, Tel-Aviv University, Tel Aviv, 69978 Israel) 
26 July I999 
References 
1 POOLE, c.D., and WAGNER, R.E.: 'Phenomenological approach to 
polarisation dispersion in long single-mode fibres', Electron. Lett., 
1986, 22, (19), pp. 1029-1030 
2 POOLE, c.D., and GILES, c.R.: 'Polarization dependent pulse 
compression and broadening due to polarization dispersion in 
dispersion shifted fiber', Opt. Lett., 1988, 13, pp. 155-157 
PRIETO, F., and ZIMMER, c.w.: 'Second order polarization mode 
dispersion: Impact on analog and digital transmissions', J. 
Lightwave Technol., 1998, 16, (5 ) ,  pp. 751-771 
4 JONES, R.c.: 'A new calculus for the treatment of optical systems 
(I)', J.  Optical Soc. Am;, 1941, 31, pp. 488493 
5 HEFFNER, B.L.: 'Automated measurement of polarization mode 
dispersion using Jones matrix eigenanalysis', ZEEE Phoronics 
Technol. Lett., 1992, 4, (9), pp. 1066-1069 
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Surface plasmon resonance sensing in 
capillaries 
T.M. Chinowsky and S.S. Yee 
A capillary-based surface plasmon resonance sensor is presented. 
Glass capillaries internally coated with Au are radially illuminated 
with focused laser light that strikes the interior capillary surface 
and exits the capillary at angles that vary with the interior 
incident angle. Measurement of the exit light provides reflectivity 
against angle spectra useful for chemical sensing. 
11. 35 NO. I9 1659 


Representation of second-order polarisation mode dispersion

22, (19), pp. 1029-1030

35 NO. I9 1659

A new calculus for the treatment of optical systems

Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis',

HEFFNER, B.L.: 'Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis', ZEEE Phoronics Technol. Lett., 1992, 4, (9), pp. 1066-1069

JONES, R.c.: 'A new calculus for the treatment of optical systems (I)', J. Optical Soc. Am;, 1941, 31, pp. 488493

Phenomenological approach to polarisation dispersion in long single-mode fibres',

Polarization dependent pulse compression and broadening due to polarization dispersion in dispersion shifted fiber',

POOLE, c.D., and WAGNER, R.E.: 'Phenomenological approach to polarisation dispersion in long single-mode fibres', Electron. Lett.,

https://authors.library.caltech.edu/475/1/EYAel99.pdf

Representation of second-order polarisation mode dispersion

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