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Modelling of long-period fibre grating response
to refractive index higher than that of cladding
ARTICLE in MEASUREMENT SCIENCE AND TECHNOLOGY · OCTOBER 2001
Impact Factor: 1.43 · DOI: 10.1088/0957-0233/12/10/314
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2. INSTITUTE OF PHYSICS PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 12 (2001) 1–5 PII: S0957-0233(01)23287-1
Modelling of long-period fibre grating
response to refractive index higher than
that of cladding
R Hou1,3
, Z Ghassemlooy1
, A Hassan1
, C Lu1,2
and K P Dowker1
1
Optical Communications Research Group, School of Engineering,
Sheffield Hallam University, Pond Street, Sheffield S1 1WB, UK
2
School of Electrical and Electronics Engineering, Nanyang Technological University,
Singapore 639798
E-mail: r.hou@braggphotonics.com, z.f.ghassemlooy@shu.ac.uk and k.p.dowker@shu.ac.uk
Sunrise Setting
Marked Proof
MST/123287/PAP
23885ap
Printed on 21/08/01
at 11.29Received 22 March 2001, in final form and accepted for publication
14 August 2001
Published
Online at stacks.iop.org/MST/12
Abstract
In this paper we introduce a new mathematical model, which can be used to
investigate the response of a long-period fibre grating to changes in ambient
refractive indices of values both greater and less than that of the cladding.
A numerical method to calculate the HE core and cladding modes is
presented. The results show that even when the ambient refractive index is
higher than that of the cladding, the coupling (resonance) wavelengths
experience a measurable shift (a few nm). The result obtained suggests that
LPGs coated with a material of higher refractive index than the cladding
may be used as index sensors.
Keywords: long-period grating, core mode, cladding mode, leaky mode,
index sensors
1. Introduction
A long-period fibre grating (LPG), which couples light from a
fundamental guided core mode into co-propagating cladding
modesatvariouswavelengths, wasfirstreportedbyVengsarkar
and co-workers in1996 [1]. LPGs have also been used as
gain-flattening filters for erbium-doped fibre amplifiers [2],
and as optical fibre polarizers [3]. As in the fibre Bragg
grating (FBG), the LPG is sensitive to measurands such as
temperature or strain, which may alter the period of the grating
or the refractive index of the core or cladding. Unlike FBG
the cladding mode configuration of the LPG is extremely
sensitive to the refractive index of the medium surrounding
the cladding, thus allowing it to be used as an ambient index
sensor. Compared to other optical sensors, LPGs offer many
advantages such as low insertion losses, low back reflection
and relatively simple fabrication, but they do show high
temperature sensitivity. However, temperature sensitivity can
normally be controlled by using a second grating as a control,
3 Currently with Bragg Photonics Inc., 880 Selkirk, Pointe-Claire, Montreal,
Quebec, H9R 3S3, Canada.
or by coating the cladding with a suitable material with a
refractive index designed to compensate for this [6]. The latter
method may not be possible when the LPG is to be used as
an ambient index sensor as this compensating coating may
insulate the cladding from the surroundings to be monitored.
An LPG is a photo-induced periodic structure written into
the core of a germanium doped silica fibre by exposing it to
UV light via an amplitude mask. The wavelength at which the
coupling from core to cladding modes takes place is directly
dependent on the difference between the core and cladding
indices, the dimensions of the core and cladding and the grating
period. Any change in these values can shift the transmission
spectral profile. The cladding mode profiles are also very
sensitive to changes in the refractive index of the ambient
(surrounding) environment, particularly when the value of
the ambient refractive higher than the cladding index (ncl) is
referred to as a leaky or hollow di-electrical wave-guide [7].
Experimental data published show that the LPG transmission
spectrum changes when it is submerged into various solutions
having refractive indices in the range of 1.40 to 1.72 [4]. Lee
and co-workers have adopted a graphical method to investigate
0957-0233/01/010001+05$30.00 © 2001 IOP Publishing Ltd Printed in the UK 1

3. R Hou et al
Core nco
Cladding ncl
Outer layer nam
LPG
Λ
Cladding mode
Fundamental
guided mode
Grating length
Jacket
Figure 1. Long-period grating.
thedisplacementoftheresonantpeaksbychangingtheambient
index [8]. Duhem and co-workers have investigated the LPG’s
response experimentally in different fibres for the case when
the ambient refractive index is higher than that of the cladding
in order to determine the coupling intensities [9]. Stegall
and Erdogan have estimated the intensity of the coupling
losses into leaky cladding modes by using a one-dimensional
(planar) waveguide approximation [10]. Patrick has studied
the response of LPGs to changes in external refractive index,
but over a very limited range. In this paper for the first time,
to the best of our knowledge, we fully model the ambient
refractive index response of an LPG over a much wider index
range. Here we present a numerical method to calculate
the HE cladding modes. In developing this model we have
assumed that the dispersion due to the core, cladding and
its surrounding material are negligible and therefore have
been ignored. The model considers three separate conditions
at the cladding/ambient interface, nam < ncl, nam ≈ ncl
and nam > ncl. The LPG coupling wavelength shift is
examined for all three conditions, in particular when the
surrounding index is larger than that of the cladding. Finally,
the calculated wavelength shift is compared with measured
data from published literature.
2. Theory
LPGs are transmission gratings in which the coupling is
between forward propagating core and cladding modes.
Typically in a single mode fibre (see figure 1) an LPG
couples the fundamental guided core mode to a co-propagating
cladding mode at a coupling (or resonance) wavelength given
by [11]
λn
i = (neff.co(λi) − nn
eff.cl(λi)) (1)
where λn
i is the nth coupling wavelength, neff.co(λi) is the
effective index of the core at the wavelength λi and neff.cl(λi)
is the effective index of the nth cladding mode at the
wavelength λi. is the LPG period, which is much longer
for copropagating coupling at a given wavelength than for the
counterpropagating coupling.
Coupled mode theory is a powerful and straightforward
tool that has been used to obtain quantitative information about
spectral properties of fibre gratings [10, 12]. Here our analysis
simply follows that given by Okoshi [12]. The core mode
is not affected by the surrounding index change, therefore at a
specific wavelength λi the HE core mode is obtained from [12]
u1
Jm(u1)
Jm−1(u1)
= w1
Km(w1)
Km−1(w1)
(2)
where J(.) is the Bessel function of the first kind, K(.) is the
modified Bessel function of the second kind and m is the mode
number.
For the HE11 core mode with m = 1, the normalized
transverse phase constant is given as
u1 = a1(k2
n2
co − β2
co)1/2
(3)
and the normalized transverse attenuation constant in the core
is given as
w1 = a1(β2
co − k2
n2
cl)1/2
(4)
where k = 2π/λi, ai is the core radius and βco is the core mode
propagation constant.
The core effective refractive index is given as
neff.co(λi) =
βco
k
(λi). (5)
The effective indices of the cladding modes are dependent on
the cladding index and the index of the surrounding (outer)
environment (nam) (see figure 1). This means that the nth
cladding mode coupling wavelength will change as the index
of the surrounding environment changes. Cladding modes
are most accurately calculated using the three-layer model
[13] for the case of nam < neff.cl < ncl. However, as the
literature states, the three-layer model is only effective while
nam < ncl. For the case where nam > ncl, the two-layer
model is used to calculate the cladding modes, which provides
reasonably accurate values as discussed in the results section.
The two-layer model basically treats the cladding and core as
one multimode fibre and the surrounding environment as the
new cladding only in the LPG region. Here we discuss the
three cases previously described and show the analysis for the
case when nam > ncl as outlined below.
Case 1: nam < ncl. This is a total internal reflection
condition, which is best dealt with using a three-layer model
developed by [13]. As nam increases from unity to ncl, the
principaleffectonthetransmissionwavelengthhasbeenshown
to be a blue shift, with little change in the extinction ratios of
the modes, about a few dB [10].
Case 2: nam
∼= ncl. As nam approaches ncl, each
mode will experience a maximum displacement, which is
finite. When the surrounding index is approximately equal
2

4. Long-period fibre grating
nam
ncl
a2
Figure 2. Cross-section of hollow circular waveguide where
nam > ncl.a2 is the radius of the cladding.
to the cladding index the core is effectively surrounded by
an infinitely thick cladding, thus preventing the existence of
any cladding mode structure. As there are no cladding modes
to which the core modes can couple the distinct loss bands
disappear as the cladding modes are converted to radiation
modes [5]. There are a few dB of broadband radiation-mode
coupling losses, but no distinct resonances [10]. In this case
equations (1) to (9) were used to calculate a wavelength shift
of +2.0525 nm, as depicted in figures 3 and 4.
Case 3: nam ncl. As the ambient index increases
beyond the cladding index, experimental data shows that the
coupling wavelengths reappear with both spectral shift and
amplitude changes evident [10]. The amplitude changes, over
a small refractive index range, have been investigated [10].
However, no analysis has been reported for the spectral shift,
which is the subject of this study. For the case of nam > ncl, the
cladding modes no longer experience total internal reflection
and are referred to as leaky modes. We have used the two-layer
model (see figure 2), in order to calculate the cladding modes,
as outlined above.
The HE cladding mode expression is obtained from the
well known eigenvalue equation given by [14] and was derived
from Maxwell’s equations for EM radiation:
Jm−1(u2)
u2Jm(u2)
= −
(n2
cl + n2
am)Km(w2)
2n2
clw2Km(w2)
+
m
u2
2
−
(n2
cl − n2
am)Km(w2)
2n2
clw2Km(w2)
2
+
mv2
βcl
u2
2w2
2nclk
1
2
. (6)
For HE1n cladding modes, m = 1 and the normalized
transverse phase constant is given as
u2 = a2(k2
n2
cl − β2
cl)1/2
. (7)
The normalized transverse attenuation constant in the cladding
region is given as
w2 = a2(β2
cl − k2
n2
am)1/2
(8)
where the normalized frequency is given as v = (u2
2 + w2
2)1/2
.
Note that the quantities βcl, u2, w2 and v have complex
values, therefore the Bessel functions may have complex
arguments [15]. If the propagation constant βcl becomes
complex, then it is necessary to consider the lossy or leaky
modes. For complex βcl, the real part βr is equal to the phase
velocity of the specified mode, whereas the imaginary part βi
represents its loss rate [16]. Using MATLAB, both graphical
and numerical methods were used to calculate the real and
imaginary parts of propagation constants of leaky cladding
modes.
-120
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
Index of ambient environment, n am
Resonancewavelenthshift,(nm)
Figure 3. Simulated coupling wavelength shift of the seventh mode
versus nam on a wide range.
The cladding effective refractive index is defined as
nn
eff.cl(λi) =
βn
cl
k
(λi). (9)
However, for circular waveguides, Adams [15] has shown that,
by making certain approximations, βcl can be expressed in
terms of its real part and power attenuation coefficient α as
given by
βcl = βr −
iα
2
(10)
where
βr = kncl 1 −
1
2
(l + 0.25)π
akncl
2
(11)
α ∼=
(n2
am + n2
cl)
akn3
cl n2
am − n2
cl
l +
1
4
π for HE modes (12)
where l = 1, 2, 3, . . ..
3. Results and discussion
Erdogan [13] has shown that a more accurate method may
be used for calculating cladding modes using the three-layer
model for nam < neff < ncl, but not for nam > ncl. In
this paper we focus on modelling of the leaky configuration
(i.e. when nam > ncl) and its special properties. The two-
layer model, equations (1) to (9), was used to create figure 3,
the wavelength shift of the seventh order cladding mode being
calculated over a wide range of nam. The plot shows that the
most sensitive regions are when nam is close to ncl and when
nam changes from 1.449 997 609 to 1.449 997 61 (i.e. n =
1 × 10−9
). The coupling wavelength shift shows a dramatic
change from a sharp decrease of −120 nm to a sharp increase
of +2.0525 nm. This ‘switching property’, in this sensitive
region, has a potential application in optical communications
and optical sensors. The calculation process, in a flow chart
form, is shown in figure 5, and the parameters used are shown
in table 1.
When nam > ncl, coupling occurs at slightly longer
wavelengths than the initial coupling wavelengths when the
index of surrounding material nam = 1 (in air). This is
best illustrated by enlarging the right-hand side of figure 3,
3

5. R Hou et al
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8
Index of ambient environment, n am
Resonancewavelengthshift,(nm)
Figure 4. Simulated coupling wavelength shift of the seventh
resonance mode versus nam > ncl.
Set counter = 0
Determine βco (λi) (equ 2)
Determine βcl
n
(λi) (equ 6)
Determine design ΛD to couple
core mode into required cladding
mode (7th
) (equ 1)
λ = 1550nm
nam = 1
Yes No
Calculate new value of ΛC using
(7th
) cladding mode (equ 1)
NoYes
nam > ncl
λi = λi +δλ λi = λi - δλ
NoYesPrint: λi will couple
to coupling mode 7
Increment nam by
δnam
Increment counter by 1
Counter = 0
ΛD = ΛC
Figure 5. A flow chart showing the calculation process for
modelling the LPG coupling wavelength shift when nam changes.
as shown in figure 4. At nam > 1.45 (the index of the
cladding) the wavelength shift is non-linear, increasing first
to a maximum of δλ = 5.1 nm at nam ≈ 1.52, then
decreasing gradually as nam increases further. In the non-
linear region (1.475 < nam < 1.54), the wavelength shift
is too small to be calculated or measured, perhaps due to the
effect of the transition from TIR mode structure to radiation
mode structure in the cladding causing slight and irregular
effects. At nam = 2.666, the wavelength shift is equal
to +1 nm, see figure 4. Although these spectral shifts are
small they can be measured using a high resolution (better
than 0.05 nm) optical spectrum analyser (e.g. ANDO, AQ
6317B, resolution 0.015 nm, or better). The predicted results
obtained show good agreement with the published practical
data [4, 11].
Table 1. Simulation parameters used in the modelling.
Parameters Values
Core refractive index 1.4566
Cladding refractive index 1.45
Core radius 2.5 µm
Cladding radius 62.5 µm
Grating period 425.0713 µm
These characteristics make LPG an alternative candidate
for sensing. Unlike fibre Bragg grating sensors, where the
fibre cladding needs to be removed before applying any coating
around the grating, in LPGs the coating can be applied directly
onto the cladding, thus making the fabrication process much
simpler. Exposure of a suitably coated LPG to an external
agent (gases, liquids) should result in the coating material
absorbing the gas, thus changing its refractive index, which
in turn will change the LPG coupling wavelengths. Currently
we are fabricating an LPG sensor for monitoring CO and CO2,
the results of which will be published in due course.
4. Conclusions
In this paper we have presented a detailed mathematical model
used to simulate the coupling wavelength shift in an LPG
by changing the ambient refractive index over a wide range
(1–2.8). This is the first time that such results have been
obtained through mathematical modelling, while previous
literature has tackled the problem on an experimental basis
only. These results, presented for the first time, reveal that
when the ambient refractive index is higher than that of the
cladding the coupling wavelength experiences a noticeable
shift, which can easily be measured using a high resolution
optical spectrum analyser. These results suggest that the LPG
may be used as an index sensor, when coated with a suitable
material.
Acknowledgments
The authors gratefully acknowledge Dr Lin Zhang, Dr Tom D P
Allsop and Dr Wei Zhang (in the Photonics Research Group
of Aston University, UK) for all their help.
References
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[2] Qian J R and Chen H E 1998 Gain flattening fibre filters using
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[3] Ortega B, Dong I, Lic W E, Sandro J P, Reekie I, Tsypina S I,
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[4] Patrik H J, Kersey A D and Bucholtz F 1998 Analysis of the
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[5] Vikram B and Vengsarkar A M 1996 Optical fibre long-period
grating sensors Opt. Lett. 21 692–4
4

6. Long-period fibre grating
[6] Jang J N, Kim S Y, Kim S W and Kim M S 2000 Novel
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[9] Duhem O, Henninot J F and Douay M 2000 Study of in fibre
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[10] Stegall D B and Erdogan T 1999 Leaky cladding mode
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[11] Patrick H J, Kersey A D, Bucholtz F, Ewing K J, Judkins J B
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CThQ5, pp 420–1
[12] Okoshi T 1982 Optical Fibres (New York: Academic)
[13] Erdogan T 1997 Cladding-mode resonances in short- and
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[14] Yariv A 1996 Optical Electronics in Modern Communications
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[15] Adams M J 1981 An Introduction to Optical Waveguides
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[16] Tsao C 1992 Optical Fibre Waveguide Analysis (Oxford:
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5

7. Annotations from 23885ap.pdf
Page 1
Annotation 1
Are the affiliations for Lu OK, or should there just be one to the second address?
Page 3
Annotation 1
We have changed Erdogen to `Erdogan' to match ref. list. Is this correct?
Annotation 2
Figure 5 mentioned before figure 4. Is this OK?