Page 1

Optimization of sensitivity in Long Period Fiber

Gratings with overlay deposition

Ignacio Del Villar, Ignacio R. Matías and Francisco J. Arregui

Departamento de Ingeniería Eléctrica y Electrónica, Universidad Pública de Navarra, 31006 Pamplona, Spain.

ignacio.delvillar@unavarra.es, natxo@unavarra.es , parregui@unavarra.es

Philippe Lalanne

Institut d'Optique Centre National de la Recherche Scientifique BP 147 Orsay 91 403 France.

philippe.lalanne@iota.u-psud.fr.

Abstract: The deposition of an overlay of higher refractive index than the

cladding in a Long Period Fiber Grating (LPFG) permits to improve the

sensitivity to ambient refractive index changes in a great manner. When the

overlay is thick enough, one of the cladding modes is guided by the overlay.

This causes important shifts in the effective index values of the cladding

modes, and henceforward fast shifts of the resonance wavelength of the

attenuations bands in the transmission spectrum. This could be applied for

improving the sensitivity of LPFG sensors. The problem is analysed with a

numerical method based on LP mode approximation and coupled mode

theory, which agrees with so far published experimental results.

2004 Optical Society of America

OCIS codes: (050.2770) Gratings, (060.2430) Fibers, single-mode, (260.2110) Electromagnetic

theory, (310.1860) Deposition and fabrication.

References and links

1.

J. R.Qiang and H. E. Chen, “Gain flattening fibre filters using phase shifted long period fibre grating,”

Electron. Lett. 34, 1132-1133 (1998).

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long-period fiber

gratings as Band Rejection Filters,” J. Lightwave Technol. 14, 58-65 (1996).

B. J. Eggleton, R. E. Slusher, J. B. Judkins, J. B. Stark and A. M. Vengsarkar, “All-optical switching in

long period fiber gratings,” Opt. Lett. 22, 883-885 (1997).

K. W. Chung, S. Yin, “Analysis of widely tunable long-period grating by use of an ultrathin cladding layer

and higher-order cladding mode coupling,” Opt. Lett. 29, 812-814 (2004).

V. Bhatia and A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692-694 (1996).

V. Bhatia, “Applications of long-period gratings to single and multi-parameter sensing,” Opt. Exp. 4, 457-

466 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-11-457.

Y. G. Han, S. B. Lee, C. S. Kim, J. U. Kang, U. C. Paek and Y. Chung, “Simultaneous measurement of

temperature and strain using dual long-period fiber gratings with controlled temperature and strain

sensitivities,” Opt. Exp. 11, 476-481 (2003),

http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-476.

8.

C. C. Ye, S. W. James and R. P. Tatam, “Simultaneous temperature and bend sensing using using long-

period fiber gratings,” Opt. Lett. 25, 1007-1009 (2000).

9.

H. J. Patrick, A. D. Kersey and F. Bucholtz, “Analysis of the response of long period fiber gratings to

external index of refraction,” J. Lightwave Technol. 16, 1606-1612 (1998).

10. R. Hou, Z. Ghassemlooy, A. Hassan, C. Lu and K. P. Dowker, “Modelling of long-period fibre grating

response to refractive index higher than that of cladding,” Meas. Sci. Technol. 12, 1709-1713 (2001).

11. S. T. Lee, R. D. Kumar, P. S. Kumar, P. Radhakrishnan, C. P. G. Vallabhan, V. P. N. Nampoori, “Long

period gratings in multimode optical fibers: application in chemical sensing,” Opt. Comm. 224, 237-241

(2003).

12. N. D. Rees, S. W. James, R. P. Tatam and G. J. Ashwell, “Optical fiber long-period gratings with

Langmuir-Blodgett thin-film overlays,” Opt. Lett. 27, 686-688 (2002).

13. S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,”

Meas. Sci Technol. 14, R49-R61 (2003).

2.

3.

4.

5.

6.

7.

(C) 2005 OSA 10 January 2005 / Vol. 13, No. 1 / OPTICS EXPRESS 56

#5675 - $15.00 USReceived 8 November 2004; revised 20 December 2004; accepted 20 December 2004

Page 2

14. E. Anemogiannis, E. N. Glytsis and T. K. Gaylord, “Transmission characteristics of long- period fiber

gratings having arbitrary azimutal/radial refractive index variation,” J. Lightwave Technol. 21, 218-227

(2003).

15. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).

16. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber gratings filters,” J. Opt. Soc. Am.

A, 14, 1760-1773 (1997).

17. D. B. Stegall and T. Erdogan, “Leaky cladding mode propagation in long-period fiber grating devices,”

IEEE Photon. Technol. Lett. 11, 343-345 (1999).

18. Y. Koyamada, “Numerical analysis of core-mode to radiation-mode coupling in long-period fiber gratings,”

IEEE Photon. Technol. Lett. 13, 308-310 (2001).

19. I. Del Villar, M. Achaerandio, I. R. Matías and F. J. Arregui, “Deposition of an Overlay with Electrostactic

Self-Assembly Method in Long Period Fiber Gratings,” Opt. Lett. In press.

20. K. Morishita, “Numerical analysis of pulse broadening in grated index optical fibers,” IEEE Trans.

Microwave Theory Tech. 29, 348-352 (1981).

21. D. Gloge, “Weakly guiding fibers,” App. Opt. 10, 2252-2258 (1971).

22. A. K. Ghatak, K. Thyagarajan, and M. R. Shenoy, IEEE J. Lightwave Technol. 5, 660- 667(1987).

23. A. W. Snyder and J. D. Love, Optical waveguide theory (London U.K: Chapman and Hall, 1983).

24. K. Skjonnemand, “Optical and structural characterisation of ultra thin films,” Ph.D. disseration (Cranfield

University, Bedford, UK, 2000).

25. G. Decher, “Fuzzy nanoassemblies: toward layered polymeric multicomposites,” Science, 277, 1232-1237

(1997).

26. S. A. Khodier, “Refractive index of standard oils as a function of wavelength and temperature,” Optics &

Laser Tech., 34, 125-128 (2002).

1. Introduction

Long Period Fiber Gratings (LPFGs) consist of a periodic index modulation of the refractive

index of the core of a single mode fiber (SMF), with a much longer period than Fiber Bragg

Gratings (FBGs). They have found many applications during the nineties in optical

communications and sensors fields. In optical communications many devices have been

developed, such as gain equalizers [1], band rejection filters [2], tunable filters [3] and optical

switches [4]. In sensors field, if compared with FBGs, they are also sensitive to measurands

such as strain or temperature [5-8], which may alter the period of the grating or the refractive

of the core or cladding. Nonetheless, modes couple in a different way with respect to FBGs,

which improves the characteristics of sensors in a great manner. LPFGs are highly sensitive to

the surrounding media, which also includes the drawback of a dependence on temperature.

Anyway, there exist techniques for avoiding this problem [7, 8], which permits at the same

time multi-parameter sensing [6]. They also present low background reflections and insertion

losses, and demodulation schemes are economical. All these good properties make LPFGs

adequate for more purposes than strain or temperature detection. They can be used as

refractometers [9, 10], or for detection of chemical substances in the ambient [11].

Furthermore, if an overlay is deposited on the cladding, its refractive index will modify the

coupling of modes [12]. If the material selected is sensitive to a specific parameter, highly

sensitive and specific devices will be obtained.

Regarding the fabrication, LPFGs can be obtained with several techniques, being

ultraviolet (UV) irradiation the most extended one. Others are ion implantation, irradiation by

femtosecond pulses in the infrared, irradiation by CO2 lasers, diffusion of dopants into the

core, relaxation of mechanical stress, and electrical discharges. A good review on these

techniques can be found in [13].

Typically, the periodicity of LPFGs ranges between 100µm to 1mm. As a result, dips are

created in the transmission spectrum at wavelengths where there is a coupling between the

core and copropagating cladding modes, unlike in FBGs, where there is a coupling between

contrapropagating modes. Each attenuation band presents a minimum, notated as resonance

wavelength. This wavelength value is in close relation with the one that satisfies the Bragg

condition between the coupled modes. A much better approximation can be obtained if the

influence of the self coupling coefficient of the modes is included in the formulation [14], as it

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will be explained in next section. The third possibility is to solve the coupled mode equations.

The consideration of all coupling coefficients will permit to obtain more exact values for the

resonance wavelengths and the transmission spectrum. The drawback of this option is a higher

computational effort in comparison with the other two, where no differential equation has to

be solved.

Regarding the depth of the attenuation bands, there are many factors that influence this

value. The two most important ones are the cross coupling coefficient between the core mode

and the cladding mode that couples at that resonance wavelength, and the length of the

grating. A simple formula permits to have an approximate idea of the exact value [13, 15]:

T

i

sin1−=

where ki is the coupling coefficient of the ith cladding mode, L the length of the grating and Ti

minimum transmission of the attenuation band created by coupling between the core mode

and the ith cladding mode.

Henceforward, the modulation of the grating is critical, because the coupling coefficients

are directly proportional to this value. Consequently, LPFGs can be classified into weak and

strong LPFGs [16].

Regarding the analysis of LPFGs, two different cases have been studied so far. Until now

we have considered for the explanations the first one, where it is assumed that the ambient

refractive index is lower than cladding [16]. As the ambient refractive index approaches that

of the cladding, the sensitivity of the resonance wavelength to variations of the ambient

refractive index is higher. Then, the second case starts when the ambient refractive index

exceeds that of the cladding. The core couples with radiation modes [10,17,18] and the

dependence of the resonance wavelength on the ambient refractive index is not so accused.

Instead, the resonance depth is more dependent on this parameter for values close to the

refractive index of the cladding [9]. In both cases, the region of highest sensitivity is located

around the refractive index of the cladding.

In this work, a third possibility is presented. In the previous two cases, the cladding was

surrounded by a medium of infinite thickness. Now a thin overlay of higher refractive index

than the cladding is deposited between the cladding and the infinite thickness surrounding

media. One of the cladding modes will be guided by the overlay if it is thick enough. This

causes a reorganization of the effective indices of the modes of the cladding. As a result, there

are important variations of the Bragg condition, which leads to dramatic shifts of the

resonance wavelengths if we work around the thickness value where there is a transition to

guidance of a mode in the overlay. The aim will be to select an adequate refractive index and

thickness of the overlay, in order to increase the sensitivity of the effective index of the

cladding modes for a specific application.

The shift is maximum when the effective index of the mode is half way between its

original effective index and the original effective index before deposition of the next lower

cladding mode. Consequently, there is an optimum overlay thickness (OOT). The value of the

OOT depends mainly on the refractive index of the material deposited and on the ambient. In

first place, the refractive index of the overlay is fixed. This fact defines the amount of material

that is necessary to deposit and the sensitivity of the refractometer. Secondly, an optimum

thickness value is calculated for the middle value of the ambient refractive index range where

the refractometer will operate. In this way, highly sensitive refractometers can be designed for

small refractive index ranges. On the other hand, if the OOT is calculated for a material whose

refractive index varies with some parameter, the sensitivity of LPFG can be improved in many

sensor fields: chemical sensors, biosensors, inmunoassays and so on.

So far, Electrostatic Self-Assembly (ESA) [19] and Langmuir Blodgett (LB) [12]

techniques have been applied for the deposition of an overlay of tens of nanometers. In this

work it is assumed that the refractive index of the material deposited is purely real, which is

not the case in the case in ESA and LB techniques. However, this assumption permits to

understand more easily the effects of deposition of an overlay on an LPFG, and it still predicts

()

Lk

i

2

(1)

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Page 4

adequately experimental results in [12] and [19]. In a future work it will be analysed the

deposition of an overlay with losses, which is more complex.

The numerical method used for determining the wavelength that satisfies the Bragg

condition between the core mode and each cladding mode, and the transmittance of the LPG is

based on that described in [14]. Coupled mode theory is the basis for the calculation of LP

modes in a cylindrical multilayer waveguide and it is explained in next section. In section 3

the explanation of the effects of a thin overlay on an LPFG and its applications are presented.

Finally, some conclusions are given in section 4.

2. Theory

So far coupled mode theory has proved to be a powerful tool for simulation of LPFG

structures. In [16] a three layer model is presented, where the transmission can be accurately

obtained provided the ambient refractive index is lower than the cladding. If the ambient

refractive index is higher than the cladding, other approximations are necessary [10, 17, 18].

The method used in this chapter is based on that presented in [14] and it includes three steps.

2.1 Calculation of the propagation constants of LP modes

The problem analysed in this work (see Fig. 1) presents four layers. The calculation of the

modes in a cylindrical multilayer waveguide becomes a difficult and computational expensive

task.

To avoid this problem, in [14, 20] a theoretical model is described. It is based on scalar

approximation analysis to obtain the LP modes of a cylindrical dielectric waveguide. LP

modes are adequate for description of a cylindrical waveguide under assumption of weak

guidance [21]. But in [14] it is proved that the high contrast between ambient and cladding

refractive indices plays no important role in the results. Consequently, the presence of an

overlay of a refractive index which shows not important contrast with the cladding will not

affect to the results in a great manner, provided low LP modes are analysed. As the material

deposited on the cladding shows a higher contrast, error present in the results will increase,

but results will remain qualitatively correct. LPνj modes (ν=0) are everywhere polarized in

the same direction, or in other words the fields are plane polarized. Nonetheless, the fields of

higher order LPνj modes are not plane polarized and it is proved in [14] that they play a role

in the results obtained in LPFGs. The reason is that the irradiation for the generation of the

grating is not symmetric around the fiber. However, for the sake of simplicity, it will be

assumed that the structure simulated presents no azimutal variation of the perturbed index

profile after exposure to UV radiation. In this way, there are only interactions between LP0j

modes, and each mode is not treated as two independent modes, as it is the case for ν>0

modes.

deposited regiondeposited region

r1r1

r1

r1

r2

r2

r3

r3

claddingcladding

corecore

ambient ambient

period period

nh

nh

nl

nl

11

223 4 3 4

Fig. 1. Transversal and longitudinal section of LPFG structure

deposition of an overlay on the cladding.

After solving the following scalar equation [22]:

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( )

r

[]

0

~

n

222

0

2

2

=−+

uk

dr

ud

β

(2)

where k0 = 2π/λ, λ is the freespace wavelength, β is the propagation constant of a mode, r is

the radius, n~is defined by:

( )

r

22

0

2

22

4

1

~

n

rk

n

−

−=

ν

0

=ν

(3)

where n is the refractive index as a function of the radius and ν is the azimutal order, and u is

defined by:

r

rR

ru

)(

)(

=

(4)

where R(r) is the radial variation of the modal field.

The transverse electric field component propagating along the z-axis is given by:

(

−=

jzrU

j i j

0,0

exp,,

βφ

(

(

i j

rIA

0,0

γ

where β0j is the propagation constant of the LP0j mode,

)

()

()

() ( )

Φ

β

( )=

r

nk

0

−=

Y

Ψ

Rzj

)

)

rz

r

ijjij

)

)

,0

<

0,0

exp

(

(

r

0

,

φβ

φ

B

0

)

(

+

+

×−=

i j

,

i jij

iji

K

j i j

,

ij

j

B

rJA

zj

0,0,0

,00,00,0

0

exp

γ

γγ

β

when

ij

ij

nk

00

0

>β

(5)

2

0

2

i

2

0,0

jij

nk

βγ−=

is the magnitude of

the transverse wavenumber, φ is the azimutal angle, and A0j,i and B0j,i are not normalized field

expansion coefficients determined by the boundary conditions within the cylindrical layer i.

(

ij

rJ

,0

ij

rY

,0

are the ordinary Bessel functions of first and second kind of order

0, while

(

ij

rI

,0

and

ij

rK

,0

are the modified Bessel functions of first and second

kind of order 0.

After this, the Transfer Matrix Method (TMM) [14,20] is applied for the calculation of the

propagation constants of the four layer cylindrical waveguide problem. At the same time the

coefficients A0j,i and B0j,i in each layer will be calculated and normalized so that each mode

carries the same power P0:

β

)

0 γ

0 γ

and ()

0 γ

)()

0 γ

( )

r

( )rdrrRRdP

r

r

jj

j

j

∫∫

==

=

1

0

00

2

0

0

0

0

2

π

φ

φ

ωµ

(6)

2.2. Derivation of coupling coefficients

According to coupled mode theory [14, 16, 23], the interaction between optical modes is

proportional to their coupling coefficient. The contribution of longitudinal coupling

coefficient in coupled mode analysis can be neglected [14, 15, 16]. Consequently we will refer

to the transversal coupling coefficient as the general coupling coefficient. In cylindrical

coordinates the coupling coefficient between each two modes can be expressed as:

ω

µν

P

∫∫

0

4

where Ψ(r,φ) is the transverse field for an LP mode as expressed in Eq. (5) and ∆ε(r,φ,z) is

the permittivity variation. There is no azimutal variation of the perturbed index profile, and

there is weak guidance between the core and the cladding of the fiber. Consequently, the

permittivity can be expressed as:

()()()

φφ

,

φ

,

φ

,

ε

µν

π

2

φ

rdrdrrzrK

r

kjkj

∞

==

ΨΨ∆×=

00

,

,

(7)

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()() z

,

rnrnzr

)(2,

00

∆≈∆εε

(8)

where ε0 is the freespace permittivity, n0(r) is the refractive index profile of the structure

without the perturbation, and ∆n(r,z) is the variation of the refractive index. This last variable

is the product of a perturbation constant and two other functions:

(

zrn

,

=∆

where p(r) is the transverse refractive index perturbation. σ(z) is the apodization factor, and

S(z) is the longitudinal refractive index perturbation factor. It will be approximated by a

Fourier series of two terms:

( )

sszS

+=

0

where Λ is the period of the grating.

After these approximations, the coupling coefficients will be expressed as:

Λ

∫

P

0

2

+=

)( ) ( ) zSz

σ )

rp

(

(9)

()() z

Λ

/2 cos

1

π

(10)

( ) ( )

pr

()()

=ΨΨ×

+=

∫

∞

==

φφ

,

φ

,

ωε

π

2

µν

π

2

φ

µ

,

ν

rdrdrrrnzssK

r

kjkj

0

0

0

0

10

cos

kj

zss

µ

,

ν ς

π

2

Λ

10

cos

(11)

where ςνj,µk is the coupling constant, even though it can show a slow ‘z’ varying dependence

[16]. From now on, this value will be considered the coupling coefficient and Kνj,µk will not be

used in the formulation for coupled mode differential equations.

Under the assumption of a uniform index perturbation within the core of the fiber, the

interactions occur between LP0k modes. As a result, there is no azimutal dependence and

expression (8) of [14] can be simplified for the coupling coefficient:

( ) ( )

rprn

P

∫

0

2

Otherwise, the LP modes of higher azimutal order should be treated as two independent

modes. An important reduction in computation time can be obtained if only half of the

coupling coefficients are calculated. This can be obtained if the symmetry property is used:

ς0j,0k = ς0k,0j.

2.3. Coupled mode equations

( )

r

( )rdrrRRd

r

r

kjkj

∫

==

=

1

0

00

2

0

110

0

0 ,0

π

φ

φ

ωε

ς

(12)

After deduction of the propagation constants of the modes and the coupling coefficients,

coupled mode theory will be introduced. Unlike in FBGs, backward propagating modes will

be neglected. In this way, the generalized coupled mode equations describing an LPG can be

expressed as [14, 16]:

( )

( )

∑

=

j

dz

1

This can be expressed in a matrix form in the following way:

)(

0201

.

QV

zF

?

?

()()

−−−=

M

kjjkj

k

zjzFKj

z dF

000 0 ,0

0

exp

ββ

for k=1,2,…M. (13)

=

)(

)(

)(

)(

)(

0

02

01

00 ,020 ,01

02 ,0 02

?

02,01

01,001,

.

0

02

.

01

zF

zF

zF

QVV

V

VVQ

zF

zF

N

NNN

N

?

N

N

?

?

?

?

?

(14)

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where F0j is the normalized amplitude of the j mode, and the differential equation matrix

elements are defined as:

jQ

0

σ

−=

s

zjV

jjj

s

[

z

0 ,

(

00

)(

ζ

)

]

Λ

±−−−=

π

2

ββζσ

exp

2

)(

000 ,

j

0

1

0 ,

j

0

kjkk

jz

(15)

where σ(z) is the slowly varying envelope of the grating, s0 and s1 are the coefficients of the

first two Fourier components of the grating function S(z), β0j is the propagation constant of the

j mode, and Λ is the period of the grating.

The ± sign in the exponential function depends on the sign of the difference between the

propagation constants of the modes, which permits the coupling between each pair of them. If

β0j > β0k, the minus sign is selected; otherwise the plus sign is chosen.

The transmission can be found by assuming that only one mode is incident (F01(0)=1 and

F02(0)=·····=F0N(0)=0) and solving the differential equation. The transmission power at the end

of the LPFG can be expressed as:

F

2

01

2

01

) 0 (

)(

F

L

(16)

where L is the length of the grating.

If only the self coupling coefficients and the cross coupling coefficients between the core

mode and the cladding modes are considered, the computational effort is one order of

magnitude lower, but additional errors can be caused for gratings with strong refractive index

modulation [14]. For this reason, the full matrix formulation is used.

Finally, important reduction in computational effort can be obtained if adequate selection

of modes included in the coupled mode equations is done. A rule suggested [14] is to select

those modes that satisfy the Bragg condition to within a:

( )

001

−

j

λβλβ

( )

13

10

2

Λ

−−

≤−

m

µ

π

(17)

However, it has been proved that in some cases important modes are discarded, which may

lead to important errors. Consequently, this rule must be widened depending on the structure

analysed. Other modes that can be discarded are those whose coupling coefficient

the core mode is lower than 10-7µm-1. Their contribution is negligible for the transmission

spectrum.

kj 0 ,0

ζ

with

2.4. Methods for calculation of resonance displacement

In the analysis of Section 3, one of the main purposes is to see the displacement of the

resonance wavelengths as well as the parameter to measure (i.e., ambient refractive index),

experiments a variation. For this reason, alternative solutions to the application of coupled

mode equations can be used if it is only necessary to analyse the displacement of the

resonance wavelength. In this way computational effort is reduced.

The first one is the calculation of the resonance wavelength with the Bragg condition:

−

βλβ

)(

01

Λ

=

π

2

λ

()

0

j

(18)

where β01 and β0j are the propagation constants of the core and the j cladding modes

respectively, and Λ is the period of the grating. Results obtained present appreciable variations

related to those values calculated with transmission curves. However, if the modified first-

order Bragg condition is applied, errors are lower than 0.1% [14]:

(

−+

λζλβ

)()(

s

)

Λ

=+

π

2

λ

(

ζ

0

s

λ

(

β

))

0 ,

j

0001,01001

jj

(19)

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If this error is compared with fabrication tolerances, it can be concluded that this

approximation offers great advantages in terms of computational effort. Henceforward it will

be used in some cases in next section. After seeing the better accuracy of this formulation

compared to that of expression (18) we can conclude that self-coupling coefficients are more

related to the modification of the resonance wavelength of the attenuation bands, whereas in

expression (1) the cross-coupling coefficients modify the depth.

3. Analysis of LPFG structures with deposition of an overlay

A commercial LPFG is selected for the analysis performed in this section. A modulation of

the core refractive index is induced in a SMF28 single mode fiber. The parameters of the

LPFG are: core diameter of 8.3 um, cladding diameter 125 um, core refractive index 1.5362,

cladding refractive index 1.5306, overlay refractive index 1.67 ([PDDA+/PolyS-119-]), period

of the grating is 276 µm, and the length of the grating is 25 mm. The modulation is considered

sinusoidal. Consequently σ(z) = s0 = s1 = 1. The amplitude of the modulation is 3×10-4.

If an overlay of higher refractive index than the cladding is deposited on this LPFG, as the

thickness of the overlay increases, cladding modes shift their effective index to higher values.

When the overlay is thick enough, one of the cladding modes is guided by the overlay. It is

exactly the highest state of energy that jumps to the overlay, which means that the highest

effective index mode (lowest order cladding mode) becomes guided. This causes a

reorganization of the effective index of the rest of modes. Higher order cladding modes than

the one that is guided by the overlay will shift their effective index value towards the effective

index of the immediate lower order cladding mode. As more material is deposited, the

effective index distribution before deposition is recovered. The effective index of the eight

cladding mode will be now that of the seventh one, the effective index of the seventh cladding

mode will be that of the sixth mode, and so forth. The same is true for the resonance

wavelength values. The phenomenon repeats as more material is deposited. This means that if

the thickness continues to be increased, more modes are guided by the overlay and new

reorganizations of cladding modes takes place. In Fig. 2 the effective index of the core mode

and the first ten cladding modes is represented as a function of the overlay thickness for a

fixed wavelength of 1200 nm. LP02, LP03, LP04, LP05, become guided at 300, 1200, 2100 and

3000 nm.

Around the thickness value at which each of these modes is guided by the overlay, higher

cladding modes shift their effective index value to cover the energy state left by their

predecessors.

This phenomenon can be understood in terms of reorganization of modes. There exist

allowed states for the effective indices of the modes. When the structure is perturbed by the

deposition of an overlay, there exist not-allowed states that coincide with the transition to

guidance of a cladding mode in the overlay. This is confirmed by the fact that fields of

cladding modes present maxima or minima at the interface between the cladding and the

overlay. In other words, the mode profile is mutating to the profile of the lower mode. In Fig.

3, the transverse field as a function of the radius is represented for the LP05 mode at a

wavelength of 1200 nm. Three cases are considered: no overlay, overlay of 300 nm (transition

to guidance of a mode at the overlay), and overlay of 750 nm. It can be appreciated the shift of

the field profile towards that of the LP04 before deposition started.

The immediate consequence of the shift in effective index is that it leads to a displacement

in all the attenuation bands. The attenuation band corresponding with the eighth mode shifts

the wavelength to that of the seventh mode; the same is true for the seventh mode that shifts

the wavelength to the attenuation band of the sixth mode, and so forth. Furthermore, there is

an optimal deposition thickness where the central wavelength shift as a function of the

ambient refractive index will be highest. This is the optimum overlay thickness (OOT). This

value depends mainly on two variables: the refractive index of the overlay, and the ambient

refractive index. Consequently, a good choice for a high sensitive device to the external

refractive index is to stop the deposition when the effective index value of a mode is located

between the effective index of the mode itself before deposition, and that of the next lower

(C) 2005 OSA10 January 2005 / Vol. 13, No. 1 / OPTICS EXPRESS 63

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cladding mode before deposition. This is an approximate solution. To calculate a more exact

value, either the modified Bragg condition or the couple mode equations explained in section

2 will be used.

0 500 100015002000250030003500

1.527

1.5275

1.528

1.5285

1.529

1.5295

1.53

1.5305

1.531

Overlay thickness (nm)

Effective index of LP modes

LP0,3

LP0,4 LP0,5

LP0,6

LP0,7

LP0,8

LP0,9

LP0,11

LP0,10

LP0,2

a)

0500 100015002000 25003000 3500

1.52

1.54

1.56

1.58

1.6

1.62

1.64

1.66

1.68

Overlay thickness (nm)

Effective index of LP modes

LP0,2

LP0,1

LP0,3

LP0,4

LP0,5

b)

Fig. 2. Effective index as a function of the overlay thickness of a) first ten cladding modes and

b) core mode and first four cladding modes.

The modified Bragg condition is much more effective in terms of computation than the

coupled mode equations. So, it is used for calculating the shift of the resonance wavelengths

as a function of the overlay thickness. Three different refractive index materials are analysed

in Fig. 4: 1.57 (tricosenic acid [24]), 1.62 ([PDDA+/PolyR-47-]), and 1.67 ([PDDA+/PSS-]).

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As expected, the guiding starts faster if the overlay refractive index is higher. There are two

important advantages in the usage of a higher refractive index overlay. The first one is that a

lower amount of material is needed for the deposition. For adequate methods for

nanodeposition, such as Electrostatic Self Assembly (ESA) [25], and Langmuir Blodgett (LB)

this saves much time. Secondly, the transition to guidance of a cladding mode in the overlay is

faster. This implies a higher variation of the effective index of the cladding modes as a

function of the ambient refractive index, and a higher shift in the attenuation bands. For these

reasons, the analysis will be continued with an overlay refractive index 1.67. The highest

variation of the resonance wavelength is obtained at 278.5 nm, which is different from the

value calculated for the classical Bragg condition: 267.5. Furthermore, there is a slight shift of

the OOT for each mode considered. For instance, the fourth presents an optimum at 284 nm.

Obviously it is not possible to obtain an optimum for each mode at the same time. However,

the fact that the shift is not very important is proved in the next examples. Even if the

optimum overlay thickness is not exactly fixed for a cladding mode, there is a range of values

that also permits high sensitivities as a function of the ambient refractive index.

1

0 10 2030 4050 6070

-1

-0.5

0

0.5

Transversal field (a.u.)

no overlay

01020304050 6070

-1

-0.5

0

0.5

1

Transversal field (a.u.)

overlay of 300 nm

0 1020 3040 506070

-1

-0.5

0

0.5

1

radius (micrometers)

Transversal field (a.u.)

overlay of 750 nm

Fig. 3. Transverse electric field of the fifth cladding mode for three overlay thickness values: 0,

300, and 750 nm.

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00.10.20.3 0.40.50.60.70.80.91

1.1

1.11

1.12

1.13

overlay thickness (µm)

Wavelength (µm)

00.10.2 0.30.40.50.60.7 0.80.91

1.1

1.11

1.12

1.13

Wavelength (µm)

00.10.20.30.40.50.60.70.80.91

1.1

1.11

1.12

1.13

Wavelength (µm)

refractive index = 1.57

refractive index = 1.62

refractive index = 1.67

Fig. 4. Resonance wavelength shift in third cladding mode, as a function of the thickness of the

overlay. Refractive indices of overlay: a) 1.57, b) 1.62 c) 1.67. Ambient index: 1.

Once the refractive index of the overlay is fixed, the versatility of this phenomenon is

proved by the dependence of the OOT on a second parameter: the ambient refractive index.

This permits to design refractometers sensitive to specific ranges of refractive indices. In Fig.

4 the design has been aimed for instance for gases. The OOT for the third mode is 278.5 nm

for ambient refractive index 1. The purpose of using the OOT of the third cladding mode is to

show that even for the case of lowest wavelength shift, the phenomenon is appreciable. In Fig.

5a, the change for three different ambient refractive indices in the transmission plot of an

LPFG without overlay is compared. The shift in resonance wavelength is nearly

unappreciable: 0.01, 0.03 and 0.05 nm respectively for the third fourth and fifth mode. On the

other hand, if an overlay is added, in Fig. 5(b), dynamic ranges of 4.63 nm, 9.33 nm, and 8.34

nm for the third, fourth and fifth cladding mode resonance are obtained. These values are

obtained even for a non optimum overlay thickness in the case of fourth and fifth resonance.

As a second example an important application is shown: the detection of oils. In this case,

an optimum around an ambient refractive index 1.468 is selected. In Fig. 6 it is shown the

evolution of the resonance wavelength caused by coupling between third, fourth and fifth

cladding modes with the core mode. The decrease in the optimum thickness, as the ambient

refractive index increases, is explained from slab waveguide theory. A slab between two

different media starts guiding for a lower thickness as the refractive index of both media is

more similar. When both media are the same it always guides. This theory can be extrapoled

to cylindrical waveguides. There is a slight influence of the number of the mode in the

optimum overlay thickness, and the range of displacement is higher for higher order modes.

That is the reason for selecting in this case the optimum overlay thickness of the fifth mode:

159 nm. It is remarkable that the shape of the plot is very similar to the experimental results of

[12] (Fig. 3 of such reference). In [12] the shift in fifth and sixth mode resonances of zone C is

actually the sixth and seventh modes shift respectively, and the relative shift should be shifted

to fit the values of zone A. Furthermore, in [19] it is proved theoretical results fit experimental

ones.

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1.081.11.121.141.161.18 1.21.221.24

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Wavelength (µm)

Transmission (dB)

1

1.05

1.1

a)

1.081.11.121.141.161.18 1.21.221.24

-8

-7

-6

-5

-4

-3

-2

-1

0

Wavelength (µm)

Transmission (dB)

1

1.05

1.1

b)

Fig. 5. Transmission spectra of an LPFG as a function of three ambient refractive indices: 1,

1.05 and 1.1 a) without overlay b) with overlay of 278.5 nm and refractive index 1.67.

In Fig. 7(a) it is shown the shift in resonance wavelength of the fifth mode for an LPG

where no overlay is added. Three different oils with refractive indices 1.461 (tee), 1.481 (cod)

and 1.518 (tung) are analysed [26]. In Fig. 7(b), the same is done for an LPFG with overlay

optimized for ambient refractive index 1.468. It can be clearly appreciated that the LPFG with

overlay is also highly more sensitive than the other one in this case where the ambient

refractive index is close to that of the cladding. There is an improvement in the sensitivity of

the device of Fig. 7(a) in comparison with that of Fig. 5(a), but it is still far from the

sensitivity obtained in Fig. 7(b). Between the first and the third refractive index there is a shift

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for the fifth mode of 22.98 nm, whereas for the LPFG without overlay it is of 1.77 nm. A

factor 12.98 of improvement has been obtained. Furthermore, a factor of 70.5 is obtained if

we only consider tee and cod oils, because their refractive indices are closer to the refractive

index the OOT has been calculated for.

00.05 0.10.15 0.20.25 0.30.35 0.40.45 0.5

1.1

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.2

Wavelength (µm)

overlay thickness (µm)

LP05 resonance

LP03 resonance

LP04 resonance

Fig. 6. Resonance wavelength shift in third, fourth and fifth cladding modes, as a function of

the thickness of the overlay. Overlay refractive index: 1.67. Ambient index 1.468.

4. Conclusions

Calculation of the LP modes in a multilayer waveguide and coupled mode equations permits

to obtain the transmission spectrum in an LPFG with an overlay. The approximation is valid

for low order LP modes. This structure completes the other two cases studied so far for LPGs:

ambient refractive index lower than the cladding [14, 16], and higher than the cladding

[10, 17, 18]. Now a thin overlay is placed between the ambient and the cladding.

If the overlay presents a higher refractive index than the core, as its thickness is increased,

it starts guiding the cladding modes with the highest effective index. In the transition of each

of these modes to guidance in the overlay, there is a fast shift of the attenuation bands

obtained in the transmission spectrum. This phenomenon has been analysed in terms of fields

and allowed states of energy. Losses are not considered for the sake of simpler explanation of

the phenomenon. In a next work the effect of losses in the overlay will be presented.

Many consequences are extracted from this. After selecting an overlay with an adequate

refractive index, there is an optimum overlay thickness for each ambient refractive index. This

optimum permits to obtain high sensitivity to low variations of the ambient refractive index

itself. This avoids the limitation of LPFGs without cladding, where the highest sensitivity is

obtained for those indices close to the cladding refractive index. Two examples have proved

this fact. In the second one, even at optimum conditions for the LPG without overlay, the LPG

with overlay offers a much better sensitivity. Other possible applications are the deposition of

a material sensitive to a specific parameter. For instance a material whose refractive index is

sensitive to oxygen will detect much lower concentrations if an overlay is deposited with an

optimal thickness in air ambient conditions than typical LPFGs without deposition.

This technique will permit to extend even more the applications of LPFGs. Furthermore,

the theory can be extrapoled to many sensor fields. So far dielectric structures studied are

static. Here a dynamic structure has been analysed.

(C) 2005 OSA 10 January 2005 / Vol. 13, No. 1 / OPTICS EXPRESS 68

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1.1751.181.185

Wavelength (µm)

1.191.1951.2

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Transmission (dB)

1.461

1.481

1.518

a)

1.141.1451.151.1551.161.1651.171.1751.181.185

-7

-6

-5

-4

-3

-2

-1

0

Wavelength (µm)

Transmission (dB)

1.461

1.481

1.518

b)

Fig. 7. Transmittance spectra for the fifth cladding mode resonance of an LPFG for three oil

refractive indices: 1.461, 1.481, 1.518. a) without overlay b) with overlay of 159 nm and

refractive index 1.67.

Acknowledgments

This work was supported by Spanish Ministerio de Ciencia y Tecnologia and FEDER

Research Grants CICYT-TIC 2003-00909, Gobierno de Navarra and FPU MECD Grant.

(C) 2005 OSA10 January 2005 / Vol. 13, No. 1 / OPTICS EXPRESS 69

#5675 - $15.00 USReceived 8 November 2004; revised 20 December 2004; accepted 20 December 2004