Dynamic evolution of the spectrum of long-period
fiber Bragg gratings fabricated fromhydrogen-loaded optical fiber by
ultraviolet laser irradiation
Keio Fujita, Yuji Masuda, Keisuke Nakayama, Maki Ando, Kenji Sakamoto,Jun-pei Mohri, Makoto Yamauchi, Masanori Kimura, Yasuo Mizutani,
Susumu Kimura, Takashi Yokouchi, Yoshifumi Suzaki, and Seiki Ejima

Long-period fiber Bragg gratings fabricated by exposure of hydrogen-loaded fiber to UV laser light exhibit
large-scale dynamic evolution for approximately two weeks at room temperature. During this time two
distinct features show up in their spectrum: a large upswing in wavelength and a substantial deepeningof the transmission minimum. The dynamic evolution of the transmission spectrum is explained quantitatively by use of Malo’s theory of UV-induced quenching [Electron. Lett. 30, 442 (1994)] followed byrefilling of hydrogen in the fiber core and the theory of hydrogen diffusion in the fiber material. The
amount of hydrogen quenched by the UV irradiation is 6% of the loaded hydrogen. © 2005 OpticalSociety of America
OCIS codes: 050.2770, 060.2310, 060.2340, 160.5340.

1. Introduction

Long-period fiber Bragg gratings (LP-FBGs), whichhave a periodic structure in the refractive index in
the longitudinal direction of the fiber with a period of⬃0.5 mm, convert incident light in the fundamental
LP01 mode into higher-order LP0m modes throughmode coupling. Because the higher-order modes cannot propagate in a single-mode fiber but are lost outside the fiber, a LP-FBG works as an attenuator.
Owing to the wavelength dependence of the attenuation, LP-FBGs are utilized extensively for applications such as equalizing light sources and opticalamplifiers to flatten the wavelength dependence of
their outputs.

K. Fujita, Y. Masuda, K. Nakayama, M. Ando, K. Sakamoto,
J.-P. Mohri, T. Yokouchi, Y. Suzaki, and S. Ejima (ejima@
eng.kagawa-u.ac.jp) are with the Faculty of Engineering, Kagawa
University, 2217-20 Hayashi-cho, Takamatsu, Kagawa 761-0396
Japan. M. Yamauchi, M. Kimura, Y. Mizutani, and S. Kimura arewith the Shinko Electric Wire Company, Ltd., 1298-12 Shido,
Shido-cho, Sanuki, Kagawa 769-2101 Japan.
Received 19 May 2005; revised manuscript received 8 August
2005; accepted 13 August 2005.
0003-6935/05/337032-07$15.00/0© 2005 Optical Society of America
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LP-FBGs are commonly produced by exposure ofan optical fiber to UV laser light in a manner similar
to that used for fabricating short-period fiber Bragg
gratings (SP-FBGs). Their optical spectrum is known
to exhibit large scale dynamic evolution after fabrication. At room temperature, their wavelength increases by several tens to hundreds of nanometersand decrease thereafter. Also, the transmission minimum becomes deeper over that time as if thestrength of the grating modulation were increasing.
Bakhti et al. have shown that the large upswing in
the LP-FBG wavelength is due to diffusion of thehydrogen loaded into the fiber before exposure to UV
laser light to enhance the fiber’s UV photosensitivity.1 The wavelength of the LP-FBG is given by␭ ⫽ 共neff01 ⫺ neff0m兲⌳,(1)where neff01 and neff0m are the effective refractive indices for the LP01 and LP0m modes and ⌳ is the grating period. Because the intensity of fundamental
mode LP01 is concentrated in the fiber core, neff01
should be close to the index of the core, ncore. Similarly, as the intensity of higher mode LP0m is distributed over the fiber’s cross section, neff 0m reflects theindex of the fiber cladding, nclad because the crosssectional area is dominated by the cladding.
The hydrogen loaded into the fiber starts to diffuse
out from the fiber when it is exposed to the laser light.
During this outdiffusion the hydrogen concentration
decreases initially in the outer portion of the fiber
cladding and later in the central fiber core. Consequently, as the hydrogen concentration evolves, thecomponent of the refractive index that is due to thepresence of hydrogen causes first neff0m and then neff01
to decrease. In Eq. (1), the second term decreases
early in time and the first term decreases later on;
this means that the wavelength undergoes an upswing. Bakhti et al. have demonstrated that the
wavelength evolves in this way by comparison withtheoretical calculations.1 The behavior of the transmission minimum of the spectrum has not, however,been explained.
An upswing of the wavelength is also seen in SP-FBGs
but for a different physical reason. The wavelength of a
SP-FBG fabricated by exposure of hydrogen-loaded fiber
to UV laser light exhibits an increase within ⬃5 h and adecrease by 0.7–1 nm to the initial value after ⬃200h at room temperature. Malo et al. explained thisupswing in the wavelength in terms of quenching ofhydrogen at the fiber core.2 The UV irradiation induces photodissociation of molecular hydrogen. Someof the dissociated atoms react in the fiber core to forma Ge–OH bond followed by creation of oxygen deficiencies, while the rest of the atoms recombine intomolecules. The consumption of hydrogen in Ge–OH
formation has the effect of quenching hydrogen molecules in the fiber core. In the cladding region thehydrogen concentration does not change because thedissociated atoms simply recombine there without
reacting otherwise. Then the hydrogen in the cladding surrounding the core starts to diffuse inward torefill the depleted core region. Thus the wavelength of
the SP-FBG, which is given by␭ ⫽ 2neff01⌳,(2)

increases through effective refractive index neff 01 by
an amount that corresponds to the replacement of thehydrogen in the depleted core. Later the wavelength
decreases as all the hydrogen diffuses out of the fiber.
We have found experimentally and theoreticallythat the evolution of both the wavelength and thetransmission minimum of the spectrum of a LP-FBG
can be explained in terms of the quenching and replacement of hydrogen in the core. The amount ofhydrogen depleted by quenching in the core turns out
to be small, as low as 6%, in LP-FBGs, unlike Malo’s
result for SP-FBGs, which was as high 80%.
In this paper, after describing the conditions for
LP-FBG fabrication, we present experimental observations of the evolution of their transmission spectrum over a period of 500 h at room temperature. We
then explain our physical model for the dynamic evolution of the spectrum. Using theoretical calculationsof hydrogen diffusion and comparing the experiments

Fig. 1. Transmission spectrum of a LP-FBG. The evolution of theLP04 mode is tracked and analyzed in these experiments.

with a fiber Bragg grating’s simulation code, we showthat the dynamic evolution can be explained entirely
by our model. The fraction of the hydrogen that isquenched in the fiber core by UV irradiation is discussed briefly.
2. Experiment

The optical fiber used in these experiments is standard germanium-doped single-mode fiber (SMF-28,
Corning). To increase its UV photosensitivity, we
loaded hydrogen gas into the fiber at 100 atm at room
temperature for 10 days and then stored it in afreezer at ⫺70 °C before use. The resin coating wasremoved just before exposure to the UV laser light.
We fabricated a LP-FBG by exposing the fiber to KrF
excimer laser light through multiple slits in thinstainless steel with a 50% opening factor and a
518 ␮m period. The fiber holder together with themultiple slits was moved across the laser beam in thelongitudinal direction of the fiber. The beam size was
15 mm ⫻ 15 mm, and the total length of the LP-FBG
was 40 mm, with 77 periods, each exposed for 45 s.
The laser fluence was 280 共mJ兾cm2兲兾pulse at the position of the fiber with a 20 Hz repetition rate. A weaktension of 30 g at. wt. was applied to the fiber to keep
it stretched.
After exposure of the fiber to the UV laser, we kept
the fiber at room temperature and tracked the dynamic evolution of its spectrum for more than 500 h.
Figure 1 shows the transmission spectrum for wavelengths of 1300–1700 nm, measured with a whitelight source after completion of the experiments
described below. In this study we tracked the LP04
mode at ⬃1480 nm after identifying the mode number as described previously,3 using a wideband strong
light source (Thorlabs-ASE-7701-AP).
The LP-FBG had a transmission minimum of ⫺2.2 dB
at a wavelength of 1485 nm just after fabrication by
exposure to UV laser light, as shown by curve (A) of
Fig. 2. The spectrum evolved from (A) to (B) in
160 min, to (C) after 1500 min 共25 h兲, and reached asteady-state spectrum (D) after 30,000 min 共500 h兲.
The upswing in the wavelength is large, nearly
80 nm. The initial transmission minimum of ⫺2.2 dBlater reached ⫺5 to ⫺6 dB. The wavelength and thetransmission minimum are plotted in Figs. 3(a) and3(b), respectively.
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7033

and in the cladding by an amount that is due to theloaded hydrogen. The profile shape is the same asthat without hydrogen loading.
B.

Fig. 2. Dynamic evolution of the transmission spectrum of LPFBG. The trajectory of its minimum is shown by closed-circle data
points.

3. Model for the Evolution of the LP-FBG Spectrum

Our physical model for the dynamic evolution of the
LP-FBG is illustrated schematically in Fig. 4. The
radial distributions of the molecular-hydrogen concentration are depicted on the left-hand side of the
figure for six stages, along with the corresponding
radial distributions of the refractive index at theright. The magnitudes in the figure are not to scale
but are drawn for ease in understanding of the physical model. The stages in the evolution of the spectrum depicted in Fig. 4 are as follows:A.

Stage 1

Stage 1 is before exposure to UV laser light. Hydrogen molecules are distributed uniformly across thefiber as shown in (H-1) at the left in Fig. 4. The
refractive index is raised uniformly both in the core

Stage 2

Immediately after exposure of the fiber to UV laser
light, which we refer to as time slice A, the refractive
index is raised at the core. In this stage, hydrogenmolecules are dissociated by the UV laser light. In thefiber core a fraction of the dissociated atoms reacts
with dopant germanium to form a Ge—OH bond,while in the fiber cladding they just recombine toform hydrogen. Thus hydrogen is consumed in thefiber core and its concentration decreases there; thatis, quenching takes place as shown by (H-2) in Fig. 4.
The refractive index increases in the core owing to thephotosensitivity of germanium to UV irradiation. At
the same time, the index simultaneously decreases by
an amount proportional to the amount of hydrogenquenched at the fiber core [indicated by the dottedline (R-2)]. The longitudinal distributions of the hydrogen concentration and the refractive index in thefiber core are shown in Fig. 5.
C.

Stage 3

Hydrogen molecules migrate into the depleted fibercore from the surrounding region of the fiber cladding
by diffusion. The amplitude of modulated refractive
index ⌬nmod increases until it returns to the level thatexisted before the quenching of molecular hydrogenby the UV laser light. The time after which hydrogen
filling has been completed corresponds to time slice B.
D.

Stages 4 and 5

After completion of hydrogen refilling into the core,
the hydrogen molecules in the cladding and in thecore start to diffuse out of the fiber. We define timeslice C as the time of the maximum difference between on-axis hydrogen concentration and that averaged over the cross section, as described in Section 4
below.
E.

Stage 6

When the hydrogen molecules have all diffused outward, the refractive index settles down to its value
without hydrogen, as depicted by (R-6) in Fig. 4.
Next we examine this model quantitatively withthe aid of hydrogen diffusion theory and a simulation
code for fiber Bragg gratings.
4. Theory of Hydrogen Diffusion

The diffusion of hydrogen molecules can be describedby the diffusion equation⭸C⫺ Dⵜ2C ⫽ 0,⭸t
Fig. 3. (a) Wavelength and (b) transmission minimum of the LPFBG spectrum plotted against time. Filled diamonds A–D correspond to the time slices shown in Fig. 4. Their magnitudes are theresults of computer simulation.
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APPLIED OPTICS 兾 Vol. 44, No. 33 兾 20 November 2005

(3)

where C is the hydrogen concentration and D is the
diffusion coefficient. Coefficient D for hydrogen in silica glass is given by Lemaire4 asFig. 4. Model for the evolution of the spectrum. The radial profiles of the hydrogen concentration (left) and the refractive index (right)
are shown for six stages. A–D correspond to time slices discussed in the text.

冉D 关cm2兴兾s ⫽ 2.83 ⫻ 10⫺4 exp ⫺冊

40.19 关kJ兾mol兴
,
RT
(4)

where R ⫽ 8.31 共J兾K兲 mol is the gas constant and T
is given in degrees kelvin. A rigorous solution in cylindrical coordinates is given by Crank.5 For boundary conditions of
C ⫽ C1,C ⫽ C0,r ⫽ a,r ⬍ a,t ⱖ 0,t ⫽ 0,(5)

the solution is2 ⬁ J0共r␣n兲
C ⫺ C0⫽1⫺ 兺
exp共⫺D␣n2t兲,C1 ⫺ C0a n⫽1 ␣nJ1共a␣n兲

(6)where ␣n is determined from the zero point of theBessel function, J0共␣na兲 ⫽ 0. For C1 ⫽ 0 and C0 ⫽ 1 we
get the solution for outward diffusion from a profile
with an initial uniform distribution. For C1 ⫽ 1 andC0 ⫽ 0 we obtain the solution for inward diffusion.
Calculated profiles for the radial distribution are
shown for several values of Dt兾a2 in Figs. 6(a) and6(b) for outdiffusion and indiffusion, respectively.
The amount of hydrogen that diffuses out from thefiber, M共t兲, can be obtained by integration of the concentration with respect to time and is given by⬁4
M共t兲⫽ 1 ⫺ 兺 2 2 exp共⫺D␣n2t兲,M共⬁兲n⫽1 a ␣n(7)

where M共⬁兲 is the total amount that diffuses out.
The time to refill the depleted core after quenching(stage 3) can be estimated as the inward diffusion
time trefill ⬃ acore2兾D. For a core radius acore ⫽4.5 ␮m, trefill becomes 157 min. Similarly, for a cladding radius aclad ⫽ 62.5 ␮m, the time to complete theoutward diffusion, that is, to finish stage #6, istoutdiff ⫽ aclad2兾D ⬃ 30,300 min. Here diffusion coefficient D is 2.15 ⫻ 10⫺15 m2兾s according to Eq. (4) for
22 °C. Refilling time trefill ⫽ 157 min corresponds totime slice B, and the time for outward diffusion,
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7035

Fig. 7. Hydrogen concentrations during diffusion: Con-axis is the
on-axis value, Caverage is the value averaged over the fiber’s crosssection, and Cdifference is the difference between the two.
Fig. 5. Hydrogen concentration and refractive index along theaxis of the fiber in our model.

toutdiff ⫽ 30,300 min, corresponds to time slice D in
our model of Fig. 4.
The transmission minimum becomes deeper as theamplitude of the refractive-index modulation increases from that of (R-2) in Fig. 4 to that of (R-3)
during refilling of the core until to time B. Thus, thetransmission minimum started at ⫺2.2 dB (time sliceA), deepened to ⫺5 to ⫺6 dB at time slice B, andremained nearly the same thereafter, as shown in
Fig. 3(b).
When hydrogen molecules diffuse out of the fiber instages #4 and #5, the concentration decreases first at
the outside of the fiber cladding and then in the core.
The evolution of the radial profile of the hydrogenconcentration can be analyzed by use of Eqs. (6) and(7). The result is shown in Fig. 7, which shows theaverage concentration Caverage calculated from Eq. (7)and the on-axis concentration according to Eq. (6).
Cdifference, the difference between the two, reaches amaximum at the time when Dt兾aclad2 ⫽ 0.08 and decreases thereafter. The wavelength of the LP-FBG isgiven by Eq. (1). neff01 follows closely the refractive
index of the fiber core, and neff0m follows the averageof the refractive index over the cross section. Therefore the effect of hydrogen on the wavelength should
yield a wavelength evolution similar to that of thetime dependence of Cdifference shown in Fig. 7. The
time that corresponds to Dt兾aclad2 ⫽ 0.08 gives tpeak⫽ 2430 min. After this tpeak (time slice C), the wavelength should decrease.
The results of this analysis of the temporal behavior are shown in Figs. 3(a) and 3(b). There the filled
diamonds show the calculations for times A–D, which
agree well with the experimental times (Fig. 8). In
Section 5 we analyze the magnitudes of the wavelength shifts and changes in the transmission minimum, using a computer simulation code.
5. Simulation

We used the IFO_grating code from the Optiwave
Corporation.6 This code calculates the transmission
and reflection spectra for specified fiber conditions,
i.e., the refractive-index distributions in the fiber coreand cladding and the Bragg grating conditions, i.e.,
grating period ⌳, modulated refractive index in thecore ⌬nmod, and grating length L. The fiber conditions
are defined by the specifications of SMF-28 singlemode fiber. The specific difference of the refractive

Fig. 6. Radial profiles of hydrogen concentration during outward
diffusion and inward diffusion.
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APPLIED OPTICS 兾 Vol. 44, No. 33 兾 20 November 2005

Fig. 8. Radial profiles of the refractive index. A–D correspond to
the time slices in Table 1 and Fig. 4.

Table 1. Refractive Indices for Four Time Slices, A–D

Refractive Index
Time
Slice
A
B
C
D

Stage (Time)stage 2(t ⫽ 0 min)end of stage 3(t ⫽ 157 min)stage 5(t ⫽ 2430 min)stage 6(t ⫽ 30,300 min)Conditionncore, 0 ⬍ r ⬍ 4.5 ␮mncladding, 4.5 ⬍ r ⬍ 62.5 ␮m⌬nmodulation

1.449,050

1.444,300

0.000,075

Dt兾acore2 ⫽ 1

1.449,050

1.444,300–1.443,700

0.000,111

Dt兾aclad2 ⫽ 0.08

1.449,000

1.444,250–1.443,700

0.000,111

Dt兾aclad2 ⫽ 1

1.448,450

1.443,700

0.000,111

Difference between
A and DncoreA–D⫽ 0.000 600

index is ⌬ ⫽ 共ncore ⫺ nclad兲兾ncore ⫽ 0.36%, with nclad⫽ 1.4437 at 1550 nm. The core and cladding diameters are 2acore ⫽ 9 ␮m and 2aclad ⫽ 125 ␮m, respectively. Outside the fiber, air cladding with n ⫽ 1 isassumed. Rectangular modulation of the grating was
assumed, and we adjusted its amplitude by fitting toexperiment. The grating’s period is ⌳ ⫽ 518 ␮m, andits total length is L ⫽ 40 mm.
The refractive indices ncore and nclad just after the
fabrication (time slice A) were determined by fittingof the computed wavelength to that of experiments.
We determined ⌬nmod by fitting the transmission minimum to experimentally observed values. These values are listed in Table 1 for time slice A, and thespectrum is shown in Fig. 9.
Two parameters must be clarified here. One is nH2,
the refractive index caused by hydrogen. The other is␦nH2⫺quench the fraction of nH2, lost by quenching during exposure to the UV laser light. The radial distribution of hydrogen can be obtained from Fig. 6. At
time A, the refractive indices ncore and nclad include
nH2, and ⌬nmod is reduced by the quenched fraction␦nH2⫺quench. By increasing ⌬nmod from the value fortime A, i.e., 0.075 ⫻ 10⫺3, we adjusted the transmission minimum to fit the experimental value at time B,
which yields ⌬nmod ⫽ 0.111 ⫻ 10⫺3. This value should
remain constant thereafter in our model. The increase in ⌬nmod from A to B raised the wavelength by⬃20 nm. The contribution to the refractive index ow-

ncladdingA–D⫽ 0.000 600⌬ncoreA–D⫽ 0.000 036

ing to outdiffusion of hydrogen is given by
nclad共r, t兲 ⫽ nclad D ⫹ nH2共r, t兲⫽ nclad A ⫹ nH2关C共r, t兲 ⫺ C0兴兾C0,(8)

where ncladD is the cladding index without hydrogen,i.e., at time D. Using the radial distribution of thehydrogen concentration C共r, t兲兾C0 and adjusting its
magnitude by varying nH2 to fit the wavelength attimes C and D consistently, we obtainnH2 ⫽ 0.600 ⫻ 10⫺3.

(9)In this calculation, ⌬nmod is the value for time B, i.e.,
0.111 ⫻ 10⫺3. This value of nH2 agrees well with ourprevious result 共6.70 ⫻ 10⫺4兲.3 It should be noted thatthe change in the refractive index caused by the hydrogen loading, nH2, is large, approximately 5–6 times
greater than ⌬nmod induced by UV irradiation. These
results are summarized in Table 1. The differences inncore and nclad between times A and D both correspond
to nH2. The increase in ⌬nmod from time slice A to B
corresponds to the amount of quenching by the UV
irradiation, that is,␦nH2⫺quench ⫽ 0.036 ⫻ 10⫺3 ⬵ 6% ⫻ nH2.

(10)

6. Discussion

Fig. 9. Transmission spectra obtained by computer simulation.

The evolution of LP-FBGs after fabrication by exposure to UV lasers after hydrogen loading is characterized by two distinct features: a large wavelength
upswing and a deepening of the transmission minimum as if the modulation of the grating had gotten
stronger. Bakhti et al. have proved that the former isdue to outdiffusion of the hydrogen that was loaded
into the fiber to enhance its UV photosensitivity. But,
to the best of our knowledge, the behavior of thetransmission minimum was not explained before. Using the theory of Malo et al.,2 according to which thehydrogen in the fiber core gets quenched by UV
irradiation and is subsequently refilled by inward
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7037

diffusion from surrounding cladding, we have demonstrated that the increase in ⌬nmod attributable tohydrogen refilling causes the transmission minimum
to decrease.
The amount of the refractive index of hydrogenmolecules has turned out to be nH2 ⫽ 0.600 ⫻ 10⫺3 in
our analysis as described in Eq. (9). This is the index
increased by the dissolved hydrogen saturated by its
loading at 100 atm. In Malo’s experiment2 the index
increase is 0.05% of the effective refractive index
of the optical fiber used, which becomes 1.444 ⫻
0.05%, or 0.7 ⫻ 10⫺3 with 100 atm loading 共1 mol. %saturation) and with SMF28 optical fiber. The agreement of our index increase with Malo’s is quite
remarkable.
The fraction of hydrogen molecules quenched by
UV irradiation is roughly 6% relation [Eq. (10)] of
the hydrogen load. Malo’s experiment for SP-FBGs
yielded a fraction much larger than ours, i.e., approximately 70–80%, as shown in Fig. 1(a) of Ref. 2. The
difference is likely caused by the magnitude of theirenergy of UV irradiation, which appears to be muchlarger than ours. Because the index change owing to
the loaded hydrogen, i.e., nH2 in our analysis, is approximately 5–6 times greater than ⌬nmod induced by
the UV irradiation, if approximately 70–80% of hydrogen were quenched, ⌬nmod could easily become
negative, which would means that the grating phase
could be inverted in the longitudinal direction of thefiber. If this occurred, the transmission minimum
should go to zero first and grow later. Of course, nosuch evolution has been observed.
The value of 6% for the quenching must be closely
related to the efficiency of the photoinduced refractiveindex increase. Theoretical analysis evaluating thisparameter quantitatively is needed for an understanding of the efficiency of UV irradiation.
7. Conclusions

The transmission spectra of LP-FBGs fabricated byexposure of hydrogen-loaded fiber to UV laser light
manifest a large-scale evolution in both their wavelength and their transmission minima during approximately two weeks at room temperature. The
wavelength experiences a large upswing, and the7038

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transmission minimum deepens as if the grating
strength had increased. Bakhti et al.1 have shownthat the evolution of the wavelength is caused byoutdiffusion of hydrogen from the fiber, but the evolution of the transmission minimum was not explained. Using the theory of Malo et al.2 forquenching and refilling of hydrogen in SP-FBGs, wehave explained the dynamic evolution of the transmission minimum for LP-FBGs. That is, just after
exposure to the UV laser, the modulation of refractive
index ⌬nmod is reduced by an amount induced by hydrogen quenching in the fiber core owing to UV irradiation. Subsequently the hydrogen is replaced byinward diffusion of hydrogen from the surrounding
cladding. This refilling effectively increases ⌬nmod
enough to recover the previous value without hydrogen. This causes a substantial deepening of the transmission minimum. We have found that the amount ofhydrogen quenched by the UV irradiation is 6% of thehydrogen load.
We thank D. H. McNeill for his careful proofreading of our manuscript.
References
1. F. Bakhti, J. Larrey, P. Sansonetti, and B. Poumellec, “Impact
of hydrogen in-fiber and out-fiber diffusion on central wavelength of UV-written long period gratings,” in Bragg Gratings,
Photosensitivity and Poling in Glass Fibers and Waveguides:
Fundamentals and Applications, Vol. 17 of OSA Technical Series (Optical Society of America, 1997), pp. 55–57, paper
BsuD4-1.
2. B. Malo, J. Albert, K. O. Hill, F. Bilodeau, and D. C. Johnson,“Effective index drift from molecular hydrogen diffusion inhydrogen-loaded optical fibres and its effect on Bragg grating
fabrication,” Electron. Lett. 30, 442– 444 (1994).
3. Y. Masuda, M. Nakamura, C. Komatsu, K. Fujita, M. Yamauchi, M. Kimura, Y. Mizutani, S. Kimura, Y. Suzaki, T. Yokouchi,
K. Nakagawa, and S. Ejima, “Wavelength evolution of fiber
Bragg gratings fabricated from hydrogen loaded optical fiber
during annealing,” J. Lightwave Technol. 22, 934 –941 (2004).
4. P. J. Lemaire, “Reliability of optical fibers exposed to hydrogen:prediction of long-term loss increases,” Opt. Eng. 30, 780 –789
(1991).
5. J. Crank, The Mathematics of Diffusion (Clarendon, 1975),
p. 73.
6. “IFO_Grating,” integrated and fiber optical gratings design software program (Optiwave Corporation, Ottawa, 2001).