PECULIARITIES OF THE THERMO-OPTIC COEFFICIENT
AT HIGH TEMPERATURES IN FIBERS
CONTAINING BRAGG GRATINGS
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Igor Fedin
August, 2011 PECULIARITIES OF THE THERMO-OPTIC COEFFICIENT
AT HIGH TEMPERATURES IN FIBERS
CONTAINING BRAGG GRATINGS
Igor Fedin
Thesis
Approved: Accepted:
______Advisor Dean of the College Dr. Sergei F. Lyuksyutov Dr. Chand K. Midha
______Faculty Reader Dean of the Graduate School Dr. David S. Perry Dr. George R. Newkome
______Department Chair Date Dr. Robert R. Mallik
ii ABSTRACT
The temperature dependence of thermo-optic coefficient in silica-based fibers containing fiber Bragg gratings (FBGs) includes thermal instability of chemical composition gratings, non-linear temperature dependence of FBGs written in different fibers, quadratic behavior of FBGs, and long-term stability of silica-based FBGs.
Experimental measurements of the thermo-optic coefficient for the temperature interval
50 – 7800C in fused silica fiber containing FBGs were conducted while the temperature shift of the Bragg’s peak was monitored between 1300 and 1311 nm with sub-Angstrom precision. Numerical computations were focused on the FBG’s diffraction efficiency calculations accounting for the temperature drift of the gratings and found to be in excellent agreement with obtained experimental data. It has been found that the thermo- optic coefficient changes between 0.79×10-5 and 1.45×10-5 K-1 and undergoes a minimum in the vicinity of 440°C. Additional observation indicates a negative sign of the second- order thermo-optic coefficient. The experiments reveal that the grating reflectivity decays at temperatures higher than 6600C which correlates with calculated decay of the refractive index modulation. It suggests that an FBG is erased at high temperatures.
Based on the energy dispersive spectroscopy it has been determined that thermal erasing of the FBGs at a temperature around 780°C correlates well with germanium sublimation
(apparently in the form of germanium monoxide) out of silica-based fiber cores.
iii ACKNOWLEDGEMENTS
I acknowledge the attention and valuable comments of Professor Robert R. Mallik and
Professor David S. Perry related to this project.
I would like to thank my co-workers that helped me during this project: Ujitha
Abeywickrema, Dr. Ivan Dolog (UA), and Dr. Mindaugas Rackaitis (Bridgestone
Americas)
I am very thankful and grateful to my scientific advisor Dr. Sergei F. Lyuksyutov for his guidance during this project, help, and support throughout two years of my studies and research. Without him this project would never be possible.
iv TABLE OF CONTENTS
Page
LIST OF TABLES ...... vii
LIST OF FIGURES ...... viii
CHAPTER
I. INTRODUCTION ...... 1
II. BACKGROUND ...... 3
2.1. Optical Fibers ...... 3
Geometrical Optics Prospectives ...... 3
Wave Optics Prospectives...... 4
Connection to Geometrical Model ...... 9
2.2. Fiber Grating: Historical Background ...... 10
2.3. Photosensitivity in Fibers ...... 12
Origins of Photosensitivity ...... 12
Ways to Enhance Fiber Photosensitivity ...... 17
Models for the Photoinduced Refractive Index Change ...... 19
2.4. Methods for External Grating Inscribing ...... 21
Interferometric Fabrication Technique ...... 21
Phase-Mask Technique ...... 24
Other Methods of Grating Fabrication ...... 25
v 2.5. Quantitative Description of Gratings ...... 26
Definition of Gratings and Resonance Conditions ...... 26
Coupled Wave Equations ...... 29
Bragg Gratings as Temperature Sensors ...... 32
2.6. Thermal Decay of FBGs ...... 33
III. EXPERIMENTAL PROCEDURE ...... 35
3.1. Fiber Optics Setup...... 35
3.2. SEM and EDS of FBGs ...... 36
IV. CALCULATION METHOD ...... 38
4.1. Calculation of Thermo-optic Coefficient ...... 38
4.2. Calculation of Other Fiber, Modal, and Grating Parameters ...... 40
V. RESULTS AND DISSCUSION ...... 42
5.1. Grating Response as a Function of Temperature ...... 42
5.2. Results for Thermo-Optic Coefficient ...... 43
5.3. Results for Fiber and Mode Parameters ...... 47
5.4. Grating Decay at Elevated Temperatures ...... 50
5.3. Suggested Mechanism of Grating Decay ...... 51
VI. CONCLUSIONS ...... 55
REFERENCES ...... 57
vi LIST OF TABLES
Table Page
5.1.1 The values of the fitting parameters, and the calculated parameters Bi΄...... 43
5.3.1 The fiber and mode parameters for the two sustained modes: LP01 and LP11...... 47
5.3.2 The resonant wavelengths for possible mode couplings ...... 49
vii LIST OF FIGURES
Figure Page
2.1.1 A schematic of optical fiber is shown...... 3
2.1.2 A diagram for a ray launched into a fiber is shown...... 4
2.1.3 Cartesian (x, y, z) and cylindrical (r, φ, z) coordinates in optical fiber are introduced...... 4
2.1.4 Electric field patterns for several lowest LP modes are shown...... 9
2.1.5 A vector model for the eigenvalue u and propagation constant β ...... 9
2.3.1 Absorption spectrum of germania-silica glasses in the UV ...... 15
2.3.2 The diagram exhibits the way of GeE΄, Ge(1) and Ge(2) centers formation from the Ge-Si “wrong bonds” ...... 15
2.3.3 Schematic energy diagram showing relevant defect levels and UV– induced photochemical reactions in GeO2-SiO2 glasses...... 16
2.3.4 Illustration of the dipole model ...... 20
2.4.1 A schematic of amplitude-splitting interferometer for grating formation ...... 22
2.4.2 Schematic of the prism interferometer for the grating fabrication ...... 23
2.4.3 Schematic of Lloyd interferometer for grating formation ...... 23
2.4.4 Schematic of the phase-mask technique for grating fabrication ...... 24
2.4.5 Setup for point-by-point grating fabrication ...... 26
2.5.1 A grating inscribed in the fiber core is shown ...... 26
2.5.2 A schematic of the refractive index perturbation is shown...... 27
viii 2.5.3 Diffraction of light by a grating is shown ...... 28
2.6.1 Diagram for the proposed model for grating decay ...... 33
3.3.1 Schematic presentation of experimental setup ...... 36
5.1.1 The variation of grating response with temperature ...... 42
5.2.1 The experimental data for the temperature range 50 – 7800C and the polynomial (cubic) fit to it for the temperature range 50 – 6600C ...... 44
5.2.2 Calculated refractive index as a function of temperature ...... 46
5.2.2 Calculated thermo-optic coefficient as a function of temperature ...... 46
5.3.1 Colormaps of intensity distribution in LP01 (top) and LP11 (bottom) modes ...... 48
5.4.1 Experimental and calculated FBG spectra for two different temperatures ...... 50
5.4.2 Decay of the effective “AC” refractive index modulation 0 Δnac at temperatures higher than 700 C is shown...... 50
5.5.1 (a) SEM image of a FBG core; (b) Distribution of Si, (c) O, and (d) Ge inside the fiber core with the FBG...... 52
5.5.2 EDS images of extracted spectra for a sample (a) before the thermal treatment and (b) after the sample was annealed at 1000°C for 30 minutes...... 53
ix CHAPTER I
INTRODUCTION
It has been more than 30 years since grating formation in a germanium-doped fiber core as a result of fiber photosensitivity was first described by K. Hill and co- workers [1, 2]. Grating fabrication at the industrial level became possible after the discovery of the external imprinting technique done by G. Meltz et al. in 1989 [3]. Over the last 30 years various efficient grating fabrication techniques have been elaborated [4–
6], industrial grating manufacturing has been streamlined, possible mechanisms of grating formation have been extensively studied [6], formal quantitative description of gratings has been provided [7], and various applications of fiber grating sensors have been established [8–10].
Fiber gratings have been widely used as strain, temperature, pressure and acoustic, current and dynamic magnetic field sensors etc. [8–10].
Temperature sensors based on fiber gratings are light, simple, sensitive, highly adaptable, intrinsic sensing elements. For this reason they can be used in aeronautics as to sense the engine temperature [9]. Variation of fiber, mode, and grating parameters with temperature causes changes in grating response, which is the idea of temperature sensing.
In order for the sensors to efficiently measure temperatures inside engines, they must tolerate prolongated exposures to harsh environments, namely, temperatures of 10000C
1 and higher. However, under such conditions, some chemical and structural changes in the core can cause dramatic refractive index changes and/or grating distortion and disappearance. In fact, until recently, the problem of grating distortion had been a serious limitation in high temperature sensing [11,12].
The goals and objectives of this work are to investigate the peculiarities of the thermo-optic coefficient as a function of temperature; to determine fiber and mode parameters from Bragg wavelength vs temperature data; to describe quantitatively grating behavior at elevated temperatures; to investigate factors that influence grating distortion at high temperatures.
2 CHAPTER II
BACKGROUND
2.1. Optical Fibers
2.1.1. Geometrical Optics Prospectives
An optical fiber (Fig. 2.2.1) consists of an inner cylinder of certain refractive index n1, called the fiber core, and an outer cylinder of a smaller refractive index n2 < n1, called the fiber cladding.
air (vacuum) n0
cladding n2
core n 1 2a
Figure 2.1.1. A schematic of optical fiber is shown. The refractive indices of the core, cladding and surroundings (air or vacuum) are indicated as n1, n2, n0 respectively; n0 < n2
< n1. The radius of the core is a.
In the geometrical model, a light beam injected into the fiber at an angle smaller or equal ζ0 (2.1.1) due to the total internal reflection propagates inside the fiber core only
(Fig. 2.1.2).
3
n0
n2
θ n 2 1 θ1
θ0
Fig. 2.1.2. A diagram for a ray injected into a fiber is shown. The total inner reflection occurs at the boundary between the core and the cladding when
1 2 2 0 arcsin n1 n2 (2.1.1) n0
2.1.2. Wave Optics Prospectives
n 2
n1 φ z
y r x Figure 2.1.3. Cartesian (x, y, z) and cylindrical (r, φ, z) coordinates in optical fiber are introduced. For the step profile fiber n1 and n2 are independent of the coordinates.
We consider the propagating light beam as a traveling wave. The electric and magnetic field vectors E and H are solutions to Maxwell’s equations. The spatial dependent parts E and H , and the oscillatory term eit are separated (2.1.2, 2.1.3).
4 Ex, y, z,t ReEx, y, zeit (2.1.2) Hx, y, z,t ReHx, y, zeit (2.1.3) The energy density flow is given by the Poynting vector S (2.1.4). Intensity I is defined as z component of the time averaged Poynting vector (2.1.5, 2.1.6). Power P is simply the
intensity integrated over the area A perpendicular to the propagation direction (2.1.7). S E H (2.1.4)
1 S ReE H * (2.1.5) t 2 I S zˆ (2.1.6) t P S dA (2.1.7) t A
In fact, for the optical fiber configuration, there are distinct solutions to
Maxwell’s equations called fiber modes. Thus, the electric and magnetic field vectors can be represented as a linear combination of forward propagating (subscribed by j), backward propagating (subscribed by – j) and radiation (rad) modes (2.1.8, 2.1.9). Ex, y, z a j E j x, y, z a j E j x, y, z Erad x, y, z (2.1.8) j j Hx, y, z a j H j x, y, z a j H j x, y, z H rad x, y, z (2.1.9) j j
Any two different forward propagating modes (j and j΄) are orthogonal as well as any two backward propagating modes (2.1.10). Radiation modes are orthogonal to both forward and backward propagation modes (2.1.11). For two identical modes j the normalization coefficient Nj is calculated (2.1.10).
5 1 * E H zˆdA N (2.1.10) 2 j j j jj A
1 1 E H * zˆdA E H * zˆdA 0 (2.1.11) 2 rad j 2 j rad A A
Hence, (2.1.8) and (2.1.9) can be viewed as an expansion of the fields in a set of orthogonal modes. Furthermore, because of the modal orthogonality, the power is transferred by every mode individually, and in the absence of radiation the net bound power flow is given by (2.1.12).
2 2 P a j N j a j N j (2.1.12) j j
In the quantitative description of fiber modes we define a normalized frequency of the fiber, or so called V-number:
2 2 2 2 2 2 2 V a n1 n2 k0a n1 n2 a k1 k2 , (2.1.13) 0 where λ0 is the wavelength in vacuum, k0, k1 and k2 are the wavenumbers in vacuum, core and cladding respectively. As we can see, the V-number depends on the fiber parameters and the light wavelength. It determines the number and types of allowed modes in a fiber.
For an infinitely long fiber with uniform core and cladding refractive indices,
Maxwell’s equations can be solved analytically and the relevant expressions for the modes can be found in closed form. We, however, shall consider approximate linearly polarized (LP) modes for weakly-guiding fibers (n1 ≈ n2). In this approximation the wave propagates nearly parallel to the fiber axis, the fields are nearly transverse, and their longitudinal components are nearly zero (2.1.14, 2.1.15).
6 E 0 (2.1.14) z H z 0 (2.1.15) The transverse component of electric field Elm can be expressed as a product of the (r,
ilm z φ)-dependent real factor elm and the propagation term e (2.1.16). ilm z Elm r,, z elm r,e (2.1.16)
The radial part of is given by the Bessel function of the first kind inside the core, and the modified Bessel function of the second kind in the cladding. The continuity of the transverse electric field at the core-cladding boundary for the weak-guiding approximation is imposed in (2.1.17).
r E0 J l U lm cosl r a a elm (2.1.17) J U r E l lm K W cosl r a 0 l lm Kl Wlm a
For a particular V-number Ulm is a distinct number: it is the mth root of the transcendental equation (2.1.18), which is also called the eigenvalue equation [13]:
J U K W l l , (2.1.18) U J l1 U W Kl1 W where W is given by (2.1.19).
W V 2 U 2 (2.1.19)
The propagation constant β that appears in (2.1.16) is related to the eigenvalue U by
(2.1.20).
2 2 U k1 (2.1.20) a
7 The fraction of the modal power in the fiber core (the power confinement factor) ε is define as the ratio of the confined modal power to the total modal power, and for the LP modes can be expressed in closed form (2.1.21).
1 * elm hlm zˆdA 2 U 2 W 2 K 2 W core l (2.1.21) 1 2 2 * V U Kl1 W Kl1 W elm hlm zˆdA 2 A
For isotropic linear media, at every point the magnitude of the magnetic field is proportional to that of the electric field, the direction is set by the cross product (2.1.22):
n H lm zˆ Elm (2.1.22) c0 with n = n1 for the core and n = n2 for the cladding.
As we can see, fiber modes are labeled with two subscripts l and m. l is a non- negative integer that, for a particular V-number, sets the eigenvalue equation (2.1.18) and the order of the Bessel function in (2.1.17). m is a positive integer that determines which root of (2.1.18) is used as the eigenvalue. The eigenvalue Ulm not only appears under the
Bessel functions but also determines the propagation constant βlm.
Not all modes with all possible l and m are sustained in the fiber core. In fact, the number of allowed modes is determined by the V-number. An LPlm mode is sustained in the core if the V-number is greater than the cutoff Vc which is the mth for l > 0 and the
(m-1)st for l = 0 nonzero root of the cutoff equation (2.1.23). The fundamental LP01 mode has no cutoff. If V < 2.4048 (the first root of J0(U)), then only the fundamental LP01 mode is sustained in the fiber.
J l1 U 0 (2.1.23)
8
Figure 2.1.4. Electric field patterns for several lowest LP modes are shown. Loci with positive values of electric field are shown in red, and ones with negative values are shown in blue. This figure is taken from Encyclopedia of Laser Physics and Technology. 9 As we can see from Fig. 2.1.4, every LPlm has l azimuthal nodes and m – 1 radial nodes in its electric field and, therefore, intensity pattern.
Besides the eigenvalue Ulm and the propagation constant βlm, another important characteristic of every allowed mode is the effective refractive index defined by (2.1.24)
[13].
2 U 2 neff n1 (2.1.24) k0 k0a
2.1.3. Connection to Geometrical Model
n0
n2 k2
n1 k1 θ u
Figure 2.1.5. A vector model for the eigenvalue u and propagation constant β. The wave vectors in the core and cladding are shown.
The diagram above is a convenient tool to demonstrate the relationships between various fiber and mode parameters. The wave number is proportional to the refractive index of the medium, and is indicated by the length of the wave vector in the figure. The
10 core wave vector k1 points in the direction of the light ray. The propagation constant is the longitudinal component of k1 . Relation (2.1.20) is simply the Pythagorean theorem in the right triangle made up of u U / a , and k1 . From the trigonometric relations in the right triangle (2.1.25, 2.1.26) follow.
k1 sin (2.1.25)
neff n1 sin (2.1.26)
In order that the light beam be sustained in the core, the inequalities (2.1.27, 2.1.28) must hold true [14].
k2 k1 (2.1.27)
n2 neff n1 (2.1.28)
2.2. Fiber Gratings: Historical Background
Grating formation was first performed by K. Hill and coworkers at the
Communication Research Center in Canada in 1978. In their experiment the light from an
Argon blue laser (488 nm) was launched into a short piece of germania-doped fiber core, and the reflected light intensity was measured. Initially the fiber was almost totally transmissive and only 4% of the reflected power from the cleaved end was detected. But eventually the transmittance went down, and in several minutes almost all the injected light was back-reflected. This non-linear photo-refractive effect, called photosensitivity of the fiber, enables formation of permanent gratings [1, 2].
The mechanism of this phenomenon is the following. The forward propagated light beam forms a weak standing wave pattern with the coherent Fresnel reflected beam.
At the loci of high intensity the refractive index of the fiber core undergoes an 11 irreversible change (according to the mechanisms described in sections 2.3.1 and 2.3.3 of this Thesis). This results in formation of a weak periodic longitudinal perturbation of the refractive index of the core (i.e. the grating), which exactly reproduces the standing wave pattern caused this perturbation. In fact, there is no phase shift between the recorded grating and the standing wave. The newly formed grating starts to serve as a reflector enhancing, therefore, the reflected beam, which enhances the standing wave pattern, which, in its turn, enhances the grating. Thus, under the prolonged exposure of the fiber core to the laser output beam, a permanent self-organized grating is formed.
The described effect is called “photosensitivity”. As we shall see, germanium dopants are vitally important for this effect. Germania-doped silica fiber gratings produced this way were called “Hill gratings”. They worked as selective mirrors for a certain wavelength of the input light. The fact that these gratings are very narrow-band reflectors is disadvantageous on the other hand, since their reflectivity interval is mostly limited to a single wavelength that is exactly the laser output light wavelength used in the grating formation. But this method doesn’t allow fabrication of gratings operating in the
IR region, which are important for optical communications. Another limitation is the relatively low refractive index modulation and, therefore, reflectivity of the grating produced by this method due to the two-photon mechanism involved [15].
In 1989, Meltz et al. performed an experiment, in which a fiber was irradiated from the side with two coherent intersecting ultraviolet light beams. The wavelength used was 244 nm that is one half of that one used by Hill et al. a decade earlier. This time the grating formation involved a single photon process with the energy about 5 eV corresponding to the absorption band due to the germanium oxygen-vacancy defect
12 discussed in section 2.3 of this Thesis. The refractive index change turned out to be orders-of-magnitude larger in this experiment. Furthermore, the period of the interference maxima as well as the index change was determined by not only the wavelength of the light used, but also the angle between the beams, which enabled the fabrication of gratings with the desired grating period. The practical convenience is that the gratings can be formed in the core without the glass cladding removal [3].
Another grating imprinting technique was described by Hill et al. in 1993. The method is photolithographic and involves a use of a special phase mask grating made of silica glass. A KrF laser beam (249 nm) is incident normally on the mask. The fiber, which is oriented parallel to the mask grating, is exposed to the diffraction pattern produced. Usually zero order maximum is suppressed, and the fiber is located in the fringe pattern of the first order maximum that results in formation of a grating that resembles this pattern. As in previous case, the desired grating period is not fixed but can be chosen by varying the phase mask period and the distance from the mask to the fiber
[16].
2.3. Photosensitivity in Fibers
2.3.1. Origins of Photosensitivity
The principal component of silica fibers is silicon dioxide (silica) SiO2. However, there are a number of dopants added to silica in fiber manufacturing. Germanium dioxide
(germania) GeO2 raises the refractive index of silica and is, therefore, used in the core production. Boron trioxide B2O3 lowers the refractive index of silica and, thus, may be added to the cladding. Phosphorus pentoxide P2O5 is added to decrease silica melting
13 point and increase its malleability [17]. Some of the listed oxides are produced from the corresponding chlorides at high temperatures:
SiCl4 + O2 → SiO2 + 2Cl2
GeCl4 + O2 → GeO2 + 2Cl2
4POCl3 + 3O2 → 2P2O5 + 6Cl2
Although fibers may have various dopants, germanium is highly important for the photosensitivity. Fiber drawing is held at 14000C – 16000C [17]. Germania is known to decay to the monoxide and oxygen at high temperatures. In fact, at 11000С the equilibrium of the following reaction is 90% displaced to the right [18]:
GeO2 (l.) ↔ GeO(g.) + ½O2 (g.)
Germanium monoxide was first synthesized and characterized by L. M. Dennis and R. E.
Hulse in 1930. Pure GeO oxidizes at 5500C in dry air, sublimates at 7100C in a closed chamber in pure nitrogen, and begins to disproportionate at 600 - 7000C [19]. If a drawn fiber is rapidly cooled down, some GeO molecules get trapped in the fiber core, and, thus, in a fiber core there are somehow embedded GeO molecules and other defects. Even though there has not been proposed a unique mechanism of photosensitivity, great work has been done in investigating the role of these defects in photosensitivity.
The UV absorption studies of diverse germania-doped (GeO2/SiO2,
GeO2/P2O5/SiO2, GeO2/B2O3/SiO2) fibers by M.J. Yuen (1982) revealed the presence of certain absorption bands in all three of them that were referred to germanium species.
Namely, the 185-nm band was assigned to GeO2, and the 242-nm and 325-nm bands were assigned to the singlet-singlet and the forbidden but nonetheless occurring singlet- triplet transitions in GeO respectively [20].
14 Silica has three-dimensional tetrahedral structure. In case of low concentration of
GeO2 doping, Ge atoms simply substitute Si atoms in the structure. In 1986 Friebele and
Griscom proposed the color-center model. According to this model, the “wrong” bonds are Ge – Si bonds in oxygen-deficient centers. As a result of UV irradiation, these
“wrong” bonds break releasing electrons. The electrons dwell across the glass matrix, and eventually get trapped at one of germanium centers. This gives rise to the color centers: paramagnetic GeE΄ in places of the broken “wrong” bonds, and Ge(1) and Ge(2) centers in places of trapped electrons.
O O O Ge O O O + + Si + e- Si Ge O O O O O O "wrong bond" GeE' neutral oxygen vacancy (NOV)
-. OE OE - SiO SiO Ge + e Ge SiO OSi SiO OSi
Ge(1)- with E = Si - Ge(2) with E = Ge
In 1990 P. St. J. Russel et al. assigned electronic transitions at 213 nm, 240 nm and 281 nm to Ge(2), “wrong” bond Ge – X, and Ge(1) respectively (Fig. 2.3.1). The hypothesis was invented to confirm the “wrong-bond” model. Unpaired electrons at germanium atoms are indicated with arrows in Fig. 2.3.2. Most likely, the figure suggests the change of the hybridization state of germanium atoms due to the trapped electrons that results in a change of the bond directions around germanium atoms.
15
Figure 2.3.1. Absorption spectrum of germania-silica glasses in the UV. An additional very weak band at 325 nm is not shown. This is Figure 1 in [21].
Figure 2.3.2. The diagram exhibits the way of GeE΄, Ge(1) and Ge(2) center formation from the Ge-Si “wrong bonds”. This is Figure 2 in [21].
Further studies by R.M. Atkins et al. did not reveal an absorption band at 185 nm, instead a sharply rising peak was detected at 170 nm. Therefore, the 185-nm absorption 16 band described in [20] might have happened due to other impurities in the fiber. When the fiber is exposed to the KrF UV laser radiation (248 nm), the 242-nm (5.1 eV) band is being bleached, whereas several new bands arise. Gaussian deconvolution of the induced spectrum showed among others a strong band at 195 nm which was assigned to the induced GeE΄ color center absorption [22].
In 1995 J. Nishii et al. broadened the range of considered defects by introducing
Ge lone-pair centers (GLPC), which are bivalent germanium atoms carrying a lone pair of electrons (-O–Ge–O-), as another representative of germanium oxygen-deficient centers (GODC). The discussed above “wrong” bonds were named neutral oxygen vacancies (NOV). In addition, self-trapped hole (STH) centers were introduced [23].
Figure 2.3.3. Schematic energy diagram showing relevant defect levels and UV–induced photochemical reactions in GeO2-SiO2 glasses. GEC stands for germanium electron center. This is Figure 6 in [23].
More recent studies by M. Fujimaki et al. involved photoinduced absorption, electron spin resonance (ESR), photoluminescence (PL) and thermally stimulated
17 luminescence (TSL) of variously Ge-doped glasses exposed to four different UV radiation sources. Both GODC have an absorption band at 5.1 eV, however, only NOVs are responsible for the GeE΄ centers formation. The Ge(1) band was assigned to the both described in [21] germanium electron centers (GEC), and the Ge(2) band was assigned to the (GODC)+. It was shown that electrons released at NOV centers cannot be trapped at the fourfold coordinated germanium center, and only GLPC are the sources of such electrons. Therefore, the (GODC)+ observed in ESR corresponds to the cation-radical formed from GLPC [24].
+ Ge Ge + e- O O O O
- . OE OE e- Ge Ge + SiO OSi SiO OSi OSi OSi
- - Ge(1) and Ge(2)
2.3.2. Ways to Enhance Fiber Photosensitivity
Photosensitivity can be enhanced by increasing the concentration of the oxygen deficiency centers. One such way is hydrogen loading (or hydrogenation), which is saturating an optical fiber with molecular hydrogen gas, usually carried out by means of placing the fiber in a chamber containing hydrogen at 1 atm and elevated temperature, followed by UV irradiation. IR spectra revealed formation of – OH bonds, in the treated fibers, corresponding to Si – OH and Ge – OH [6]. Other sources [17] claim Ge – H bond formation The reduction of Ge – O – Si bonds with molecular hydrogen is likely to occur according to the following reaction scheme:
18 O O O O O O Ge Si O + H2 Ge H + HO Si O O O O O O
One advantage of hydrogen loading is formation of sufficient number of defects even in fibers with low germanium concentration. Another advantage is that the changes in refractive index occur only in parts of the fiber exposed to UV, and the extra hydrogen slowly diffuses out. One serious limitation of the hydrogen loading method is the presence of broad absorption peaks at 1.39 and 1.41 μm due to the presence of OH groups, which lowers the fiber transmittance in this IR region. One more disadvantage is the low thermal stability of such gratings [6].
Another way of photosensitivity enhancement is brushing the fiber in a hydrogen rich flame at about 17000C for about 20 min. Hydrogen readily diffuses in and creates Ge oxygen-deficient centers [17]. The increased photosensitivity in this case is permanent; however, the high-temperature flame weakens the fiber [6].
O O O O O O Ge Si O + H2 Ge Si + H2O O O O O O O O NOV Si O O O O O + H Ge + 2 HO Si Si Ge 2 O O O O O O O O O GLPC
Other ways to enhance the photosensitivity include boron codoping, which also increases the refractive index difference between the core and the cladding, and use of different wavelengths for grating fabrication such as 193 nm (ArF) or 248 nm (KrF) rather than traditional 245 nm corresponding to the germania absorption band [6].
19 2.3.3. Models for the Photoinduced Refractive Index Change
The color center model is based on the relation among the local refractive index n, absorption index κ and dielectric constant ε, which consists of the real (εr) and imaginary
2 (εi) parts: ε = εr + iεi = (n + iκ) . The real part is related to the imaginary part according to the Kramers-Kronig relationship:
1 i d (2.3.1) r
The generated Ge(1) and Ge(2) “color centers” are more polarizable, and have distinct absorption coefficient affecting, thus, the local refractive index that depends only on the local concentration and orientation of these defects. Therefore, if the fiber core is exposed to the fringe pattern, high intensity areas will produce larger concentration of these “color centers” causing the local refractive index change. A supporting evidence for the color center model is the reversibility of absorption changes upon heating the fiber at 9000C for
60 min [6,17].
The dipole model is based on the creation of local electric fields inside the fiber core due to the charge separation. At maxima of the fringe pattern the fiber is exposed to the germanium oxygen deficiency centers release electrons forming, thus, positively charged layers. These electrons get trapped at low intensity regions forming negatively charged layers of anion-radicals. Therefore, the charge separation pattern arises that repeats the fringe pattern used for the grating inscribing. The local refractive index changes are proportional to the square of the local electric field of the induced dipolar layers. This model works well for photorefractive crystals but in case of photosensitive
20 fibers very high density of the defects is required if to assume that the described is the only mechanism of the refractive index change [6].
Figure 2.3.4. Illustration of the dipole model. This is Figure 10 in [6].
According to the compaction model, density changes occur in the areas exposed to the fringe pattern maxima leading to the local refractive index changes. When a fiber is held at elevated temperature (9500C), the density becomes uniform across the fiber core and the grating is erased [6].
The stress-relief model assumes that the fiber core is under the tension due to the difference in the thermal expansion coefficients of the core and cladding and the significant temperature decrease in cooling down the fiber from the drawing to the room temperature. The breakage of the “wrong bonds” exposed to the pattern maxima causes some local relief in the structure affecting, thus, the local refractive index [6].
21 2.4. Methods for External Grating Inscribing
Unlike the method of grating formation proposed by Hill et al. in 1978, which involved the standing wave pattern inside the fiber core produced by counter propagating core modes, ways of external grating writing imply an exposure of a fiber to an external interference or diffraction pattern or point-by-point fabrication technique.
2.4.1. Interferometric Fabrication Technique
The idea of this means of grating imprinting is to split the UV light into two coherent beams using an interferometer, recombine them to form an interference pattern, and expose the fiber core to the pattern fringes. Both amplitude-splitting and wave-front- splitting interferometers can be used.
First, consider the amplitude-splitting interferometer. A UV light beam is split into two beams of equal intensity using an interferometer as shown in Fig. 2.4.1. The beams travel different optical paths and, therefore, form an interference pattern inside the fiber core. The grating period Λ exactly equals the fringe period of the pattern, and is determined by both wavelength of the light used and the half-angle between the intersecting UV beams, but is independent of the refractive index of the fiber core:
(2.4.1) 2sin
22
Figure 2.4.1. A schematic of amplitude-splitting interferometer for grating formation.
This is Figure 12 in [6].
One important advantage of the amplitude-splitting interferometric technique is the ability to get the desired grating period by means of varying angle φ even if the same wavelength is used. This method also enables forming gratings of different length and of diverse complexity by using curved reflecting surfaces in the interferometer. The disadvantages are basically due to the high sensitivity of the interference pattern to various conditions such as mechanical trembling, local refractive index stability, temporal and spatial coherence and laser output stability.
Examples of the named above interferometers used are the prism interferometer and the Lloyd's interferometer shown in Fig. 2.4.2 and Fig. 2.4.3. In the prism interferometer the beam expands inside a prism, and part of the beam is reflected due to the total internal reflection. The two half-beams recombine at the output face of the prism, and with help of the lens the interference pattern is positioned along the fiber core.
23 The dielectric mirror of the Lloyd interferometer reflects half of the UV beam to the fiber that is perpendicular to the mirror interfering with the other half-beam. A cylindrical lens is used to focus the interference pattern inside the fiber core.
The main advantage of the wave-front-splitting method is that only one optical element is used, which significantly reduces the sensitivity of the interference pattern to the mechanical vibrations. The grating period can still be tuned by means of varying the beam intersection angle. The disadvantages include the grating length limitation to the half of the beam width and grating period range limitation due to the beam coherence length limits [6].
Figure 2.4.2. Schematic of the prism interferometer for the grating fabrication. This is
Figure 13 in [6].
24
Figure 2.4.3. Schematic of Lloyd interferometer for grating formation. This is Figure 14 in [6].
2.4.2. Phase-Mask Technique
Both described above internal grating writing and holographic technique were superseded by the phase mask technique [6]. The phase mask grating is a diffractive optical element that consists of a one-dimensional surface-relief structure fabricated in a high-quality fused silica flat transparent to the UV writing beam. When a UV beam is incident on the phase mask, it results in a number of diffracted beams. The phase mask profile is chosen to suppress the zero order maximum (down to 5%) and to enhance the plus and minus first order maxima (up to 35% each). As a result of interference of these two beams a fringe pattern, with the period one-half that of the mask, is formed. The fiber is placed behind the phase mask, and the grating is imprinted in the core exposed to the fringe pattern [4 – 6].
25
Figure 2.4.4. Schematic of the phase-mask technique for grating fabrication. This is
Figure 18 in [6].
The main advantage of the phase mask techniques is the use of only one optical element, which makes it a stable method of reproducing fiber gratings. The sensitivity to mechanical vibrations is minimized in this method as well. Finally, due to the problem geometry, the writing efficiency is not affected by the low temporal coherence. The low spatial coherence is, however, disadvantageous, and requires positioning the fiber right behind the mask. The farther the fiber is placed, the lower refractive index modulation and, therefore, the lower diffraction efficiency is induced [6].
2.4.3. Other Methods of Grating Fabrication
In a point-by-point technique (Fig. 2.4.5) a light beam is passes through a mask slit, a lens then images a slit on the fiber and, thus, the local refractive index change in
26 the fiber core occurs. The fiber is then translated by a distance Λ corresponding to the desired grating period and the next local refractive index perturbation is induced. This technique enables producing gratings of diverse types by simply varying the grating period and the refractive index modulation from point to point. The disadvantage is that this time and effort consuming method is that very accurate fiber translating is required, and since the procedure is very tedious, the grating length is typically limited [6].
Figure 2.4.5. Setup for point-by-point grating fabrication. This is Figure 21 in [6].
Another means of grating fabrication is high-resolution mask image projection.
The transmission mask consists of a series of UV opaque line spaces. The UV beam is incident on the mask, and the transmitted beam is imaged on the fiber core. The advantages of this technique are in its simplicity and capability of producing complicated
(blazed, chirped) grating structures [6].
27 2.5. Quantitative Description of Gratings
2.5.1. Definition of Gratings and Resonance Conditions
Λ
air (vacuum) n0
cladding n K 2 core n 1
Figure 2.5.1. A grating inscribed in the fiber core is shown. The grating period Λ and the vector K are indicated.
A fiber grating is a periodic refractive index variation in the fiber core. If the grating period is Λ, then we can define the grating vector K normal to the grating (2.5.1).
2 K nˆ (2.5.1)
The effective refractive index perturbation Δneff as a function of coordinate along the fiber axis z is given by:
2 neff (z) ndc(z) nac(z) cos z (z), (2.5.2) where Δndc(z) is the “dc” average change in refractive index, Δnac(z) is the “ac” refractive index modulation, Λ is the grating period, z is the period chirp.
28 nac
n dc
Figure 2.5.2. A schematic of refractive index perturbation is shown. The “dc” and “ac” refractive index changes, and the grating period Λ are shown. z is constant.
The effective refractive index change Δneff is related to the induced index change Δn1 created uniformly across the core through the power confinement factor ε according to:
neff n1 (2.5.3)
Fiber grating acts as a regular phase diffraction grating. For a light beam of wavelength λ incident on the grating at an angle ζ΄ and diffracted from the grating an angle ζ˝ (Fig 2.5.3), the condition for the constructive interference reads:
n sin n sin m , (2.5.4) 1 1 with m = 0, ± 1, ± 2…
29 ΄
΄
˝
Figure 2.5.3. Diffraction of a light wave by a grating is shown [7].
Using the relationships discussed in section 2.2 one can rewrite this condition in terms of the relevant refractive indices (2.5.5).
n n m (2.5.5) eff eff
For two forward propagating modes with propagation constants β΄ and β˝ the resonance condition is given by (2.5.6). Since the values of the propagation constants are normally close, relatively large values of Λ satisfy the resonance condition. For this reason, this type of gratings is call long-period, or transmission gratings.
2 m (2.5.6)
For two modes propagating in opposite directions the resonance condition is given by
(2.5.7). In this case the values of Λ that satisfy this condition are smaller than the wavelength of the light. For this reason, this type of gratings is called short-period, or reflection, or simply Bragg gratings.
30 2 m (2.5.7)
For two identical modes propagating in opposite directions the first order constructive interference can be expressed in terms of the effective refractive index by the Bragg condition (2.5.8).
B 2neff (2.5.8)
2.5.2. Coupled Wave Equations
Coupled-mode theory is used to obtain information about the diffraction efficiency and spectral response of fiber gratings. For two identical modes propagating in opposite directions the total electric field vector is given by (2.5.9). A(z) and B(z) stand for amplitudes of the forward and backward propagating waves respectively. E(x, y, z) A(z)eiz B(z)eiz e(x, y) (2.5.9)
The detuning δ is defined as the difference between the actual (β) and resonant
/ propagation constants (2.5.10).
1 1 2neff (2.5.10) B
The period averaged (“dc”) coupling coefficient ζkj between any two modes k and j can be evaluated by integrating the scalar product of the transverse mode fields over the plane perpendicular to the principal fiber axis. For our particular case the “dc” coupling coefficient is given by (2.5.12).
n (z) 1 n (z) e (x, y) e * (x, y)dxdy (2.5.11) kj 2 1dc kt jt core 2 n (2.5.12) dc
31 The “ac” coupling coefficient is defined by (2.5.13), and for our case is given by (2.5.14).
1 n (z) ac (z) (2.5.13) kj 2 n kj dc n (2.5.14) ac
d For uniform gratings 0 the general “dc” self-coupling coefficient is simply a sum dz of the “dc” coupling coefficient and the detuning factor (2.5.15).
ˆ (2.5.15)
Let us now define slowly varying amplitudes R(z) and S(z) by absorbing the standing wave terms:
Rz Azexpiz (2.5.16) 2
Sz Bzexpiz (2.5.17) 2
The coupled wave equations then take form of:
dR iˆR(z) iS(z) (2.5.18) dz
dS iˆS(z) i * R(z) (2.5.19) dz
For a grating of length L the amplitude ρ and the power r reflection coefficients are defined by and can be calculated from (2.5.20, 2.5.21).
L S 2 sinh 2 ˆ 2 L L ˆ 2 ˆ 2 2 ˆ 2 2 ˆ 2 (2.5.20) R sinh L i cosh 2
32 2 2 2 2 sinh ˆ L r (2.5.21) ˆ 2 cosh 2 2 ˆ 2 L 2
When ˆ vanishes, the reflectivity r acquires its maximum value and is simply given by:
2 rmax tanh L (2.5.22)
It is worth mentioning that it is not the detuning being zero that maximizes the reflectivity. Therefore, the wavelength corresponding to rmax is slightly different from the
Bragg wavelength λB (2.5.23) [7].
n 1 dc (2.5.23) max B neff
2.5.3. Bragg Gratings as Temperature Sensors
The Bragg wavelength λB is given by λB = 2nΛ (from this point forward we shall omit index eff). Then the total differential of λB is given by (2.5.24).
B B n n dB dL dT 2 n dL 2 n dT (2.5.24) L T L L T T
The first part of the differential corresponds to the strain effect, which we will not take into account, the second part of the differential can be rewritten as
1 n 1 dB 2n dT 2n t dT, (2.5.25) n T T
1 n 1 with being the thermo-optic coefficient of the fiber and being the n T t T coefficient of the thermal expansion of the fiber [6].
The numeric values of these coefficients depend on the composition of the fiber, and the thermo-optic coefficient depends on the light wavelength as well. For fused silica
33 typical values for δ from measurements are about 0.83 . 10-5 K-1 [26 – 27]. In fact, the behavior of the refractive index with temperature change is non-linear, especially in wide temperature ranges, and the thermo-optic coefficient is a function of temperature itself
[27]. The value for the coefficient of thermal expansion is 0.055 . 10-5 K-1 (which remains fairly constant in the temperature range from 00C to 10000C) [25 – 27].
If the values of the coefficients are known for a certain fiber, the temperature can be determined from the peak wavelength λB, therefore, FBG can serve as a temperature sensor. However, if the temperature exceeds certain value (6000C – 10000C depending on the composition of the fiber and the grating type), other mechanisms, besides thermal expansion and thermo-optic effect, weaken the grating reflectivity and affect the grating response.
2.6. Thermal decay of FBGs
Thermal decay of UV-induced refractive index changes as a result of slow temperature ramp was first reported by Meltz and Morey in 1991 [28]. Later, in 1994,
Erdogan et al. analyzed thermal stability of germanium-rich FBGs over time for different temperatures, and fit the experimental data with a “power-law” function of time with a small exponent [29]. The refractive index change drops rapidly first, and then decays in time slowly. In their model, upon exposure to the UV radiation, electrons get excited to the conduction band, and get trapped in continuous distribution of energy states rather than in a single trap level. At high temperatures electrons in states of energy values beyond some threshold value Ed get excited back to the conduction band and repopulate the original deep level prior to UV excitation.
34
Figure 2.6.1. Diagram for the proposed model for grating decay. (a): Upon UV excitation electrons get trapped in continuous distribution of traps. (b) shows thermal depopulation of traps with energies higher than the threshold Ed [29].
This model explains the experimental data acquired and supports the fitting. Also, the model enables extrapolation of the results to lower temperatures (300 K) to predict the long-term FBG stability under ambient conditions. In addition, it enables preannealing FBGs to wipe out the part of the refractive index change that would decay over the lifetime of the fiber and to keep the very stable portion of the refractive index change [29].
ESR studies of grating behavior at elevated temperatures by M. Ferraris et al. showed that ESR signals due to UV-induced paramagnetic “color centers” decay if the fiber is held at 4000C [30]. It suggests that gratings undergo distortions under such conditions.
35 CHAPTER III
EXPERIMENTAL PROCEDURE
3.1. Fiber Optics Setup
The experimental setup for measuring grating reflectivity contains two light sources, as shown in Figure 3.1.1. One light source is a 20 mW LED operating at 550 nm wavelength and the other one is a superluminescent laser diode (SLD) emitting 1 – 5 mW of infrared radiation with the central wavelength of about 1310 nm and a minimum bandwidth about ±20 nm. Light from both sources is coupled into two identical pieces of a commercial optical fiber with core diameter of about 9.15 µm using an optical distribution board. One of the fibers contains an optical Bragg grating (FBG) written into it. The peak wavelength of the FBG at 20 °C is about 1300.135 nm and bandwidth is between 16.8 and 25.7 GHz for 1 and 3 dB respectively. Both fibers are placed in ceramic capillary tubes and put inside a 24’’ long 800°C split hinge tube furnace in such a way that the FBG is approximately in the middle of the furnace heating tube. The temperature of the tube furnace is controlled in the range 20 – 800°C.
36
Figure 3.1.1. Schematic presentation of experimental setup. This is Figure 1 in [31].
A portion of the infrared radiation from the SLD is reflected from the FBG and is detected by a photodetector incorporated inside of an optical spectrum analyzer (OSA).
As the temperature in the furnace changes, the wavelength of the radiation reflected by the FBG and detected by the OSA changes as well. Those changes are displayed on the screen of the OSA.
Radiation at the visible wavelength passes through both fibers and is then observed on a screen in the form of two patterns. One pattern is generated by a portion of that radiation passed through a fiber containing the FBG and the other one by radiation passed through the fiber without the FBG. The transmitted visible light is also collected, analyzed and recorded with a CCD camera to monitor distribution of the observed linearly polarized (LP) modes in the samples with and without the FBG.
3.2. SEM and EDS of FBGs
Two fiber samples containing FBGs were embedded in epoxy. Samples for the micro-toning procedure were extracted from that epoxy containing the embedded gratings. Each prepared sample was in the form of a small block that was removed using
37 a diamond knife in such a way that visual and mechanical access was available to the fibers. The cutting rate was set to 0.3 mm/s with the material feed rate set to 100 nm/s.
These rates were maintained until the fiber core was reached.
The SEM procedure was performed using the electron acceleration voltage between 5 and 25 kV, current of about 10 A, and working distance between 4 and 9 mm.
The surface of the sample was too smooth to perform useful SEM analysis in that region, although some chipping of a silica fiber was clearly observed. Based on the SEM analysis of the unexposed sample, it was found that the major elements of the sample (Si, O, and
Ge) were distributed uniformly in the area where an FBG was located as presented in Fig.
5.5.1 (b-d).
The EDS analysis using an ultra dry silicon drift detector with 30 mm2 active area was conducted on the samples before and after temperature treatment [31].
38 CHAPTER IV
CALCULATION METHOD
4.1. Calculation of Thermo-optic Coefficient
For moderate temperatures the variations in Bragg wavelength λB with temperature happen primarily due to the temperature dependence of the refractive index n and the thermal expansion of fiber leading to increase in grating period Λ. For the former one can expand the refractive index as a function of temperature T about some point T0:
n 2n 3n n(T) n(T ) (T T ) (T T )2 (T T )3 ... (4.1.1) 0 0 2 0 3 0 T T T T 0 T0 T0
If we are interested in changes of our parameters with respect to their zero temperature values, then set T0 = 273.15 K, and consequently T – T0 = t is simply the temperature in
Celsius degrees that will be used further. Note that since the magnitudes of one Celsius degree and one Kelvin degree are equal, it follows that dT = dt, and we will use them interchangeably. We can, therefore, rewrite the expansion in the following form:
2 3 n(t) n0 1t t t ..., (4.1.2)
1 n 1 2 n 1 3n with n = n(T ), , , . The thermo- 0 0 2 3 n0 t t0 n0 2! t t0 n0 3! t t0 optic coefficient in this case becomes a function of temperature itself:
1 n 1 t n 2t 3t 2 ... (4.1.3) n t n 0
39 Assuming linear thermal expansion for a broad range of temperatures we express the grating period Λ as:
t 0 1t t, (4.1.4)
1 with t . 0 t t0
Then, we can express the Bragg wavelength at arbitrary temperature t in terms of that one at t = 00C, which we call (4.1.5). B0
2 3 B 2n 2 n0 1 t t t ... 0 1 t t 2 3 2n0 0 1 t t t ... 1 t t (4.1.5) 1 t t 2 t 3 ... B0 t t t
The polynomial fit performed by some program comes out in the following form:
2 3 2 3 B t B t B t ... 1 B t B t B t ... (4.1.6) B B0 1 2 3 B0 1 2 3
Comparing (4.1.6) and (4.1.7) we identify:
B 1 B (4.1.7) 1 t B0 B 2 B (4.1.8) 2 t B0 B 3 B (4.1.9) 3 t B0
As shown above, having done the polynomial fit we end up with the zero temperature
Bragg wavelength λB0 and a set of coefficients Bi along with uncertainties for all of them.
Given the zero temperature grating period Λ0 one can calculate the effective refractive index n0 (4.1.10), and calculate the relevant uncertainty by means of error propagation
(4.1.11).
40 B 0 n0 (4.1.10) 2 0
2 2 2 2 n n 0 2 0 2 B0 0 n n0 , (4.1.11) 0 B0 0 B 0 0 B0 0 where ζi stands for the uncertainty of i. For all further calculated quantities the error propagation is performed in a similar way. After that we calculate a set of Bi΄s, and use their values and the value for the thermal expansion coefficient of silica from found in
. -5 -1 [25] αt = 0.055 10 K to calculate the expansion coefficients α, β and γ (4.1.12 –
4.1.14). The number of significant figures is set by the calculated uncertainties: we keep two significant figures in the calculated uncertainty values and the relevant number of figures in the corresponding quantities. Error propagation is not quite applicable for the U values because the latter are bounded between 0 and V. Therefore, we keep just two decimals for the values of U, W, and V.
B1 t (4.1.12)
B2 t (4.1.13)
B3 t (4.1.14)
Now the effective refractive index and the thermo-optic coefficient can be calculated for any temperature with help of (4.1.2 a) and (4.1.3 a) [31].
2 3 neff (t) n0 1t t t ... (4.1.2 a)
1 t n 2t 3t 2 ... (4.1.3 a) n 0
41 4.2. Calculation of Other Fiber, Modal and Grating Parameters
Propagation constant β can be calculated from the effective refractive index and the wavelength according to (4.1.16). If the refractive index of the core is known, the eigenvalue U can be calculated by (4.1.17). W is then can be found from the eigenvalue equation (2.1.18). From U and W one can calculate the V-number using (4.1.18). With this information the refractive index of the cladding n2 can be calculated by (4.1.19) and the fraction of modal power in the core can be found by (2.1.21).
2 k n n (4.1.16) 0 eff eff
2 U a n2 n2 (4.1.17) 1 eff
J U K W l l (2.1.18) U J l1 U W Kl1 W
V U 2 W 2 (4.1.18)
1 n V 2n2 W 2n2 (4.1.19) 2 U eff 1
U 2 W 2 K 2 (W ) l (2.1.21) 2 2 V U Kl1 (W )Kl1 (W )
From the values at the peak of reflection spectrum rmax and λmax the “ac” refractive index modulation can be calculated using (4.1.20).
n max Atanh r (4.1.20) ac L max
42 CHAPTER V
RESULTS AND DISCUSSION
5.1. Grating Response as a Function of Temperature
Graph 5.1.1 displays the reflectivity of the FBG as a function of temperature. Up until 6600C the grating response is pretty stable, so the spectral changes happen due to the refractive index change and thermal expansion. It suggests calculating grating, fiber and mode parameters for the temperature interval from 500C to 6600C. Above that the
reflected power drops: the grating is likely to undergo distortion.
Reflectivity
Figure 5.1.1. The variation of grating response with temperature. This is Figure 2 in [31].
43 5.2. Results for Thermo-optic Coefficient (500C – 6600C)
From the fact that the reflectivity of the grating starts to drop dramatically for temperatures above 6600C it follows that other mechanisms become involved in the variation of the grating period at high temperatures, and the expression (4.1.5) is not valid anymore. Consequently, we fit the data only in the temperature range from 500C to
6600C, where our reflectivity is slightly deviating from the average. As we increase the number of terms in the expansion, the correlation coefficient R2 approaches 1 but the uncertainties of the fitting coefficients vary significantly. The shape of the experimental graph (Fig. 5.2.1), namely, the curvature change suggests the cubic fit as the lowest possible one. At the same time the cubic fit is the optimal one because the uncertainties in the fitting coefficients do not reach 10%, whereas for the quadratic fit the errors exceed
50%.
Table 5.1.1. The values of the fitting parameters, and the calculated parameters Bi΄.
Temperature range, 0C 50 - 660
Correlation coefficient R2 0.99967
Intercept λB0, nm 1299.098 ± 0.033
B1, nm/K 0.02168 ± 0.00037
2 . -5 B2, nm/K (- 2.43 ± 0.12) 10
3 . -8 B3, nm/K (- 1.86 ± 0.11) 10
44 -5 -1 B1 t , 10 K 1.669 ± 0.028
-8 -2 - 1.871 ± 0.089 B2 t , 10 K
-11 -3 1.429 ± 0.083 B3 t , 10 K
Figure 5.2.1. The experimental data for the temperature range 50 – 7800C and the polynomial (cubic) fit to it for the temperature range 50 – 6600C.
Finally, we end up with the values for the coefficients: n0 = 1.4646 ± 0.0017
45 α = (1.614 ± 0.029) . 10-5 Κ-1
β = (-1.872 ± 0.089) . 10-8 Κ-2
γ = (1.430 ± 0.083) . 10-11 Κ-3
Effective refractive index as a function of temperature:
5 8 2 11 3 neff 1.4646 11.614 10 t 1.872 10 t 1.430 10 t (5.2.1)
Thermo-optic coefficient as a function of temperature:
1.614 105 3.744 108 t 4.29 1011 t 2 (5.2.2) 11.614 105 t 1.872 108 t 2 1.430 1011 t 3
The determined values for thermo-optic coefficient at low temperatures are slightly higher than those mentioned in literature [6, 26 –27]. One possible reason for that is the following. The thermo-optic coefficient is calculated not for the refractive index of the core or cladding but for the effective refractive index corresponding to the fundamental mode. The effective refractive index for a certain mode depends on the normalized frequency of the fiber, which, in its turn, is determined not only by refractive indices of the core and cladding but also by the radius of the core (2.1.13). From Graph
5.1.4 we can see that thermo-optic coefficient initially decays, then at approximately
4360C reaches the minimum value of 0.793 . 10-5, and rises at higher temperatures.
46 1.476
1.474
1.472
1.47
1.468 Effective refractive index refractive Effective 1.466
1.464 0 100 200 300 400 500 600 700 0 Temperature, C
Figure 5.2.2. Calculated refractive index as a function of temperature.
-5 x 10 1.5
1.4
1.3
1.2
1.1
1
0.9 Thermo-optic coefficient, 1/K coefficient, Thermo-optic 0.8
0.7 0 100 200 300 400 500 600 700 0 Temperature, C
Figure 5.2.3. Calculated thermo-optic coefficient as a function of temperature.
47 5.3. Results for Fiber and Mode Parameters
For the fundamental mode using equation (4.1.17) we find the eigenvalue U01 to
0 be 1.82 at t = 0 C. Using (2.1.18) we find that W01 = 2.73. Given the values for U01 and
W01, and using (4.1.18) we find the V-number to equal 3.28. The refractive index of the cladding (4.1.19) equals 1.4594. The cutoff value for LP11 mode is 2.4048, and for the next modes (LP21 and LP02) it equals 3.8317. Therefore, there can be two modes sustained in the core: LP01 and LP11. We calculate the relevant parameters for the latter mode, and organize them in the following table.
Table 5.3.1. The fiber and mode parameters for the two sustained modes: LP01 and LP11.
Parameter Value for the LP01 mode Value for the LP11 mode
Refractive index of the core n1 1.4669 1.4669
Effective refractive index neff 1.4646 1.4614
Refractive index of the cladding n2 1.4594 1.4594
Eigenvalue U 1.82 2.81
W 2.73 1.68
V-number 3.28 3.28
Power confinement ε 0.917 0.719
48 The intensity plots of the allowed modes are shown in Fig. 5.3.1 below. As we can see, LP01 mode has radial symmetry whereas LP11 mode has one azimuthal node.
LP 01
LP 11
Figure 5.3.1. Colormaps of intensity distribution in LP01 (top) and LP11 (bottom) modes.
49 The LP01 ↔ LP11 mode conversion in untilted gratings is forbidden by symmetry
[14]. The allowed mode conversions LP01 ↔ LP02, LP01 ↔ LP03 etc. cannot occur since the higher modes are not sustained in the fiber. The resonant wavelengths for all possible mode couplings upon reflection and transmission are shown in Table 5.3.2.
Table 5.3.2. The resonant wavelengths for possible mode couplings.
Coupling Resonant wavelength, nm
LP01 – LP01, reflection 1299.1
LP11 – LP11, reflection 1296.3
LP01 – LP11, reflection 1297.7
LP01 – LP11, transmission 1.4
The calculated transmission resonant wavelength makes sense since one needs long- period gratings to satisfy the interference condition upon transmission whereas we deal with a short-period grating. As we can see from the experimental spectrum in Graph
5.4.1, there is a single peak that corresponds to the LP01 – LP01 reflection resonance but there are no peaks next to it. It suggests that the fraction of LP11 mode is very low or essentially zero. It makes sense since the number of allowed modes depends on the source output intensity distribution and the angle of light beam incidence. And it is possible to adjust the settings so that the fiber operates in a single-mode regime.
50 5.4. Grating Decay at Elevated Temperatures
As we can see from the experimental spectrum in Figure 5.4.1, the sharp peak at
6600C corresponding to the maximum reflectivity turns into a broad band at 7500С. The
. -7 calculated refractive index modulation of the core Δn1ac equals 3.0 10 for T1 = 660°C
. -7 and 2.0 10 for T1 = 750°C.
-4 x 10 2 T 1 1.8
T1 =660°C T2 =750°C 1.6
1.4
1.2
1 T 2
0.8 Reflectivity
0.6
0.4
0.2
0 1308 1308.5 1309 1309.5 1310 Wavelength, nm
Figure 5.4.1. Experimental and calculated FBG spectra for two different temperatures.
This is Figure 3 in [31].
Using equation (4.1.20) we can calculate the “ac” refractive index modulation for and plot it against temperature (Graph 5.4.2). Steep grating decay occurs at temperatures
7000C and higher. 51
Figure 5.4.2. Decay of the effective “AC” refractive index modulation Δnac at temperatures higher than 7000C is shown.
5.5. Suggested Mechanism of Grating Decay
Since the described above germanium defects are vitally important for grating formation, and GeO has a sublimation temperature of 7100C, we decided to determine the elementary composition of the fiber core before and after annealing (prolongated exposure to high temperature). In order to determine the position and diameter of the core scanning electron microscope (SEM) image of the fiber was taken (Fig. 5.5.1 (a)). Fig.
5.5.1 (b – d) shows distribution of elements in the core. To analyze composition of the core energy dispersive spectroscopy (EDS) was utilized. The cladding was removed, the 52 analyzed area of the core was 25*20 μm2, the depth was about 5 μm, which is bigger than the radius of the core. The results of EDS analysis are shown in Fig. 5.5.2.
(a) (b)
Core with FBG Cladding layer
(c) (d)
Figure 5.5.1. (a) SEM image of a FBG core; (b) Distribution of Si, (c) O, and (d) Ge inside the fiber core with the FBG. This is Figure 7 in [31].
53
Figure 5.5.2. EDS images of extracted spectra for a sample (a) before the thermal treatment and (b) after the sample was annealed at 1000°C for 30 minutes. This is Figure
8 in [31]. 54 As we can see from the Figure 5.5.1, Ge atoms are distributed fairly uniformly. In addition, the analyzed volume is big enough to judge about Ge distribution in the entire core. The numerical data shown in Figure 5.5.2 is a direct evidence of absence of germanium lines in the spectrum of the sample after annealing. Even though EDS is not always accurate, this technique is quite sensitive. In particular, it can sense as low as ten germanium atoms per million. The absence of germanium lines in the spectrum of the annealed sample implies that the concentration of Ge in the core is less than 10-3%.
Therefore, during the annealing, germanium is likely to diffuse out of the core. Since the defective centers are forms of bivalent germanium, and GeO sublimates at 7100C, it is reasonable to conclude that at high temperatures germanium leaves the core in the form of germanium monoxide sublimate. Alternatively, germanium can simply spread over the core and the cladding resulting in an undetectably low concentration in the core. Both mechanisms can lead to the grating distortion and refractive index modulation decay.
55 CHAPTER VI
CONCLUSIONS
Fiber Bragg grating output from 500C to 7800C was analyzed. From the maximum reflectivity wavelength (essentially, Bragg wavelength) vs temperature fit, the effective refractive index corresponding to the fundamental LP01 mode was found to be
5 8 2 11 3 neff 1.4646 11.614 10 t 1.872 10 t 1.430 10 t . There are two peculiarities of the thermo-optic coefficient. First, the low temperature values for the thermo-optic coefficient are about 1.4 . 10-5, which are quite higher than those mentioned in literature. Second, the thermo-optic coefficient decreases as temperature increases and reaches its minimum value of about 0.73 . 10-5 (slightly lower than values mentioned in literature) at t = 4360C. For temperatures higher than that, thermo-optic coefficient rises again. It is worth noting that the sufficient accuracy in the results for coefficients in the expansion up to the third power are possible thanks to sufficient number of data points taken accurately.
The normalized frequency of the fiber was found to be 3.28. The effective index of the cladding was 1.4594, which is 0.0074 lower than the refractive index of the core.
Besides the fundamental LP01 mode, the next LP11 mode with the effective refractive index of 1.4614 was allowed in principle. From the absence of peaks corresponding to
LP11 – LP11 and LP01 – LP11 resonances upon reflection, we conclude that LP11 mode wasnot sustained in the fiber. This may have been due to the output intensity distribution 56 and the angle of incidence of the light beam. The mode conversion was forbidden by symmetry. Therefore, it was legitimate to treat the fiber as a single-mode one.
The experiments revealed that the grating reflectivity decays at temperatures higher than 6600C. The calculated refractive index modulation exhibits a steep decay from the value of 3.0 . 10-7 at 6600C to about 2.0 . 10-7 at 7800C . It suggests that the grating gets distorted upon exposure to high temperatures. Comparison of energy dispersive spectra of a fiber with FBG before and after annealing showed that germanium diffused out of the core. It was concluded that possible mechanisms of grating decay are either germanium sublimation in the form of germanium monoxide or germanium spreading over the core and the cladding.
There were several sources of discrepancies. Firstly, the fiber had finite width and might have had bends; it, however, has rather minor effect on the results since the wavelength used was much shorter compared to the lengths of the fiber and grating.
There could have been some internal structures even at low and medium temperatures, then the data fitting would not be quite legitimate. Possible non-uniformities of the grating, possible radial dependence of the refractive index could have affected the results.
Dispersion (dependence of refractive index on the wavelength) was not taken into account since the wavelength range was quite narrow (1290 – 1310 nm).
Possible suggestions for future work involve UV and ESR studies of fiber cores annealed under diverse regimes, modeling of GeO diffusion in the glass matrix, calculations of spectra of germanium defects imbedded in the glass structure, invention of some GeO sensor to monitor germanium monoxide effusion while fiber annealing [31].
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