2.1 Couple mode theory [22]
The relation between the spectral dependence of a fiber Bragg grating and the corresponding grating structure is usually described by the coupled mode theory.
Coupled mode theory is described in a number of texts and the detailed analyses can be found in [10, 18, 19, 20, 21]. The axial refractive index of the fiber Bragg grating is represented by
as shown in Fig2.1. The coupled mode equation can be written as
u
where u and v represent the forward and backward propagating modes respectively,
− Λ
=β π
δ is the wavenumber detuning, Λ is the grating period, and
) 2 modulus of q is proportional to the index modulation amplitude or the peak to peak modulation of the index variation. The phase of q corresponds to the excess optical
phase or phase envelope of the grating. We can get q(z) from the structure parameter of the fiber Bragg grating and then use q(z) to simulate the reflection and transmission spectra by (2.2).
2.2 Transfer matrix method [22]
We divide the grating into a sufficient number of N sections so that each section can be approximately treated as a uniform grating. Let the section length be
LN
=
∆ . By
applying the appropriate boundary conditions and solving the coupled mode equations similar to the procedure above. We find the following transfer matrix relation between the fields at z and at z+Δ
Hence, we can connect the fields at the two ends of the grating through
1
where T is the overall transfer matrix. The matrix Tj is the transfer matrix written in (2.4) with q=qj =q(j∆), the coupling coefficient of the jth section. As a result, T is a 2*2 matrix with elements
⎥⎦
Once T is found, the reflection coefficient and the transmission coefficient are calculated by the relations
22 22
21 , ( ) 1
)
( t T
T
r δ =−T δ = , (2.7)
which are obtained by the substitution of the appropriate boundary conditions into (2.5)
2.3 Discrete layer peeling method [22]
One can replace the matrix T by the product of two transfer matrixes T∆Tρ. One of them (Tρ) describes a discrete reflector, and the other (T∆) describes the pure propagation of the fields,
⎥⎦
here the discrete, complex reflection coefficient is given by
q
The discrete model of the entire grating is thus a series of N discrete, complex reflectors with a distance ∆ between all reflectors. We will define the forward and backward propagating fields before the jth section as uj(δ) and vj(δ) ( see Fig(2.2)). For numerical implementation, the spectral dependence must also be discretized, and hence the calculation of ρ by the inverse fourier transform of 1
)
1(δ
r can be achieved by the discrete fourier transform
∑
=Eq(2.10) is valid for all layers by substituting 1→j in the subscripts, because the reference plane is transferred to the actual layer through
)
The discrete layer peeling algorithm can be summarized in the following simple steps:
i) Start with the target reflection intensity spectrum(R(δ)) and reflection phase spectrum(φ(λ))
ii) Compute ρ1 from Eq(2.10) iii) Propagate the fields using Eq(2.11)
iv) Repeat step ii) until the entire grating structure is detemined.
v) Compute the coupling coefficient q by j
2.4 Experimental setup and analysis for side diffraction interference [23][24][25]
2.4.1 Experimental setup
Figure 2.3 and 2.4 shows the two experimental setups demonstrated in the thesis for the characterization of the fiber Bragg gratings. The difference between the two setups is that the setup1 has a higher spatial resolution than the setup2, but in contrast we can detect the variation of the grating period only by using setup2. Although there is little difference between these two setups, the principles of them are the same. The light source is a cw laser (here a He-Ne laser operating at a wavelength of λ = 632.8nm).
A polarization beam splitter is used to split the laser beam into a probe beam and a reference beam. Two half-wave-plates produce nearly perfect s-polarized beams and allow the ratio of the intensities of the two beams to be adjusted when the first half wave plate is rotated. In this way, substantially more intensity can be provided to the probe beam, which will generate the first order diffracted beam with quite low efficiency, depending on the grating strength. In addition, an attenuator can be used to achieve an even larger ratio of Ip/Ir to yield a high fringe visibility. In the setup1 the probe beam is focused by a spherical lens of 20cm focal length onto the side of the fiber Bragg grating, so the FWHM width of the probe beam measured along the fiber axis is 80μm. In the setup2 the probe beam is focused by a cylindrical lens of 30cm focal length onto the side of the fiber Bragg grating, so the FWHM width of the probe beam measured along the fiber axis is 1.5mm which is identical to the beam width of the He-Ne laser source. Because the probe beam width of setup1 measured along the fiber axis is much smaller than that of setup2. the spatial resolution of the setup1 is larger than the spatial resolution of the setup2. The angle of the incidence satisfies the
matching the grating vector K with the difference of the axial components of the incident (i) and reflected (r) wave vectors,
= Λ and the reflected beams leave the fiber at the opposite sides of the fiber at equal angles ±θr, we have
Here NB is the effective index of the fiber at the retroreflected Bragg wavelength
B
B = 2ΛN
λ . The calculated angle of incidence is θi ≒ 36.2568(deg). The fiber Bragg grating is held horizontally in a pair of clamps separated by 30 cm, which can be translated along the fiber axis by a translation stage and PZT. The probe beam produces first order diffracted beam after it passes through the grating. In the setup1 the diffracted beam is refocused by another spherical lens of 20cm focal length, but not in the setup2. The purpose of using another spherical lens in the setup1 is to produce a plane wave in the horizontal plane and than the period of interference pattern will be a constant. A monochrome CCD camera of 7.15-um pixel width is used to record the fringe pattern. A computer data acquisition system controlled by the Labview program is used to save the images of the interference pattern and to perform the required analysis. By scanning the fiber Bragg grating we can measure the ac index profile and the variation of the grating period. As described in 2.3.2, the ac index modulation of the FBG in the probed area, Δnac, can be inferred from the total radiation power detected by the CCD or from the amplitude of the interference pattern.
The change in the grating period, ΔΛ, can be measured from the change in the period of the interference fringes.
2.4.2 Analysis [23][24][25]
We assume that the fiber Bragg grating to be measured is approximately uniform across the fiber core of diameter 2a and has a sinusoidal index variation of amplitude Δnac along the fiber axial direction z. The fringes of the grating are represented by a spatial modulation of the core refractive n(z) as
2 ) the ac index profile, Λ is the grating period. The intensity of the first order diffracted beam (with intensity Ip) is denoted Id, and the intensity of the reference beam is denoted Ir. The diffraction efficiency η of the grating for the first order diffracted beam is
when Bragg condition is satisfied, where c is a constant of proportionality defined for convenience. The power of the diffracted beam is denoted Pd.
2 ac d
d I n
P ∝ ∝∆ . (2.16)
The interference of the diffracted beam and the reference beam is given by
where f is the spatial frequency of the interference fringe pattern and
er
f
int
1λ
= .
Here λinter is the period of the interference fringe pattern. We perform the Fourier transform of the interference fringe patterns, then use a filter to select the interference term 2 c2∆nac2IpIr cos(2πfx). The amplitude of this term is proportional to Δnac
and the angle α between the two interference beams is given by
2)
With the phase matching condition along the axis of the FBG, the variation of the grating period can be represented by
tan )
The requirement that α<<1, which also corresponds to λf 2 <<1, simply implies that the two beams are nearly parallel and hence give rise to interference fringes that are spaced by the distance much larger than the wavelength of the He-Ne laser. Using Eq.(2.28), we can then write the local grating period with respect to the central period Λ0 as
)f tan ( 0 2
0 θ
Λ λ
− Λ
≈
Λ . (2.20)
Therefore the relative variation in the grating period, ΔΛ, is given by the expression
θ f λ tan 2
= Λ
∆Λ . (2.21)
Equation (2.16) and (2.17) form the basis for evaluating the ac index modulation Δnac, and equation (2.21) forms the basis for evaluating grating period variation ΔΛ.
The whole analysis process is shown below
2.5 Experimental setup and analysis for phase measurement[26]
For performing analyses for the whole optical properties of fiber Bragg gratings, we not only have to obtain the reflection intensity spectrum, but also the phase spectrum of the reflection light. To get the reflection intensity spectrum we just have to use a broad band source and an optical spectrum analyzer. On the other hand, the phase information can not be performed simply. Figure 2.5 shows the experiment setup demonstrated in this thesis for the phase spectrum measurement of the fiber Bragg grating. The setup is a typical Michelson interferometer. The broad-band light from the Er-fiber ASE (Amplified Spontaneous Emission) source is split after passing through a 50/50 coupler. The signal beam is reflected by a fiber Bragg grating and carries the phase information of the grating. The reference is reflected by a reflection mirror after emerging from a collimator. The two beams form the interference pattern after passing through 50/50 coupler. We use an optical spectrum analyzer to observe the interference fringe. The optical path reflected from the grating is denoted path1.
The optical field of the path1 can be represented as
))
Igrating = , Φ is the phase delay induced by grating, which is also the phase
of the r = r exp(iΦ) in Equation (2.7). L1 is the length of the path1. The optical field of the path2 is represented by
2 )
Let ~22 E
Ireference = , The reflection mirror is almost dispersionless near the
wavelength of 1550nm. L2 is the length of the path2. The interference of the two beams can be represented as
2 ) interference fringe pattern is shown in Fig2.6. We then make the fourier transform of the Iinter and plot the absolute value of it, as shown in Fig2.7. The phase information is encoded in the ac signals. In order to get ϕ, we use a filter window to filter out the positive frequency ac signal. The bandwidth of the window (filter) will affect the measured group time delay. Generally speaking, the group time delay will oscillate when the bandwidth is large, whereas the result will be smooth when the bandwidth is small. The reason is that the use of a large bandwidth filter will include in more high frequency noises. We shift the ac signal to dc and perform the inverse Fourier transform to get IreflectIgrating exp(iϕ). By extracting the phase term we obtain the phase spectrum of the fiber Bragg grating. The length of the path1 should be longer than the length of the path2. If the length of the path2 is longer than the length of the path1, Equation (3.3) would become
))
Then the phase we obtain is − . If the length of the path1 is equal to the length of ϕ the path2, after we make the fourier transform of Iinter, Igrating, Ireflect and
)
separate them apart. After getting the phase spectrum of the grating, we can evaluate the group delay by the expression below.
λ
The whole process of analysis is shown below I i n t e r ( λ )
Fig.2.1 Ac and dc index modulation of two fiber Bragg gratings
Δ ρ1
Δ
ρ2 ρ3
v1
u2 v2 u1
Fig.2.2 The discrete model of fiber Bragg grating
H e - N e l a s e r M
H W P
S L
P B S
M
H W P
S L
B S M
S L
A S L 3 X
C C D c a m e r a
F i g e r B r a g g g r a t i n g
α Θ B
Fig.2.3 Setup1 for side diffraction interference
H e- N e l as er M
H WP
C L
P BS
M
H WP
C L
B S
M A
α C CD
c am e ra
F ig e r B ra gg gr a ti n g
α S L
S L 8 X
ΘB
Fig.2.4 Setup2 for side diffraction interference
PC FBG
Collimator
Mirror 50/50 Coupler
ASE
OSA
Fig.2.5 Setup for the phase measurement of the fiber Bragg grating
1546.0 1546.5 1547.0 1547.5 1548.0 1548.5 0.00000
0.00001 0.00002 0.00003 0.00004 0.00005 0.00006
interference fringe
wavelength(nm)
interference fringe
Fig.2.6 Interference fringe from optical spectrum analyzer