Discrete Layer-Peeling Method - Synthesis of Fiber Bragg Gratings

Chapter 2. Synthesis of Fiber Bragg Gratings

2.5 Discrete Layer-Peeling Method

The modulus of q is proportional to the ac-index modulation amplitude. The termθ(z) is the spatial grating phase and the integral term in (2.7) is the modification to the spatial phase due to the increased dc index.

2.5 Discrete Layer-Peeling Method

This layer-peeling method for synthesis of fiber gratings comes from an inherently discrete model [21]. It was first developed by geophysicists like Goupillaud and Robinson, and was extended by Bruckstein et. al. [22,23]. In this thesis, the described discrete layer-peeling (DLP) method was developed by J. Skaar et. al. [24]. This method is an inverse method for finding the structure parameters of the complex coupling coefficient of a fiber grating from the

complex reflection spectrum. The description of the inverse and the forward methods for the synthesis of fiber gratings. are summarized in Fig. 2.7.

The discretized LP method can be described as shown in Fig. 2.8. The starting point for the numerical modeling of FBGs is based on the transfer matrix formulation, which connects the field at point z+Δ with the field at z by the following relation :

is the wavenumber detuning compared to a Bragg design wavenumber β_B. We refer to (2.8) as the piecewise uniform model since the grating is considered uniform in the interval [z, z+

Δ]. The matrix T can be replaced by the product of two transfer matrixes T_∆*T . Here _ρ

is the pure propagation transfer matrix obtained by letting q Æ 0, and

( )

is the discrete reflector matrix obtained by letting q Æ ∞ and holding qΔ constant. The discrete reflection coefficient is given by

( )

It is straightforward to show that the result of transferring the fields using T_∆⋅T_ρ is equivalent to the recursive formulation given below

( ) ( ) ( ) ( )

ρ δ

To obtain an explicit expression for the determination of ρ1 by the inverse Fourier transform, we note that the spectrum r₁(δ) can be written as a discrete-time Fourier transform of the impulse response h1(τ), Since the impulse response forτ=0 is the same as if only the first reflector is present, we can see thatρ1 is simply the zeroth order Fourier coefficient of the series in (2.13) :

( ) ∫

^∆

( )

For numerical implementation, the spectral dependence must also be discretized, and hence the calculation ofρ1 by the inverse Fourier transform of r1(δ) can be achieved by the discrete Fourier transform

∑ ( )

M is the number of wavelengths in the spectrum. After determiningρj , we can get the complex coupling constant q(z) by

2.6 The Least Square Method

From the DLP method, the normalized refractive-index modulation amplitude and phase profiles can be reconstructed from the target spectrum. In order to realize this complicated FBG by the proposed fabrication methods, it is important to include the overlap-step-scan effects in the design [25]. We use the least square (LS) fitting method [26] to find the best experimental exposure parameters.

In our experiment we will use a small gaussion beam to fabricate fiber gratings by an overlap-step-scan exposure method. It is assumed that Aid(z) is the refractive index envelope reconstructed from the Layer-Peeling method and A_m(z) is the refractive index envelope of the m-th small gaussion beam. Therefore the following merit function can be used for determining the optimum parameters of the overlap-step-scan exposure.

( { } ) ∫ ∑ ( )



where Cm represents the amplitude of the m-th small gaussion beam, and Am(z) can be expressed as

( )

z =Cm⋅^exp

(

−

(

z−zm

)

² ws²

)

(2.18) In the above expression ws is the width of the gaussion beam and zm is the central position of the m-th exposure gaussion beam. We want to let

∑

m z

A ( ) be as close to Aid(z) as possible.

This can be achieved by the least square method, which solves the following equations to find the optimum values.

2.7 Summary

In this chapter we have described the characteristics of fiber Bragg gratings with various types of refractive index profiles. It is helpful for the design of various complex FBG structures.

The discrete layer-peeling method has been adopted for the design of advanced fiber gratings.

We also combine this synthesis method with the least square method to find the best experimental parameters for sequential UV exposure. These developed methods will be applied to the design of advanced FBG structures and can be realized by using the proposed fabrication methods to be described in the following chapter.

References for Chapter 2:

[1] P. L. Swart, M. G. Shlysgin, A. A. Chtcherbakov, and V. V. Spirin, “Photosensitivity measurement in optical fibre with Bragg grating interferometers,” Electron. Lett., vol. 38, pp. 1508-1510, 2002.

[2] P. J. Lemaire, R. M. Atkins, V. Mizrahi, and W. A. Reced, ”High pressure H₂ loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett., vol. 29, pp. 1191-1193, 1993.

[3] G. Meltz and W. W. Morey, SPIE, vol. 1515, pp. 185, 1991.

[4] L. Dong, J. L. Cruz, L. Reekie, M. G. Xu and D. N. Payne, “Enhanced photosensitivity in tin-codoped germanosilicate optical fibers,” IEEE Photon. Technol. Lett.,vol. 7, pp.

1048-1050, 1995.

[5] Thomas A. Strasser, “photosensitivity in phosphorus fibers,” in OFC’96, invited TuO1.

[6] T. Erdogan and V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B, vol. 11, pp.

2100-2105, 1994.

[7] K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: Application to the external writing of rocking filters,” Electron.

Lett., vol. 27, pp. 1548–1550, 1991.

[8] K. O. Hill, Y. Fujii, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: Application to reflection filter fabrication,” Appl. Phys. Lett., vol. 32, pp.

647–649, 1978.

[9] C. G. Askins, T.-E. Tsai, G. M. Williams, M. A. Putnam, M. Bashkansky, and E. J.

Friebele, “Fiber Bragg reflectors prepared by a single excimer pulse,” Opt. Lett., vol. 17, pp. 833-835, 1992.

Lett., vol. 21, pp. 1550-1552, 1996.

[11] H. A. Haus and C. V. Shank, “Anti-symmetric type of distributed feedback lasers,” J.

Quantum Electron, vol. 12, pp. 532-539, 1976.

[12] R. Kashyap, P. E. McKee, and D. Armes, “UV written reflection grating structures in photosensitive optical fibres using phase-shifted phase-mask,” Electron. Lett., vol. 30, pp.

1977-1978, 1994.

[13] L. Wei and W. Y. Lit, “Phase-shifted Bragg grating filters with symmetrical structures,”

IEEE J. Lightwave Tech., vol. 15, pp. 1405-1410, 1997.

[14] K. C. Byron, K. Sugden, T. Bircheno, and I. Bennion, “Fabrication of chirped Bragg gratings in photosensitive fibre,” Electron. Lett. vol. 29, pp. 1659, 1993.

[15] B. Eggleton, P. A. Krug, and L. Poladin, “Dispersion compensation by using Bragg grating filters with self induced chirp,” in Tech. Digest of Opt. Fib. Comm. Conf., OFC’94, pp. 227.

[16] F. Ouellette, “The effect of profile noise on the spectral response of fiber grating,” in Bragg Gratings, Photosensitivity, Poling in Glass Fibers and Waveguides: Applications and Fundamentals, vol. 17, OSA Technical Digest Series (Optical Society of America, Washington, DC, 1997), paper BMG13, pp. 222-224.

[17] A. Boskovic, M. J. Guy, S. V. Chernikov, J. R. Taylor, and R. Kashyap, “All-fiber diode pumped, femtosecond chirped pulse amplification system,” Electron. Lett., vol. 31, pp.

877-879, 1995.

[18] R. Kashyap, R. Wyatt, and P. F. McKee, “Wavelength flattened saturated erbium amplifier using multiple side-tap Bragg gratings,” Electron. Lett., vol. 29, pp. 1025, 1993.

[19] A. D. Kersey and M. A. Davis, “Interferometric fiber sensor with a chirped grating distributed sensor element,” Proc. OFS’94, pp. 319-322, Glasgow UK, 1994.

[20] A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

[21] R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fibre Bragg gratings,” J. Quantum El., vol. 35, pp. 1105-1115, 1999.

[22] A. M. Bruckstein and T. Kailath, “Inverse scattering for discrete transmission-line models,” SIAM Rev., vol. 29, pp. 359-389, 1987.

[23] A. M. Bruckstein, B. C. Levy and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math, vol. 45, pp. 312-335, 1995.

[24] J. Skaar, L. Wang, and T. Erdogan , “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron., vol. 37, pp.165-173, 2001.

[25] L. G. Sheu, K. P. Chuang, and Y. Lai, “Fiber Bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photon. Technol. Lett., vol. 15, pp.

939-941, 2003.

[26] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, New York, 1992).

Fig. 2.1 The refractive-index modulation along the length of waveguides.

Fig. 2.2 The common types of fiber Bragg gratings; (a) uniform FBGs, (b) phase-shift FBGs, (c, d) apodized FBGs, and (e) chirped FBGs.

1549.6 1549.8 1550.0 1550.2 1550.4

1549.6 1549.8 1550.0 1550.2 1550.4 -200

Fig. 2.3 (a) The spectra of three Bragg gratings with different coupling constants of 2, 4, and 8.; (b) time delay of the same gratings as in Fig. 2.3(a).

1549.2 1549.6 1550.0 1550.4 1550.8 -30

-25 -20 -15 -10 -5 0 5

Wavelength (nm)

Reflectivity (dB)

-30 -25 -20 -15 -10 -5 0 5

Transmission (dB)

Fig. 2.4 A π-phase shift uniform fiber Bragg grating. (the parameters are abs(qL)=1.6 and L=5 mm)

1549.2 1549.6 1550.0 1550.4 1550.8

-40

Fig. 2.5 (a) A Gaussian apodized FBG with non-constant dc index; (b) a pure apodization FBG with a Gaussian index profile.

1548.5 1549.0 1549.5 1550.0 1550.5 1551.0 1551.5

1548.5 1549.0 1549.5 1550.0 1550.5 1551.0 1551.5 0

Fig. 2.6 A chirped FBG with the grating length = 30 mm, chirped rate = 0.1667 nm/cm, and a tanh index profile. (a) The reflection and transmission spectra; (b) the phase and time delay.

coupling coefficient q(z) ( index ∆ nand phase θ )

reflection coefficient r(λ) or t(λ)

forward inverse

Fig. 2.7 The description of the inverse and the forward methods for the synthesis and analysis of fiber gratings.

u

z

ρ

v

section 1 section 2

ρ

Optical Waveguide

Fig. 2.8 A discretized model of the LP method.

Chapter 3. Characterization of Fiber Bragg Gratings

3.1 Introduction

The characterization of a fiber Bragg grating involves the determination of the complex reflection coefficient or the complex coupling coefficient of the grating. The mostly direct method for finding FBG reflection and transmission spectra is by using the optical spectrum analyzer (OSA). This measurement can help us to find the performances of a fiber grating initially and is not enough to get the grating structures for improving the design and fabrication of advanced fiber gratings. Accurate characterization of the amplitude (refractive-index modulation) and phase (dispersion) of FBGs are needed for many applications. In the literature, there have been a lot of different approaches based on the interferometry [1-4], side-scattering [5-7], heat-scan [8-9] or phase modulation methods [10-11]. In this chapter, we describe two measurement methods that we developed for measuring the spectral phase, index amplitude and period change of the FBGs. The two methods are based on the balanced Michelson interferometer technique for measuring the spectral phase information and the side-diffraction technique for measuring the refractive index change and the grating period change.

3.2 The Balanced Michelson Interferometer Method

The characterization of fiber Bragg gratings is very important for their applications in high-bit rate fiber communication systems. It is not easy to measure the phase information of a fiber grating. In reference 4, Skaar used a lossless Fabry-Perot-like system consisting of two reflectors, where one reflector has known characteristics, and the other has not. The first

reflector is a FBG with unknown characteristics, and the other reflector is quite broad-band when compared to the FBG as shown in Fig. 3.1. They obtained the complex reflection spectrum of the FBG, or equivalently the reflectivity and group delay spectra, from the measurement of the reflectivity of the Fabry-Perot structure.

In this paper, we analyze a similar structure, namely a balanced Michelson interferometer consisting of two general reflectors, where one of them is the FBG to be characterized, and the other is a reference reflector [12]. The experimental setup is shown in Fig. 3.2 and can be analyzed as follows. In the FBG path, we obtain the path 1 signal

2 ( )) the reference path, we obtain the path 2 signal

2 ) where L2 is the path length from coupler. When these two signal are combined at the output port of the coupler which is connected to an Optical Spectrum Analyzer (OSA), the measured output signal spectrum can be expressed as

2 ( ))

From equation (3.3) the interference spectrum I_inter(λ) will look like that in Fig. 3.3. One can take the inverse Fourier transform of Iinter(λ) to get the result shown in Fig. 3.4. By multiplying the result by a filtering window function centered at the peak corresponding to the angular frequency +τand then taking a Fourier transform back to the spectral domain, we can get the phase response φ(λ) from the result and the group delay τg=dφ/dω can also be calculated. The whole algorithm is summarized in Fig. 3.5.

linear chirped FBG with the length L=5mm, the chirp rate = 0.5 (nm/cm), and the maximum grating reflection = 96.84%. The measured interference pattern I_inter(λ) is shown in Fig.

3.6(a). Fig. 3.6(b) shows the result of the calculated phase spectrum φ(λ).

We have also calculated the group delay time τg=dφ/dω from the result of the phase response )φ(λ . For comparison, we experimentally measure the group delay of this grating sample using a commercial dispersion measurement equipment (“Advantest Q7760”). The measured result (Fig. 3.7(a)) is compared with the result from our measuring method (Fig.

3.7(b)). One can see that we can get good measured results and the measurement resolution is about 5 ps, which is probably limited by the resolution of the OSA as well as the environmental vibration.

The presented method has been carried out experimentally, and has been shown to be easy and accurate for the measurement of the grating phase. Some of the fabricated advanced fiber Bragg gratings shown in the next chapter will be measured by using this method.

3.3 The Side-Diffraction Technique

The spatial quality of a FBG will strongly affect the performance of the FBG device. By using the side-diffraction technique, we can directly measure the refractive index modulation and the grating period profile. In this section we describe how this side-diffraction method can be used to get the information of grating index and period changes. And we also proposed an improved method for increasing the measuring resolution when compared with the prior methods in the literature.

The side-diffraction technique can be divided into two categories of without/with interference as shown in Fig. 3.8 and Fig. 3.9. First, a side-diffraction method for measuring the refractive index modulation is shown in Fig. 3.8. A He-Ne laser as a probe beam is

focused through the side of the fiber into its core. The first-order Bragg diffraction of the probe beam occurs at the Bragg condition:

λ the free-space wavelength of the probe beam which is 632.8 nm in our case. The FBG index structure can be expressed as

, sin )

(z n n Kz

n = _dc +∆ (3.5) where n is the dc refractive index, K is the grating vector and Δn is the refractive index _dc modulation. We can express the first-order diffracted power P by means of a scattering _d cross section σ_i in the form of

i i

d p

P =σ × ˆ , η =P /_d P_i (3.6) where P is the total input power and _i η is the diffracted efficiency. The scattering cross section can be express as

i angle of polarization vector with respect to the diffraction plane. When the s-polarized light is used for the probe beam (i.e. the electric vector is perpendicular to the plane, γ =π /2), the amount of the diffracted energy is maximum. From equation (3.5)~(3.7), we can get the refractive index modulation ∆n with respect to the first-order diffracted power P , which _d has the relation of

n∝

∆ . (3.8) The side-diffraction method provides a very accurate measurement of the index profile which have demonstrated simultaneous measurement of the fiber grating refractive index modulation

-6

The other method is the side-diffraction interference method as shown in Fig. 3.9. We can measure the grating period change and the refractive-index modulation at the same time [see Ref. 6]. If we use the CCD camera to record the interference pattern, the intensity distribution of the interference fringe can be express as

) α is the angle between the reference and the first-order diffracted beams. Based on the phase match condition, the local grating period can be written as

[

sinθ sin(θ α)

]

The local grating period Λ with respect to the central period can also be expressed as

 From equation (3.12), we can find that the grating period variation ∆Λ can be got from the spatial frequency of the interference pattern.

From equation (3.8), we can find that the refractive-index modulation ∆n is proportion to P or _d I . In order to get the intensity information of the first-order diffracted beam _d

I , we take the Fourier transform of the interference pattern I expressed in equation (3.9) d

to get the amplitude of the positive beating frequency pattern. The positive beating frequency

amplitude is proportion to 2(I_rI_d)¹^/². By using this method, we can measure the distribution of the refraction-index modulation along the whole grating.

Fig. 3.10 shows the improved side-diffraction interference method for measuring the grating period variation and the ac-index modulation. In order to improve the resolution of both measured parameters, we use a single-polarization He-Ne laser with a larger output power of 25mW as a light source. We used a CCD camera to record the interference pattern.

When a larger interference period is used, we can have a better period change resolution. For example, if the grating period Λ =535nm, the wavelength of a He-Ne laser ₀ λ =632.8nm, the interference angle θ_i=36.2⁰, the interference period on CCD camera dx=100pixels=1/ f (where one pixel width = 7.15um), we can calculate the resolution of the measured grating period change to be about 0.0032nm. If the dx=50pixels=1/ f is used, the resolution of the measured grating period change is about 0.0128nm.

3.4 Characterization of Fiber Gratings

Complete characterization of fiber gratings including the complex reflection spectrum and the complex coupling coefficient is required for the analysis of fiber grating properties or for improving the fabrication methods. In this section, we propose an analysis method for determining the complete characteristics of the fiber gratings including the dc-index modulation. This analysis method is based on the discrete layer peeling method (described in section 2.5), the balanced Michelson interferometer method (described in section 3.2) and the side-diffraction method (described in section 3.3). One example is proposed to demonstrate the applicability of this analysis method.

I. Analysis Process

The flow chart of the proposed characterization method of fiber gratings is shown in Fig.

3.11. In general, if the grating structure has been known, one can use the transfer-matrix approach to compute the complex spectrum. Conversely, the complex coupling coefficient can be obtained from the complex spectrum by the use of the inverse scattering algorithm, like the discrete layer peeling method. For the inverse algorithm, when the reflectivity of the FBG is small (R<30%), the structure of the FBG can be directly calculated from the complex reflection spectrum using a Fourier transform [13,14]. When the grating reflectivity is high, it does not give the accurate grating structure due to multiple reflections that propagate along the grating. An inverse scattering algorithm should be used for the fiber gratings with high reflectivities [15-18]. In our work the discrete layer-peeling algorithm [18] was used to extract the complex coupling coefficient of a fiber grating from the measured complex reflection spectrum. In the measurement of the complex reflection spectrum, we used the balanced Michelson interferometer method described in section 3.2. However, for highly reflecting fiber gratings, the forward propagating wave strongly attenuates along the grating and, therefore, the reflection from the region located close to the grating end is very low. Since the measured reflection signals contain noise, the reconstruction of the region located close to the grating end becomes inaccurate. In order to measure the accurate complex reflection spectrum using the balanced Michelson interferometer method, an improved method described in reference [19] can be used. The experimental setup is shown in Fig. 3.12. The complex reflection spectrum of the grating can be extracted by measuring the impulse response from both sides of the grating. Then the grating structure (q(z) and argq(z)) can be calculated from the complex reflection spectrum by the use of the DLP algorithm. By using the side-diffraction technique described in section 3.3, we can measure the ac-index modulation

nac

∆ and the grating period variation ∆Λ . From the results of the DLP algorithm and the side-diffraction technique, we can get the grating structure parameters which include ∆n_ac,

∆Λ and ∆n_dc. In the above parameters, the ∆n_dc distribution of the grating can be extracted from the equation (2.7) of argq(z).

II. Experimental Results

A chirped Bragg grating has been studied with the proposed analysis method. This chirped Bragg grating was fabricated by the use of a chirped phase mask with the chirp rate of 1 nm/cm which corresponds to the grating period change of 0.5 nm/cm. A frequency-doubled argon-ion laser launches a CW 244 nm single-polarization UV beam into a FBG fabrication system to write the grating. The grating with a gaussion-apodized index profile was written in a Fibercore photosensitive fiber, which has the maximum reflectivity of 5 dB. In order to get all of the grating structure parameters, we firstly used the balanced Michelson interferometer method to find the spectrum phase information of the grating. Fig. 3.13 shows the reflection spectrum and the interference pattern of the grating. From the interference pattern, we can calculate the spectral phase by using the algorithm as shown in Fig. 3.5. The calculated phase information of the grating is shown in Fig. 3.14. In the next step, we use the reflection

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