## Polarization-resolved spatial characterization of birefringent Fiber Bragg Gratings

Optics Express, Vol. 14, Issue 10, pp. 4221-4236 (2006)

http://dx.doi.org/10.1364/OE.14.004221

Acrobat PDF (364 KB)

### Abstract

A method that enables polarization-resolved spatial characterization of fiber Bragg gratings is presented. The polarization-resolved reflection spectrum of the grating is measured using optical-frequency domain reflectometry. A polarization-resolved layer-peeling algorithm is used to compute the spatial profile, including the local birefringence and the local polarization-dependent index modulation. A strain-tuned distributed feedback fiber laser is used as source. With closed-loop control of the laser sweep, 0.14 % maximum deviation from constant sweep rate is achieved, which is much better than commercial available tunable lasers. The polarization of the source is modulated synchronous with the laser sweep by passing the light through a three-armed Mach-Zehnder-type interferometer having different retardation. The method is used to investigate the polarization-dependence of the index modulation amplitude of a fiber Bragg grating.

© 2006 Optical Society of America

## 1. Introduction

1. D.-W. Huang and C.-C. Yang, “Reconstruction of fiber grating refractive-index profiles from complex Bragg reflection spectra,” Appl. Opt. **38**, 4494–4498 (1999). [CrossRef]

2. S. Keren and M. Horowitz, “Interrogation of fiber gratings by use of low-coherence spectral interferometry of noiselike pulses,” Opt. Lett. **26**, 328–330 (2001). [CrossRef]

3. P. Giaccari, H. Limberger, and R. Salathé, “Local coupling-coefficient characterization in fiber Bragg gratings,” Opt. Lett. **28**, 598–600 (2003). [CrossRef] [PubMed]

4. D. Sandel, R. Noé, G. Heise, and B. Borchert, “Optical network analysis and longitunal structure characterization of fiber Bragg gratings,” IEEE J. Lightwave Technol. **16**, 2435–2442 (1998). [CrossRef]

6. O. Waagaard, “Spatial characterization of strong fiber Bragg gratings using thermal chirp and optical-frequency-domain reflectometry,” IEEE J. Lightwave Technol. **23**, 909–914 (2005). [CrossRef]

7. W. Eickhoff and R. Ulrich, “Optical frequency-domain reflectometry in single-mode fiber,” Applied Physics Letters **39**, 693–695 (1981). [CrossRef]

8. U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” IEEE J. Lightwave Technol. **11**, 1377–1384 (1993). [CrossRef]

9. J. von der Weid, R. Passy, G. Mussi, and N. Gisin, “On the characterization of optical network componenents with optical frequency domain reflectometry,” IEEE J. Lightwave Technol. **15**, 1131–1141 (1997). [CrossRef]

11. K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. **27**, 1548–1550 (1991). [CrossRef]

12. T. Erdogan and V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B **11**, 2100–2105 (1994). [CrossRef]

13. S. Pereira, J. E. Sipe, R. E. Slusher, and S. Spälter, “Enhanced and suppressed birefringence in fiber Bragg gratings,” J. Opt. Soc. Am. B **19**, 1509–1515 (2002). [CrossRef]

14. B. Soller, D. Gifford, M. Wolfe, and M. Foggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express **13**, 666–674 (2005). [CrossRef] [PubMed]

15. O. Waagaard and J. Skaar, “Synthesis of birefringent reflective gratings,” J. Opt. Soc. Am. A **21**, 1207–1220 (2004). [CrossRef]

4. D. Sandel, R. Noé, G. Heise, and B. Borchert, “Optical network analysis and longitunal structure characterization of fiber Bragg gratings,” IEEE J. Lightwave Technol. **16**, 2435–2442 (1998). [CrossRef]

16. Luna Technologies white paper, “Optical vector network analyzer for single scan measurements of loss, group delay and polarization mode dispersion,” http://www.lunatechnologies.com/products/ova/files/OVAwhitePaper.pdf (Luna Technologies, 2005).

17. Agilent Technologies white paper, “Agilent 81910A Photonic All-Parameter Analyzer User Guide,” http://www.home.agilent.com/agilent/facet.jspx?kt=1&cc=US&lc=eng&k=81910 (Agilent, 2005).

15. O. Waagaard and J. Skaar, “Synthesis of birefringent reflective gratings,” J. Opt. Soc. Am. A **21**, 1207–1220 (2004). [CrossRef]

## 2. Theory

18. R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. **35**, 1105–1115 (1999). [CrossRef]

19. J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. **37**, 165–173 (2001). [CrossRef]

20. J. Skaar and O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. **39**, 1238–1245 (2003). [CrossRef]

*(*

**R***ν*) and the transmission Jones matrix is denoted

*(*

**T***ν*). Let

*E*

_{s}(

*t*)=

*Ê*

_{s}e

^{-i2πνt}be the electric field vector of light from the TLS, where

**Ê**_{s}=[

*E*

_{1}

*E*

_{2}]

^{T}is the Jones vector describing the state of polarization (SOP) and the superscript T represents the matrix transpose operation. The polarization reflected from the FBG and the reflector is altered by the birefringence of the fibers in the interferometer. Let the retardation Jones matrices

**Φ**

_{s},

**Φ**

_{r},

**Φ**

_{m}and

**Φ**

_{d}describe the birefringence of the four fibers into and out of the coupler as shown in Fig. 1. We assume that these fiber sections are lossless so that these corresponding matrices are unitary.

*E*

_{m}(

*t*) and

*E*

_{r}(

*t*) reaching the reflection measurement detector after reflection from the FBG and the mirror, respectively, are given by

*τ*

_{m}and

*τ*

_{r}account for the optical delays of the two paths, and

*k*

_{m}and

*k*

_{r}are scalar constants given by the coupling ratios of the coupler and the reflectivity of the mirror. (Due to reciprocity, a retardation Jones matrix in the backward direction is given by the transpose of the retardation Jones matrix in the forward direction [15

15. O. Waagaard and J. Skaar, “Synthesis of birefringent reflective gratings,” J. Opt. Soc. Am. A **21**, 1207–1220 (2004). [CrossRef]

*τ*

_{0}=

*τ*

_{r}-

*τ*

_{m}is the imbalance of the interferometer. The term e

^{i2πντ0}, may be viewed as a carrier signal oscillating versus optical frequency. The interference part of

*P*

_{r}(

*ν*) can be extracted in the Fourier domain (i.e. the optical delay-domain) around delay

*τ*

_{0}:

*k*

_{m}

*k*

_{r}. Note that

*P*

_{i}is complex since we have only extracted the signal for positive delay. Since

**Φ**

_{d}is unitary, we have used that

*I*, where

*I*is the identity matrix. The Jones matrices

**Φ**

_{o}and

**Φ**

_{i}are the products of Jones matrices before and after

**(**

*R**ν*) in the second line in Eq. (4) and represent the effective birefringence from the TLS to the grating and from the grating to the detector.

*P*

_{i}(

*ν*) can be found by writing Eq. (4) in terms of the components of

*Ê*

_{s}and

**=**

*R*̃**Φ**

_{o}

**(**

*R**ν*)

**Φ**

_{i}:

*C*

_{s}=[|

*E*

_{1}|

^{2}

*E*

^{∗}2

*E*

_{1}

*E*

_{2}|

*E*

_{2}|

^{2}]

^{T}is the coherency vector [21] of

*Ê*

_{s}(similar to the coherency matrix, except that the elements are collected into a 4×1 column vector rather than a 2×2 matrix). Note that there is a linear and invertible relationship between the coherency vector

*C*

_{s}and the 4-component Stokes vector [21].

*from Eq. (5), requires four measurements of*

**R**̃*P*

_{i}with different source SOPs. These 4 measurements can be put together by writing

*P*

_{i,j}(

*ν*),

*j*=1,…, 4 are the detected interference signal with the four different coherency vectors

*C*

_{s,j}. By inverting the matrix [

*C*

_{s,1}

*C*

_{s,2}

*C*

_{s,3}

*C*

_{s,4}]

^{T},

*R̃*can be found from [Pi,1(ν) Pi,2(ν) Pi,3(ν) Pi,4(ν)]T. In order to do that, the matrix must be invertible. This means that the four coherency vectors

*C*

_{s,j,}

*j*=1,…, 4 (or the four corresponding Stokes vectors), must be linearly independent. One example of such a set of SOPs is horizontal polarization, vertical polarization, linear 45° polarization and right circular polarization.

*R̃*have different carrier frequencies [16

16. Luna Technologies white paper, “Optical vector network analyzer for single scan measurements of loss, group delay and polarization mode dispersion,” http://www.lunatechnologies.com/products/ova/files/OVAwhitePaper.pdf (Luna Technologies, 2005).

*.*

**R**̃*R̃*. From Fig. 1 we find that the SOP of the light reaching the transmission measurement detector is given by

*k*

_{t}accounts for loss and detector responsivity. The detected power becomes

_{t}=

*I*.

**21**, 1207–1220 (2004). [CrossRef]

**21**, 1207–1220 (2004). [CrossRef]

*ν*. This is not the case since the reflection Jones matrix seen from the detector,

*(*

**R**̃*ν*)=Φ

_{o}

*(*

**R***ν*)

**Φ**

_{i}, is only symmetric when

**Φ**

_{o}=

*h*(

*τ*)=FT{

*(*

**R***ν*)}, as

*h*(

*τ*)=

**Φ̃**†

_{o}

*h̃*(

*τ*)

*h̃*(

*τ*)=FT{

*(*

**R**̃*ν*)}. Here, FT denotes the Fourier transform. The matrices

*h*(

*τ*) is symmetric for all τ, which means that

_{o}=Φ

_{o}

_{i}=

^{T}Φ

_{i}where

_{o}and

_{i}, select a delay

*τ*=

*τ*

^{′}where the singular values of

*h̃*(

*τ*

^{′}) do not degenerate. Using singular decomposition we may factorize

*h̃*(

*τ*

^{′})=

*U*Σ

*V*, where

*U*and

*V*are unitary and Σ is diagonal and non-negative. Since the singular values are different, the

*U*and

*V*are unique up to a matrix

*θ*is real but otherwise arbitrary, since

*UDΣD*

^{∗}

*V*=

*U*Σ

*V*. We choose

_{o}=

*UD*and

_{i}=

*D*

^{∗}

*V*, which gives

*ĥ*

_{nm}are the components of the matrix

*U*

^{†}

*h̃*(

*τ*)

*V*

^{†}. By evaluating this equation for all

*τ*≠

*τ*

^{′}, the phase constant

*θ*can be found as the phase in

*D*that minimizes the difference

*ĥ*

_{12}e

^{-i2θ}-

*ĥ*

_{21}e

^{i2θ}for all τ.

**21**, 1207–1220 (2004). [CrossRef]

*h*(

*τ*). In Ref. [15

**21**, 1207–1220 (2004). [CrossRef]

*N*layers, where each layer is a cascade of a retardation section, a discrete reflector and a time-delay section. From

*h*(0), the retardation and the discrete reflector of the first layer can be found. Once the first layer is characterized, the transfer matrix of the layer can be computed, and the effect of the layer on the FBG can be removed using the calculated transfer matrix. Then, the retardance and the discrete reflector of the next layer can be found from the reduced grating. This procedure is repeated until all layers are characterized.

_{j}represent the reflection response from layer

*j*alone. γ

_{j}should be symmetric due to reciprocity, and given by

*ρ*

_{j}Φ

_{j}, where Φ

_{j}is the Jones matrix describing the retardation from layer

*j*-1 to layer

*j*and

*ρ*

_{j}the reflection from discrete reflector

*j*.

**21**, 1207–1220 (2004). [CrossRef]

**γ**

_{j}when it is not symmetric. The asymmetry is handled by adding an extra retardation

**Φ**

_{as,j}for the forward propagating field, so that

**γ**

_{j}=

*ρ*

_{j}

**Φ**

_{j}

**Φ**

_{as,j}.

## 3. Measurement of the grating impulse response

### 3.1. Optical frequency modulation

^{TM}and Matlab

^{TM}.

22. E. Rønnekleiv, “Frequency and Intensity Noise of Single Frequency Fiber Bragg Grating Lasers,” Opt. Fiber Technol. **7**, 206–235 (2001). [CrossRef]

23. P. Oberson, B. Hutter, O. Guinnard, L. Guinnard, G. Ribordy, and N. Gisin, “Optical frequency domain reflectometry with a narrow linwidth fiber laser,” IEEE Photon. Technol. Lett. **12**, 867–869 (2000). [CrossRef]

22. E. Rønnekleiv, “Frequency and Intensity Noise of Single Frequency Fiber Bragg Grating Lasers,” Opt. Fiber Technol. **7**, 206–235 (2001). [CrossRef]

8. U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” IEEE J. Lightwave Technol. **11**, 1377–1384 (1993). [CrossRef]

*ϕ*

_{4}(

*t*)=2π

*ν*(

*t*)

*τ*

_{trig}, where

*ν*(

*t*) is the optical frequency and

*τ*

_{trig}is the dual pass delay difference of the interferometer. The fringe output D4 is connected to an ac-coupled comparator, which produces 0V or 5V dependent on the sign of the fringe signal. The PLD produces a 30 ns short trigger pulse from this signal. This pulse train can be used as a sampling clock for D1-3. However, in order to make the system more flexible, this pulse train is fed to a counter integrated on the NI-6052 card. This counter enables software-selectable down-sampling so that D1-3 can be sampled equidistant in optical frequency with a sampling interval that is any multiple of 1/(2

*τ*

_{trig})~0.5 MHz.

*k*Δ

*ν*+

*δν*(

*k*), where

*k*is an integer and

*δν*(

*k*) is the sampling jitter. Then the detected interference power in Eq. (3) is given by

*δν*

_{rms}. The sampling jitter is a function of the product of the fluctuations in laser sweep rate and the difference in delay to the AD converter between the trigger pulses and the sampled signal. If the laser sweep is completely linear and the delay is constant, this delay is unimportant since the delay transforms to a constant

*δν*. If the laser sweep is not completely linear, it can be compensated by adding fiber before detector D4. However, the delays have to be very well matched when the fluctuations in sweep rate is large. In addition, group delay ripple in filters may give a delay difference that is not constant. The laser sweep should therefore be as linear as possible.

24. S. Kakuma, K. Ohmura, and R. Ohba, “Improved uncertainty of optical frequency domain reflectometry based length measurement by linearizing the frequency chirping of a laser diode,” Opt. Rev. **10**, 182–183 (2003). [CrossRef]

*dν*(

*t*)/

*dt*of the DFB-FL. This pulse train is low-pass filtered using a 4-pole Butterworth active filter with 60 kHz bandwidth. The resulting voltage amplitude is proportional to the sweep rate. Using a PID controller, we lock the measured sweep rate to a reference sweep rate. The optical frequency of the DFB-FL is a (nonlinear) function of the voltage applied to the solenoid. The output of the PID controller is therefore integrated to generate a signal that is proportional to optical frequency. This signal is applied to the solenoid.

### 3.2. Polarization modulation

*τ*

_{1},

*τ*

_{2}and

*τ*

_{3}, respectively. These delays are chosen so that

*τ*

_{a}=

*τ*

_{2}-

*τ*

_{1}=10 ns and

*τ*

_{b}=

*τ*

_{3}-

*τ*

_{1}=15 ns. The SOP out of the PolIF depends on the birefringence of the paths and the optical frequency.

*Ê*

_{1},

*Ê*

_{2}and

*Ê*

_{3}are the Jones vectors describing the SOP out of each of the three paths, respectively, and

*ê*

_{1}=[1 0]

^{T}.

**Φ**

_{1}is a unitary matrix with the first column equal to

*Ê*

_{1}, so that

**Φ**

_{1}

*ê*

_{1}=

*Ê*

_{1}. We write,

**Φ**

_{1}must be orthonormal to the first column, but can have an arbitrary common mode phase. This common mode phase will be chosen so that

*β*

_{a}=-

*β*

_{b}=

*β*. When the coupler has 50/50 % coupling ratio and there is no loss neither in the couplers nor the fibers within PolIF,

*k*

_{a}=

*k*

_{b}=1/2.

**Φ**

_{1}may be absorbed into the matrices

**Φ**

_{i}and

**Φ**

_{o}, and

*k*

_{1}can be absorbed into

*k*

_{m}and

*k*

_{r}, which is left out in Eq. (4). Without loss of generality we may therefore set

*k*

_{1}=1,

**Φ**

_{1}=

*I*and

*τ*

_{1}=0.

*τ*

_{a}≠

*τ*

_{b}≠

*τ*

_{b}-

*τ*

_{a}≠0,

*k*

_{a},

*k*

_{b}≠0 and

*has linearly independent rows.*

**M***will only have linear dependent rows when*

**M**

**Ê**_{a},

**Ê**_{b}‖

*ê*

_{1}. The optimum configuration is when

*Ê*

_{a},

*Ê*

_{b}⊥

*ê*

_{1}, which gives

*h̃*(

*τ*) be the Fourier transform of

*(*

**R**̃*ν*), and ζ(

*τ*) be the Fourier transform of

*P*

_{r}(

*ν*). Even though the impulse response

*h̃*(

*τ*) is infinitely long, we assume that the amplitude is insignificant outside the range [0,

*τ*

*l*]. The Fourier transform of the leftmost vector in Eq. (20) is a column of delta functions

*δ*(

*τ-τ*

^{′}) with different delays

*τ*

^{′}. In the delay-domain, this factor is convolved with the Fourier transform of the remaining factors to produce the signal ζ(

*τ*), where the matrix components of

*h̃*is divided into separable bands starting at

*τ*

_{0},

*τ*

_{0}±(

*τ*

_{b}-

*τ*

_{a}),

*τ*

_{0}±

*τ*

_{a}and

*τ*

_{0}±

*τ*

_{b}. To avoid overlap between these bands, we must require that

*τ*

_{l}<

*τ*

_{a}-

*τ*

_{b}=5 ns. We may extract these bands and shift them to zero delay, yielding the vector

*τ*≤

*τl*. The components of

*h̃*(

*τ*) can be found this equation. Note that there is 4 unknowns and 7 equations, i.e. the system is over-determined. A least-square solution can be found by multiplying each side of Eq. (21) with the pseudo-inverse

*M*

^{+}of

*M*

^{T}, which satisfy

*M*+

*M*

^{T}=

*I*.

### 3.3. Calibration

*. The Jones vector at detector D3 is proportional to*

**M***ê*

_{1}-

*k*

_{c}(

**Ê**_{a}exp[

*i*2

*πντ*

_{a}]+

**Ê**_{b}exp[

*i*2

*πντ*

_{b}], where

*k*

_{c}is given by the power coupling ratio of the coupler at the output of the PolIF. Thus, the detected power becomes,

*k*

_{D3}is a scaling constant.

_{D3}(

*τ*)=FT{

*P*

_{D3}(

*ν*)}. PC1 should be adjusted until ζ

_{D3}(

*τ*

_{a})=0 and PC2 so that ζ

_{D3}(

*τ*

_{b})=0. This will ensure that

**Ê**_{a},

**Ê**_{b}‖[0 1]

^{T}, which is the wanted mode of operation.

*in Eq. (18).*

**M***k*

_{a},

*k*

_{b}and kc are known (see below),

*k*

_{D3}can be found from ζ

_{D3}(0). We may then calculate

*.*

**M***k*

_{a},

*k*

_{b}and

*k*

_{c}depend only on the coupling ratios of the couplers and the loss in the interferometer path, and we can therefore assume that they do not fluctuate with time.

*k*

_{c}=-1. The constant

*k*

_{c}can therefore be found by comparing the measured responses at D3 and D2.

_{D3}(τ)=ζ

_{D3}(τ)/ζ

_{D3}(0),

*k*

_{c}

*k*

_{a}and

*k*

_{c}

*k*

_{b}. Measuring the response at D3 two or more times with different settings of PC1 and PC2, and inserting the measured responses into Eq. (24), gives a set of nonlinear equations, from which the constants

*k*

_{a}and

*k*

_{b}can be found.

## 4. Results

### 4.1. Swept fiber laser with high sweep linearity

### 4.2. Reconstruction of the polarization-dependent spatial profile of an FBG

25. A. Asseh, H. Storøy, B. Sahlgren, S. Sandgren, and R. Stubbe, “A writing technique for long fiber bragg gratings with complex reflectivity profiles,” IEEE J. Lightwave Technol. **15**, 1419–1423 (1997). [CrossRef]

_{D3}(

*τ*

_{a})| and |ζ

_{D3}(

*τ*

_{b})|, so that the SOP out of path 2 and 3 is approximately orthogonal to the SOP out of path 1 of the PolIF. Then, the DFB-FL was swept from 1550 nm to 1557 nm, which provides a theoretical spatial resolution of 0.15 mm. The interference signal at D1 and the PolIF calibration output at D3 were sampled equidistant in optical frequency as described in Sec. 3.1. Both signals were then multiplied with a Blackman window, which reduced the effective spatial resolution to 0.45 mm. The PolIF calibration output signal was used to calculate the matrix

*using Eq. (23). The transmission spectrum was not measured, instead the maximum reflectivity was set equal to the reflectivity measured with an optical spectrum analyzer in transmission mode.*

**M***τ*)|, which is the Fourier transform of the measured interference pattern

*P*

_{r}(

*ν*) at detector D1. There are in total ten signal bands. The peaks at 5, 10 and 15 ns corresponds to the imbalances of the PolIF, and is the Fourier transform of the sum of the (non-interfering) reflected power from the mirror and the FBG. Note that the peaks at 5 ns, 10 ns and 15 ns are actually beyond the vertical scale in this figure. This is not shown since the other signal bands are much weaker. The amplitudes in these bands were 12.5 · 10

^{-3}, 3.0·10

^{-3}and 2.9·10

^{-3}, respectively. The remaining seven signal bands originate from the interference between the reflections from the reference mirror and the FBG, and are the signal bands that will be used for calculation of the spatial profile of the FBG. The delay to the center band equals

*τ*

_{0}=21.3 ns, which is the imbalance of the measurement interferometer. If PolIF is replaced by a patch-cord, this would be the only signal band in Fig. 4. The remaining signal bands at

*τ*

_{0}±(

*τ*

_{b}-

*τ*

_{a}),

*τ*

_{0}±

*τ*

_{a}and

*τ*

_{0}±

*τ*

_{b}have offsets from

*τ*

_{0}that equals the imbalances of the PolIF.

*τ*) are shown to the left in Fig. 5. From these signals, the pseudo-inverse

**M**^{+}of

**M**^{T}is used to calculate

*h̃*(

*τ*). The grating impulse response matrix

*h*(

*τ*) is calculated from

*h̃*(

*τ*), using the method described in the paragraph before Eq. (10) to remove the effect of the matrices

**Φ**

_{i}and

**Φ**

_{o}. The resulting impulse response is shown on the right. This procedure will make

*h*

_{12}(

*τ*

^{′})=

*h*

_{21}(

*τ*

^{′})=0 for a chosen delay

*τ*=

*τ*

^{′}. We observe that |

*h*

_{12}(

*τ*)| and |

*h*

_{21}(

*τ*)| is nearly equal to zero over the whole length of the grating. This means that the grating has almost constant orthogonal eigenpolarizations. Beating between the intrinsic and uv-induced birefringence of the fiber may cause non-orthogonal eigenpolarizations [15

**21**, 1207–1220 (2004). [CrossRef]

*h*

_{12}(

*τ*),

*h*

_{21}(

*τ*)≈0,

*h*

_{11}(

*τ*) and

*h*

_{22}(

*τ*) are the impulse response of the two eigenpolarizations. There is an observable difference between |

*h*

_{11}(

*τ*)| and |

*h*

_{22}(

*τ*)|, indicating polarization-dependent index modulation.

*n*

_{ac,x}and

*n*

_{ac,y}versus position calculated by the polarization-resolved layer-peeling algorithm. The impulse response amplitudes in Fig. 5 has a clear negative slope versus position, and there is significant amplitude at delays larger 0.1 ns, which is the dual-pass delay through the grating. The index modulation amplitudes in Fig. 6 are reasonable flat, as one should expect for a uniform grating. In addition, the relative amplitude beyond the grating is reduced. This indicates that the polarization-resolved layer-peeling algorithm removes higher-order multiple reflections from the scattering data.

12. T. Erdogan and V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B **11**, 2100–2105 (1994). [CrossRef]

*n*

_{ac,x}-

*n*

_{ac,y}of 1·10

^{-6}(2.9 % of common mode index modulation), which is independent of uv-polarization. The middle plot shows the orientation of the eigenaxis that corresponds to

*n*

_{ac,x}. We find that the eigenaxis is approximately constant throughout the whole grating. One explanation of the photosensitivity of Ge-doped fiber without H

_{2}-loading, is the modification of the glass structure leading to a volume change of the glass. These volume changes lead to uv-induced birefringence [26

26. F. Kherbouche and B. Poumellec, “UV-induced stress fields during Bragg grating inscription in optical fibers,” J. Opt. A **3**, 429–439 (2001). [CrossRef]

*n*

_{ac}=1.7·10

^{-4}. The effective average birefringence across the mode field should be smaller than this. It should be noted that the resolution in differential index modulation depends on PDL in couplers and detectors. The couplers used had a PDL as low as 0.2 %, while the detectors are specified to have a maximum PDL of 2 %. Thus, the uncertainty in the measurement of differential index modulation amplitude is lower than 2 %.

^{-6}. It is tempting to assume that these fluctuations are due to the dependence of the uv-polarization. However, by closer inspection, we find that the fluctuations are proportional to the derivative of common mode index modulation amplitude. This may be caused by a horizontal offsets between the reconstructed

*n*

_{ac,x}and

*n*

_{ac,y}. When extracting the seven bands from ζ(

*τ*), the bands may have offsets that are fractions of a sample. Such small offsets may give an offset between the two index modulation profiles. Assume that the two index modulation profiles are equal but the second is slightly shifted in vertically an amount

*∊*of from the position

*z*. Then,

^{-6}for this fiber.

^{-5}. When integrating along the grating length, this corresponds to a grating phase variation of 60 mrad. In comparison, the relative peak-to-peak variation in index modulation amplitude between 8 and 10 mm, where the uv-polarization was constant equal to s, was about 1.5 %. By combining the index modulation and phase into a phasor, one should expect that white measurement noise contributes equally in all directions around this phasor. In this case, the relative noise in index modulation should be approximately equal to the phase noise. Since the grating phase variations are 4 times larger, we can conclude that measured dc-index are originating from the writing setup or the fiber. The measured difference in grating phase between the two axes is in the same range as the relative variations in index modulation amplitude. Thus, the low birefringence of this fiber could not be measured with this setup. However, the birefringence is less than 1·10

^{-5}, corresponding to wavelength shift of less than 10 pm. The rapid variations of the orientation of the dc-index eigenaxis shown in the middle plot, also indicates that this measurement is dominated by noise.

## 5. Conclusions

^{-6}was observed, but insignificant dependence on uv-polarization was found.

### A. Factorization of γ

**21**, 1207–1220 (2004). [CrossRef]

**γ**can be written

*U*

^{T}Σ

*U*, where

*U*is unitary and Σ is diagonal and nonnegative.

**γ**that can be used in the layer-peeling algorithm to handle non-reciprocity.

**γ=**

*V*_{1}Σ*V*_{2}, where

*V*

_{1}and

*V*symmetric. By evaluating

_{2}is unitary and Σ is diagonal and nonnegative. Let the matrix Φ

_{as}be a unitary matrix with detΦ

_{as}=1, and is such that

**=**γ ˜

**V**_{1}Σ

**V**_{2}

**γ̃**

^{†}

**γ̃**)

^{T}=

**γ̃**

**γ̃**

^{†}, we find that

*D*Σ^{2}=Σ

^{2}

*, where*

**D****1. Since**D = V 2 ∗ Φ as T V

*commutes with*

**D****Σ**

^{2},

**is diagonal and unitary when the singular values are unequal. We may write**

*D*V 2 ∗ V

*W*=_{1}. Since det

**Φ̃**

_{as}=detΦas=1, det

*=det*

**D***. Thus,*

**W****Φ**

_{as}can be calculated from Eq. (26) where

*θ*is real but otherwise arbitrary.

**should be chosen so that the phase eigenvalues of**

*D***Φ**

_{as}are minimum. This can be done by maximizing the real parts of its eigenvalues. The matrices

**Φ**

_{as}and

**Φ̃**

_{as}have the same eigenvalues, which is given by

_{as,11}} gives e

^{iθ}=

*W*

_{11}/|

*W*

_{11}|.

## References and links

1. | D.-W. Huang and C.-C. Yang, “Reconstruction of fiber grating refractive-index profiles from complex Bragg reflection spectra,” Appl. Opt. |

2. | S. Keren and M. Horowitz, “Interrogation of fiber gratings by use of low-coherence spectral interferometry of noiselike pulses,” Opt. Lett. |

3. | P. Giaccari, H. Limberger, and R. Salathé, “Local coupling-coefficient characterization in fiber Bragg gratings,” Opt. Lett. |

4. | D. Sandel, R. Noé, G. Heise, and B. Borchert, “Optical network analysis and longitunal structure characterization of fiber Bragg gratings,” IEEE J. Lightwave Technol. |

5. | O. Waagaard, E. Rønnekleiv, and J.T. KringlebotnV. Pruneri, R. Dahlgren, and G. Sanger, “Spatial characterization of strong fiber Bragg gatings,” in |

6. | O. Waagaard, “Spatial characterization of strong fiber Bragg gratings using thermal chirp and optical-frequency-domain reflectometry,” IEEE J. Lightwave Technol. |

7. | W. Eickhoff and R. Ulrich, “Optical frequency-domain reflectometry in single-mode fiber,” Applied Physics Letters |

8. | U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides,” IEEE J. Lightwave Technol. |

9. | J. von der Weid, R. Passy, G. Mussi, and N. Gisin, “On the characterization of optical network componenents with optical frequency domain reflectometry,” IEEE J. Lightwave Technol. |

10. | G. Meltz and W. W. Morey, “Bragg grating formation and germanosilicate fiber photosensitivity,” in International workshop on photoinduced self-organization effects in optical fiber, Proc. Soc. Photo-Opt. Instrum. Eng. |

11. | K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters,” Electron. Lett. |

12. | T. Erdogan and V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B |

13. | S. Pereira, J. E. Sipe, R. E. Slusher, and S. Spälter, “Enhanced and suppressed birefringence in fiber Bragg gratings,” J. Opt. Soc. Am. B |

14. | B. Soller, D. Gifford, M. Wolfe, and M. Foggatt, “High resolution optical frequency domain reflectometry for characterization of components and assemblies,” Opt. Express |

15. | O. Waagaard and J. Skaar, “Synthesis of birefringent reflective gratings,” J. Opt. Soc. Am. A |

16. | Luna Technologies white paper, “Optical vector network analyzer for single scan measurements of loss, group delay and polarization mode dispersion,” http://www.lunatechnologies.com/products/ova/files/OVAwhitePaper.pdf (Luna Technologies, 2005). |

17. | Agilent Technologies white paper, “Agilent 81910A Photonic All-Parameter Analyzer User Guide,” http://www.home.agilent.com/agilent/facet.jspx?kt=1&cc=US&lc=eng&k=81910 (Agilent, 2005). |

18. | R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. |

19. | J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. |

20. | J. Skaar and O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. |

21. | R. Azzam and N. Bashara, |

22. | E. Rønnekleiv, “Frequency and Intensity Noise of Single Frequency Fiber Bragg Grating Lasers,” Opt. Fiber Technol. |

23. | P. Oberson, B. Hutter, O. Guinnard, L. Guinnard, G. Ribordy, and N. Gisin, “Optical frequency domain reflectometry with a narrow linwidth fiber laser,” IEEE Photon. Technol. Lett. |

24. | S. Kakuma, K. Ohmura, and R. Ohba, “Improved uncertainty of optical frequency domain reflectometry based length measurement by linearizing the frequency chirping of a laser diode,” Opt. Rev. |

25. | A. Asseh, H. Storøy, B. Sahlgren, S. Sandgren, and R. Stubbe, “A writing technique for long fiber bragg gratings with complex reflectivity profiles,” IEEE J. Lightwave Technol. |

26. | F. Kherbouche and B. Poumellec, “UV-induced stress fields during Bragg grating inscription in optical fibers,” J. Opt. A |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(230.1480) Optical devices : Bragg reflectors

(260.1440) Physical optics : Birefringence

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 9, 2006

Revised Manuscript: April 27, 2006

Manuscript Accepted: April 28, 2006

Published: May 15, 2006

**Citation**

Ole H. Waagaard, "Polarization-resolved spatial characterization of birefringent Fiber Bragg Gratings," Opt. Express **14**, 4221-4236 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4221

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### References

- D.-W. Huang and C.-C. Yang, "Reconstruction of fiber grating refractive-index profiles from complex Bragg reflection spectra," Appl. Opt. 38, 4494-4498 (1999). [CrossRef]
- S. Keren and M. Horowitz, "Interrogation of fiber gratings by use of low-coherence spectral interferometry of noiselike pulses," Opt. Lett. 26, 328-330 (2001). [CrossRef]
- P. Giaccari, H. Limberger, and R. Salathe, "Local coupling-coefficient characterization in fiber Bragg gratings," Opt. Lett. 28, 598-600 (2003). [CrossRef] [PubMed]
- D. Sandel, R. Noe, G. Heise, and B. Borchert, "Optical network analysis and longitunal structure characterization of fiber Bragg gratings," IEEE J. Lightwave Technol. 16, 2435-2442 (1998). [CrossRef]
- O. Waagaard, E. Rønnekleiv, and J.T. Kringlebotn, "Spatial characterization of strong fiber Bragg gatings," in Proceedings of SPIE, Fiber-Based Components Fabrication, Testing, and Connectorization, V. Pruneri, R. Dahlgren, and G. Sanger, eds., vol. 4943, pp. 16-24 (2003).
- O. Waagaard, "Spatial characterization of strong fiber Bragg gratings using thermal chirp and optical-frequency-domain reflectometry," IEEE J. Lightwave Technol. 23, 909-914 (2005). [CrossRef]
- W. Eickhoff and R. Ulrich, "Optical frequency-domain reflectometry in single-mode fiber," Applied Physics Letters 39, 693-695 (1981). [CrossRef]
- U. Glombitza and E. Brinkmeyer, "Coherent frequency-domain reflectometry for characterization of single-mode integrated-optical waveguides," IEEE J. Lightwave Technol. 11, 1377-1384 (1993). [CrossRef]
- J. von der Weid, R. Passy, G. Mussi, and N. Gisin, "On the characterization of optical network componenents with optical frequency domain reflectometry," IEEE J. Lightwave Technol. 15, 1131-1141 (1997). [CrossRef]
- G. Meltz and W. W. Morey, "Bragg grating formation and germanosilicate fiber photosensitivity," in International workshop on photoinduced self-organization effects in optical fiber, Proc. Soc. Photo-Opt.Instrum. Eng. 1516, 185-199 (1991).
- K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, "Birefringent photosensitivity in monomode optical fibre: application to external writing of rocking filters," Electron. Lett. 27, 1548-1550 (1991). [CrossRef]
- T. Erdogan and V. Mizrahi, "Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers," J. Opt. Soc. Am. B 11, 2100-2105 (1994). [CrossRef]
- S. Pereira, J. E. Sipe, R. E. Slusher, and S. Spalter, "Enhanced and suppressed birefringence in fiber Bragg gratings," J. Opt. Soc. Am. B 19, 1509-1515 (2002). [CrossRef]
- B. Soller, D. Gifford, M. Wolfe, and M. Foggatt, "High resolution optical frequency domain reflectometry for characterization of components and assemblies," Opt. Express 13, 666-674 (2005). [CrossRef] [PubMed]
- O. Waagaard and J. Skaar, "Synthesis of birefringent reflective gratings," J. Opt. Soc. Am. A 21, 1207-1220 (2004). [CrossRef]
- Luna Technologies white paper, "Optical vector network analyzer for single scan measurements of loss, group delay and polarization mode dispersion," http://www.lunatechnologies.com/products/ova/files/OVAwhitePaper.pdf (Luna Technologies, 2005).
- Agilent Technologies white paper, "Agilent 81910A Photonic All-Parameter Analyzer User Guide," http://www.home.agilent.com/agilent/facet.jspx?kt=1&cc=US&lc=eng&k=81910 (Agilent, 2005).
- R. Feced, M. N. Zervas, and M. A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999). [CrossRef]
- J. Skaar, L. Wang, and T. Erdogan, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quantum Electron. 37, 165-173 (2001). [CrossRef]
- J. Skaar and O. H. Waagaard, "Design and characterization of finite length fiber gratings," IEEE J. Quantum Electron. 39, 1238-1245 (2003). [CrossRef]
- R. Azzam and N. Bashara, Ellipsometry and polarized light (North-Holland, 1977).
- E. Rønnekleiv, "Frequency and Intensity Noise of Single Frequency Fiber Bragg Grating Lasers," Opt. Fiber Technol. 7, 206-235 (2001). [CrossRef]
- P. Oberson, B. Hutter, O. Guinnard, L. Guinnard, G. Ribordy, and N. Gisin, "Optical frequency domain reflectometry with a narrow linwidth fiber laser," IEEE Photon. Technol. Lett. 12, 867-869 (2000). [CrossRef]
- S. Kakuma, K. Ohmura, and R. Ohba, "Improved uncertainty of optical frequency domain reflectometry based length measurement by linearizing the frequency chirping of a laser diode," Opt. Rev. 10, 182-183 (2003). [CrossRef]
- A. Asseh, H. Storøy, B. Sahlgren, S. Sandgren, and R. Stubbe, "A writing technique for long fiber bragg gratings with complex reflectivity profiles," IEEE J. Lightwave Technol. 15, 1419-1423 (1997). [CrossRef]
- F. Kherbouche and B. Poumellec, "UV-induced stress fields during Bragg grating inscription in optical fibers," J. Opt. A 3, 429-439 (2001). [CrossRef]

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