## 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

Spatial characterization of fabricated fiber Bragg gratings (FBGs) is a useful tool because it provides direct information of the noise in the fabrication of the FBG or fiber properties. The spatial profile of the FBG can be found by first measuring the complex reflectivity spectrum, and then reconstruct the spatial profile using an inverse scattering algorithm such as the layer peeling algorithm [1, 2, 3, 4, 5, 6].

A method for measuring the complex reflectivity spectrum is the optical frequency domain reflectometry (OFDR) [7, 8, 9]. In OFDR, the optical frequency of a highly coherent tunable laser source (TLS) is swept, and the light is launched into a measurement interferometer consisting of one arm containing the FBG and one reference arm. The interference signal from the interferometer is detected, and the complex reflection spectrum can be extracted from the interference signal.

In previous methods the grating is assumed to be polarization independent, that is the reflection and transmission spectra are treated as scalar functions. However, the fiber in which the FBG is fabricated will always have some intrinsic birefringence, and the fabrication using uv-light may also induce birefringence as well as polarization-dependent index modulation [10, 11, 12, 13].

If the intrinsic birefringence is constant and there is no uv-induced birefringence or polarization-dependent index modulation, the FBG can be represented as two decoupled scalar spectra in the two eigenaxes of the fiber. Thus, as long as the interrogation state of polarization (SOP) is along one of the eigenaxes of the fiber, only a scalar measurement of the reflection spectrum is required. The interrogation SOP can be oriented along one of the eigenaxes of the fiber by maximizing or minimizing the transmissivity at an edge of the FBG stop band [5]. Alternatively, a polarization diversity receiver can be used [14]. Here, one take advantage of that the sum of the power reflectivities in any two orthogonal polarization states equals the square sum of the two decoupled scalar spectra.

In cases where both intrinsic and uv-induced birefringence contribute, the combined birefringence may have eigenpolarizations that depends on position [15]. Then, the reflectivity spectrum of the FBG can no longer be treated as two decoupled scalar spectra, and a measurement of the full reflection Jones matrix using a polarizationresolved characterization method is required.

In Ref. [4], a polarization-resolved spectral characterization method is presented, where polarization controllers are placed in both arms of the measurement interferometer. This method has the disadvantage that 16 laser sweeps are required with different settings for the polarization controllers. There are also commercial instruments that are able to measure the reflection Jones matrix, such as the Luna Technologies Optical Vector Analyzer [16] and the Agilent 81910A Photonic All-parameter Analyzer [17]. Both these instruments include a polarization-diversity receiver.

In this paper we will present and demonstrate a new a method for spectral measurements of the reflection Jones matrix of an FBG using a polarization modulation interferometer between the TLS and the measurement interferometer. In this may a polarization-diversity receiver is not required. This polarization modulation interferometer has three paths with different delays, where the SOPs out of two of the paths are parallel, while the third is orthogonal to the two other SOPs. The polarization modulation interferometer spreads the information about the reflection Jones matrix elements into multiple frequency bands in the detected signal. The polarization-dependent layer-peeling algorithm [15] can be used to find spatial distribution of the index modulation and phase, and also the birefringence and the polarization-dependence of the index modulation from the measured spectrum of the reflection Jones matrix.

## 2. Theory

Figure 1 shows the basic setup for measuring complex reflection spectrum of an FBG using an OFDR measurement [5]. The tunable laser source (TLS) sweeps the wavelength across the bandwidth of the FBG. The FBG constitutes one of the reflectors in a Michelson interferometer and a highly reflective mirror the other. The light that has been reflected from the FBG and the mirror will interfere. By extracting the phase and amplitude of this interference signal, the complex reflection spectrum can be found, which serves as an input to the layer-peeling algorithm [18, 19, 20] that calculates the spatial profile of the grating. The light that is transmitted through the grating can also be measured, and can be used to scale the measured reflectivity so that the sum of power reflectivity and the power transmissivity is unity across the whole grating bandwidth [5].

We will now analyze the requirement for spectral measurement of all Jones matrix elements of the FBG reflection and transmission. The optical frequency dependent reflection Jones matrix of the grating is denoted * 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.

The electric field vectors *E*
_{m}(*t*) and *E*
_{r}(*t*) reaching the reflection measurement detector after reflection from the FBG and the mirror, respectively, are given by

where *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].) The two light signals interfere, and the detected power is given by

$$+2{k}_{m}{k}_{r}\mathrm{Re}\left\{{\mathit{E}}_{m}{\left(t\right)}^{\u2020}{\mathit{E}}_{r}\left(t\right)\right\}$$

$$={k}_{m}^{2}{\hat{\mathit{E}}}_{m}^{\u2020}{\hat{\mathit{E}}}_{m}+{k}_{r}^{2}{\hat{\mathit{E}}}_{r}^{\u2020}{\hat{\mathit{E}}}_{r}+2{k}_{m}{k}_{r}\mathrm{Re}\left\{{\hat{\mathit{E}}}_{m}^{\u2020}{\hat{\mathit{E}}}_{r}{\text{e}}^{i2\pi \nu {\tau}_{0}}\right\}$$

where †=T∗ is the transpose conjugate operation, and *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}:

$$={\hat{\mathit{E}}}_{s}^{\u2020}{\mathbf{\Phi}}_{s}^{\u2020}{\mathbf{\Phi}}_{m}^{\u2020}{\mathbf{\Phi}}_{m}^{*}{\mathbf{\Phi}}_{r}^{T}\mathit{R}\left(\nu \right){\mathbf{\Phi}}_{r}{\mathbf{\Phi}}_{s}{\hat{\mathit{E}}}_{s}{e}^{i2\pi \nu {\tau}_{0}}$$

$$={\hat{E}}_{s}^{\u2020}{\mathbf{\Phi}}_{o}\mathit{R}\left(\nu \right){\mathbf{\Phi}}_{i}{\hat{\mathit{E}}}_{s}{\text{e}}^{i2\pi \nu {\tau}_{0}},$$

Here, we have left out the constant *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 ${\mathbf{\Phi}}_{\mathrm{d}}^{\ast}$${\mathbf{\Phi}}_{\mathrm{d}}^{\mathrm{T}}$=*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.

A more explicit expression for *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}:

$$={C}_{s}^{T}\left[\begin{array}{c}{\tilde{R}}_{11}\\ {\tilde{R}}_{21}\\ {\tilde{R}}_{12}\\ {\tilde{R}}_{22}\end{array}\right]{\text{e}}^{i2\pi {\tau}_{0}\nu},$$

where *C*
_{s}=[|*E*
_{1}|^{2}
*E*
^{∗}2*E*
_{1}
${E}_{1}^{\ast}$
*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].

Extracting the four components of * R
̃* from Eq. (5), requires four measurements of

*P*

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

where *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.

The four measurements can be made by sweeping the laser four times with different polarization settings. However, this may lead to offsets in optical frequencies between the measurements, due to the uncertainty in absolute wavelength of the TLS. An alternative is to continuously modulate the SOP so that components of *R̃* have different carrier frequencies [16]. This method is more elaborated in Sec. 3. Using this method, only one sweep is required and there will be no problems with frequency offsets between the components of * R
̃*.

The transmission measurement is not used directly to calculated the spatial profile of the FBG, but can be used for proper scaling of *R̃*. From Fig. 1 we find that the SOP of the light reaching the transmission measurement detector is given by

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

$$={k}_{t}^{2}{\hat{\mathit{E}}}_{s}^{\u2020}{\mathbf{\Phi}}_{i}^{\u2020}T{\left(\nu \right)}^{\u2020}T\left(\nu \right){\mathbf{\Phi}}_{i}{\hat{\mathit{E}}}_{s}={k}_{t}^{2}{\hat{\mathit{E}}}_{s}^{\u2020}\tilde{\mathit{T}}{\left(\nu \right)}^{\u2020}\tilde{\mathit{T}}\left(\nu \right){\tilde{\mathit{E}}}_{s}$$

where we have used that ${\mathrm{\Phi}}_{\mathrm{t}}^{\u2020}$Φ_{t}=*I*.

Assuming that the FBG is lossless [15],

This relation is used to find *k*
_{m}
*k*
_{r} in Eq. (3) so that * R
̃* is scaled correctly.

The polarization-resolved layer-peeling algorithm in Ref. [15] will be used to calculate the spatial profile of the grating from the grating impulse response (the Fourier transform of the reflection spectrum.) However this algorithm requires that the grating is reciprocal and measured in a linear basis, which means that the reflection Jones matrix is symmetric for all optical frequencies *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}=${\mathbf{\Phi}}_{\mathrm{i}}^{\mathrm{T}}$. We calculate the grating impulse response

*h*(

*τ*)=FT{

*(*

**R***ν*)}, as

*h*(

*τ*)=

**Φ̃**†

_{o}

*h̃*(

*τ*)${\stackrel{\mathbf{~}}{\mathbf{\Phi}}}_{\mathrm{i}}^{\u2020}$, where

*h̃*(

*τ*)=FT{

*(*

**R**̃*ν*)}. Here, FT denotes the Fourier transform. The matrices $\tilde{\mathbf{\Phi}}$ o and $\tilde{\mathbf{\Phi}}$ i must be chosen so that

*h*(

*τ*) is symmetric for all τ, which means that $\tilde{\mathbf{\Phi}}$

_{o}=Φ

_{o}$\tilde{\mathbf{\Phi}}$ and $\tilde{\mathbf{\Phi}}$

_{i}=$\tilde{\mathbf{\Phi}}$

^{T}Φ

_{i}where $\tilde{\mathbf{\Phi}}$ is an arbitrary and constant unitary matrix. To remove the effect of $\tilde{\mathbf{\Phi}}$

_{o}and $\tilde{\mathbf{\Phi}}$

_{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

where *θ* is real but otherwise arbitrary, since *UDΣD*
^{∗}
*V*=*U*Σ*V*. We choose $\tilde{\mathbf{\Phi}}$
_{o}=*UD* and $\tilde{\mathbf{\Phi}}$
_{i}=*D*
^{∗}
*V*, which gives

where *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.

The polarization-resolved layer-peeling algorithm [15] can now be used to calculate the polarization-dependent spatial profile of the FBG from *h*(*τ*). In Ref. [15], the grating is discretized into *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.

In the polarization-resolved layer-peeling algorithm, let γ_{j} represent the reflection response from layer *j* alone. γ_{j} should be symmetric due to reciprocity, and given by ${\mathbf{\Phi}}_{j}^{\mathrm{T}}$
*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*.

When the grating reflection response is measured, and not synthesized as in Ref. [15], measurement noise and calibration errors may lead to a measured response that is not symmetric. In Appendix A it is shown how to factorize **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}=${\mathbf{\Phi}}_{j}^{\mathrm{T}}$
*ρ*
_{j}
**Φ**
_{j}
**Φ**
_{as,j}.

## 3. Measurement of the grating impulse response

#### 3.1. Optical frequency modulation

The setup used for measuring the optical-frequency dependent reflection Jones matrix of a fiber Bragg grating (FBG) is shown in Fig. 2. The reflection of the FBG is detected at detector output D1, while detector output D2 gives the transmissivity. These two waveforms are sampled with a National Instrument NI-6052 data acquisition (DAQ) card, and processed in LabView^{TM} and Matlab^{TM}.

A distributed-feedback fiber laser (DFB-FL) at 1549 nm is used as the tunable laser source. This source provides both low frequency and intensity noise and mode-hop free operation [22]. The linewidth is typically in the order of 1–10 kHz. By using this source, superior sensitivity and length range can be obtained [23]. The DFB-FL is arranged in a master oscillator power amplifier (MOPA) configuration. The laser is pumped by a 100 mW 1480 nm laser diode, and an EDFA boosts up the power from the laser and removes the remaining pump light. The power at the output of the MOPA is detected with D5. The detected power provides negative feedback to the laser diode current to minimize the intensity noise of the laser output [22].

One end of the DFB-FL is fixed, while the other end is mounted to the armature (the moving part) of a solenoid. When voltage is applied to the solenoid, the fiber is strained. This provides tuning of the wavelength of the laser across the grating bandwidth. We apply at maximum 1.2% stain to the fiber, and the available wavelength range becomes 1549 nm to 1563 nm.

A trigger Michelson interferometer with ~100 m imbalance is used to keep track of the exact evolution of the optical frequency during the sweep of the laser. We use the fringe output D4 of this interferometer to generate a sampling clock to the NI-6052 card for sampling the detector outputs D1-3 equidistant in optical frequency [8]. The phase of the fringe signal at detector D4 is *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.

In practice, the interference signal at D1 will not be sampled exactly equidistant in optical frequency; there will always be some sampling jitter associated with the generation of trigger pulses. Let the interference signal be sampled at a frequency *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

Here, we assume for simplicity that 2*k*
_{m}
*k*
_{r}=1. With sufficiently large trigger interferometer delays, the last term in this equation will typically give a noise contribution to the measured value for *P*
_{i}(*ν*) that exceeds the quantum noise with several orders of magnitude [8]. It is therefore essential to minimize this sampling jitter in order to minimize noise in the characterization results. Assuming that Re{*P*
_{i}(*k*Δ*ν*)} and *δν*(*k*) are uncorrelated, the rms of this noise contribution is given by

Improved signal-to-noise ratio (SNR) can be obtained by reducing *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.

To linearize the sweep, the trigger interferometer enables closed-loop control of the laser sweep [24]. The PLD in Fig. 2 produces a 120 ns long pulse every time the fringe signal at D4 crosses zero. The delay between subsequent pulses is given by the actual sweep rate SR=*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

The polarization modulation interferometer (PolIF) comprises four 3 dB couplers and the three paths 1, 2 and 3 with delays *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.

The electric field vector at the output of the PolIF is given by

$$={k}_{1}{\mathbf{\Phi}}_{1}{e}^{-i\left(2\pi \nu \left(t-{\tau}_{1}\right)\right)}\left({\hat{\mathit{e}}}_{1}+{\hat{\mathit{E}}}_{a}{e}^{i2\pi \nu {\tau}_{a}}+{\hat{\mathit{E}}}_{b}{e}^{i2\pi \nu {\tau}_{\text{b}}}\right)\text{},$$

where *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}.

The matrix **Φ**
_{1} is a unitary matrix with the first column equal to *Ê*
_{1}, so that **Φ**
_{1}
*ê*
_{1}=*Ê*
_{1}. We write,

The second column of **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.

By inserting Eq. (14) into Eq. (4) we find that the matrix **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.

The coherence vector found by inserting Eq. (15) and Eq. (16) into Eq. (14) then becomes,

where

In Sec. 2 we found that the reflection Jones matrix can be extracted if the SOP in Eq. (17) is modulated through 4 SOPs represented by 4 linearly independent coherence vectors (or equally 4 linearly independent Stokes vectors). This will be the case as long as *τ*
_{a}≠*τ*
_{b}≠*τ*
_{b}-*τ*
_{a}≠0, *k*
_{a}, *k*
_{b}≠0 and * M* has linearly independent rows.

*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

Inserting Eq. (17) into Eq. (5) gives the detected interference power at D1:

Now, let *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

which is valid for 0≤*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

Detector D3 is used for monitoring of the polarization modulation and calculation of the matrix * M*. The Jones vector at detector D3 is proportional to

*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,

$$={k}_{D3}[1+{k}_{c}^{2}\left({k}_{a}^{2}+{k}_{b}^{2}+2\mathrm{Re}\left\{\left({s}_{a}^{*}{s}_{b}+{c}_{a}^{*}{c}_{b}\right){e}^{i2\pi \nu \left({\tau}_{b}-{\tau}_{a}\right)}\right\}\right)$$

$$-2{k}_{c}\left(\mathrm{Re}\left\{{s}_{a}{e}^{i2\pi \nu {\tau}_{a}}\right\}+\mathrm{Re}\left\{{s}_{b}{e}^{i2\pi \nu {\tau}_{b}}\right\}\right)],$$

where *k*
_{D3} is a scaling constant.

The PolIF can be set in the optimum mode of operation by adjusting polarization controllers PC1 and PC2, while repeatedly sweeping the laser, measure the response at detector D3, and calculate the Fourier transform ζ_{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.

Even though the PolIF is calibrated so that the SOP out of the second and third path is orthogonal to the SOP out of the first path, the birefringence in the paths may fluctuate between measurements. Instead of readjusting PC1 and PC2 to compensate for the drift in birefringence, the response at detector D3 is measured simultaneously with the responses at detectors D1 and D2 to find the matrix * M* in Eq. (18).

Provided that the constants *k*
_{a}, *k*
_{b} and kc are known (see below), *k*
_{D3} can be found from ζ_{D3}(0). We may then calculate

Thus, we have obtained all the components required for calculation of * M*.

The constants *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.

When the FBG replaced is replaced by a patch-cord, the resulting detected power at D2 will be proportional to Eq. (22) but with *k*
_{c}=-1. The constant *k*
_{c} can therefore be found by comparing the measured responses at D3 and D2.

Equation (22) provides measurement of seven independent real parameters, whereas Eqs. (23) only use six. The remaining independent parameter gives rise to the equation

$$=\frac{{k\prime}_{a}^{2}{k\prime}_{b}^{2}}{{\left(1+{k\prime}_{a}^{2}+{k\prime}_{b}^{2}\right)}^{2}}+2\left(1+{k\prime}_{a}^{2}+{k\prime}_{b}^{2}\right)\mathrm{Re}\left\{{\zeta}_{D3}^{\prime}\left({\tau}_{b}-{\tau}_{a}\right){\zeta}_{D3}^{\prime}\left({\tau}_{a}\right){\zeta \prime}_{D3}^{*}\left({\tau}_{b}\right)\right\},$$

where ζ′_{D3}(τ)=ζ_{D3}(τ)/ζ_{D3}(0), ${k}_{\mathrm{a}}^{\prime}$=*k*
_{c}
*k*
_{a} and ${k}_{\mathrm{b}}^{\prime}$=*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

The output SR in Fig. 2 provides measurements of the laser sweep rate. Fig. 3 shows the measured sweep rate for a commercial tunable laser (Ando AQ4320B), and our fiber laser both with and without closed-loop control of the laser sweep. The Ando laser has large high-frequency fluctuations in sweep rate, but with almost constant mean. The sweep rate fluctuations for the open-loop swept fiber laser is large but much more low frequent than the Ando laser. The sweep rate has a increasing trend, which is caused by the increased force applied to the armature of solenoid when a larger part of the armature is within the coil. The more rapid fluctuations are probably caused by varying friction between the armature and the inner walls of the solenoid.

When the sweep of the fiber laser is controlled using the PID-controller, most of the sweep rate fluctuations are eliminated. The sweep rate has an overshoot of 3.5 THz/s the first 30 ms (not shown), but is constant throughout the rest of the sweep with a rms variation of 0.63 GHz/s or 0.14 % of mean sweep rate. The large bandwidth of the measured sweep rate, may indicate that a larger part of the remaining fluctuations in the measured sweep rate are measurement noise, since the bandwidth of the mechanical response is low. Nevertheless, the sweep rate fluctuations of the closed-loop controlled fiber laser are more than two orders of magnitude lower than that of the Ando laser (16 %) and the open-loop controlled fiber laser (30 %).

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

The measurement setup was tested on a 10 mm uniform grating at 1554.5 mn with a reflectivity of 35 % written by scanning the uv-light [25] along a Nufern GF3 photosensitive fiber. The polarization of the uv-light was varied between p- and s-polarization during production.

Before measuring the grating response, the polarization controllers PC1 and PC2 were adjusted to minimize |ζ_{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 * M* 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.

Figure 4 shows a segment of |ζ(*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.

The signal bands extracted from ζ(*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]. This is therefore an indication that such beating does not take place. Since, *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.

The upper plot in Fig. 6 shows the reconstructed index modulation eigenmode amplitudes *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.

The index modulation amplitudes have a pronounced common mode dependence of the uv-polarization. At the end of the grating, s-polarization was used. We find that the common mode index modulation for p-polarization is about 7 % lower. Most of this difference is due to the dependence of visibility on the angle between the uv-beams. The reduction in visibility for p-polarization becomes 5.2 % for a grating at 1550 nm [12]. Additional common mode amplitude difference may be caused by polarization-dependent loss of the bulk optics in the writing setup and difference in photosensitivity due to stress effects.

The upper plot of Fig. 6 shows an average differential index modulation amplitude *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], and birefringence has been calculated to be 6 % of the index modulation amplitude at the fiber center with *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 %.

The fluctuations in differential index modulation amplitude is ±0.2·10^{-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,

The reconstructed differential index modulation amplitude in Eq. (25) shows that it will be proportional to the derivative of the common mode index modulation amplitude. A scaled version of the derivative of the common mode index was calculated so that the best fit to the differential index modulation amplitude was obtained. This signal was subtracted from the differential index modulation amplitude. Assuming that none of the components in the differential index modulation amplitude that are proportional to derivative of the common mode index modulation amplitude are real, we then find that the uv-dependence of the index modulation difference is ~0.1·10^{-6} for this fiber.

The lower plot in Fig. 6 shows the measured change in dc-index in the two birefringence eigenaxes. We find a common mode peak-to-peak variation of about 9·10^{-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

We have developed a method for polarization-resolved characterization of fiber Bragg gratings using OFDR. A strain-tuned fiber laser was used as source. With closed-loop control of the sweep rate, the sweep in optical frequency was made highly linear, with a rms variation of 0.14 % from constant sweep rate. The state of polarization of the light interrogating the FBG was modulated using a polarization modulation interferometer. In this way, the different components of the grating impulse response was spread in the optical delay-domain. From these bands, the grating impulse response matrix was calculated. This grating impulse response was fed into the layer-peeling algorithm to calculate the birefringence, the common mode index modulation amplitude and differential index modulation amplitude. The measurement setup was tested on a grating that was written with varying uv-polarization. A differential index modulation amplitude of 1·10^{-6} was observed, but insignificant dependence on uv-polarization was found.

#### A. Factorization of γ

In Appendix D.2 in Ref. [15] it is shown that a symmetric matrix **γ** can be written *U*
^{T}Σ*U*, where *U* is unitary and Σ is diagonal and nonnegative.

We must find a relevant factorization of a general non-symmetric matrix **γ** that can be used in the layer-peeling algorithm to handle non-reciprocity.

Using singular value decomposition (svd), **γ= V_{1}ΣV**

_{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

**$\tilde{\gamma}$**=${\mathbf{\gamma}\mathbf{\Phi}}_{\text{as}}^{\u2020}$=

**V**_{1}Σ

**V**_{2}${\mathbf{\Phi}}_{\text{as}}^{\u2020}$ is symmetric. By evaluating (

**γ̃**

^{†}

**γ̃**)

^{T}=

**γ̃**

**γ̃**

^{†}, we find that

*D*Σ^{2}=Σ

^{2}

*, where*

**D****${D\mathit{=}V}_{2}^{\ast}$${\mathbf{\Phi}}_{\text{as}}^{\mathrm{T}}$V**1. Since

*commutes with*

**D****Σ**

^{2},

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

*D*where *W*=${V}_{2}^{\ast}$V_{1}. Since det **Φ̃**_{as}=detΦas=1, det* D*=det

*. Thus,*

**W****Φ**

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

and *θ* is real but otherwise arbitrary.

The matrix ** D** should be chosen so that the phase eigenvalues of

**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 $\mathrm{Re}\left\{{\tilde{\mathbf{\Phi}}}_{\mathrm{as},11}\right\}i\sqrt{1-{\mathrm{Re}\left\{{\tilde{\mathbf{\Phi}}}_{\mathrm{as},11}\right\}}^{2}}$, where the index 11 denotes the upper left component. Maximizing Re{$\tilde{\mathbf{\Phi}}$

_{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. **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]

**5. **O. Waagaard, E. Rønnekleiv, and J.T. KringlebotnV. Pruneri, R. Dahlgren, and G. Sanger, “Spatial characterization of strong fiber Bragg gatings,” in *Proceedings of SPIE, Fiber-Based Components Fabrication, Testing, and Connectorization*, eds., vol. 4943, pp. 16–24 (2003).

**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]

**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. **1516**, 185–199 (1991).

**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]

**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. **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]

**21. **R. Azzam and N. Bashara, *Ellipsometry and polarized light* (North-Holland, 1977).

**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]

**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]

**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]

**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]