1416 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 4, APRIL 2011 On the Measurement of Fiber Bragg Grating’s Phase Responses and the Applicability of Phase Reconstruction Methods María José Erro, Alayn Loayssa, Member, IEEE, Santiago Tainta, Rubén Hernandez, David Benito, María J. Garde, and Miguel A. Muriel, Senior Member, IEEE Abstract—A signal processing algorithm is used to reconstruct the phase response of fiber Bragg gratings from easy-to-obtain module data. The algorithm is applied to optical spectrum analyzer measurements, avoiding the use of costly and complex equipment for the direct measurement of the complete frequency response. Reconstructed results, both in transmission and reflec- tion, are then validated by comparison with three of the most common experimental techniques (interferometric, modulation phase shift, and single-sideband modulation). Index Terms—Fiber Bragg gratings (FBGs), frequency response measurement system, phase reconstruction. I. I NTRODUCTION F IBER BRAGG gratings (BGs) (FBGs) are wavelength- selective devices widely used in the sensor field and in optical communications. They are wavelength-selective filters, directly photowritten in the core of an optical fiber by means of an ultraviolet fringe pattern [1]. Likewise, periodic struc- tures like BGs have always been a hot topic for microwave research [2]. In many of the applications for BGs fabricated in fiber or in microwave substrates, the phase response of these structures plays an important role in the behavior of the resulting device. For example, when high rates are used in wavelength-division-multiplexing optical communication sys- tems, the phase response of the FBG filters can determine the dispersion penalty in both transmitted and dropped channels [3]. The phase response is also crucial in the design of optical resonators for sensing applications to determine the relative loss and coupling contributions to the overall response [4]. Moreover, the reconstruction of the index profile that forms the FBG by applying inverse-scattering algorithms is possible only if a complete (amplitude and phase) measure of its frequency response is available. These data are extremely useful for Manuscript received May 26, 2010; revised September 24, 2010; accepted September 27, 2010. Date of publication November 18, 2010; date of current version March 8, 2011. This work was supported in part by the Spanish Comision Interministerial de Ciencia y Tecnologia under Projects TEC 2007- 68065-C01-02 and 03 CICYT. The Associate Editor coordinating the review process for this paper was Dr. George Xiao. M. J. Erro, A. Loayssa, S. Tainta, R. Hernandez, D. Benito, and M. J. Garde are with the Public University of Navarra, 31006 Pamplona, Spain (e-mail: mjose.erro@unavarra.es). M. A. Muriel is with the Universidad Politécnica de Madrid, 28040 Madrid, Spain (e-mail: muriel@tfo.upm.es). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2010.2087850 the characterization and improvement of the FBG fabrication setup [5]. Consequently, a characterization, including both the ampli- tude and the phase of its reflection and/or transmission co- efficients, is needed. The amplitude response for FBGs can be easily obtained with an optical spectral analyzer (OSA) or a scanning laser, and that for the BG in a microstrip can be obtained with a simple scalar network analyzer. However, measuring the phase response requires the use of more sophis- ticated systems. For the FBGs, the most popular among them is the modulation phase shift (MPS) technique [6], for which it is the group delay response what is actually measured, with the phase response being afterward obtained via the numeri- cal integration over the wavelength of the group delay data. However, there is a compromise between wavelength resolution and noise in the measurement, and hence accuracy, that are oppositely affected by the chosen modulation frequency [7]. There have been several proposals to overcome this limitation that affects particularly the characterization of fine details in the group delay of narrow-band filters, but they imply complicated and expensive setups [5], [8], [9]. Another approach is to use interferometric measurements [10]–[13], which allow us to obtain the phase response directly, but those are very sensitive to external disturbances and require, as in the case of MPS, the tuning in wavelength of an optical source, which is intrinsically constrained in resolution. To overcome this limitation, a high- resolution technique based on the one-to-one mapping between optical and radio-frequency (RF) spectral characteristics, using single-sideband (SSB) modulation of an optical carrier, has been proposed [4], [14]. This technique also offers a direct measurement of the phase response, but it is not commercially available, requiring a fairly high sideband suppression to obtain small errors in the measurements. In some particular cases, a phase-retrieval method can be applied to deduce the phase response from the amplitude mea- surement, avoiding the use of those expensive techniques. The method, first proposed for FBGs in [15] and based on the Hilbert transform, relies on the unique relationship between the amplitude and phase of the transfer function that holds for systems that are casual and minimum phase. For a filter to be minimum phase, none of the zeros of its transfer func- tion can lie on the right-half side of the s plane [16]. The transmission transfer function of an FBG can be shown [17] to have no zeros, being therefore, by definition, a minimum- phase function, no matter what the shape of the perturbation 0018-9456/$26.00 © 2010 IEEE