Abstract:
An apparatus characterizes at least one fiber Bragg grating. The apparatus includes a laser pulse source, an optical spectrum analyzer, and multiple optical paths. A first optical path includes a pulse stretcher and an attenuator. A second optical path optically coupled to the first optical path includes a mirror. A third optical path optically coupled to the first optical path includes a first fiber Bragg grating. A fourth optical path is optically coupled to the second optical path, the third optical path, and the optical spectrum analyzer. A fifth optical path optically coupled to the laser pulse source and the optical spectrum analyzer includes a delay line.

Description:
CLAIM OF PRIORITY 
   This application is a continuation of U.S. patent Ser. No. 12/116,871, filed May 7, 2008, incorporated in its entirety by reference herein, which is a divisional application of U.S. patent application Ser. No. 11/130,418, filed May 16, 2005, and incorporated in its entirety by reference herein, which claims the benefit of U.S. Provisional Application Nos. 60/571,660, filed May 15, 2004, 60/599,427, filed Aug. 6, 2004, and 60/662,684, filed Mar. 17, 2005, each of which is incorporated in its entirety by reference herein. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   This invention relates generally to apparatus and methods of characterizing the optical response of fiber Bragg gratings. 
   2. Description of the Related Art 
   Fiber Bragg gratings (FBGs) have many applications in optical communications and optical fiber sensing. The effective refractive index profile Δn(z) of the fiber core mode (e.g., the LP 01  mode, or a higher-order mode) as a function of the position z along the FBG generally varies roughly periodically with z, with an envelope that may vary along z. The effective refractive index Δn(z) determines most of the optical properties of the FBG, including but not limited to, the dispersion properties, the complex reflection impulse response h R (t), the complex transmission impulse response h T (t), the amplitudes |r(ω)|, |t(ω)| and the phases φ r (ω), φ t (ω) of the complex reflection spectrum r(ω) and the complex transmission spectrum t(ω), respectively, and the group delay in reflection 
             ⅆ       ϕ   r     ⁡     (   ω   )           ⅆ   ω           
and transmission
 
               ⅆ       ϕ   t     ⁡     (   ω   )           ⅆ   ω       ,         
where ω is the angular frequency. These functions can also be expressed as functions of the optical wavenumber k, which has a simple relationship to the optical angular frequency ω, so it is a simple matter to switch between ω and k.
 
   A first general method to determine Δn(z) of an FBG is to measure the complex reflection impulse response h R (t), which is the temporal dependence of the amplitude and phase of the signal reflected by the FBG when an extremely short optical signal is launched into the FBG. The complex reflection impulse response h R (t) can be measured directly by launching an ultra-short pulse (e.g., approximately 1 picosecond to approximately 30 picoseconds, depending on the grating length and period) into the FBG and measuring the temporal evolution of the reflected signal. This general method has the drawback of requiring that the width of the input laser pulse be much narrower than the impulse response of the FBG. In addition, interferometric techniques are used to measure both the phase and the amplitude of the complex reflection impulse response, and these techniques are complicated and inherently sensitive to noise or other fluctuations. 
   A second general technique to measure the complex reflection impulse response h R (t) is to use an interferometer to measure the wavelength dependence of both the amplitude and the phase of the optical signal reflected by the FBG (i.e., the complex reflection spectrum r(ω) or r(k)). The complex reflection spectrum r(ω) is the Fourier transform (FT) of the complex reflection impulse response h R (t), as described by A. Rosenthal and M. Horowitz, “ Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings ,” IEEE Journal of Quantum Electronics, Vol. 39, pp. 1018-1026, August 2003. The complex reflection impulse response h R (t) is then recovered from the complex reflection spectrum r(ω) by taking the inverse Fourier transform (IFT) of r(ω). As discussed below, the main difficulty of this general technique is that the measurement of the complex reflection spectrum is in general tedious, sensitive to noise, applicable to only special types of FBGs, and/or time-consuming. 
   The complex reflection spectrum r(ω)=|r(ω)|·exp(jφ r (ω)) of an FBG is measurable using various interferometric measurement systems which are generally more complex and have stronger noise sensitivities than do measurement techniques which merely provide the amplitude of the reflection or transmission spectra. For example, in Michelson interferometry (e.g., as described by D.-W. Huang and C.-C. Yang, “ Reconstruction of Fiber Grating Refractive - Index Profiles From Complex Bragg Reflection Spectra ,” Applied Optics, 1999, Vol. 38, pp. 4494-4499), a tunable laser and an optical spectrum analyzer (OSA) are used to recover the phase of the complex reflection spectrum from three independent measurements. 
   In end-reflection interferometry (e.g., as described by J. Skaar, “ Measuring the Group Delay of Fiber Bragg Gratings by Use of End - Reflection Interference ,” Optics Letters, 1999, Vol. 24, pp. 1020-1022), the FBG is characterized using a tunable laser together with an OSA by measuring the spectral reflectivity that is caused by the interference between the FBG itself and the bare fiber end. This technique, however, is generally a destructive technique, since the bare fiber end must typically be only a few centimeters away from the FBG. 
   In low-coherence time reflectometry (e.g., as described by P. Lambelet et al., “ Bragg Grating Characterization by Optical Low - Coherence Reflectometry ,” IEEE Photonics Technology Letters, 1993, Vol. 5, pp. 565-567; U. Wiedmann et al, “ A Generalized Approach to Optical Low - Coherence Reflectometry Inducing Spectral Filtering Effects ,” J. of Lightwave Technol., 1998, Vol. 16, pp. 1343-1347; E. I. Petermann et al., “ Characterization of Fiber Bragg Gratings by Use of Optical Coherence - Domain Reflectometry ,” J. of Lightwave Technol., 1999, vol. 17, pp. 2371-2378; and S. D. Dyer et al., “ Fast and Accurate Low - Coherence Interferometric Measurements of Fiber Bragg Grating Dispersion and Reflectance ,” Optics Express, 1999, Vol. 5, pp. 262-266), a Michelson interferometer is illuminated with a broadband light source, and light reflected from the FBG, placed on one arm of the interferometer, and light reflected from a moveable mirror, placed on the reference arm of the interferometer, are coupled together and directed to a detector. This technique utilizes a slow mechanical scan to retrieve the impulse response of the FBG as a function of time, which makes this type of measurement time-consuming. 
   In low-coherence spectral interferometry (e.g., as described by S. Keren and M. Horowitz, “ Interrogation of Fiber Gratings by Use of Low - Coherence Spectral Interferometry of Noiselike Pulses ,” Optics Letters, 2001, Vol. 26, pp. 328-330; and S. Keren et al., “ Measuring the Structure of Highly Reflecting Fiber Bragg Gratings ,” IEEE Photon. Tech. Letters, 2003, Vol. 15, pp. 575-577), the slow scanning process is avoided by reflecting broadband laser pulses from the FBG and temporally combining these reflected pulses with their delayed replicas. This pulse sequence is then sent to an OSA, which records the power spectrum. The pulsed laser source of this technique has an autocorrelation function which is temporally much narrower (e.g., approaching a delta function) than the impulse response of the FBG. In other words, the recovery of the impulse response of a given FBG is limited in resolution to the autocorrelation trace of the pulsed laser source. Furthermore, the delay between the reflected pulse from the FBG and the input laser pulse has to be carefully adjusted to avoid overlap in the inverse Fourier transform domain, which makes the recovery impossible due to aliasing. 
   Typically, measurement systems which measure the amplitude of the reflection spectrum or of the transmission spectrum do not provide the missing phase information (i.e., φ r (ω) and/or φ t (ω)). The amplitude measurement, which is relatively simpler than the phase measurement, involves a tunable laser and an optical spectrum analyzer (OSA). Previously, various methods have been proposed to reconstruct the missing phase spectrum or group delay spectrum from only the amplitude measurement of |r(ω)| or |t(ω)|. The phase reconstruction technique presented by Muriel et al., “ Phase Reconstruction From Reflectivity in Uniform Fiber Bragg Gratings ,” Optics Letters, 1997, Vol. 22, pp. 93-95, only works for uniform gratings and has been independently shown to be unsuited for gratings with imperfections (J. Skaar and H. E. Engan, “ Phase Reconstruction From Reflectivity in Fiber Bragg Gratings ,” Optics Letters, 1999, Vol. 24, pp. 136-138). A similar technique has been suggested to improve the noise performance of the initial technique of Muriel et al., however this technique is still limited to only uniform gratings and the processing algorithm involves adjusting of filtering parameters, which depend on the FBG being characterized (K. B. Rochford and S. D. Dyer, “ Reconstruction of Minimum - Phase Group Delay From Fibre Bragg Grating Transmittance/Reflectance Measurements ,” Electronics Letters, 1999, Vol. 35, pp. 838-839). 
   One method of recovering the phase information from the amplitude data of FBGs was previously described by L. Poladian, “ Group - Delay Reconstruction for Fiber Bragg Gratings in Reflection and Transmission ,” Optics Letters, 1997, Vol. 22, pp. 1571-1573. The technique of Poladian utilized the fact that the transmission spectra of all FBGs belong to the family of minimum-phase functions (MPF) which have their phase and amplitude related by the complex Hilbert transform. In the technique of Poladian, using the Hilbert transformation, the phase or group delay of FBGs is recovered from only the measurement of the amplitude of the transmission spectrum |t(ω)|. This technique works very well but the numerical evaluation of the principle-value Cauchy integral in the Hilbert transform is not trivial and is rather noise-sensitive, as described by Muriel et al. 
   SUMMARY OF THE INVENTION 
   In certain embodiments, a method determines a complex reflection impulse response of a fiber Bragg grating. The method comprises providing a measured amplitude of a complex reflection spectrum of the fiber Bragg grating. The method further comprises providing an estimated phase term of the complex reflection spectrum. The method further comprises multiplying the measured amplitude and the estimated phase term to generate an estimated complex reflection spectrum. The method further comprises calculating an inverse Fourier transform of the estimated complex reflection spectrum, wherein the inverse Fourier transform is a function of time. The method further comprises calculating an estimated complex reflection impulse response by applying at least one constraint to the inverse Fourier transform of the estimated complex reflection spectrum. 
   In certain embodiments, a computer system comprises means for estimating an estimated phase term of a complex reflection spectrum of a fiber Bragg grating. The computer system further comprises means for multiplying a measured amplitude of the complex reflection spectrum of the fiber Bragg grating and the estimated phase term to generate an estimated complex reflection spectrum. The computer system further comprises means for calculating an inverse Fourier transform of the estimated complex reflection spectrum, wherein the inverse Fourier transform is a function of time. The computer system further comprises means for calculating an estimated complex reflection impulse response by applying at least one constraint to the inverse Fourier transform of the estimated complex reflection spectrum. 
   In certain embodiments, a method determines a complex transmission impulse response of a fiber Bragg grating. The method comprises providing a measured amplitude of a complex transmission spectrum of the fiber Bragg grating. The method further comprises providing an estimated phase term of the complex transmission spectrum. The method further comprises multiplying the measured amplitude and the estimated phase term to generate an estimated complex transmission spectrum. The method further comprises calculating an inverse Fourier transform of the estimated complex transmission spectrum, wherein the inverse Fourier transform is a function of time. The method further comprises calculating an estimated complex transmission impulse response by applying at least one constraint to the inverse Fourier transform of the estimated complex transmission spectrum. 
   In certain embodiments, a computer system comprises means for estimating an estimated phase term of a complex transmission spectrum of a fiber Bragg grating. The computer system further comprises means for multiplying a measured amplitude of the complex transmission spectrum of the fiber Bragg grating and the estimated phase term to generate an estimated complex transmission spectrum. The computer system further comprises means for calculating an inverse Fourier transform of the estimated complex transmission spectrum, wherein the inverse Fourier transform is a function of time. The computer system further comprises means for calculating an estimated complex transmission impulse response by applying at least one constraint to the inverse Fourier transform of the estimated complex transmission spectrum. 
   In certain embodiments, a method characterizes a fiber Bragg grating. The method comprises providing a measured amplitude of a Fourier transform of a complex electric field envelope of an impulse response of the fiber Bragg grating. The method further comprises providing an estimated phase term of the Fourier transform of the complex electric field envelope. The method further comprises multiplying the measured amplitude and the estimated phase term to generate an estimated Fourier transform of the complex electric field envelope. The method further comprises calculating an inverse Fourier transform of the estimated Fourier transform of the complex electric field envelope, wherein the inverse Fourier transform is a function of time. The method further comprises calculating an estimated electric field envelope of the impulse response by applying at least one constraint to the inverse Fourier transform of the estimated Fourier transform of the complex electric field envelope. 
   In certain embodiments, a computer system comprises means for estimating an estimated phase term of a Fourier transform of the complex electric field envelope of an impulse response of a fiber Bragg grating. The computer system further comprises means for multiplying a measured amplitude of the Fourier transform of the complex electric field envelope and the estimated phase term to generate an estimated Fourier transform of the complex electric field envelope. The computer system further comprises means for calculating an inverse Fourier transform of the estimated Fourier transform, wherein the inverse Fourier transform is a function of time. The computer system further comprises means for calculating an estimated electric field envelope of the impulse response by applying at least one constraint to the inverse Fourier transform of the estimated Fourier transform of the complex electric field envelope. 
   In certain embodiments, an apparatus characterizes at least one fiber Bragg grating. The apparatus comprises a laser pulse source which generates at least one input laser pulse. The apparatus further comprises an optical spectrum analyzer. The apparatus further comprises a first optical path optically coupled to the laser pulse source. The first optical path comprises a pulse stretcher and an attenuator. A first portion of the input laser pulse propagates from the laser pulse source and is stretched by the pulse stretcher and is attenuated by the attenuator. The apparatus further comprises a second optical path optically coupled to the first optical path and comprising a mirror. A first portion of the stretched and attenuated laser pulse from the first optical path is reflected from the mirror. The apparatus further comprises a third optical path optically coupled to the first optical path and comprising a first fiber Bragg grating. A second portion of the stretched and attenuated laser pulse from the first optical path is reflected from the first fiber Bragg grating. The apparatus further comprises a fourth optical path optically coupled to the second optical path, the third optical path, and the optical spectrum analyzer. The reflected pulse from the mirror and the reflected pulse from the first fiber Bragg grating propagate to the optical spectrum analyzer. The apparatus further comprises a fifth optical path optically coupled to the laser pulse source and the optical spectrum analyzer. The fifth optical path comprises a delay line. A second portion of the input laser pulse propagates from the laser pulse source along the fifth optical path to the optical spectrum analyzer. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a flowchart of an exemplary embodiment of a method of determining the complex reflection impulse response h R (t) of a FBG. 
       FIG. 2  is a flowchart of another exemplary embodiment of a method of determining the complex reflection impulse response h R (t) of a FBG. 
       FIG. 3  is a flowchart of another exemplary embodiment of a method of determining the complex reflection impulse response h R (t) of a FBG. 
       FIG. 4A  schematically illustrates an exemplary embodiment of a method of determining the complex transmission impulse response h T (t) of a FBG. 
       FIG. 4B  is a flowchart of the exemplary embodiment of the method of  FIG. 4A . 
       FIG. 5A  is a plot of the reflection spectrum amplitude of a Gaussian-apodized symmetric FBG as a function of wavelength. 
       FIG. 5B  is a plot of the theoretical transmission group delay spectrum (solid curve) of the FBG of  FIG. 5A  as a function of wavelength and the recovered transmission group delay spectrum (dashed curve) without any noise present. 
       FIG. 5C  is a plot of the theoretical transmission group delay spectrum (solid curve) of the FBG of  FIG. 5A  as a function of wavelength and the recovered transmission group delay spectrum (dotted curve) with 5% uniform noise with a mean of 1. 
       FIG. 6A  is a plot of the reflection spectrum amplitude of an asymmetric chirped FBG as a function of wavelength. 
       FIG. 6B  is a plot of the theoretical transmission group delay spectrum (solid curve) of the FBG of  FIG. 6A  as a function of wavelength and the recovered transmission group delay spectrum (dashed curve) without any noise present. 
       FIG. 6C  is a plot of the theoretical transmission group delay spectrum (solid curve) of the FBG of  FIG. 6A  as a function of wavelength and the recovered transmission group delay spectrum (dotted curve) with 5% uniform noise with a mean of 1. 
       FIGS. 7A and 7B  schematically illustrate two exemplary measurement configurations compatible with embodiments described herein. 
       FIG. 7C  schematically illustrates an exemplary configuration of multiple FBGs compatible with certain embodiments described herein. 
       FIG. 8  schematically illustrates another exemplary measurement configuration compatible with certain embodiments described herein. 
       FIG. 9  is a block diagram of a method of characterizing an FBG in accordance with certain embodiments described herein. 
       FIGS. 10A and 10B  are plots of the theoretical electric field reflection coefficient amplitude and phase, respectively, of an exemplary asymmetric chirped FBG. 
       FIG. 11  is a plot of the amplitude of the theoretical reflection impulse response of the FBG of  FIGS. 10A and 10B . 
       FIGS. 12A and 12B  illustrates the amplitude and phase, respectively, of an exemplary temporal profile of the input laser pulse. 
       FIG. 13  is a plot of the normalized amplitude of the input laser pulse spectrum together with the reflection spectrum of the exemplary FBG of  FIGS. 10A and 10B . 
       FIG. 14  is a plot of the electric field amplitude (solid line) and phase (dashed line) for a pulse sequence inputted to an optical spectrum analyzer in accordance with certain embodiments described herein. 
       FIG. 15A  is a plot of the calculated power spectrum corresponding to the pulse sequence of  FIG. 14 . 
       FIG. 15B  is a plot of an enlarged view of the same calculated power spectrum of  FIG. 15A . 
       FIGS. 16A and 16B  show the amplitude and phase, respectively, the recovered complex reflection impulse response (dashed lines) and the original complex reflection impulse response (solid lines) of the target FBG. 
       FIGS. 17A and 17B  are plots of the reflection coefficient amplitude and the group delay spectra of the FBG calculated from the recovered complex reflection impulse response of  FIGS. 16A and 16B . 
       FIGS. 18A and 18B  are plots of the amplitude and phase, respectively, of a different input laser pulse. 
       FIGS. 19A and 19B  are plots of the amplitude and phase, respectively, for the recovered reflection coefficient amplitude and the group delay spectra of the FBG using the input laser pulse of  FIGS. 18A and 18B . 
       FIG. 20  is a plot of the results from a series of simulations, each with a different splitting ratio between the peak amplitude of the input laser pulse and the peak amplitude of the reflected impulse response. 
       FIG. 21  is a plot of a simulated noisy power spectrum. 
       FIGS. 22A and 22B  are plots of the amplitude and group delay spectra, respectively, of the reflection coefficient of the FBG calculated using the simulated noisy power spectrum of  FIG. 21 . 
       FIG. 23  is a plot of the normalized input laser pulse spectrum and the reflection spectra of two different Gaussian apodized FBGs which are characterized together in an exemplary embodiment. 
       FIGS. 24A and 24B  are plots of the amplitude and phase of the temporal electric field profile of the input laser pulse, respectively. 
       FIG. 25  is a plot of the amplitude and the phase of a pulse sequence formed by time-delaying the two reflected pulses from the two FBGs of  FIG. 23 , with respect to the leading dummy input laser pulse with a splitting ratio of 40. 
       FIG. 26  is a plot of the power spectrum for the pulse sequence of the leading dummy input laser pulse with the two reflected pulses of  FIG. 25 . 
       FIG. 27A  is a plot of the original (solid line) and recovered (dashed line) amplitude of the reflection coefficient of the first FBG of  FIG. 23 . 
       FIG. 27B  is a plot of the original (solid line) and recovered (dashed line) group delay spectra of the reflection coefficient of the first FBG of  FIG. 23 . 
       FIG. 28A  is a plot of the original (solid line) and recovered (dashed line) amplitude of the reflection coefficient of the second FBG of  FIG. 23 . 
       FIG. 28B  is a plot of the original (solid line) and recovered (dashed line) group delay spectra of the reflection coefficient of the second FBG of  FIG. 23 . 
       FIG. 29  is a plot of an FBG reflection spectrum (solid line) and the normalized input laser pulse amplitude (dotted line) in an exemplary embodiment. 
       FIGS. 30A and 30B  are plots of the amplitude and phase, respectively, of the temporal profile of the input laser pulse. 
       FIG. 31  is a plot of amplitude of the impulse response (solid line) of the FBG and the amplitude of the reflected pulse (dashed line). 
       FIGS. 32A and 32B  are plots of the amplitude and phase, respectively, of the normalized electric field envelope of the time-stretched laser pulse in an exemplary embodiment. 
       FIG. 33  is a plot of the pulse sequence of a dominant peak pulse at the leading edge followed by two weaker reflected pulses with a splitting ratio of 40 in an exemplary embodiment. 
       FIG. 34  is a plot of the output of an optical spectrum analyzer for the pulse sequence of  FIG. 33  in an exemplary embodiment. 
       FIG. 35A  is a plot of the original (solid curve) and the recovered (dashed curve) of the amplitude of the reflection coefficient of the FBG of  FIG. 29 . 
       FIG. 35B  is a plot of the original (solid curve) and the recovered (dashed curve) of the group delay spectra of the reflection coefficient of the FBG of  FIG. 29 . 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
   Certain embodiments described herein provide a simpler and less noise-sensitive technique based on an iterative process to recover the reflection phase, the reflection group delay, the transmission phase, or the transmission group delay of an FBG (e.g., chirped, asymmetric, symmetric, or uniform) from only the measurement of the reflection spectrum amplitude |r(ω)| or the transmission spectrum amplitude |t(ω)|. The measurements involved in certain embodiments are also substantially faster than existing techniques. Furthermore, in certain embodiments in which the FBG is known to be symmetric, both the reflection group delay and the transmission group delay of the FBG are uniquely determined. As described herein, numerical simulations of certain embodiments illustrate a noise sensitivity which is quite low. 
   Certain embodiments described herein are useful in computer-implemented analyses of the optical characteristics of FBGs. The general-purpose computers used for such analyses can take a wide variety of forms, including network servers, workstations, personal computers, mainframe computers and the like. The code which configures the computer to perform such analyses is typically provided to the user on a computer-readable medium, such as a CD-ROM. The code may also be downloaded by a user from a network server which is part of a local-area network (LAN) or a wide-area network (WAN), such as the Internet. 
   The general-purpose computer running the software will typically include one or more input devices, such as a mouse, trackball, touchpad, and/or keyboard, a display, and computer-readable memory media, such as random-access memory (RAM) integrated circuits and a hard-disk drive. It will be appreciated that one or more portions, or all of the code may be remote from the user and, for example, resident on a network resource, such as a LAN server, Internet server, network storage device, etc. In typical embodiments, the software receives as an input a variety of information concerning the material (e.g., structural information, dimensions, previously-measured amplitudes of reflection or transmission spectra). 
   Iterative Processing Utilizing the Reflection Spectrum Amplitude 
   Measurements of the reflection spectrum amplitude |r(ω)| can be made without using an interferometric instrument. In certain embodiments, broadband light (e.g., white light) or laser light is launched into the FBG and an optical spectrum analyzer (OSA) or a monochromator is used to record the reflected power R(ω)=|r(ω) 2  or the transmitted power T(ω)=1−R(ω) as a function of the optical angular frequency ω. In certain other embodiments, other techniques known in the art are used to measure either R(ω) or T(ω). By taking the square root of R(ω), the reflection spectrum amplitude |r(ω)| is directly obtained. However, the reflection spectrum phase φ r (ω) is still unknown. Certain embodiments described herein retrieve this missing phase information from only either the measured reflection power R(ω) or the measured transmission power T(ω), without carrying out additional measurements. 
   Certain embodiments described herein utilize an algorithm described by J. R. Fienup in “ Reconstruction of an Object from the Modulus of its Fourier Transform ,” Optics Letters, 1978, Vol. 3, pp. 27-29. This algorithm (referred to as “the Fienup algorithm” herein) is an error-reduction algorithm that involves using a known (e.g., measured) Fourier transform amplitude spectrum of an unknown function g(t), together with known properties of this function (e.g., that it is a real function or a causal function), to correct an initial guess of g(t). In certain embodiments, this correction is done iteratively. 
     FIG. 1  is a flowchart of an exemplary embodiment of a method  100  of determining the complex reflection impulse response h R (t) of an FBG. The method  100  comprises providing a measured reflection spectrum amplitude |r M (k)| of the complex reflection spectrum r(k) of an FBG in an operational block  110 . The method  100  further comprises providing an estimated phase term exp[jφ 0 (k)] of the complex reflection spectrum r(k) in an operational block  120 . The method  100  further comprises multiplying the measured reflection spectrum amplitude |r M (k)| and the estimated phase term exp[jφ 0 (k)] to generate an estimated complex reflection spectrum r′(k) in an operational block  130 . The method  100  further comprises calculating an intermediate function h′(t), which is the inverse Fourier transform (IFT) of the estimated complex reflection spectrum r′(k), in an operational block  140 . The method  100  further comprises calculating an estimated complex reflection impulse response h R   e (t) by applying at least one constraint to the intermediate function h′(t) in an operational block  150 . 
   In certain embodiments, providing the measured reflection spectrum amplitude |r M (k)| in the operational block  110  comprises measuring the reflection power spectrum |r M (k)| 2  and taking the square root to yield the measured reflection spectrum amplitude |r M (k)|. In other embodiments, a previously-measured reflection spectrum amplitude |r M (k)| is provided. 
   In certain embodiments, the measurement of the measured reflection spectrum amplitude |r M (k)| does not provide the phase term exp[jφ(k)] of the complex reflection spectrum r(k). In certain embodiments in which the method  100  is used iteratively, the choice of the initial estimated phase term exp[jφ 0 (k)] provided in the operational block  120  does not strongly impact the convergence of the method. Therefore, in certain such embodiments, the initial estimated phase term is selected to be equal to a real or complex constant (e.g., exp[jφ 0 (k)]=1). In certain other embodiments that utilize an IFT technique to provide a measured phase term exp[jφ M (k)], the estimated phase term exp[jφ 0 (k)] is selected to be the measured phase term exp[jφ M (k)]. In certain other embodiments that utilize an IFT technique to provide a measured phase term exp[jφ M (k)], the estimated phase term exp[jφ 0 (k)] is selected to be the phase term measured from another FBG which is similar to the FBG from which the reflection spectrum amplitude |r M (k)| is measured (e.g., the FBG being characterized). By providing an estimated phase term which is closer to the actual phase term, certain embodiments advantageously reduce the convergence time of the calculations. 
   In the operational block  130 , the measured reflection spectrum amplitude |r M (k)| and the estimated phase term exp[jφ 0 (k)] are multiplied together to generate an estimated complex reflection spectrum r′(k). In certain embodiments, the estimated complex reflection spectrum r′(k)=|r M (k)|·exp[jφ 0 (k)] is a complex quantity that is calculated numerically. 
   In the operational block  140 , the IFT of the estimated complex reflection spectrum r′(k)=|r M (k)|·exp[jφ 0 (k)] is calculated numerically. In certain embodiments, in the operational block  150 , the estimated complex reflection impulse response h R   e (t) is calculated by applying at least one constraint to the IFT of the estimated complex reflection spectrum r′(k). Various characteristics of the complex reflection impulse response h R   e (t) may be used as sources for the applied constraint. For example, in certain embodiments in which the complex reflection impulse response h R   e (t) is causal, the t≧0 portion of the IFT of the estimated complex reflection spectrum r′(k) is retained while the t&lt;0 portion is set equal to zero, thereby generating an estimated complex reflection impulse response h R   e (t) which is causal. In certain other embodiments, an anti-causal (or maximum-phase) constraint is applied to the IFT of the estimated complex reflection spectrum r′(k) (e.g., the t&lt;0 portion is set equal to zero and is time-reversed). In certain embodiments in which the complex reflection impulse response h R   e (t) has a known temporal duration (e.g., 300 picoseconds), the portion of the IFT of the estimated complex reflection spectrum r′(k) for times greater than the temporal duration is set equal to zero. In certain embodiments in which the FBG has a known bandwidth (e.g., 50 nanometers at wavelengths around 1550 nanometers), then the IFT of the estimated complex reflection spectrum r′(k) is adjusted to provide this known bandwidth. In other certain embodiments in which the FBG is generally uniform, that is, the FBG has an effective refractive index profile which is a periodic function with a constant (e.g., rectangular) envelope, the complex reflection spectrum r(k) is a symmetric function, which implies that the reflection impulse response h R (t) is a real function. In certain such embodiments, the applied constraint comprises using only the real portion of the reflection impulse response h R   e (t). As described more fully below, in certain embodiments in which the calculation is iterated, the application of such constraints advantageously reduces the number of iterations which achieve convergence. 
     FIG. 2  is a flowchart of another exemplary embodiment of a method  200  of determining the complex reflection impulse response h R (t) of a FBG in accordance with embodiments described herein. The method  200  comprises the operational blocks  110 ,  120 ,  130 ,  140 , and  150 , as described herein. The method  200  further comprises calculating a Fourier transform r n (k) of the estimated complex reflection impulse response h R   e (t) in an operational block  210 . The method  200  further comprises calculating a phase term exp[jφ n (k)] of the Fourier transform r n (k) of the estimated complex reflection impulse response h R   e (t) in an operational block  220 . 
   In certain embodiments, the Fourier transform r n (k) is calculated numerically in the operational block  210 . In certain embodiments, the calculated phase term exp[jφ n (k)] of this Fourier transform r n (k) is calculated numerically in the operational block  220 . In certain other embodiments in which the complex reflection spectrum is a minimum-phase function, as described more fully below, the calculated phase term exp[jφ n (k)] of this Fourier transform r n (k) is calculated analytically in the operational block by using a Hilbert transformation of the Fourier transform of the estimated complex reflection spectrum amplitude. 
     FIG. 3  is a flowchart of another exemplary embodiment of a method  300  of determining the complex reflection impulse response h R (t) of a FBG in accordance with embodiments described herein. The method  300  comprises the operational blocks  110 ,  120 ,  130 ,  140 ,  150 ,  210 , and  220 , as described herein. The method  300  further comprises using the calculated phase term exp[jφ n (k)] as the estimated phase term in the operational block  130  and repeating the operational blocks  130 ,  140 ,  150 ,  210 , and  220 . This repeat operation is denoted in  FIG. 3  by the arrow  310 . In certain such embodiments, the calculated phase term exp[jφ n (k)] provides a new estimate for the missing phase term of the complex reflection spectrum. The resulting estimated Fourier transform of the operational block  130  is the product of a measured reflection spectrum amplitude |r M (k)| and a calculated estimated phase term exp[jφ n (k)]. By repeating the operational blocks  130 ,  140 ,  150 ,  210 , and  220 , a second estimated complex reflection impulse response and a second estimated phase term are generated. 
   In certain embodiments, the operational blocks  130 ,  140 ,  150 ,  210 ,  220  as shown in  FIG. 3  are iteratively repeated a number of times. In certain such embodiments, the iterations are performed until the resulting estimated complex reflection impulse response converges. Convergence is reached in certain embodiments when the average difference between the estimated complex reflection impulse response spectra obtained after two consecutive iterations is less than a predetermined value (e.g., 0.1% of the estimated complex reflection impulse response of the iteration). For example, convergence is reached when the difference between two consecutive estimates of the function ∫|r n (k)−r n-1 (k)| 2 dt/∫|r n (k)| 2 dt is less than the predetermined value. In other embodiments, the iterations are performed a predetermined number of times (e.g., 100 times) rather than determining the differences between successive iterations. 
   In certain embodiments, the predetermined number is selected to be sufficiently large such that convergence is essentially always achieved after this number of iterations. In certain such embodiments, the predetermined number is determined by evaluating the rate of convergence for a number of FBGs. After a number of iterations, certain embodiments yield an estimated phase term which is a more accurate estimate of the actual phase term than is the initial estimated phase term. 
   In certain embodiments, the number of iterations can be reduced by using an initial estimated phase term of the complex reflection spectrum which more closely approximates the actual phase term of the complex reflection spectrum. For example, in certain embodiments in which the complex reflection spectrum is a minimum-phase function, a Hilbert transformation of the Fourier transform of the complex reflection spectrum amplitude is used as the initial estimated phase term exp[jφ 0 (k)]. In certain embodiments, only a single iteration is used to calculate the complex reflection impulse response of the FBG. In certain embodiments, the method further comprises calculating the effective refractive index profile Δn(z) as a function of the position z along the FBG using the complex reflection spectrum r(k) resulting from the calculation. Typically for non-symmetric FBGs, determining the effective refractive index profile Δn(z) utilizes both the complex reflection spectrum r(k) and the complex transmission spectrum t(k). However, in certain embodiments in which the FBG is symmetric, either the complex reflection spectrum r(k) or the complex transmission spectrum t(k) is obtained using the other. Exemplary methods of calculating the effective refractive index profile in accordance with embodiments described herein are described by A. Othonos and K. Kalli, “ Fiber Bragg gratings: fundamentals and applications in telecommunications and sensing,”  1999, Artech House, Boston; and R. Kashyap, “ Fiber Bragg gratings,”  1999, Academic Press, San Diego. 
   Iterative Processing Utilizing the Transmission Spectrum Amplitude 
     FIG. 4A  schematically illustrates an exemplary embodiment of a method  400  of determining the complex transmission impulse response h T (t) of a FBG in accordance with embodiments described herein.  FIG. 4B  is a flow diagram of the method  400 . The method  400  comprises providing a measured transmission spectrum amplitude |t M (ω)| of the complex transmission spectrum t(ω) of an FBG in an operational block  410 . In certain embodiments in which the FBG is substantially lossless, the measured transmission spectrum amplitude |t M (ω)| is derived from a measurement of the measured reflection spectrum amplitude |r M (ω)| using the relation |t M (ω)| 2 +|r M (ω)| 2 =1. 
   The method  400  further comprises providing an estimated phase term exp[jφ 0 (ω)] of the complex transmission spectrum t(ω) in an operational block  420 . In certain embodiments, the phase term exp[jφ 0 (ω)] of the complex transmission spectrum t(ω) is unknown to start with, so an arbitrary initial phase is assumed. The choice of the initial phase φ 0 (ω) does not significantly affect the convergence of the method  400 . Therefore, in certain embodiments, the initial phase φ 0 (ω) is conveniently chosen to be equal to zero. 
   The method  400  further comprises multiplying the measured transmission spectrum amplitude |t M (ω)| and the estimated phase term exp[jφ 0 (ω)] to generate an estimated complex transmission spectrum t′(ω) in an operational block  430 . The method  400  further comprises calculating an intermediate function h′(t), which is the inverse Fourier transform (IFT) of the estimated complex transmission spectrum t′(ω), in an operational block  440 . The method  400  further comprises calculating a new estimated complex transmission impulse response of the first iteration h 1 (t) by applying at least one constraint to the IFT of the estimated complex transmission spectrum, in the operational block  450 . For example, in certain embodiments, since the IFT of t(ω) is known to be causal, only the t≧0 portion of h′(t) is retained and zeros are used for t&lt;0, thereby providing a new estimated complex transmission impulse response of the first iteration h 1 (t). As described above, various other characteristics of the complex transmission spectrum t′(ω) may be used as sources for the applied constraint, which include but are not limited to, finite temporal duration, finite bandwidth of the FBG, uniformity of the FBG, and using only the real portion of the complex transmission spectrum t′(ω). 
   The method  400  further comprises calculating a Fourier transform t 1 (ω) of the estimated complex transmission impulse response h 1 (t) in an operational block  460 . The method  400  further comprises calculating a phase term exp[jφ 1 (ω)] of the Fourier transform t 1 (ω) of the estimated complex transmission impulse response h 1 (t) in an operational block  470 . The method  400  further comprises using the calculated phase term exp[jφ 1 (ω)] as the estimated phase term in the operational block  430  and repeating the operational blocks  430 ,  440 ,  450 ,  460 , and  470 , as denoted by the arrow  480 . In certain such embodiments, the calculated phase term exp[jφ n (ω)] of an interation n provides a new estimate for the missing phase term of the complex transmission spectrum. The resulting estimated Fourier transform of the operational block  430  is the product of a measured transmission spectrum amplitude |t M (ω)| and a calculated estimated phase term exp[jφ n (ω)]. By repeating the operational blocks  430 ,  440 ,  450 ,  460 , and  470 , a second estimated complex transmission impulse response and a second estimated phase term are generated. 
   In certain embodiments, the operational blocks  430 ,  440 ,  450 ,  460 ,  470  are iteratively repeated a number of times, as shown in  FIGS. 4A and 4B . In certain such embodiments, the iterations are performed until the resulting estimated complex transmission impulse response h n (t) converges. Convergence is reached in certain embodiments when the average difference between the estimated complex transmission impulse responses obtained after two consecutive iterations (e.g., |h n (t)−h n-1 (t)| 2 /|h n (t)| 2 ) is less than a predetermined value (e.g., 0.1% of the estimated complex transmission impulse response of the iteration). In other embodiments, the iterations are performed a predetermined number of times (e.g., 100 times) rather than determining the differences between successive iterations. In certain embodiments, the predetermined number is selected to be sufficiently large such that convergence is essentially always achieved after this number of iterations. In certain such embodiments, the predetermined number is determined by evaluating the rate of convergence for a number of FBGs. At the end of the n th  iteration, the phase of the Fourier transform of h n (t) is the recovered phase φ n (ω) of the FBG complex transmission spectrum t(ω). In certain embodiments, the transmission group delay 
             ⅆ       ϕ   n     ⁡     (   ω   )           ⅆ   ω           
is then calculated.
 
   In certain embodiments, the phase φ n (ω) obtained by the method  400  converges to the minimum-phase function (MPF) corresponding to a given Fourier transform amplitude. MPFs have the property that the FT phase and the logarith of the FT amplitude of an MPF are the Hilbert transform of one another. Consequently, the FT phase of an MPF can always be recovered from its FT amplitude, and an MPF can always be reconstructed from its FT amplitude alone. (See, e.g., V. Oppenheim and R. W. Schafer, “ Digital Signal Processing,”  2000, Prentice Hall, Chapter 7.) The fact that the complex transmission spectrum of any FBG is an MPF, as described more fully below, ensures the convergence of the method  400  to the unique phase φ t (ω) of the complex transmission spectrum, i.e., φ n (ω)=φ t (ω). 
   Since the IFT of the complex reflection spectrum r(ω) is not generally a minimum-phase function, the method  400  is not generally applicable to recover φ r (ω) from only the measured reflection spectrum amplitude |r M (ω)|. However, for certain embodiments in which the complex reflection spectrum r(ω) is also a minimum-phase function (e.g., for uniform FBGs), the method  400  can be used to derive the reflection group delay, as well as the transmission group delay, of the FBG. 
   In certain embodiments in which the FBG is known to be symmetric, the reflection group delay of the FBG is equal to the transmission group delay. In certain such embodiments, both the reflection group delay and the transmission group delay are recovered using the method  400 . 
   The method  400  has been used to recover the transmission group delay spectra of various exemplary FBGs to illustrate the usefulness of the method  400 .  FIGS. 5A-5C  and  6 A- 6 C schematically display two such examples.  FIGS. 5A-5C  schematically illustrate the results for a symmetric Gaussian-apodized FBG.  FIGS. 6A-6C  schematically illustrate the results for a non-symmetric chirped FBG. The amplitudes of the reflection spectra of both FBGs are shown in  FIG. 5A  and  FIG. 6A , respectively. In  FIG. 5B  and  FIG. 6B , the theoretical transmission group delays are shown with the solid curves, whereas the recovered group delays (without any noise present) recovered using the method  400  (with n=100 iterations) are shown with the dashed curves. These computations each took less then a few seconds with a 500-MHz computer. The recovery for each FBG is so good for each example that it is difficult to distinguish the theoretical group delay curve (solid line) from the recovered group delay curve (dashed line). 
   To demonstrate the noise performance of the method  400 , the theoretical |t(ω)| 2  spectrum was multiplied by 5% uniform noise with a mean of 1, and the method  400  was then applied to this noisy transmission spectrum. The results of the group delay recovery, with again n=100 iterations, are shown in  FIG. 5C  and  FIG. 6C  by dotted curves. Once again the recovery is very good, showing that the method  400  can even be quite useful under severe noise. 
   The vertical scale in  FIG. 6B  is enlarged so that the small dc offset (on the order of approximately 0.1 to 0.2 picoseconds) between the theoretical and the recovered group delay curves is more visible. With other techniques (e.g., the technique of Poladian referenced above), much stronger dc offsets, on the order of approximately 20 to 30 picoseconds, are observed. The main cause of this small dc offset for the method  400  is the usage of limited bandwidth in the simulations. For example, if a wider wavelength window (greater than 4 nanometers) were used for  FIG. 6A , the recovery results would have been much more improved. 
   Iterative Processing Utilizing Spectral Interferometry 
   Certain embodiments described herein use a pulsed input laser and an OSA to retrieve both the amplitude and the phase of the reflection or transmission spectrum of an FBG using a single power spectrum measurement. Certain embodiments are referred to herein as spectral interferometry using minimum-phase based algorithms (SIMBA). 
     FIGS. 7A and 7B  schematically illustrate two exemplary measurement configurations compatible with embodiments described herein. While these measurement configurations are slightly different, the processing of the measured quantities for the characterization of the FBG, together with the principle of operation, is identical. 
   In certain embodiments, the measurement configuration  500  of  FIG. 7A  has a pulsed laser source  510  which generates a laser pulse  520  which is split by a beam splitter  530  with a preferably uneven splitting ratio R, where R&gt;1, into a first laser pulse  522  and a second laser pulse  524 . The second laser pulse  524  is then sent to the FBG  540  being characterized. In certain embodiments, the pulsed laser source  510  comprises a mode-locked laser with a temporal width of a few picoseconds (e.g., approximately 2 picoseconds to approximately 4 picoseconds). In certain embodiments, the splitting ratio of the beam splitter  530  is between approximately 1 and approximately 200, while other embodiments have even higher splitting ratios. 
   In certain embodiments, the reflected pulse  550  from the FBG  540  is collected using circulators  575 ,  576  and temporally combined with the delayed first laser pulse  522 , thereby forming a pulse sequence  560 . The pulse sequence  560  has a sharp peak at the leading edge, due to the first laser pulse  522 , followed by a broader and much weaker pulse which is due to the reflected pulse  550  from the FBG  540 . The pulse sequence  560  is then sent to an OSA  570 , which yields the power spectrum or the square of the Fourier transform (FT) amplitude of the electric field envelope of the pulse sequence  560 . As is described more fully below, the pulse sequence  560  inputted into the OSA  570 , which has a sharp peak at its leading edge, is close to a minimum-phase function (MPF), which makes recovery of its FT phase possible from only the measurement of its FT amplitude, or vice versa. This recovery is performed by processing the measured power spectrum either analytically or iteratively, which yields both the phase and the amplitude of the reflection or transmission spectrum of the FBG  540 . 
   In certain embodiments, the optical path length between the beam splitter  530  and the FBG  540 , the optical pathlength between the beam splitter  530  and the OSA  570 , and the optical pathlength between the FBG  540  and the OSA  570  are selected to provide a predetermined time delay τ between the portions of the pulse sequence  560  received by the OSA  570  (e.g., the first laser pulse  522  and the reflected pulse  550  from the FBG  540 ). In certain embodiments, at least one of the optical paths between the beam splitter  530 , the FBG  540 , and the OSA  570  includes a delay line which can be adjusted to provide a desired time delay τ in the pulse sequence  560 . 
   In certain other embodiments, the beam splitter  530  has a splitting ratio of approximately one, and an attenuator is placed between the beam splitter  530  and the circulator  575 . In certain other embodiments, a beam splitter  530  having a splitting ratio greater than one and an attenuator between the beam splitter  530  and the circulator  575  are used. In certain embodiments, the splitting ratio of the beam splitter  530  is adjustable. In certain embodiments, the attenuation of the attenuator is adjustable. 
   In certain other embodiments, a measurement configuration  580 , such as that schematically illustrated by  FIG. 7B , utilizes a coupler  581 , an attenuator  582 , and a mirror  584 . In certain embodiments, the splitting ratio of the coupler  581  is approximately equal to one, the mirror  584  is a high reflector, and the attenuator  582  is adjusted such that the amplitude of the reflected spectrum from the FBG  540  is much smaller than the amplitude of the reflected laser pulse from the mirror  584 . In certain such embodiments, the reflected laser pulse from the mirror  584  and the reflected spectrum from the FBG  540  are combined by the coupler  581  to form a function which approximates a minimum-phase function. 
   In certain embodiments, the optical path length between the laser source  510  and the FBG  540 , the optical pathlength between the laser source  510  and the mirror  584 , the optical pathlength between the FBG  540  and the OSA  570 , and the optical pathlength between the mirror  584  and the OSA  570  are selected to provide a predetermined time delay τ between the portions of the pulse sequence  560  received by the OSA  5 . 70 . In certain embodiments, at least one of the optical paths between the laser source  510 , the FBG  540 , the mirror  584 , and the OSA  570  includes a delay line which can be adjusted to provide a desired time delay τ in the pulse sequence  560 . 
   In certain other embodiments, the coupler  581  has a splitting ratio greater than one, and the attenuator  582  between the coupler  581  and the FBG  540  is removed. In certain other embodiments, both a coupler  581  having a splitting ratio greater than one and an attenuator  582  between the coupler  581  and the FBG  540  are used. In certain embodiments, the splitting ratio of the coupler  581  is adjustable. In certain embodiments, the attenuation of the attenuator  582  is adjustable. 
   Certain embodiments provide advantages over previously-existing techniques, particularly over low-coherence spectral interferometry. In certain embodiments, the time delay τ between the reflected pulse  550  and the first laser pulse  522  can be chosen arbitrarily small as long as the two pulses temporally do not overlap. In certain embodiments, this is especially important if the OSA  570  used for the power spectrum measurement does not have enough resolution to resolve the spectral interference fringes, since the larger the time delay τ between the two pulses, the higher the maximum frequency of oscillations in the power spectrum recorded by the OSA  570 . Certain embodiments advantageously allow characterization of more than one FBG at the same time by using a single OSA measurement. Certain embodiments advantageously utilize a laser pulse  520  with a temporal profile width significantly narrower than the impulse response of the FBG  540  being characterized. In contrast, for low-coherence spectral interferometry, the autocorrelation function of the pulsed laser source is required to be much narrower than the impulse response of the FBG. In other words, for the same FBG, certain embodiments described herein give roughly two times better resolution than does low-coherence spectral interferometry. 
     FIG. 8  schematically illustrates another exemplary measurement configuration  600  compatible with certain embodiments described herein. A pulsed laser source  610  generates an input laser pulse  620  having a temporal width (e.g., 50 picoseconds). The input laser pulse  620  is split into a first laser pulse  622  which is directed to a delay line  630  and a second laser pulse which is directed to a pulse stretcher  640  (e.g., a loop of fiber optic cable). The pulse stretcher  640  broadens the temporal width of the second laser pulse by a predetermined factor (e.g., approximately 2 to 5) to produce a stretched laser pulse  624 . The stretched laser pulse  624  is sent through an attenuator  650 , and is then split into two weak pulses. One of the weak pulses is directed to the FBG  540  being characterized, while the other weak pulse is reflected from a mirror  660  (e.g., a bare fiber end, a mirrored fiber end, or a mirror placed at the end of the fiber with a collimating lens therebetween). The reflected pulse  672  from the FBG  540  and the reflected pulse  674  from the mirror  660  are temporally combined with the time-delayed version of the first laser pulse  622 . The resultant pulse sequence  680  has a dominant pulse at the leading edge followed by two weaker reflected pulses and is received by the OSA  570 , which measures the power spectrum. With proper selection of the relative amplitudes of the pulses forming the pulse sequence  680  (e.g., amplitude of the laser pulse  622  is significantly larger than the maximum amplitude of the reflected pulses  672 ,  674 ), the pulse sequence  680  approximates a minimum-phase function. In certain such embodiments, the pulse sequence  680  is processed iteratively to recover the reflected pulse  672  from the FBG  540 . The processing of the measured FT amplitude from the OSA  570 , together with the recovery algorithm described more fully below, are the same as for the measurement configurations  500 ,  580  of  FIGS. 7A and 7B . 
   Certain embodiments of the measurement configuration  600  schematically illustrated by  FIG. 8  provide the same advantages as described above in relation to the measurement configurations  500 ,  580  of  FIGS. 7A and 7B . In certain embodiments, the measurement configuration  600  of  FIG. 8  provides other advantages as well. In certain embodiments, the measurement configuration  600  advantageously does not require that the input laser pulse be temporally much narrower than the impulse response of the FBG  540 . For example, low-coherence spectral interferometry typically requires laser pulse widths of approximately 1 picosecond to accurately characterize an FBG with an impulse response of approximately 100 picoseconds duration. In contrast, certain embodiments utilizing the measurement configuration  600  advantageously avoid this constraint on the laser pulse width. For example, laser pulse widths of 50 picoseconds could be used to fully characterize the impulse response of the FBG  540 . 
   Mathematically, the reflection impulse response h R (t) and the transmission impulse response h T (t) of an FBG are simply the inverse Fourier transforms of the complex reflection spectrum r(ω) and the complex transmission spectrum t(ω), respectively. Therefore, knowing the impulse response of an FBG also means knowing the complex spectrum of the FBG. Since the impulse response of an FBG is a complex quantity, measuring the impulse response of an FBG is as equally challenging as measuring the whole complex spectrum of the FBG. 
   The transmission impulse response, h T (t), where t is the relative time, of all FBGs belong to the class of minimum-phase functions (MPFs). (See, e.g., L. Poladian, “ Group - Delay Reconstruction for Fiber Bragg Gratings in Reflection and Transmission ,” Optics Letters, 1997, Vol. 22, pp. 1571-1573; J. Skaar, “ Synthesis of Fiber Bragg Gratings for Use in Transmission ,” J. Op. Soc. Am. A, 2001, Vol. 18, pp. 557-564.) An MPF is characterized by having a z-transform with all its poles and zeros on or inside the unit circle. As a result of this property, the FT phase and the logarithm of the FT amplitude of an MPF are the Hilbert transform of one another. Consequently, the FT phase of an MPF can always be recovered from its FT amplitude, and an MPF can always be reconstructed from its FT amplitude alone. This property of MPFs is used in certain embodiments described herein to retrieve the whole complex transmission spectrum t(ω) from only the measured reflection spectrum amplitude |r(ω)| or the measured transmission spectrum amplitude |t(ω)| (since for a lossless grating |t(ω)| 2 +|r(ω)| 2 =1). This recovery of the complex reflection spectrum or the complex transmission spectrum can be achieved by either analytical or iterative techniques. (See, e.g., A. Ozcan et al., “ Group Delay Recovery Using Iterative Processing of Amplitude of Transmission Spectra of Fibre Bragg Gratings,”  Electronics Letters, 2004, Vol. 40, pp. 1104-1106.) Generally, the reflection impulse response, h R (t) of an FBG is not an MPF, so the whole complex reflection spectrum r(ω) cannot be uniquely recover from only the amplitude spectrum measurement. However, once the complex reflection spectrum r(ω) has been characterized by some means, the complex transmission spectrum t(ω) is also fully characterized using |t(ω)| 2 =1−|r(ω)| 2 , since t(ω) can fully be recovered from only |t(ω)| due to the above mentioned MPF property. Certain embodiments described herein are the first application of the concept of MPFs to the characterization of FBG spectra, either reflection or transmission. 
     FIG. 9  is a block diagram of a method  700  of characterizing an FBG in accordance with certain embodiments described herein. The method  700  includes an iterative error-reduction algorithm which uses the measured FT amplitude |E M (f)| of an unknown function e(t), together with the known properties of e(t) (e.g., being causal), and iteratively corrects an initial guess for e(t). (See, e.g., J. R. Fienup, “ Reconstruction of an Object from the Modulus of its Fourier Transform ,” Optics Letters, 1978, Vol. 3, pp. 27-29; R. W. Gerchberg and W. O. Saxton, “ Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures ,” Optik, 1972, Vol. 35, pp. 237-246.) 
   In certain embodiments, the method  700  recovers the complex electric field envelope e(t) of the impulse response of an FBG. In certain embodiments, the OSA output  710  is used to provide the measured FT amplitude |E M (f)| of a complex MPF, e(t). As shown by the block  720 , e(t) is the only quantity inputted into the method  700 . In certain embodiments, the function e(t) is an exact MPF, while in certain other embodiments, the function e(t) only approximates an MPF. (See, e.g., T. F. Quatieri, Jr. and A. V. Oppenheim, “ Iterative Techniques for Minimum Phase Signal Reconstruction from Phase or Magnitude ,” IEEE Trans. on Acoustics, Speech, and Signal Processing, 1981, Vol. 29, pp. 1187-1193; M. Hayes et al., “ Signal Reconstruction from Phase or Magnitude ,” IEEE Trans. Acoustics, Speech, and Signal Processing, 1980, Vol. 28, pp. 672-680; A. Ozcan et al., “ Iterative Processing of Second - Order Optical Nonlinearity Depth Profiles ,” Optics Express, 2004, Vol. 12, pp. 3367-3376.) 
   Prior to the method  700  being performed, the FT phase spectrum is unknown. Therefore, in certain embodiments, an arbitrary initial guess φ 0 (f) is used for the phase spectrum, as shown by the block  730 . Generally, this initial guess of the phase does not significantly affect the accuracy of the result of the method  700 . For this reason, in certain embodiments, φ 0 (f) is conveniently chosen to be equal to zero. 
   Using the measured FT amplitude |E M (f)| and the initial guess of the phase φ 0 (f), the inverse Fourier transform (IFT) e′(t)=|E M (f)|·exp(jφ 0 ) is calculated numerically, as shown by the block  740 . The method  700  further comprises applying at least one constraint to the IFT, in the operational block  750 . For example, in certain embodiments, since all MPFs are causal, only the t≧0 portion of e′(t) is retained, while all values of e′(t) for t&lt;0 are set to zero, as indicated by the block  750  of  FIG. 9 . In certain embodiments in which e(t) is known to be limited in time (e.g., finite temporal duration of less than 100 picoseconds), in the calculation of block  750 , the values of e′(t) for t&gt;100 picoseconds are also set to zero, which speeds up convergence. As described above, various other characteristics of e′(t) may be used as sources for the applied constraint, which include but are not limited to, finite bandwidth of the FBG, uniformity of the FBG, and using only the real portion of e′(t). 
   In certain embodiments, the result of the calculation of the block  750  is a new function e 1 (t), which is the first estimate of the complex MPF e(t). In certain embodiments, the FT of the first estimate e 1 (t) is calculated, as indicated by the block  760 , thereby providing a new phase φ 1 (f) and a new amplitude |E 1 (f)| for the FT of e(t). 
   Since the amplitude of the FT spectrum must be equal to the measured amplitude |E M (f)|, |E 1 (f)| is replaced by |E M (f)| and the loop is repeated using |E M (f)| and φ 1 (f) as the new input spectra to the block  740 , which provides a second function e 2 (t). This loop is repeated n times, until convergence is achieved. In certain embodiments, convergence is achieved once the difference between two consecutive estimates of the function ∫|e n (t)−e n-1 (t)| 2 dt/∫|e n (t)| 2 dt is less than a predetermined value (e.g., 0.1%). At the end of the n-th iteration, e n (t) is the recovered complex MPF, as indicated by the block  770 . Typically, approximately 100 iterations are adequate for achieving convergence, which only takes a few seconds to compute using MATLAB on a 500 MHz computer with 2 14  data points. 
   Although it has not been proven mathematically, it has been found empirically that the method  700  always converges to the minimum-phase function corresponding to a given FT amplitude. In other words, of the infinite family of FT phase functions that can be associated with the known (measured) FT amplitude, the method  700  converges to the one and only one that has the minimum phase. Since this solution is unique, if the profile to be reconstructed is known a priori to be an MPF or to approximate an MPF, then the solution provided by the error-reduction method  700  is the correct profile. 
   To understand intuitively which physical functions are likely to be minimum-phase functions, an MPF is denoted by e min (n), where n is an integer that labels sampled values of the function variable, e.g., relative time in the present case of ultra-short pulses. Because all physical MPFs are causal, e min (n) must be equal to zero for n&lt;0. The energy of a minimum-phase function, which is defined as 
             ∑     n   =   0       m   -   1       ⁢              e   min     ⁡     (   n   )            2           
for m samples of the function, satisfies the inequality
 
               ∑     n   =   0       m   -   1       ⁢              e   min     ⁡     (   n   )            2       ≥       ∑     n   =   0       m   -   1       ⁢            e   ⁡     (   n   )            2             
for all possible values of m&gt;0. In this inequality, e(n) represents any of the functions that have the same FT amplitude as e min (n). This property suggests that most of the energy of e min (n) is concentrated around n=0. Stated differently, any profile with a dominant peak around n=0 (e.g., close to the origin) will be either a minimum-phase function or close to one, and thus it will work extremely well with the iterative error-reduction method  700  schematically illustrated by  FIG. 9 . Although there might be other types of MPFs besides functions with a dominant peak, this class of MPFs is advantageously used because they are straightforward to engineer with optical pulses and because they yield exceedingly good results.
 
   In certain embodiments, the method  700  can be used to uniquely characterize the whole complex reflection and hence transmission spectra of any FBG by recovering the reflection impulse response, h R (t), using a single FT amplitude measurement. Certain embodiments described herein rely on the fact that by increasing the peak amplitude of the leading pulse in a sequence of pulses, the entire pulse sequence (even if the sequence is a complex quantity, such as the electric field envelope) becomes close to an MPF, which makes the recovery of the phase information possible from only the FT amplitude measurements. 
   Referring to the measurement configurations of  FIGS. 7A and 7B , a short laser pulse  524  with a complex electric field envelope of E(t) impinges an FBG  540  being characterized. The reflected pulse spectrum  550  can simply be calculated as E Refl (ω)=E(ω)·r(ω), where E(ω) is simply the FT of E(t). (See, e.g., L. R. Chen et al., “ Ultrashort Pulse Reflection From Fiber Gratings: A Numerical Investigation ,” J. of Lightwave Technology, 1997, Vol. 15, pp. 1503-1512.) For simplicity and convenience, the term exp(j·ω c ), where ω c  corresponds to the center frequency of the input laser  510  has been dropped. In the time domain, the reflected pulse envelope  550  can then be expressed as E Refl (t)=E(t)*h R (t), where ‘*’ stands for the convolution operation. Therefore, the complex electric field envelope of the reflected pulse  550  is simply a convolution of the reflection impulse response of the FBG  540  with the complex electric field of the input pulse  524 . For embodiments in which the input laser pulse  520  is temporally much narrower than the impulse response of the FBG  540 , the reflected pulse  550  from the FBG  540  E Refl (t) approximates or equals the reflection impulse response h R (t). Most commercially available FBGs operate at a wavelength of approximately 1550 nanometers, so the temporal width of the reflection impulse response of such an FBG is approximately 50 picoseconds to approximately 100 picoseconds, or longer. Thus, in certain embodiments, an input laser pulse  520  having a temporal width of a few picoseconds practically acts as a delta function to yield E Refl (t)=h R (t). However, in terms of measurement complexity, since both E Refl (t) and h R (t) are complex quantities, to recover these complex quantities, two independent measurements would typically be used, one for the amplitude and the other for the phase. 
   Referring to the measurement configuration  500  of  FIG. 7A , the reflected impulse response h R (t) of the FBG  540  is combined in the time domain with the second laser pulse  522 , denoted by E A (t), with a time delay of τ between the two pulses. The resulting pulse sequence  560 , expressed as E Seq (t)=E A (t)+h R (t−τ), is then sent to the OSA  570 , which yields the optical power spectrum or the square of the FT amplitude of the complex electric field envelope of the pulse sequence  560 , expressed as |E Seq (ω)| 2 , where E Seq (ω) is the FT of E Seq (t). Using only the measured FT amplitude square |E Seq (ω)| 2 , certain embodiments recover the reflection impulse response h R (t) based on the properties of MPFs. In certain embodiments, E Seq (t) is selected to approximate a true MPF by increasing the peak amplitude of E A (t), and recovering its FT phase from only its FT amplitude |E Seq (ω)|. In certain embodiments, the recovery of the FT phase spectrum is achieved by using an analytical Hilbert transformation. However, in certain other embodiments, an iterative error-reduction method, such as the method  700  schematically illustrated in  FIG. 9 , is preferably used to recover the phase spectrum, due to the simplicity and better noise performance of the method. 
   A numerical example illustrates the recovery of E Refl (t)=h R (t) from only the measured quantity, |E Seq (ω)| 2  or |E Seq (ω)| using the measurement configuration of  FIG. 7A  and the method  700  of  FIG. 9 . A strongly chirped asymmetric FBG is used in this numerical example to make the recovery a harder task. The theoretical electric field reflection coefficient amplitude and phase of the chosen chirped FBG, which was calculated using a transfer matrix formalism (see, e.g., T. Erdogan, “ Fiber Grating Spectra ,” J. of Lightwave Technology, 1997, Vol. 15, pp. 1277-1294), are shown in  FIGS. 10A and 10B , respectively. The reflection band of this FBG is approximately 4 nanometers wide, between approximately 1548 nanometers and approximately 1552 nanometers. The amplitude of the theoretical reflection impulse response of this FBG is shown in  FIG. 11 . The temporal width of the reflection impulse response is approximately 300 picoseconds. The observed broadened temporal behavior is mostly due to the strong chirp of the reflection coefficient. As expected, the reflection impulse response of the FBG shown in  FIG. 11  is causal. 
     FIGS. 12A and 12B  illustrate the amplitude and phase, respectively, of the temporal profile of the input laser pulse used in the exemplary numerical simulation of  FIGS. 10A ,  10 B, and  11 . The temporal width of the chosen laser pulse is less than approximately 6 picoseconds, such that the laser pulse approximates as a delta function for the reflection impulse response of  FIG. 11 . The normalized amplitude of the input laser pulse spectrum together with the reflection spectrum of the exemplary FBG of  FIGS. 10A and 10B  are also plotted in  FIG. 13 . In certain embodiments, not only does the input laser pulse have a temporal width much narrower than the impulse response of the FBG, whereby the input laser pulse acts as a delta function, but also the input laser pulse has a power spectrum that covers all the frequencies in the FBG reflection spectrum, as shown in  FIG. 13 . 
   Using the measurement configuration of  FIG. 7A , the amplitude (solid line) and the phase (dashed line) of the pulse sequence  560  is formed, as plotted in  FIG. 14 . The pulse sequence  560  includes the laser pulse  522  and the weaker reflected pulse  550 , which approximates the reflection impulse response of the FBG  540 . A splitting ratio of 40 was used between the peak amplitude of the laser pulse  522  and the peak amplitude of the reflection impulse response  550  of the FBG. In certain embodiments, this splitting ratio of 40 ensures that the complex electric field envelope of the pulse sequence is close to an MPF to uniquely recover h R (t) from only the measurement of the power spectrum of the pulse sequence. 
   The pulse sequence  560  is then sent to the OSA  570 , which records the power spectrum or the square of the FT amplitude of complex electric field envelope of the pulse sequence, as shown in  FIG. 14 . The calculated power spectrum is plotted in  FIGS. 15A and 15B  for a splitting ratio of 40. To calculate the power spectra of  FIGS. 15A and 15B , the resolution of the OSA  570  was assumed to be limited by approximately 10 picometers, which is a modest resolution for currently available spectrum analyzers. Other OSAs compatible with embodiments described herein have sub-picometer resolution.  FIG. 15B  shows an enlarged view of the same power spectrum of  FIG. 15A  in the range of wavelengths between approximately 1550 nanometers and approximately 1551 nanometers.  FIG. 15B  illustrates that due to the limiting 10 picometer resolution of the OSA  570 , some sharp features of the power spectrum curve are actually lost. As discussed more fully below, the whole complex spectrum of the target FBG can still be recovered accurately. The fringe pattern shown in  FIGS. 15A and 15B  is the result of interference between the input laser pulse spectra and the FBG reflection spectra. This interference is only observed in the frequency band of the reflection spectrum of the FBG, and the overall envelope of the power spectrum in  FIGS. 15A and 15B  follows the power spectrum of the input laser pulse. 
     FIGS. 16A and 16B  show the amplitude spectrum and phase spectrum, respectively, of the complex reflection impulse response of the target FBG recovered by inputting the output of the OSA  570  into the iterative error-reduction method  700  shown in  FIG. 9 . The solid and dashed curves of  FIG. 16A  correspond to the original and recovered amplitudes, respectively, of the impulse response. The solid and dashed curves of  FIG. 16B  correspond to the original and recovered phases, respectively, of the impulse response. The recovered impulse response is an excellent fit to the original impulse response. In fact, it is difficult to see any difference between the solid and dashed curves of  FIG. 16A , and the difference between the solid and dashed curves of  FIG. 16B  are small. 
   The time origin information for the recovered reflection impulse response is lost in  FIGS. 16A and 16B . In principle, the origin can be redefined by using the causality property of the impulse response, i.e., by choosing the point to the left of which the recovered impulse response is all zero, as the new time origin. Any error in the redefinition of the time origin is, however, inconsequential, since a time shift in the impulse response merely adds a linear phase to its FT phase. The recovered group delay will therefore only have a constant offset, which will be proportional to the error made in the time origin. This constant group delay offset is inconsequential for practical applications and can simply be recovered by noting the value of the group delay away from the FBG reflection spectrum. 
     FIGS. 16A and 16B  do not show the recovery of the leading input pulse electric field since it is not of interest. The input laser pulse simply acts as a dummy pulse for the recovery. The data processing does recover the leading input pulse if  FIG. 14  as well, but it does not generally recover it accurately, primarily because the pulse sequence only approximates a true MPF. In other words, in certain embodiments in which the pulse sequence only approximates a true MPF, the recovery of the leading input laser pulse&#39;s electric field is affected, but the accuracy of the recovered reflection impulse response of the FBG is not affected. However, any errors in the recovered input laser pulse profile are inconsequential since the input laser pulse profile is not the target of the analysis. In certain embodiments, the temporal profile of the input laser pulse is selected or engineered to result in a pulse sequence which is a more exact approximation of an MPF. Certain such embodiments advantageously increase the speed of convergence of the calculations or allow the application of the Hilbert transformation in the calculations. 
   From the recovered impulse response of the FBG, shown in  FIGS. 16A and 16B , the reflection coefficient amplitude and the group delay spectra of the FBG are easily computed by a single FT operation, as shown in  FIGS. 17A and 17B . The solid and dashed curves of  FIG. 17A  correspond to the original and recovered amplitudes, respectively, of the reflection coefficient. The solid and dashed curves of  FIG. 17B  correspond to the original and recovered group delay spectra, respectively. Even for a strongly chirped FBG, as shown in  FIGS. 10A and 10B , the success in the recovery is quite impressive. For this exemplary embodiment, the error in the recovery of |r(ω)|, defined as 
               ∫                     r   ⁡     (   ω   )            -            r   ^     ⁡     (   ω   )                   2     ⁢     ⅆ   ω           ∫              r   ⁡     (   ω   )            2     ⁢     ⅆ   ω           ,         
where |r(ω)| and |{circumflex over (r)}(ω)| are the original and the recovered quantities, respectively, is only approximately 0.08%.
 
   To evaluate the dependence of the success of the recovery on the temporal profile of the chosen input laser pulse, another exemplary embodiment uses a different input laser pulse having an amplitude and phase as shown in  FIGS. 18A and 18B , respectively. The laser pulse of  FIGS. 18A and 18B  is approximately 3 times narrower than the input laser pulse of the previous exemplary embodiment, as seen by comparing  FIGS. 12A and 12B  with  FIGS. 18A and 18B . For this exemplary embodiment, a splitting ratio of 120 was used between the peak amplitude of the input laser pulse and the peak amplitude of the reflection impulse response of the FBG in  FIG. 7A . 
     FIGS. 19A and 19B  show the amplitude and group delay spectra, respectively, of the reflection coefficient of the FBG. The solid and dashed curves of  FIG. 19A  are the original and recovered amplitudes, respectively, of the reflection coefficient. The solid and dashed curves of  FIG. 19B  are the original and recovered group delay spectra, respectively. The error in the recovery of |r(ω)| in this exemplary embodiment is less than 0.02%, which is reduced by more than a factor of 4 as compared to the error of the previous exemplary embodiment. The improved performance in this exemplary embodiment (0.02% versus 0.08% in the previous example) is primarily due to the narrower input laser pulse, which more closely approximates a true delta function, thereby yielding a more accurate reflected pulse that represents the true reflection impulse response of the FBG. 
     FIG. 20  is a plot of the results from a series of simulations, each with a different splitting ratio between the peak amplitude of the input laser pulse and the peak amplitude of the reflected impulse response. The two curves of  FIG. 20  correspond to the two input laser pulses shown in  FIGS. 12A and 12B  (“longer laser pulse”) and  FIGS. 18A and 18B  (“shorter laser pulse”).  FIG. 20  shows the logarithm of the error in the recovery of |r(ω)| as a function of the logarithm of the splitting ratio. 
   As shown by  FIG. 20 , the error in the recovery generally decreases for increasing splitting ratios between the peak amplitude of the input laser pulse and the peak amplitude of the reflected impulse response of the FBG. For embodiments with splitting ratios below a splitting ratio of approximately 100, the longer laser pulse has a smaller error in the recovery than does the smaller laser pulse. However, for splitting ratios larger than a critical ratio of approximately 100, using a shorter input pulse provides a much better recovery than does using a longer input laser pulse (e.g., recovery improved by a factor of approximately 4). In addition, while the baseline error for the longer laser pulse is achieved at lower splitting ratios (e.g., greater than approximately 26), it is higher than the baseline error achieved using the shorter laser pulse, even though the shorter laser pulse&#39;s baseline error is only achieved with higher splitting ratios (e.g., greater than approximately 100). 
   The reason for this observed lower baseline in error for the shorter laser pulse is that the shorter input laser pulse more closely approximates a true delta function, thereby yielding a more accurate reflected pulse to represent the true reflection impulse response of the FBG and reducing the error in the recovery. However, by narrowing the input laser pulse, the energy of the input laser pulse (proportional to the area under the temporal profile of the laser pulse) is also reduced. Therefore, higher splitting ratios are used to obtain the benefits of the reduced error from a more narrow input laser pulse. In other words, when using a shorter or narrower dummy input pulse, it is advantageous to use a stronger peak amplitude (or a higher critical ratio) for the input pulse. In certain embodiments, as shown by  FIG. 20 , the error is decreased by a factor of approximately 4 by using an input laser pulse that is temporally approximately 3 times narrower. 
   In certain embodiments, the ratio of the integral of the normalized laser field of the input laser pulse to the integral of the normalized reflection impulse response of the FBG, i.e., 
           ∫                E   Pulse     ⁡     (   t   )              max   ⁡     (            E   Pulse     ⁡     (   t   )            )         ⁢       ⅆ   t     /     ∫                h   R     ⁡     (   t   )              max   ⁡     (            h   R     ⁡     (   t   )            )         ⁢     ⅆ   t                   
is useful to select an optimum temporal width of the input laser pulse and the splitting ratio. For the shorter laser pulse of  FIG. 20 , this ratio is approximately 0.35%, and for the longer laser pulse, this ratio is approximately 1.4%. In other words, for the shorter laser pulse, the total area under the normalized electric field amplitude of the input laser pulse is only 0.35% of the total area under the normalized amplitude of the reflection impulse response of the target FBG. This low ratio is expected since the input laser pulse approximates a delta function. However, this ratio increases to 1.4% for the longer pulse (e.g., by approximately a factor of 4), while the critical splitting ratio drops by a factor of approximately 4 as well (e.g., from 100 to 26). Thus, for broader impulse response FBGs (such as strongly chirped FBGs), a generally larger critical ratio is advantageously used to achieve convergence. This behavior can also be related to the property of MPFs that most of the energy of an MPF is concentrated in proximity to the origin, as discussed above. In certain embodiments, for a broader reflection impulse response of the FBG, a higher peak amplitude of the leading dummy pulse is advantageously used to satisfy this property of MPFs for the input pulse sequence.
 
     FIG. 21  is a plot of a simulated noisy power spectrum used to illustrate how measurement errors in the power spectrum affect the accuracy of the recovery results. The power spectrum of  FIG. 21  was produced by multiplying the theoretical FT amplitude square of an input pulse sequence by a uniform random noise (e.g., 10% peak-to-peak amplitude with an average of unity). The error-reduction method  700  of  FIG. 9  was applied to the simulated noisy power spectrum of  FIG. 21  to recover the reflection impulse response of the target FBG. In this exemplary embodiment, the resolution of the OSA was assumed to be approximately 10 picometers and a splitting ratio of 27 between the optical fields was used (corresponding to a splitting ratio of 27 2  of the optical powers). 
   The calculated reflection coefficient amplitude and the group delay spectra of the target FBG, computed from the measured power spectrum of  FIG. 21 , are shown in  FIGS. 22A and 22B . The recovery is still quite good despite the strong noise added to the power spectrum used in the calculation. The large oscillations observed in the recovered group delay spectrum, especially towards the edges of the spectral window of  FIG. 22B , are due to the significant drop of the intensity of the reflection coefficient amplitude at those wavelengths, which makes the recovery of spectral phase more difficult. In the limiting case where the intensity goes to zero, the definition of phase has less meaning. However, this behavior is inconsequential, since the group delay in the more important range of wavelengths (e.g., between approximately 1548 nanometers and approximately 1552 nanometers, for the target FBG is recovered quite well. The results of  FIGS. 22A and 22B  show that certain embodiments described herein work well even with fairly noisy measurements of the power spectrum. 
   The noise sensitivity of certain embodiments described herein is also affected by the ratio of the dummy input laser pulse to reflection impulse response amplitudes. In certain embodiments in which the main source of noise in the OSA measurement system is proportional to the input intensity, a larger dummy pulse results in a larger noise intensity in the measured spectrum, and thus a larger error in the recovered FBG spectra. To maximize the accuracy of the recovery with noisy measurements, certain embodiments advantageously select an amplitude ratio close to the critical ratio. For example, for the power spectrum shown in  FIG. 21 , an amplitude ratio of 27 was used, which is close to the critical ratio of 26 for the longer laser pulse, as shown in  FIG. 20 . In certain embodiments, this choice of the amplitude ratio ensures both accurate convergence of the iterative error-reduction method and reduced sensitivity to measurement noise. In certain embodiments, the critical ratio that ensures convergence is traced by choosing two different ratio values and comparing the difference between the recovery results. If the difference is small, then convergence is achieved and the chosen ratio values are somewhere on the lower baseline of the curve in  FIG. 20 . 
   Certain embodiments described herein are conveniently used to characterize any FBG spectra uniquely. Certain embodiments also have advantages with respect to currently existing techniques, e.g., low-coherence spectral interferometry. Certain embodiments described herein provide better resolution (e.g., by a factor of two) than does low-coherence spectral interferometry using the same measurement configuration. In low-coherence spectral interferometry, by filtering in the inverse FT domain, the convolution of the impulse response with the autocorrelation function of the input laser pulse is recovered. However, in certain embodiments described herein, the convolution of the same impulse response with the input laser pulse itself is recovered, which constitutes an improved resolution by approximately a factor of 2. 
   In certain embodiments described herein, the time delay between the input laser pulse and the reflection impulse response of the FBG is made as small as possible, as long as there is no temporal overlap between the two pulses. However, low-coherence spectral interferometry requires a certain minimum delay between these two pulses to ensure individual filtering of the above-mentioned convolution term in the IFT domain. In certain embodiments in which the OSA has low resolution, the large time delay needed by low-coherence spectral interferometry can result in rapid fringes in the power spectrum that the OSA cannot resolve, which can potentially cause severe recovery errors. 
   Certain embodiments described herein are used to characterize multiple FBGs concurrently using a single OSA measurement. In certain such embodiments, the measurement configurations of either  FIGS. 7A and 7B  are used, with the additional FBGs to be characterized added in a parallel fashion next to the first FBG.  FIG. 7C  schematically illustrates an exemplary embodiment in which multiple FBGs  540   a ,  540   b ,  540   c , . . . are coupled to a 1×N coupler  590  where N is two or more. Each of the FBGs  540   a ,  540   b ,  540   c , . . . are coupled to the coupler by a corresponding optical path  592   a ,  592   b ,  592   c , . . . each having a different optical pathlength. In certain embodiments, the optical pathlengths of the optical paths  592   a ,  592   b ,  592   c , . . . are selected to avoid having the reflected pulses from the FBGs  540   a ,  540   b ,  540   c , . . . overlapping temporally when they arrive at the OSA  570 . 
   As described above for the measurement configuration for a single FBG  540 , the reflection impulse responses of all the FBGs  540   a ,  540   b ,  540   c , . . . are time-delayed with respect to the stronger dummy laser pulse. The pulse sequence of such embodiments consists of more than 2 pulses, and the pulse sequence is sent to the OSA for the power spectrum measurement. The processing of the measured power spectrum in certain such embodiments is done in the same manner as described above (e.g., using the iterative error-reduction method  700  of  FIG. 9 ). The configuration of multiple FBGs schematically illustrated in  FIG. 7C  is also used in certain embodiments with the measurement configuration of  FIG. 8  to characterize multiple FBGs concurrently. 
     FIG. 23  is a plot of the normalized input laser pulse spectrum and the reflection spectra of two different Gaussian apodized FBGs (FBG# 1  and FBG# 2 ) which are characterized together in an exemplary embodiment. The amplitude and phase of the temporal electric field profile of the input laser pulse are shown in  FIGS. 24A and 24B , respectively.  FIG. 25  shows the amplitude spectrum and the phase spectrum of the pulse sequence formed by time-delaying the two reflected pulses from the FBG# 1  and FBG# 2 , with respect to the leading dummy input laser pulse with a splitting ratio of 40. 
     FIG. 26  is a plot of the power spectrum for this pulse sequence of three pulses, as would be measured by an OSA. In this exemplary embodiment, the resolution of the OSA was assumed to a modest number, e.g., approximately 10 picometers. Using only the power spectrum measurement shown in  FIG. 26 , the simultaneous recovery of the reflection coefficient amplitude and group delay spectra for these two FBGs is achieved.  FIG. 27A  is a plot of the original (solid line) and recovered (dashed line) amplitude of the reflection coefficient of FBG# 1 .  FIG. 27B  is a plot of the original (solid line) and recovered (dashed line) group delay spectra of the reflection coefficient of FBG# 1 .  FIG. 28A  is a plot of the original (solid line) and recovered (dashed line) amplitude of the reflection coefficient of FBG# 2 .  FIG. 28B  is a plot of the original (solid line) and recovered (dashed line) group delay spectra of the reflection coefficient of FBG# 2 . 
   As in the exemplary embodiments previously described, the recovery for this exemplary multiple-FBG embodiment is very good, showing that certain embodiments can quite conveniently characterize more than one FBG all at the same time using a single power spectrum measurement. The large-scale oscillations observed in the recovered group delay spectrum, especially towards the edges of the spectral window shown in  FIG. 27B  are due to the significant drop of the intensity of the reflection coefficient amplitude at those wavelength, which makes the recovery of spectral phase more difficult. However, this is inconsequential, as discussed above in relation to  FIG. 22B . 
   Certain embodiments described herein work well if any of the electric field envelopes in the pulse sequence are time-reversed. Stated differently, certain embodiments described herein can advantageously differentiate between a pulse and its time-reversed replica. 
   Certain embodiments which utilize the measurement configuration  600  schematically illustrated in  FIG. 8  share all the attributes described above in relation to the measurement configurations  500 ,  580  of  FIGS. 7A and 7B . Certain embodiments utilizing the measurement configuration  600  advantageously provide additional desirable features over existing techniques as well. In certain such embodiments, the dummy input laser pulse does not need to be temporally much narrower than the reflection impulse response of the target FBG. As described above, the measurement configurations  500 ,  580  shown in  FIGS. 7A and 7B  advantageously provide better resolution (e.g., by a factor of two) with respect to low-coherence spectral interferometry, which uses an input dummy laser pulse that is narrower than the reflection impulse response of the target FBG by approximately 50 times or more. However, in certain embodiments utilizing the measurement configuration  600  of  FIG. 8 , the same FBG can be characterized with an input laser pulse that is only approximately 2 to 5 times narrower than the reflection impulse response of the FBG. Certain such embodiments advantageously eliminate the requirement for an ultra-short input laser pulse, which are generally more costly and more difficult to generate than longer laser pulses. 
     FIG. 29  is a plot of an FBG reflection spectrum (solid line) and the normalized input laser pulse amplitude (dotted line) in an exemplary embodiment. The amplitude and phase of the temporal profile of the input laser pulse shown in  FIGS. 30A and 30B , respectively, shows that the width of the laser pulse is approximately 18 picoseconds, where the temporal width is defined as the full width at which the field reduces to 10% of its maximum value. The temporal width of the reflection impulse response of the same FBG shown in  FIG. 29  is wider than the input laser pulse by only a factor of approximately 2. As shown by  FIG. 31 , the temporal width of the reflection impulse response is approximately 37 picoseconds, while the reflected pulse width is approximately 56 picoseconds. 
   Referring to the measurement configuration  600  of  FIG. 8 , the 18-picosecond-wide input dummy laser pulse  620  is first split into two pulses. The first pulse  622  is sent to a delay line  630 . The second pulse is sent to a pulse stretcher  640 , which broadens the temporal width of the laser pulse to produce a time-stretched laser pulse  624  which is broadened by a factor of at least approximately 2 to 5 as compared to the second pulse before being stretched. The pulse stretcher  640  of certain embodiments is a loop of single-mode fiber optic cable that broadens the width of a laser pulse through dispersion. In this exemplary embodiment, the input laser pulse  620  has an initial width of approximately 18 picoseconds, and the time-stretched laser pulse  624  is stretched in time by a factor of approximately 4, such that the width of the time-stretched pulse  624  is approximately 76 picoseconds, as shown in the plot of  FIG. 32 . The temporal electric field profile of the time-stretched pulse  624 , i.e., E s (t), is chosen in certain embodiments to be wider than the dummy input laser pulse profile by a factor of at least approximately 2 to 5. 
   The time-stretched pulse  624  is sent through an attenuator  650 , and is then split into two weak pulses, as shown in  FIG. 8 . In certain embodiments, the attenuator  650  is approximately 32 dB (in power), while in other embodiments, the attenuator  650  has other values (e.g., greater than approximately 20 dB). The reflected weak pulse  672  from the target FBG  540  is no longer the reflection impulse response of the FBG  540 , since the incident pulse is much broader than the impulse response of the FBG  540 , as shown in  FIG. 31 . The reflected pulse  672  is the convolution of the impulse response of the FBG  540  with the temporal profile of the time-stretched pulse  624 , i.e., h R (t)*E s (t). The back reflections from the FBG  540  and from the mirror  660  (e.g., a bare fiber end) are then temporally combined with the time-delayed version of the unattenuated initial dummy laser pulse  622 , as shown in  FIG. 8 . This pulse sequence  680  has a dominant peak pulse at the leading edge followed by two weaker reflected pulses, as shown in  FIG. 33 . The pulse sequence  680  is then sent to the OSA  570 , which measures its power spectrum. Assuming a typical OSA  570 , with a resolution of approximately 10 picometers, the theoretical power spectrum of the pulse sequence of  FIG. 33  is shown in  FIG. 34 . 
   As discussed above, the error-reduction method  600  of  FIG. 9  is used in certain embodiments to process the measured output of the OSA. Because the input pulse sequence is close to an MPF, both E s (t) and h R (t)*E s (t) can simultaneously be recovered using only the OSA output. The reflection spectrum of the target FBG is computed in certain embodiments by taking the FTs of both E s (t) and h R (t)*E s (t), i.e., 
             r   ⁡     (   ω   )       =         FT   ⁢     {         h   R     ⁡     (   t   )       ⋆       E   s     ⁡     (   t   )               FT   ⁢     {       E   s     ⁡     (   t   )       }         .           
In certain embodiments, the power spectrum of the input laser pulse covers the frequency band of the target FBG, as shown in  FIG. 29 . The result of this exemplary embodiment is shown in  FIGS. 35A and 35B . Once again the recovery is very good. In this exemplary embodiment, a Gaussian apodized FBG with an impulse response temporal width of approximately 37 picoseconds has been fully characterized using a dummy laser pulse that has a temporal width of approximately 18 picoseconds by using only a single OSA measurement. The whole computation involved in the exemplary embodiment took only a few seconds to run using MATLAB 5 on a 500-MHz computer.
 
   In certain embodiments, various ultrashort pulse-shaping techniques (see, e.g., M. M. Wefers and K. A. Nelson, “ Analysis of Programmable Ultrashort Waveform Generation Using Liquid - Crystal Spatial Light Modulators ,” J. Opt. Soc. Am. B, 1995, Vol. 12, pp. 1343-1362; A. Rundquist et al., “ Pulse Shaping with the Gerchberg - Saxton Algorithm ,” J. Opt. Soc. Am. B, 2002, Vol. 19, pp. 2468-2478) are used to modify the temporal profile of the dummy pulse in order to achieve a true MPF for the electric field of the pulse sequence, which can potentially improve the recovery speed of certain such embodiments dramatically. By using a true MPF, certain embodiments can converge in less than 5 iterations, thus cutting down the computation time to a fraction of a second, even when using a relatively slow programming environment such as MATLAB 5.1. 
   Various embodiments of the present invention have been described above. Although this invention has been described with reference to these specific embodiments, the descriptions are intended to be illustrative of the invention and are not intended to be limiting. Various modifications and applications may occur to those skilled in the art without departing from the true spirit and scope of the invention as defined in the appended claims.