1. Introduction

Fiber systems based on low-coherence interferometry (LCI) have been used over the years for the high-resolution characterization of sources, components, and sensors [1]. Fourier-transform (FT) spectrometers take advantage of the Fourier duality between the interferogram and the optical spectrum for the analysis of sources [2,3] and wavelength encoded sensors [4], and Optical Low-Coherence Reflectometers (OLCR) [5–9], Optical Coherence Tomography (OCT) systems [10,11] and displacement sensors [12] locate reflective events by use of the sharp interference of low-coherence light. A major endeavor in these systems is to accurately record the interferogram’s phase, since this allows to determine the spectrum of an optical wave or the vector information of the device or system under consideration. For this purpose, phase sensitive LCI systems conventionally use an auxiliary laser to monitor the optical path difference (OPD) scan [2,3,8,9,13–15].

As a basic interference phenomenon, LCI continues to attract attention in different contexts, and also in Microwave Photonics (MWP), where there has recently been an increasing interest in optical metrology systems for optical fiber sensors [16,17]. In this regard, MWP offers opportunities for the analysis of optical waves through the use of wideband radio-frequency (RF) modulation, custom definition of optical circuits using standard splitting, filtering and propagation technologies, high stability of the RF interference, and ease of reconfigurability by purely electronic means. Within this context, the use of a well known MWP architecture, namely an analog-modulated dispersive link fed with a broadband source (BBS) [18], has been identified as a simple means to characterize fiber Bragg grating (FBG) sensors [19–22], interferometers [23,24] or multilayered samples [25]. In [26] we demonstrated the equivalence of this architecture with a self-referenced double-interferometer LCI system [12,27], where a first, static interferometer is used to probe the device under test (DUT), and the second interferometer, built on MWP concepts, is used to scan the OPD through standard electrical vector network analysis (VNA). The approach is attractive since high-resolution OPD scans are performed without the need of mechanical parts, the double interferometer allows for absolute distance measurements and, as an additional benefit, the relationship between path delays and scanning RF tones can be adjusted through the value of dispersion. Also, and due to the high accuracy of the electronic oscillators’ frequency, the corresponding accuracy in path delays is in practice only limited by the stability of the imparted dispersion. We also identified, however, a number of shortcomings limiting LCI range and resolution: the broadening, distortion, and fading of the RF resonance representing the LCI peak was due to the link’s higher-order dispersion (HOD) and to the carrier suppression effect (CSE) inherited from the amplitude modulation format used. Subsequent efforts have been targeted to overcome these issues with single-sideband modulation [28], alternative MWP architectures [29], a double pass in the interferometer [30], and also to provide increases in scanning speed [31] and sensitivity [32]. In all these implementations, the analysis is focused in the interferogram’s amplitude, typically locating the position of test reflectors by use of the RF resonance, and little attention seems to have been paid to the stability of the interferogram’s phase extracted by this procedure.

In this paper we present, for the first time to the best of our knowledge, a phase sensitive MWP LCI system based on double-sideband with suppressed carrier (DSB-SC) modulation, dispersive propagation in a chirped FBG (CFBG), and demodulation by electro-optical frequency down-conversion, thus benefiting from the common-path propagation of the modulated and dispersed waves responsible of the optical interference. Using only 20 GHz of RF bandwidth and a direct normalization of the link’s amplitude and phase response, our system retrieves the amplitude and phase of interferograms in an OPD range of 16.3 mm, outperforming previously reported values based on related MWP approaches. This range allows to explore a number of LCI applications targeted to sources and components in the C band, such as a direct analysis of interferograms, FT spectroscopy, and OLCR characterization of FBG. Built from a combination of frequency up- and down-conversion, the system, when operated with monochromatic light, can be understood as a Frequency-Shifted Interferometer (FSI) [33–35]. As will be shown below, however, it is the inclusion of a dispersive element together with broadband illumination what provides the possibility to record low-coherence interferograms.

These results have been attained by a number of improvements with respect to our initial demonstration [26]. DSB-SC modulation shows no CSE and provides a twofold increase in range for given dispersion and modulation bandwidth. The broadband CFBG is a non-standard device providing stable 10-km dispersion compensation in the C band with negligible HOD [36]. This element has previously been used to demonstrate FT spectroscopy using conventional amplitude modulation [37]. Additionally, the use of a second electro-optical modulator to perform homodyne down-conversion avoids the use of a harmonic VNA (i.e., a VNA whose receiver is tuned to a harmonic of the stimulus frequency). The low bandwidth detection enables the cost-effective use of high-sensitive detectors and high-depth digitizers. Finally, impairments due to modulators’ harmonic distortion and bias drifts have been numerically compensated for by suitable signal processing techniques.

This paper is organized as follows. In Section 2 we analyze the MWP architecture, the normalization procedure leading to the interferogram, and the signal processing techniques used. Section 3 introduces the experimental setup and the normalization traces. In Section 4 we present the characterization of several C-band sources and reflectors and the determination of performance parameters in FT spectroscopy. Finally, we end in Section 5 with our conclusions.

2. Principle of operation

Referring to Fig. 1, the system can be presented as follows. A polarized laser line from a tunable laser source (TLS) with optical frequency $\omega _0$, electric field $\mathcal {E}_0(t)=E_0\exp (j\omega _0 t)$ and power $P_0=|E_0|^2$ is injected in a first push-pull electro-optic Mach-Zehnder modulator (MZM1). The modulator is biased at minimum transmission and driven by a RF tone at frequency $\Omega$, so that the output optical field is given by:

 

Fig. 1. Schematics of the system. The box (ABC) refers to different measurement configurations, shown in detail on the right and not used in the normalization procedure.

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The modulated optical wave $\mathcal {E}_1(t)$ shows two optical sidebands at frequencies $\omega _0\pm \Omega$. The CFBG acts as a dispersive delay line (DDL) imparting on these sidebands propagation phases $\phi (\omega _0\pm \Omega )=\phi (\omega _0)\mp \Omega \tau _0$, with $\tau _0$ the group delay at frequency $\omega _0$. The dispersed wave $\mathcal {E}_1^{'}(t)$ is subsequently remodulated by a second modulator (MZM2), synchronously driven at the same tone $\Omega$ to produce a second DSB-SC modulation $\mathcal {E}_2(t)=a_2(\Omega )\cos [\Omega t+\varphi _2(\Omega )]\mathcal {E}_1^{'}(t)$. The resulting wave is given by:

Hence, if the photodiode’s (PD) bandwidth is lower than $2\Omega$, the voltage $\mathcal {V}_0(\Omega )$ recorded by the digitizer is proportional to the average power of $\mathcal {E}_2(t)$, which is simply the sum of the power at the carrier and those of the two sidebands. According to Eq. (2), the optical power at the carrier wavelength is $\frac {1}{8}T A(\Omega )P_0 [1+\cos (2\Omega \tau _0 -\varphi (\Omega ))]$, with $A(\Omega )=a_1^2(\Omega )a_2^2(\Omega )$ the product of modulators’ power transfer functions, $\varphi (\Omega )=2[\varphi _1(\Omega )-\varphi _2(\Omega )]$ the double of their difference in phase response, and $T$ the system’s insertion loss when $P_0$ is referred to the input of the first modulator. Also, each of the sidebands at $\omega _0\pm 2\Omega$ provides an additional power $\frac {1}{16}T A(\Omega )P_0$. The recorded voltage is thus:

In Fig. 2 we illustrate Eq. (2) using the experimental system described in the following section. In this plot we show the optical spectrum, measured at the output of the second modulator, of a 1550 nm carrier modulated around $\Omega /(2\pi )\simeq$10 GHz when this frequency is fine tuned to reach the interference’s maximum (blue trace) and minimum (orange trace) at the carrier wavelength. At the minimum, the power decreases to a value determined by the attenuation and the suppression of the carrier in the two modulators biased at minimum transmission. This value can be determined by switching off the modulation, as is shown with a yellow trace. The measured total extinction ratio, relative to the carrier power measured before the first modulator, is $-$67.6 dB, including an insertion loss $T$ of 10 dB. Also observable are residual sidebands at $\pm 10$ GHz, generated by the first modulator and suppressed only by the second. The relative carrier power at the interference’s maximum, $\frac {1}{4}T A(\Omega )$, is $-$36.3 dB, as so the modulation and down-conversion loss factor $A(\Omega )$ is $-$20.3 dB at 10 GHz.

 

Fig. 2. Optical spectrum of the modulated, dispersed and down-converted waves at $\Omega /(2\pi )\sim 10$ GHz over a 1550 nm carrier under constructive (blue) and destructive (orange) interference. Yellow: power spectrum when the modulation is off. The spectrum is referred to the carrier power $P_0$ measured at the input of the first modulator.

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Voltage $\mathcal {V}_0(\Omega )$ suffers from a background component due to the modulators’ finite extinction ratio, and also from variations due to their bias drifts, which were not controlled in our experiments. After typically half an hour of warm-up time, these effects are sufficiently slow to allow for stable measurements in time spans of the order of minutes. Inclusion of modulators’ harmonic distortion in the analysis results in the appearance of spurious tones in $\mathcal {V}_0(\Omega )$ of the form $\cos [n\Omega \tau _0-n\varphi (\Omega )/2]$ with $n\neq 2$. These terms are bandpass signals centered at a period in frequency $1/(n\tau _0)$, and can be removed by Fourier transforming $\mathcal {V}_0(\Omega )$ to a new signal $v_0(t)$ since they are concentrated at $t=n\tau _0$. The second term in (3) representing the interferogram can be isolated by filtering $v_0(t)$ around $t=2\tau _0$.

When the system is fed with a BBS, described by an electric field $\mathcal {E}(t)$ and an optical spectral density $S(\omega )$, each of its spectral components undergoes the same transformation as that written in Eq. (3), yielding a voltage:

The raw temporal step, $\delta u$, or one-point resolution of our low-coherence interferograms, depends on the frequency step $\Delta \Omega$ used to sequentially scan the temporal range, and is given by $\delta u=2\Delta \Omega \tau _0'$. In LCI systems, ranges are expressed as OPD, or single-pass distance in free space ($\textrm {OPD}=cu$, with $c$ the speed of light in vacuum). The LCI (two-point) spatial resolution equals the source’s coherence length $\ell _c$, which is the FWHM of the amplitude peak that reflects the low-coherence interference. The coherence length is inversely proportional to the source’s spectral width $\Delta \nu$, with a proportionality constant that depends on the form of the spectrum. In our LCI experiments, we use a BBS which has a roughly square-like spectrum. For a perfect square spectrum, the autocorrelation $\Gamma (u)$ is sinc-like and results in a relationship $\ell _c=1.21c/\Delta \nu$. In OLCR systems, the scan is conventionally expressed in terms of double pass distance, $z=cu/(2n_g)$ with $n_g$ the fiber’s group refractive index. The standard two-point OLCR resolution in fiber is thus given by $\delta z_2=\ell _c/(2n_g)$ [1]. As will be shown below, however, in our OLCR characterization of the impulse response of FBGs the relevant resolution is the one-point spatial resolution, given by $\delta z=c\delta u/(2n_g)$. Finally, in FT spectroscopy, the inteferograms are extended in complex form to negative delays using the hermiticity condition $\Gamma (-u)=\Gamma (u)^\ast$. The spectral resolution $\delta \nu$ is given by $\delta \nu =1.21\kappa /(2u_{\textrm {max}})$, where $\kappa > 1$ describes the increase in $\delta \nu$ when a window is applied to $\Gamma (u)$ before the FT and $\kappa =1$ in the absence of windowing. The wavelength range in FT spectroscopy, however, is here limited by the reflection band of the CFBG grating used to implement the DDL.

It is apparent from the previous discussion that the ability to record the complex interferogram requires that the involved dispersive delays are stable during the normalization and measurement procedures. The implementation of the DDL with a fiber coil may fail in meeting this requirement due to thermal variations and mechanical stress in the fiber. The same shortcoming occurs, for instance, in the distribution of RF tones through fiber lines [38]. Other, more costly, alternatives include the use of programmable optical phase filters [39]. In our setup, the DDL used was a low-ripple, athermally-packaged, C-band CFBG, described in the following section. We finally stress that the recovery of the BBS autocorrelation $\Gamma (u)$ relies on the fact that the dispersive element shows negligible HOD $\tau _0''$. Conventional CFBG for dispersion compensation of standard single-mode fiber (SMF) would distort the interferograms and, although they can be numerically corrected [40,41], this has not been necessary here.

3. Experimental setup and normalization

In our implementation, and referring again to Fig. 1, two push-pull Mach-Zehnder electro-optic modulators of bandwidth 20 GHz (JDSU AM150 and Avanex SD20) were used to implement the DSB-SC modulation and down-conversion stages, both at an input RF power of +16 dBm and without active bias control. The CFBG (Proximion) was a broadband device (1540-1560 nm) with a dispersion at 1550 nm of $-$170.04 ps/nm ($\tau '_0 =216.87 \, \,{\textrm {ps}}^2/{\textrm {rad}}$), negligible HOD ($\tau ''_0 =-0.05 \, \,{\textrm {ps}}^3/{\textrm {rad}}^2$) and low ripple (rms phase ripple <0.02 rad). Additional details can be consulted in [36]. As mentioned before, the system’s insertion loss $T$ measured from the first modulator was 10 dB and includes a non-uniform CFBG loss of 1.8 $\pm$ 0.2 dB. The modulators were driven by a RF or a microwave (MW) signal generator, to be described below. The detector was a standard 1-GHz pin photodiode (Thorlabs DET01CFC, $\mathcal {R}=0.95$ A/W) whose bandwidth was limited to a few tens of kHz by the 1-M$\Omega$ load imposed by a 14-bit digitizer (Digilent Analog Discovery 2) operated at a sampling rate of 100 kHz.

In a first test we fed the system with a monochromatic laser line at 1550 nm extracted form our TLS (Keysight N7711A) and programmed a narrowband RF sweep from $\Omega /(2\pi )=$ 10 GHz to 10.05 GHz in steps of 25 kHz to exemplify a few cycles of the interference pattern given by Eq. (3). The result is plotted in Fig. 3. In the first plot, Fig. 3(a), the frequency sweep shown in blue and starting at 0.6 s is governed by the MW source (Anritsu MG3690C). The frequency period between interference’s maxima is $1/(2\tau _0)$ = 9.25 MHz, and is due to a delay $\tau _0$ in a SMF length between modulators of 10.8 m. The equivalent OPD step is as low as 4 nm. This experiment also permits to visualize the interferometer’s short-term stability: the constant voltage level at 10 GHz shown in orange has a standard deviation of 0.59 mV. In terms of the inteferometer’s phase, $\Phi (\Omega )=2\Omega \tau _0-\varphi (\Omega )$, this corresponds to a phase deviation of 0.14 deg or to a deviation in OPD of 0.6 nm. The yellow trace is the reading when the laser is switched off and represents the system’s noise floor, with a standard deviation of 0.18 mV. The excess in standard deviation of the 10-GHz interference level is attributed to thermal effects within the light path between modulators. Figure 3(b) is a zoom view showing the structure of the frequency steps and the stability of the 10-GHz level.

 

Fig. 3. (a) Blue: digitized voltage $\mathcal {V}_0(\Omega )$ corresponding to a laser line at 1550 nm with $P_0=$ 3.5 dBm. The initial value of the RF is $\Omega /(2\pi )=10$ GHz, increasing in 25 kHz steps of duration $\sim$10 ms up to 10.05 GHz. Orange: constant RF level at 10 GHz. Yellow: noise floor when the laser is switched off. (b) Zoom of the blue and orange traces in (a) showing the structure of the 25-kHz RF steps. The glitches are due to the commutation of the MW source.

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The system was then completed to the LCI setup, as described in Eqs. (5)–(7). The BBS was a polarized ASE source with a roughly flat spectral density in the CFBG reflection band. The available RF bandwidth was scanned from 2 MHz up to $\Omega _{\textrm {max}}/(2\pi )=$20 GHz to display a total temporal range $u_{\textrm {max}}=2\Omega _{\textrm {max}}\tau _0'=54.5$ ps. This corresponds to an OPD of 16.3 mm in free space or 10.9 mm in fiber, ranges that are halved in reflectometric measurements. The frequency step was $\Delta \Omega /(2\pi )=$2 MHz, and had a duration of $\sim$3 ms. Since the lowest frequency delivered by the MW signal generator is 10 MHz, it was necessary to use a second RF source (TTi TG4001) whose scanning range was partially coincident with the MW signal generator, from 2 MHz to 30 MHz, including an additional tone at 75 kHz to help interpolate $\mathcal {V}(\Omega )$ for $\Omega \to 0$. The temporal step of our interferograms is thus $\delta u=2\Delta \Omega \tau _0'=5.45$ fs (1.63 $\mu$m in free space, single pass). It was limited by the number of frequency steps that can be sequentially delivered by our MW generator ($10^4$) which were chosen to evenly sweep the total available temporal range. For each frequency $\Omega$, the constant output of the digitizer was averaged over the step duration by first detecting the commutation glitches, which were gated by a synchronization signal from the RF/MW source, and then averaging the level over the interval between glitches. The $\Omega \to 0$ value of signal $\mathcal {V}(\Omega )$ was then interpolated and, after Hilbert transform, it was divided in complex form by a 1550-nm normalization trace $\mathcal {V}_0^c(\Omega )$ taken using the same range and frequency steps. This normalization was previously filtered around frequency $1/(2\tau _0)$. The scan time was about 3 seconds per mm of OPD.

Figure 4 shows the plot of the amplitude $|\mathcal {V}_0^c(\Omega )|/(Z_L\mathcal {R}P_0)$ and phase $\varphi (\Omega )$ of a normalization trace at $P_0=-5$ dBm, in this last case after the subtraction of the linear phase term describing the propagation delay $\tau _0$ between modulators. According to Eq. (6), the amplitude equals $\frac {1}{8}TA(\Omega )$ and represents the effective optical loss experienced by a given input power. It shows a peak value of $-$30.6 dB at 230 MHz followed by a roll-off due to the bandwidth limitations of modulators, cables and RF splitter, which results in a decrease in signal-to-noise ratio for increasing delays in all our interferograms. It the middle of this roll-off, the curve reproduces the value $-$39.3 dB at 10 GHz inferred from our analysis of Fig. 2. Below 230 MHz, the amplitude is limited by the modulators’ response, whereas the ripples observed in the phase are likely due to small impedance mismatches within the RF path. We also compared six of these normalization traces taken in a span of four weeks in standard laboratory conditions. The manual polarization alignment before modulators (not shown in Fig. 1) and bias drifts result in global variations in the amplitude response of $\sim$1 dB. Once normalized to its maximum value, the remaining rms amplitude variation, averaged over the whole bandwidth, was $<0.2$ dB for all the analyzed traces. The global delay $\tau _0$ showed a maximum variation among traces of $\Delta \tau _0= 7$ ps. According to Eq. (5), this variation translates into a variation in the interferogram’s frequency of $\Delta \omega =\Delta \tau _0/\tau '_0$, with value 7 GHz (56 pm in the C band), a figure that represents an estimate of the accuracy in the determination of optical frequencies due to environmental changes. The remaining phase variation after subtraction of the linear phase, in turn, was lower than 2 deg rms.

 

Fig. 4. (a) Amplitude $10\log _{10}[|\mathcal {V}_0^c(\Omega )|/(Z_L\mathcal {R}P_0)]$ and (b) phase $\varphi (\Omega )$ of a normalization trace.

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4. Results

Our first experiment was targeted to confirm the system’s OPD range. The BBS was followed by a Michelson interferometer placed before the first modulator and built from a 3 dB splitter, fiber collimators, and dielectric mirrors, as shown by scheme A in Fig. 1. The interferogram’s amplitude, depicted in Fig. 5, showed different peaks with a uniform FWHM $\delta \ell$ of $142 \pm 3~\mu$m as one of the mirrors was displaced in steps of 1 mm, confirming the expected OPD range and the absence of peak broadening and fading due to CSE or HOD. The small distortion in the peaks’ structure, visible as an asymmetry in its low-power sidelobes, is ascribed to the CFBG’s residual HOD.

 

Fig. 5. Amplitude $10\log _{10}|\Gamma (u)|$ of the complex low-coherence interferogram describing the incidence of a BBS in an interferometer for different path imbalances, increasing from 2.2 mm in 2-mm steps.

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We then analyzed the typical BBS used in our experiments, directing the broadband light to the first modulator, retrieving its autocorrelation $\Gamma _{\textrm {BBS}}(u)$, and computing the FT spectrum. We also measured the optical spectrum using a standard dispersive grating-based OSA (Agilent 86142B, 60 pm resolution). The measurement was taken from the detector’s side so that both spectra suffer from the same non-uniform CFBG loss and are therefore comparable. We finally computed the inverse Fourier transform (IFT) of the OSA spectrum to compare it with the experimental interferogram. The resolution of this IFT was increased by zero padding the spectrum, a procedure which is enabled here since the BBS spectra is bandlimited due to its pass through the CFBG.

The optical spectrum, together with the amplitude and phase of the interferogram, are presented in Fig. 6; in orange those corresponding to the OSA spectrum and its IFT, and in blue the interferogram’s amplitude and phase together with its FT spectrum. The agreement of both approaches is excellent. The optical spectrum in Fig. 6(a) is square like and contained in the $\sim$20-nm CFBG reflection bandwidth. The interferogram’s amplitude, shown in Fig. 6(b), presents a coherence length $\ell _c$ of 144 $\mu$m, in agreement with Fig. 5 and also with the expected value of a square optical spectrum with width $\Delta \nu =2.5$ THz. The interferogram’s amplitude indeed shows, for low values of the OPD, the typical sinc functional form. As for the phase in Fig. 6(c), the expected behaviour of $\arg \Gamma _{\textrm {BBS}}(u)$ for low values of time lag $u$ or OPD is proportional to the BBS’s central frequency $\overline {\omega }=\int \omega S(\omega )d\omega /P_0$, with $S(\omega )$ the optical spectral density and $P_0=\int S(\omega )d\omega$ the total BBS power. This can be deduced by expanding the exponential term in the IFT to first order:

 

Fig. 6. (a) FT (blue) and OSA (orange) spectra of the BBS. (b) Interferogram’s amplitude (blue), and amplitude of the OSA spectrum’s IFT (orange). (c) Interferogram’s phase (blue), and phase of the OSA spectrum’s IFT (orange, shifted by $-2\pi$), both referred to the BBS’s central frequency. Gray line: linear fit of the asymptotic phase.

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After an OPD of 380 $\mu$m, and therefore well above the coherence length, the phase shows a linear decreasing trend that describes a behaviour of the autocorrelation of the form $\Gamma _{\textrm {BBS}}(u) \propto \exp (j\omega _l u)$, with $\omega _l$ corresponding to the wavelength 1560.82 nm of the steepest edge of the BBS spectrum. The linear fit to this decreasing phase has been depicted with a gray line in Fig. 6(c). This change in the phase behaviour is a consequence of the asymptotic limit for large $|u|$ of the IFT that defines $\Gamma _{\textrm {BBS}}(u)$ in terms of the spectrum [42], which is determined by the different spectral density at the spectrum’s edges:

In a further set of experiments we explored the system’s ability to perform FT spectroscopy. The raw spectral resolution is $\delta \nu =1.21/(2u_{\textrm {max}})=11$ GHz (90 pm), which is lower by a factor of two with respect to previous demonstrations based on conventional amplitude modulation and the same RF bandwidth [37]. This value increases up to 14 GHz after windowing the interferogram with our default $\beta =3$ Kaiser-Bessel window, introduced to reduce the ripples around sharp spectral features.

We first analyzed, using scheme B in Fig. 1, a commercial FBG WDM array (FBGS) built in low bend loss fiber and composed of five weak uniform FBG with length 4 mm, FWHM of 0.18 nm, and reflectivities in the range 7.4% to 8.7%, starting at a wavelength of $\sim$1546 nm, increasing in nominal steps of 2 nm, and mutually separated by 1 m. The amplitude and phase of the complex interferogram is depicted in Fig. 7(a), and its FT spectrum plotted in blue in Fig. 7(b). In this last plot we also show with an orange trace its OSA spectrum. The interferogram is thus the amplitude and phase of five oscillations with approximately equal power and frequency spacing. When these frequencies are in phase, the interferogram’s amplitude shows clear peaks that allow for the observation of an envelope of triangular shape, extending up to the nominal length of the individual FBG. This triangular shape reflects the fact that, being weak, the FBG’s amplitude reflection coefficient $r(\omega )$ follows a sinc function, and so its reflectivity $R(\omega )=|r(\omega )|^2$ is a sinc$^2$. In the central peaks of Fig. 7(a), the phase returns to the initial zero value after periodic excursions, but the pattern is progressively lost due to the imbalance of reflected power and frequency spacing between reflectors. This experiment also permits the estimation of the displayed noise floor: the BBS used for illumination had an approximately flat spectral density of +14 dBm in 20 nm and so, and taking into account the FBG reflectivity, the noise floor of Fig. 7(b) is about $-$35 dBm/nm.

 

Fig. 7. (a) Amplitude and phase of the complex interferogram $\Gamma (u)$ of the WDM FBG array. A linear phase term accounting for the array’s spectral center has been added to help its visualization. (b) FT (blue) and OSA (orange) traces of the spectrum reflected in the WDM FBG array. The differences in the spectral peaks are mostly due to the non-uniform spectral density of the BBS used for illumination.

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We then performed some additional characterizations of our FT spectra. In Fig. 8(a) we show the spectra of monochromatic lines at 1540, 1545, 1555 and 1560 nm extracted from our TLS and directed to the first modulator with a power of 2 dBm. Here, the complex interferograms were windowed with a Kaiser-Bessel window with $\beta =6$ (spectral resolution of 19 GHz) to display the full dynamic range. The line at the normalization wavelength of 1550 nm represents the window used. In these measurements we observe a dynamic range of $\sim$23 dB and extracted a maximum variation in power of $0.6$ dB, mostly due to the non-uniform insertion loss of our CFBG. We also found a wavelength deviation with respect to the nominal TLS value $<$40 pm in all cases, of the same order of our previous analysis of the normalization trace in Fig. 4.

 

Fig. 8. (a) FT spectrum of laser lines at 1540, 1545, 1555, and 1560 nm. The line at 1550 nm represents the window used. (b) FT (blue) and OSA (orange) traces of the BBS spectrum reflected in a microscope slide.

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Second, we compared in Fig. 8(b) the reflected spectra of a 1.1-mm thick microscope slide, illuminated by our BBS and placed again in position B of Fig. 1, with an OSA measurement. The OSA trace, taken from the detector’s side, undergoes the same non-uniform CFBG loss as the FT spectrum, and therefore allows for a comparison of the relative precision in wavelength and power by comparing the multiple peaks. We found rms deviations of 0.3 dB in power and 30 pm in wavelength, slightly lower than our estimates due to environmental changes.

In a final example we performed an OLCR experiment targeted to the analysis of a commercial 1-mm long apodized uniform FBG written in SMF (Technica T10, peak wavelength 1550.20 nm, FWHM 1.6 nm, reflectivity 51%). The BBS was used to feed a composite fiber/free space interferometer whose arms are built from a dielectric mirror and the FBG, placed in the position of the DUT in scheme C of Fig. 1. The interferometer’s output field $\mathcal {E}_{\textrm {T}}(t)$ is the sum of the BBS field reflected from the mirror, $\mathcal {E}_{\textrm {BBS}}(t)$, and from the FBG, $\mathcal {E}_{\textrm {FBG}}(t)$, so that the autocorrelation $\Gamma _{\textrm {T}}(u)=\langle \mathcal {E}_{\textrm {T}}(t)^\ast \mathcal {E}_{\textrm {T}}(t+u)\rangle$ is given by:

The results are presented in Fig. 9. In Fig. 9(a) we depict both the FT and OSA spectra and in Fig. 9(b), with a blue trace, the inteferogram’s amplitude as a function of distance in fiber, $z=cu/(2n_g)$. The autocorrelation terms of this last trace were subtracted by performing two additional measurements, each with one of the interferometer’s arms blocked. After this subtraction, the cross-correlation amplitude is isolated as shown by the orange trace. The sub-micron resolution permits the observation of the standard structure of the apodized FBG’s impulse response [43–45], namely a main smooth peak due to the strong reflection of wavelengths close to the Bragg wavelength, followed by a secondary peak due to the more internal reflections of wavelengths far from the Bragg condition. Finally, we recall that the cross-correlation $\langle \mathcal {E}_{\textrm {BBS}}(t)^\ast \mathcal {E}_{\textrm {FBG}}(t-t_0+u)\rangle$ is proportional to the phase term describing the OPD difference between interferometer’s arms, so that the cross-correlation amplitude, in contrast to the phase, does not depend on the interferometer’s stability [26]. In our case, unfortunately, our external interferometer used to place the DUT was not sufficiently stable to provide a complete, vector characterization of the FBG in the time domain.

 

Fig. 9. (a) Reflected optical spectrum in the 1-mm, apodized FBG: blue, FT spectrum; orange, OSA spectrum. (b) Blue: amplitude of the low-coherence interferogram. Orange: cross-correlation term. The trace’s resolution $\delta z=c\delta u/(2n_g)$ is 550 nm in fiber.

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5. Conclusions

We have presented a simple and versatile phase sensitive LCI system built on MWP concepts, with C-band applications in spectral analysis and reflectometric characterization of components. The control of the interferogram’s phase is attained by a direct normalization of the MW amplitude and phase transfer in a dispersive MWP link based on electro-optic modulation and down-conversion. This architecture can be simplified further by operating a single modulator bidirectionally [46–48], so that the tasks of modulation and down-conversion are performed by the same device, or by using a frequency chirp instead of the sequential delivery of frequencies used here. The LCI system provides high resolution scans (5.45 fs in the time domain or 1.63 $\mu$m in free space) of a total temporal range of 54.5 ps, or 16.3 mm of OPD, range that can be increased by both dispersion and RF bandwidth. For OLCR experiments in fiber, the obtained one-point resolution is 0.55 $\mu$m in a range of 5.5 mm.

With respect to the spectral analysis, the present MWP LCI system shows a 14-GHz spectral resolution after windowing, a figure which is close to commercial FT spectrometers or grating-based dispersive OSA and can be increased by extending the temporal range $u_{\textrm {max}}$. The wavelength range, however, is limited by the CFBG’s reflection band. This shortcoming can be partially overcome by use of DDL based on dispersive fiber, since in that case the wavelength range could be extended to the operation range of the electro-optic modulators. This would require, however, HOD compensation and the monitoring of the dispersive delays used for the OPD scan. The moderate value of the obtained dynamic range ($\sim$23 dB) also deserves further study and, in particular, the limitations imposed by the CFBG’s phase ripple, which have not been addressed in the present investigation. The noise floor is expected to decrease by use of electro-optical heterodyne down-conversion [49], as similar approaches have demonstrated sensitivity values down to $-$90 dB in incoherent optical frequency-domain reflectometers (OFDR) [50]. Similar considerations apply to the comparison with conventional LCI and OLCR instruments, which outperform the values of OPD range and sensitivity presented here at the expense of overall system’s complexity.

In comparison with OFDR instruments devoted to the optical vector analysis of C-band components, systems which are based on the wavelength scan of a tunable laser at $\sim$pm steps [51], the present demonstration cannot compete in terms of spectral resolution, and also shows the limitation of dynamic range and sensitivity mentioned above. As a reflectometer, OFDR offers resolutions that can be as low as 5 $\mu$m [52,53] in ranges of tens of meters [51], ranges that are typically beyond the reach of low-coherence reflectometers. In this regard, the main advantage of the present approach, in addition to its simplicity, is its high spatial or temporal resolution. The analysis of interferograms with fs resolution and the sub-$\mu$m characterization of components represent examples of this advantage. The proposed LCI approach, in summary, constitutes a potentially compact and versatile technique for the high-resolution characterization of components and devices using MWP.