We propose and demonstrate an ultrahigh-speed optical frequency domain reflectometry (OFDR) system based on optical frequency-to-time conversion by pulse time stretching with a linearly chirped fiber Bragg grating (LCFG). This method will be referred to as OFDR based on real-time Fourier transformation (OFDR-RTFT). In this approach the frequency domain interference pattern, from which the desired axial depth profile is reconstructed, can be captured directly in the time-domain over the duration of a single stretched pulse, which translates into unprecedented axial line acquisition rates (as high as the input pulse repetition rate). We provide here a comprehensive, rigorous mathematical analysis of this new OFDR approach. In particular, we derive the main design equations of an OFDR-RTFT system in terms of its key performance parameters. Our analysis reveals the detrimental influence of nonlinear phase variations in the input optical pulse (including higher-order dispersion terms and group delay ripples introduced by the LCFG stretcher) on the system performance, e.g. achievable resolution. A simple and powerful method based on Hilbert transformation is successfully demonstrated to compensate for these detrimental phase distortions. We show that besides its potential to provide ultrahigh acquisition speeds (in the MHz range), LCFG-based OFDR-RTFT also offers the potential for performance advantages in terms of axial resolution, depth range and sensitivity. All these features make this approach particularly attractive for imaging applications based on optical coherence tomography (OCT). In our experiments, single-reflection depth profiles with nearly transform-limited ≈ 92.8 μm (average) axial resolutions over a remarkable 18 mm depth range have been obtained from OFDR-RTFT interferograms, each one measured over a time window of ≈50 ns at 20 MHz repetition rate. Improved sensitivities up to -61 dB have been achieved without using any balanced detection scheme.
©2007 Optical Society of America
Optical frequency-domain reflectometry (OFDR) has found many interesting applications in optical telecommunications, metrology and biomedical imaging [1–3]. OFDR is based on the measurement of the interference pattern from the sample to be characterized in the frequency (wavelength) domain, in contrast to optical low-coherence reflectometry (OLCR), where the interferometric signal is measured in the time domain, or equivalently in the delay length domain. In OFDR, the measured spectral-domain information is converted into the desired length-domain information using discrete Fourier transformations (DFTs) with digital signal processors.
A relevant application example of coherence reflectometry is for biomedical imaging, i.e. so-called optical coherence tomography (OCT) [4–14], in which a 2-D or 3-D scanning version of a reflectometry measurement system is implemented. Specifically, the use of OFDR has enabled high-resolution cross-sectional OCT imaging of biological tissues with faster acquisition rates and higher sensitivities than those achievable with OLCR techniques [5–8]. In conventional OFDR, the information acquisition rate is typically limited by the speed with which the interferometric signal spectrum can be recorded (using e.g. a spectrometer). There has been much recent effort to overcome this limitation, especially in the context of OCT applications, where faster image acquisition rates (i.e. faster A-line scan rates) are highly desired, e.g. to enhance the quality of the reconstructed 2-D and 3-D images by reducing motion artifacts. Specifically, it has been shown that increased imaging speeds can be achieved by use of high-speed, wavelength-swept lasers combined with OFDR [8–12]. Recently, a record axial frame rate of 370 kHz in OCT has been obtained by employing an advanced wavelength-swept laser technique, i.e. so-called buffered Fourier-domain mode-locking technique [11,12].
A new technique for ultrahigh-speed OFDR has been recently proposed [14,15]. This technique is based on the use of optical pulse stretching for mapping the spectral information into the time domain so that this information can be captured in real time. This frequency-to-time mapping is achieved by simply inducing a large amount of chromatic dispersion over the input optical pulse (so-called real-time Fourier transformation, RTFT [16–18]). This new OFDR approach, which will be referred to as OFDR based on RTFT (OFDR-RTFT), has been demonstrated using (i) a long section of dispersive optical fiber acting as the temporal pulse stretcher  (specifically referred to as stretched pulse – OCT, SP-OCT, technique) or (ii) a more compact linearly chirped fiber Bragg grating (LCFG) operated in reflection . OFDR-RTFT offers several key advantages for application in OCT systems. First, as mentioned above, the interferogram associated with the object depth profile is obtained along the duration of a single stretched pulse; as a result, the axial line (A-line) acquisition rate can be as high as the pulse repetition rate from the seeded optical source (up to a few tens of MHz). Second, only a completely passive and linear pulse stretching process (e.g. linear dispersion) is required as the wavelength sweep mechanism; this avoids the need for an active mechanical motion device for precise synchronous tuning of a narrowband filter. Third, the peak power of the pulse incident upon the sample is effectively reduced by the input fiber (or LCFG) dispersion, which is also a potentially attractive feature to prevent damage in the tissues or components under test. Finally, as it will be shown here, OFDR-RTFT has the potential to provide the level of performance that is desired for high-quality OCT imaging [8–12] also in terms of sensitivity, axial resolution and axial depth range. It should be mentioned that the SP-OCT system demonstrated in Ref.  already presented an unprecedented frequency sweep rate of 5MHz and an improved axial resolution of ≈ 8 μm (even though this resolution was far from the transform-limited resolution for this system, according to the used input supercontinuum pulse bandwidth > 200 nm). This system exhibited however a poorer performance than that of the currently employed OCT systems (e.g. wavelength-sweep OCT [8–12]) in terms of key parameters such as depth range and sensitivity.
In this report, we provide the first comprehensive, rigorous theoretical analysis of OFDR-RTFT. It should be noted that the previous analysis of SP-OCT has been based on assuming a simple frequency-to-time amplitude mapping . The analysis reported here provides a fundamental, deep understanding of the OFDR-RTFT technique and allows us to clearly establish the actual limitations and capabilities of this innovative technique. This analysis can be readily applied to define the system specifications so that to optimize the key performance parameters of the OFDR-RTFT method, including sensitivity, axial spatial resolution and axial depth range. For instance, our theory reveals that in contrast to conventional OFDR, the axial resolution in an LPFG system can be degraded by variations in the spectral phase of the input broadband optical source. This includes the spectral phase profile of the input optical pulse itself (if this is not a transform-limited pulse) as well as the spectral phase variations induced by (i) higher-order dispersion terms and group delay ripples of the used dispersive medium, e.g. LCFG, and (ii) dispersion imbalance in the fiber interferometric system. A simple but powerful technique based on Hilbert transform is applied here to compensate for this degradation in a practical system. This so-called Hilbert transformation compensation method (HTCM)  allows us to determine the precise numerical time-to-frequency mapping, including all the phase deviations mentioned above, to be applied on the measured time-domain spectral interferograms. This accurate time-to-frequency mapping can be recovered from a single interferogram measurement. The application of the HTCM is fundamental to optimize the performance of the OFDR technique, e.g. in terms of achievable axial resolution and axial depth range. We note that a somehow similar approach has been recently reported for nonlinear frequency sweep compensation in OCT systems ; this previous method is however based on Fourier transformation (instead of Hilbert transformation).
Based on the presented theoretical analysis, we experimentally demonstrate an optimized OFDR-RTFT system, which offers several distinctive features as compared with the previously reported SP-OCT system . First, nearly transform-limited spatial resolutions (average ≈ 92.8 μm) are achieved over the whole axial depth range, thus taking full advantage of the relatively narrow input pulse bandwidth used in our proof-of-concept experiments (SPM-broadened optical pulse bandwidth of ≈10 nm). Second, an unprecedented axial depth length > 18 mm is obtained (this compares very favorably with the axial depth range achieved in the previous SP-OCT system, which was limited to < 1 mm ). Third, the system sensitivity is also significantly improved up to ≈-61 dB (a sensitivity below -40dB was previously reported ).
Central to these key improvements in our OFDR-RTFT system is the use of a highly-dispersive LCFG instead of a long section of optical fiber. This translates into various important advantages. An LCFG allows stretching optical pulses with a much higher input peak power than in the equivalent optical fiber scheme. This is associated with the fact that an LCFG can provide a very high dispersion in significantly more compact forms than a conventional optical fiber (for instance, the dispersion introduced by the 10-meter long LCFG used in the experiments reported here is equivalent to that of 120 km of conventional SMF fiber). In a long dispersive optical fiber, the input pulse may need to be strongly attenuated to avoid the power leakage associated with nonlinear optical wave breaking as well as additional unwanted nonlinear effects. A high input peak power in this system is however desired to be able to improve both the system sensitivity and the axial depth range. As another advantage, a fiber Bragg grating (e.g. LCFG) can be specifically designed to achieve a desired group delay curve (e.g. linear group delay) over a pre-specified bandwidth; thus, these two specifications (dispersion and bandwidth) can be independently customized in the fiber grating device according to the targeted system requirements [16,17]. It is worth noting that our theoretical analysis has revealed that a high, linear pulse stretching rate is essential to achieve a uniform sensitivity and resolution over a longer depth range. An LCFG can provide the desired nearly linear group delay over a very broad bandwidth. Specifically, LCFG technology has evolved to the point that several meters long, high-quality gratings can be readily fabricated. This should easily allow scaling the technique for operation over input pulse bandwidths > 100 nm . In contrast, it is extremely difficult to obtain a linear group delay over such broad bandwidths using a long section of conventional SMF or dispersion-shifted optical fiber.
The remainder of this paper is structured as follows. The operation principle of the proposed reflectrometry scheme (OFDR-RTFT) is described in Section 2. Section 3 presents the detailed theoretical analysis of OFDR-RTFT, including the Hilbert-transform method for precise calibration of the required numerical frequency-to-time conversion law. The mathematical expressions governing the performance parameters (axial resolution, depth range and sensitivity) of an LPFG system are given in Section 4. Proof-of-concept experiments are presented and detailed discussed in Section 5. Finally, we summarize and conclude our work in Section 6.
2. Basic operation principle of OFDR-RTFT
A schematic of OFDR-RTFT is shown in Fig. 1. It consists of three main sub-systems: a coherent pulse source with a broad spectral bandwidth, an LCFG-based optical pulse stretching system, and a fiber-optic interferometer including ultrafast sampling electronics. The key feature is to exploit the light coherence along the duration of an optical pulse after being temporally stretched by a highly-dispersive LCFG operated in reflection (notice that the LCFG needs to be incorporated in an optical circulator, OC, to retrieve the reflected signal). This temporal stretching process induces a linear mapping of the input complex spectrum into the time coordinate (real-time optical Fourier transformation) [16,17], which in turn allows us to capture the conventional OFDR spectral interferogram directly in the time domain within the stretched pulse duration. The sample’s axial profile can then be recovered from this measured interferogram using a simple DFT-based signal processing algorithm .
In our implementation, the optical pulses directly generated from a fiber mode-locked laser were spectrally broadened by self-phase-modulation (SPM) in a highly-nonlinear (HNL) optical fiber section, with the aim of improving the system axial resolution. The SPM-broadened pulses exhibited a highly stable performance with extremely low relative intensity noise (RIN) characteristics. However, the induced nonlinear instantaneous phase in the SPM-broadened pulses together with the higher-order dispersion terms and group delay ripples introduced by the LCFG are expected to degrade the axial resolution considerably. As mentioned above, in our work, this degradation is numerically compensated by use of an accurate high-resolution numerical time-to-frequency mapping process, which is determined by means of the so-called Hilbert transformation compensation method (HTCM) . In this method, a single reflection point measurement is required to determine the precise numerical time-to-frequency mapping to be applied in a given reflectometry configuration. Moreover, the HTCM method is not only useful for accurately determining the LCFG-induced dispersion and nonlinear chirp induced by the SPM process, but it can be also used for simultaneously compensating for the dispersion imbalance between the interferometer arms . It is important to note that for optimal depth profile recovery (in terms of axial resolution and sensitivity), the spectral phase of the input pulse should be single-valued at each wavelength; this is not the case when the optical pulse is broken into several individual temporal components, for instance as a result of strong nonlinear processes induced by propagation through an optical fiber (e.g. in the case of supercontinuum generation ).
3. Theoretical analysis of OFDR-RTFT
Figure 2 shows a schematic of the proposed OFDR-RTFT technique, presenting the evolution of the involved time waveforms and corresponding spectra for each of its three main stages (according to the experimental set-up in Fig. 1). A detailed explanation of this schematic is given in what follows.
We define the input pulse source centered at the angular frequency ω 0 as an electromagnetic wave, a 0, with a complex slowly varying envelope, â 0 as shown in Fig.2(a) i.e. a 0 (t) = â 0 (t) ∙ exp(jω 0 t). In practice, an LCFG operated in reflection can provide a flat spectral response and a (nearly) linear group delay over the input pulse spectral bandwidth [23, 24]. Specifically, the spectral transfer function of the LCFG is assumed to be centered at ω 0 and can be mathematically described using the Taylor series expansion at ω 0, 
where Φ0=Φ(ω 0) is a phase constant, Φ0 = [∂Φ(ω)/∂ω)ω=ω0 is the group delay, 0 = [∂ 2Φ(ω)/∂ω 2]ω=ω0 is the first-order dispersion coefficient and δΦ is the phase deviation (including higher-order dispersion terms and group-delay ripples of the LCFG). An LCFG is predominantly a first-order dispersive element. The spectral phase of an ideal LCFG is shown in Fig. 2(b). As a result, the phase deviation δΦ is expected to be much smaller than the phase component of the main transfer function Ĥ́(ω) (transfer function of an ideal first-order dispersive element). Notice that in the above notation ω= ωopt - ω 0, where ωopt is the optical frequency variable and ω is the baseband frequency variable. For the sake of simplicity, the amplitude ripples on the spectral transmission of the LCFG has not been considered in the theoretical analysis. It is well known that the impulse response corresponding to the transfer function Ĥ́(ω) can be written as 
where tR = t - Φ0 and . In this time-invariant linear system, the spectrum of the reflected wave, Â 2 (ω), and that of the input wave Â 0 (ω) (which is assumed to be spectrally narrower than the LCFG reflection bandwidth) can be related as follows:
where the phase deviation of the LCFG from its ideal characteristics is included in the “intermediate” pulse spectrum Â 1(ω) = Â 0(ω) ∙ exp(jδΦ) = |Â 0(ω)| ∙ exp[j(ϕ + δΦ)], ϕ(ω) being the spectral phase profile of the input optical pulse â 0. The corresponding time-domain expression of Eq. (3) is as follows: â 2(tR) = â 1(tR)*ĥ(tR), where â 1 and â 2 are the inverse Fourier transforms of Â 1 and Â 2, respectively. Introducing Eq. (2) into this last expression, the reflected pulse can be written in the following integral form:
where Δt is the time window limiting the duration of â 1(t). Assuming that the LCFG phase deviation δΦ is sufficiently small, the time duration of â 1(t) should be similar to that of the input optical pulse â 0. In fact, if this time duration is sufficiently short such that the following condition is satisfied 
then the reflected pulse can be approximated from Eq. (4) by the following expression:
where ℑ denotes the Fourier transform and ώ = tR/ 0 is the transformed frequency variable, which is scaled by the first-order dispersion term (frequency-to-time conversion ratio). Thus, Eq. (6) indicates that under the conditions of Eq. (5), the amplitude spectrum of the input optical pulse is efficiently mapped into the time domain (i.e. the output pulse time intensity is directly proportional to the input pulse power spectrum, |â 2 (tR)|2 ∝ |Â 1 ( )|2 ∝ |Â 1 ( )|2) as shown in Fig 2(b). This operation is usually referred to as RTFT and can be interpreted as the time-domain equivalent of Fraunhofer spatial diffraction [16,17].
Similarly, the reflection (and/or scattering) of the stretched optical pulse â 2(t) from a sample is assumed to be approximated by the convolution of this pulse with the characteristic time-flying impulse response, f(tR), of the sample in the moving time frame, tR. Notice that the function f(tR) is proportional to the desired sample (amplitude and phase) depth profile with a finite frequency bandwidth restricted by the bandwidth of the input light source. In the frequency domain, this relation can be written as
where F̂ is the spectral transfer function associated with the sample impulse response. A similar approach was previously used to model the problem of light scattering in conventional OFDR . The temporal representation of Eq. (7) can be similarly approximated as done in Eqs. (4)–(6) by
The above equation implies that the sample transfer function is replicated along the duration of the stretched pulse when this pulse is scattered and/or reflected from the sample. In other words, the sample spectral transfer function (in amplitude) can be directly captured in the temporal domain by simply measuring the time intensity envelope of the reflected/scattered pulse using a high-speed photodetector. This is so because the time intensity envelope of the reflected/scattered pulse is directly proportional to its optical power spectrum, i.e. |âs(tR)|2 ∝ ∙|Â 1 ( )|2 ∙|F̂( )|2. However, it is well known that this measurement only allows one to recover the sample power spectrum, i.e. it provides information only about the autocorrelation of the sample impulse response but does not allow one to fully reconstruct the sample depth profile (f̂(tR)). As previously discussed in detail (see for instance Ref. [2,4]), this can be solved by employing coherent interference with a reference pulse, i.e. a delayed version of the stretched optical pulse, â 2(tR - 2δt), where 2δt is the relative time delay in a Michelson interferometer (see Fig. 1). Using Eq. (6) and Eq. (8), the signal resulting from the interference of âs (tR) and â 2 (tR - 2δt) can be written as
where Ω = 2δt/ 0 is the frequency shearing induced by the time delay, 2δt, in the frequency-to-time converted coordinate. In this derivation, the same Fraunhofer approximation as that in Eq. (5) has been used, i.e. δt 2/(2 0)≪1. Thus, the corresponding temporal intensity (measured with a high-speed photodetector) is given by the following expression:
where we recall that ϕ represents the spectral phase profile of the input optical pulse â 0. The first term, IDC, on the right-hand-side of the above equation is considered as a DC term composed by two contributions, namely the power spectrum of the optical pulse reflected/scattered from the sample and the spectrally sheared reference spectrum. The second and the third terms, IAC and IAC * are the interference terms from which one can directly reconstruct the complex impulse response of the sample under test (i.e. depth complex profile) by use of the DFT.
In order to achieve an optimal reconstruction of the sample impulse response, (i) the spectral amplitude |Â 0(ώ)||â 0(ώ - Ω)| must be properly scaled out (this factor is known “a priori” from the direct measurement of the input pulse power spectrum) and (ii) the differential spectral phase factor Δϕ(ώ) must be properly compensated for. We recall that this detrimental phase factor includes the nonlinear spectral phase profile of the input optical pulse (before temporal stretching) as well as the phase deviations induced by higher-order dispersion terms and group delay ripples in the LCFG spectral response. A schematic of the recorded interferogram in the OFDR-RTFT technique is shown in Fig. 2(c). As evidenced by Eq. (10), if uncompensated, this differential phase factor may significantly affect the axial resolution of the recovered sample depth profile. In other words, the optimal (transform-limited) axial resolution in the OFDR-RTFT system can be achieved only if the intrinsic phase distortions in the system are properly compensated for. It is important to mention that in the case of conventional OFDR, the axial resolution is not affected by a potential nonlinear spectral phase profile in the input optical pulse since the related phase variation is not present in the measured interferogram. In wavelength-swept OFDR, the axial resolution may be however degraded by instabilities in the wavelength tuning process of the input source.
3.2. Numerical time-to-frequency conversion: HTCM
Here, we use a simple and direct numerical technique for compensating for the above mentioned phase distortions which is based on pre-determining the precise time-to-frequency mapping to be applied over the measured time-domain spectral interferograms (this time-to-frequency mapping is properly calibrated “a priori” to correct the detrimental phase distortions in the system). The core of this technique is to obtain the differential spectral phase factor, Δϕ(ώ), by simply acquiring the interference pattern |âd(tR)|2 in Eq. (10) from a single reflection point, i.e. when F̂(ώ) = cons tan t, and applying the so-called Hilbert transformation compensation method (HTCM) . In this method, the complex AC interference component, IAC, can be obtained from the measured interference pattern using the following procedure. First, the real part of the AC interference component can be directly extracted from the measured interference pattern, IAC +IAC * = real(IAC), by numerically subtracting the DC component. In practice, this subtraction can be easily performed by numerically filtering the Fourier transform of the measured temporal interference pattern with a narrow bandpass filter.
In order to clearly screen out IDC using this filtering method, the shearing frequency Ω has to be higher than the spectral bandwidth of the stretched pulse envelope. Alternatively, the real part of the AC interference component could be directly obtained in the experiment using a balanced photodetector (if available) . In any case, the extracted interference pattern, real(IAC), is a truly sinusoidal function, which is apodized by the input pulse spectral envelope and whose phase variation is given by the differential spectral phase factor Δϕ(ώ) (see Eq.(10) with F̂(ώ) = constant), and is thus ideally suited for application of the Hilbert transformation method. Specifically, the phase factor Δϕ(ώ) can be easily obtained from the extracted interference pattern as the real and imaginary parts of the complex AC pattern are related by the Hilbert transform; in particular :
where ℑ-1 denotes the inverse Fourier transformation, H denotes the Hilbert transformation, Θ is a suitable window function (used to screen out the term real(IAC) by filtering in the Fourier domain) and the variable ξ, is the Fourier transform counterpart of the variable of ω’. From the calculated differential phase factor ΔΦ, the numerical time-to-frequency mapping to be applied over the measured interference patterns can be precisely recalibrated as follows:
The reader can easily prove that by applying this variable change in Eq. (10), the detrimental phase factor in the AC interference pattern (from which the depth profile will be recovered) can be fully compensated. Notice that in practice, the presented compensation method can be used for simultaneously correcting the phase distortions induced by dispersion imbalance in the fiber interferometer (for the sake of simplicity, this additional phase factor was not considered in the above mathematical derivations). We emphasize that the precise time-to-frequency recalibration given by Eq. (12) needs to be determined only from a single interferogram measurement (using a single reflection point); in other words, this recalibration process does not need to be performed for each interferogram acquisition. This is due to the fact that (i) the used pulse stretching system is almost an entirely passive, linear process (only negligible nonlinear effects may be induced by the LCFG), and (ii) the SPM-broadened input optical pulse is highly stable with very low RIN noise. It should be mentioned that a similar Hilbert transform – based technique has been previously used for compensating the interferometer dispersion imbalance and dispersion induced by the sample under test in a Fourier-domain OCT system ; in this previous work, an iterative numerical procedure was used to improve the sharpness of the obtained sample image. We anticipate that a similar iterative process may be required in OFDR-RTFT for OCT applications if the sample-induced dispersion needs to be compensated for as well.
4. Performance evaluation of OFDR-RTFT
It has been demonstrated that OFDR-RTFT [14,15] allows increasing the axial frame acquisition rate by more than one order of magnitude as compared with previous ultrahigh-speed OFDR approaches (e.g. wavelength-swept OCT systems [7,8,11,12]). In what follows we evaluate other key performance parameters of the OFDR-RTFT technique and demonstrate that this simple technique has the potential to provide a similar or improved performance than that typically achieved with previous OFDR approaches also in terms of resolution, depth range and sensitivity.
4.1. Axial resolution
As in conventional OFDR (including Fourier-domain OCT), the axial resolution that can be achieved with OFDR-RTFT essentially depends on the input pulse optical bandwidth and central wavelength . Assuming that the optical bandwidth ΔλS [nm] of the input pulse (e.g. SPM-broadened pulse in our specific implementation) is narrower than the LCFG reflection bandwidth, the optimal (transform-limited) axial resolution can be estimated as 
where λ 0 is the center wavelength of the pulse source and here, the pulse spectrum is assumed to be Gaussian-like.
It is important to note that the LCFG dispersion and input pulse bandwidth must be properly designed to ensure that the stretched individual pulse at the LCFG output is not longer than the input repetition period TR (to avoid overlapping among the individual optical pulses); mathematically, | λ|∙ΔλS < TR, where λ = -(2πc/λ02) 0 is the LCFG first-order dispersion in [ps/nm] units. We recall that an LCFG is advantageous in that it can be specifically designed to achieve a desired dispersion over a prescribed bandwidth. For instance, the LCFG reflection bandwidth ΔλLCFG can be designed to ensure that | λ|∙ΔλLCFG < TR, thus avoiding temporal overlapping of the stretched optical pulses regardless of the input optical bandwidth. In fact, if the input optical bandwidth is larger than the LCFG reflection bandwidth, then the system axial resolution will be limited by the LCFG bandwidth. A more accurate estimation of the corresponding axial resolution in this case is given by the following expression [27,28]:
which is obtained by assuming that the spectral transmission profile after the LCFG is approximately rectangular. We recall that several meters long LCFGs able to provide reflection bandwidths well above 100nm can be currently fabricated ; this implies that the OFDR-RTFT technique proposed here could be scaled to provide axial spatial resolutions at least in the micron regime.
4.2. Axial depth range
As discussed in previous works, the free-space depth range of an OFDR system essentially depends on the resolvable spectral line-width in the system ,  (a longer depth range can be achieved by narrowing the resolvable spectral line-width). In what concerns an LPFG system, this spectral line-width essentially depends on the dispersion amount introduced by the LCFG pulse stretching (i.e. stretching rate) and may be limited by either the RTFT operation itself or by the photodetector bandwidth.
For a sufficiently high LCFG dispersion, the depth range is limited by the RTFT process. Let us assume two adjacent constant-phase Gaussian-like narrow spectral features each with a FWHM bandwidth of δλ, i.e. these two features are assumed to be spectrally separated by exactly δλ. The FWHM time width of the transform-limited Gaussian pulse corresponding to each of these features is given by 
The two assumed spectral features can be resolved in the temporal domain after stretching only if this stretching process induces a temporal separation between the two corresponding temporal pulses (each associated with each of the Gaussian spectral features) longer than TFWHM; mathematically | λ|∙δλ≥TFWHM. The minimum spectral line-width δλ that can be resolved in the time domain by the implemented RTFT process can be obtained by introducing Eq. (15) into this last inequality and is given by the following expression:
According to inequalities (16) and (17), a narrower spectral line-width can be resolved (i.e. a longer depth range can be achieved) by simply using a higher amount of dispersion in the pulse stretching system.
However, if the LCFG dispersion is not sufficiently high, then the temporal impulse response of the used high-speed photodetector (estimated as 1/BW, where BW is the detector bandwidth in Hertz) may be longer than the time width corresponding to the minimum resolvable spectral-line. In this case, the free-space depth range will be limited by the photodetector bandwidth, according to the following approximate expression:
Thus, the depth range in a low stretching rate system is limited by the bandwidth of the photodetector and is directly proportional to the stretching rate (LCFG dispersion), whereas the depth range in a high stretching rate system is proportional to the square root of the introduced amount of dispersion. In any case, the axial depth range can be increased by increasing the dispersion introduced by the pulse stretching system (e.g. LCFG). We reiterate that an LCFG can be designed to achieve the desired dispersion amount according to the targeted system specifications.
The equations derived above reveal that in an OFDR-RTFT system there is a fundamental tradeoff between the achievable axial resolution and depth range. Basically, a longer depth range can be obtained by use of a higher amount of dispersion. However, the higher the dispersion introduced by the pulse stretching system, the narrower the input pulse bandwidth must be in order to avoid temporal overlapping among the stretched optical pulses (according to the input pulse repetition rate); a narrower input pulse bandwidth necessarily implies a poorer spatial resolution. Nonetheless, based on the equations given above, we estimate that an OFDR-RTFT system could be designed to simultaneously achieve micron resolutions and depth ranges > 10 mm while keeping an acquisition frame rate in the MHz regime. As a basic design example, a system with 11 μm resolution and 12 mm depth range at > 4 MHz could be accomplished using the following system parameters: 2 ns/nm dispersion, 15 GHz detector bandwidth, > 100 nm source bandwidth, and 1300 nm center wavelength of the source at > 4 MHz repetition rate. This designed system performance would be superior, especially in terms of axial frame rate and depth range, to that achievable with current wavelength-swept OCT, for which the reported record values are 290 kHz and 7 mm , respectively. In our proof-of-concept experiments, a remarkable free-space depth range > 18 mm with ~3 dB sensitivity reduction (limited by the LCFG dispersion) was achieved using an LCFG dispersion of λ ≈ 2 ns/nm at 1545 nm.
The shot-noise limited sensitivity in an OFDR-RTFT can be estimated as in a wavelength-sweep OFDR system . Specifically, we estimated a shot-noise limited sensitivity of ≈-90~95 dB in our experimental system using Eq. 11 from Ref. . As discussed in previous works, this sensitivity could be achieved in practice only if a balanced detection technique is employed. However, in the experimental results reported here, balanced detection was not used and a maximum reflectivity sensitivity of ≈-61 dB was achieved. This indicates that (i) other noise sources (e.g. relative intensity noise) should be considered to account for the difference with respect to the shot-noise limited sensitivity, and (ii) a high-frequency balanced detection should be used to further improving the system sensitivity.
It is important to point out that the sensitivity performance strongly depends onthe group-delay slope and linearity of the LCFG. As discussed in Ref.  for a wavelength-swept OCT system, the sensitivity is inversely proportional to the detector bandwidth and as a result, the optimum detection bandwidth will depend on the wavelength tuning rate (the detection bandwidth should be as narrow as possible still to be able to follow the pre-defined tuning rate). In our approach, the optimum detection bandwidth can be similarly approximated by BW ≈ 0.4Ns/( λΔλLCFG), where Ns is the sampling number, i.e. the optimum bandwidth is inversely proportional to the product dispersion-bandwidth of the LCFG, which in turns is inversely proportional to the maximum tuning rate in the OFDR system. In other words, as expected for a wavelength-swept OCT system, the optimum bandwidth is directly proportional to the wavelength tuning rate. Notice that this expression is derived from a direct application of the Nyquist criterion, assuming that the sampling bandwidth is 2.5 times wider than the detector bandwidth. From the detector BW estimation given above, it can be also inferred that the sensitivity variation as a function of the axial depth should follow the detector frequency response (responsivity) because the modulation frequency at the detector is linearly proportional to the axial depth, according to the stretched pulse temporal chirp (time-to-frequency mapping). Moreover, a constant dispersion factor is required over the operation bandwidth in order to ensure uniform sensitivity over the axial depth range (in other words, variations in the group-delay slope of the dispersive device along the operation bandwidth will translate into sensitivity variations along the depth range). While in a conventional optical fiber it is very difficult to ensure a constant dispersion factor (group delay slope) over a broad bandwidth, we emphasize that this uniformity can be easily achieved in a LCFG. As a result, the use of a LCFG in the OFDR-RTFT system allows achieving a very uniform performance of the OFDR system along the whole depth range in terms of sensitivity and resolution. These issues are experimentally investigated in the next section.
5. Experiments and discussion
For the proof-of-concept experimental demonstration, we measured single reflection depth profiles at different displacements from interferograms acquired by conventional sampling electronics. We note that in our experiments, the rate at which the axial-line (A-line) was captured and processed was essentially limited by the available sampling electronics; specifically, the A-line rate was significantly slower than the optical repetition rate of the interferograms (fixed by the input pulse repetition rate). In a practical system, the sampling electronics would need to be upgraded to fully exploit the capabilities of OFDR-RTFT and to ensure a truly real-time A-line acquisition equal to the input pulse repetition rate. The sampling electronics in the OFDR-RTFT demonstrated in Ref.  allowed for real-time A-line acquisition at 5-MHz (input pulse repetition rate) within a limited measurement time.
The input pulse in our experiments was a 700-fs/3.5-nm (FWHM) nearly transform-limited Gaussian pulse with ~2 mW average power, directly generated from a passively mode-locked wavelength-tunable fiber laser with 20 MHz repetition rate (Pritel Inc). The input pulse probe was seeded into a nonlinear fiber (Pulse Compressor, Pritel Inc.) in order to induce a spectral broadening by SPM. The pulse spectrum bandwidth was broadened to 9.2 nm (FWHM). Fig. 3 shows (a) the measured spectrum of the SPM-broadened pulse and (b) its autocorrelation trace, as measured by an optical autocorrelator.
The pulse spectrum in this experiment lied within the reflection bandwidth of the LCFG (45nm) and as a result, the image axial resolution was determined by the pulse spectrum bandwidth. Specifically, assuming a Gaussian input pulse shape with a FWHM bandwidth of 9.2 nm, the optimal (transform-limited) axial resolution is estimated to be around 110 μm (from Eq. (13)). The measured autocorrelation exhibited a well-confined pulse envelope with small sidelobes, which confirmed that the SPM-broadened pulse was not broken into several individual optical components. We remind the reader that optical pulse breaking would result into a double-valued time-to-frequency mapping, which in turn would translate into a significant degradation of the system performance, especially in terms of sensitivity and resolution.
In order to temporally stretch the SPM-broadened pulse, a 10-meter LCFG with ≈+2000ps/nm dispersion slope, 42 nm reflection bandwidth (centered at 1545 nm), and 3dB insertion loss (Proximion Inc.) was used. This commercial LCFG was designed for its application as a dispersion compensator in a wavelength-division-multiplexed fiber optics link. In particular, this LCFG can compensate a conventional SMF section of ≈ 120 km over a total bandwidth of 42-nm if its negative dispersion slope is used. However, the used LCFG was not specifically optimized for its application in an OFDR-RTFT system such as that demonstrated in this paper. For instance, the LCFG dispersion-bandwidth product is larger than the repetition period of our input pulse source (50-ns) and as a result only a fraction of the total LCFG reflection bandwidth could be used in our experiments (i.e. SPM-broadened spectrum shown in Fig. 3(a)), with the associated degradation in the obtained axial spatial resolution. Each individual SPM-broadened optical pulse was temporally stretched by the LCFG to around 40 ns. The measured temporal envelope of a single stretched pulse closely resembled the SPM-broadened pulse spectrum, thus confirming the LCFG-induced linear frequency-to-time mapping. After amplification of the stretched pulse, its time-averaged power was ~ 8.5 mW. The “time-domain” OFDR interferograms from single reflection were obtained with a fiber-optic Michelson interferometer and were acquired by an optical sampling module (detector) with 8 GHz 3dB-bandwidth (Tektronix 80C01) and a communication signal analyzer (Tektronix CSA8000).
Figure 4(a) shows a sequence of sampled interferograms corresponding to a single reflection from a broadband mirror whose reflectivity including the insertion loss from the sample arm is expected to be better than -3.8 dB. This confirms that each interferogram is generated along the duration of each stretched optical pulse (i.e. over a window of ≈ 50ns) with the same repetition rate as that of the input pulses (@20 MHz). A single interferogram for the axial image reconstruction was acquired in a single shot using the proper sampling mode of the oscilloscope, see Fig. 4(b). This interferogram consisted of 4000 data points over a 50 ns time window (12.5 ps sampling resolution). The close-up view of the interferogram, shown in the inset of this figure, exhibits a clear sinusoidal modulation associated with single reflection interference.
The instantaneous chirp of the stretched pulse, including the SPM-induced phase variations, higher-order dispersion terms and group delay ripples of the LCFG and the dispersion imbalance between the interferometer arms, was reconstructed using the HTCM described in Section 3.2. The specific interferogram used for HTCM was acquired in the frame-averaging mode using 50 multiple acquisitions, with the aim of reducing the noise in the interference pattern. Note that the averaged acquisition is only needed for the preliminary calibration of the time-to-frequency mapping using HTCM but is not necessary for capturing the interferogram(s) from which the sample depth profile(s) can be recovered (these interferograms can be captured in the single-shot mode). The interference term, real (IAC) in Eq. (10) is numerically extracted from the measured interferogram (shown in the top-right inset of Fig. 5) by application of the HTCM described in Eq. (11). The reconstructed pulse instantaneous frequency was predominantly linear and is shown in Fig. 5. The higher-order chirp of this function, which was obtained by subtracting the predominant linear term from the total instantaneous frequency, is shown in the bottom-left inset of Fig. 5. The frequency tuning rate of the OFDR-RTFT was calculated to be 62.3 THz/μs from the slope of the recovered instantaneous chirp in Fig. 5; this tuning rate is approximately 10 times faster than that of the Fourier domain mode locking swept laser source demonstrated in Ref.  (~6.8 THz/μs, as estimated from the laser swept rate and wavelength range) as well as much faster than that typically achieved in conventional OFDR [1–3].
For each measurement, the numerical time-to-frequency conversion was performed by interpolating the measured interferogram with the mapping given by the reconstructed pulse instantaneous chirp (frequency-map (F-map), shown in Fig. 5), according to Eq. (12). The spectral interference pattern so recovered was first numerically normalized in amplitude by dividing by the known pulse spectrum amplitude after reflection in the LCFG (i.e. including the LCFG amplitude ripples), following that a numerical spectral amplitude reshaping with a Hanning function was applied. The re-shaped interference pattern was then processed using the DFT algorithm for reconstructing the sample depth profile . Notice that the pulse spectral amplitude was acquired by grabbing the temporal envelope of the stretched pulse in the reference arm using the sampling oscilloscope. Because of the numerical spectral reshaping with a Hanning function, the effective spectral bandwidth after this numerical treatment was slightly wider than the input source bandwidth. This translated into slightly improved axial resolutions than that expected from the input pulse bandwidth for a high reflective sample (following estimations according to Eq. 13). Figure 6 shows the reconstructed depth profiles corresponding to a single reflection at 5.37 mm, both when the precise F-map from HTCM was used (blue curves) and when a direct linear time-to-frequency mapping according to the LCFG first-order dispersion was applied (red lines). It is clearly observed that there is significant improvement in the image resolution compared to the uncalibrated profile. In fact, the measured resolution is very close to the expected optimal (transform-limited) resolution according to the pulse optical bandwidth (after the spectral Hanning windowing discussed above). To be more concrete, The FWHM width for the blue line corresponding to a single reflection was measured to vary between 87 μm to 106 μm over the whole axial depth range (resolution of ≈ 98 μm for the single reflection at 5.37 mm shown in Fig. 6). In order to confirm that the reconstructed profile was almost transform-limited (in terms of resolution), the profile at 0 mm which corresponds to the Fourier transform of IDC in Eq. (10) is also shown in the inset of Fig. 6. The FWHM width of this profile is measured to be 86 μm.
The log scale of the reflectivity is shown in Fig. 7 (blue line), in this plot, the peak value is properly scaled to coincide with the sample reflection (-3.8 dB) and dB quotes 20log[scaled DFT amplitude]. The reflectivity sensitivity was calculated as the ratio between the reflectivity peak and the standard deviation over the mean value of the noise floor within the range of ±2mm from the peak measured when the sample arm was blocked . The maximum sensitivity in our experiment was determined to be ≈-61 dB, see Fig. 7. We emphasize that this sensitivity was achieved when considering the LCFG amplitude ripple in the numerical amplitude re-shaping process of the captured interferogram (see details above). We have observed that when the LCFG amplitude ripple correction is not applied, the floor noise level of the source and side-lobes in the recovered single-reflection depth profiles are slightly increased This sensitivity degradation has been experimentally investigated and we believe it can be attributed to the fixed pattern noise on the stretched pulse caused by the LCFG amplitude ripples which are broadly distributed from 0 GHz to 8 GHz (see Fig. 8(a)). The fixed pattern noise was experimentally confirmed by performing a persistent display mode using the sampling oscilloscope; in our experiments 1000 frames were acquired, showing very small timing errors. The acquired waveform is shown in Fig. 8 (b). The averaged DFT for each pulse intensity measurement is also shown in the inset of Fig. 8(b) confirming that the fixed noise pattern was not suppressed by the averaging. Based on this finding, we anticipated that the system sensitivity could be further improved by properly scaling the fixed amplitude ripple patterns. This is better illustrated by the results presented in Fig. 8(c)–(d), where the recovered depth profiles of a single reflection at 3 mm are shown for the case (c) when no LCFG amplitude ripple correction is used (red line) and (d) when the LCFG amplitude ripple correction is applied (blue curve). In particular, the amplitude ripple correction translates into a fairly apparent decrease of the noise floor level of ≈-2 dB, and a significant side-lobes suppression by ≈-6.5 dB for the primary side-lobe and ≈-10.6 dB for the secondary side-lobe.
In order to evaluate the system performance uniformity in terms of resolution, dynamic range, and sensitivity over the whole depth range, the single-reflection sample has been displaced with a 2 mm moving step up to the travel-range limit of our system (18 mm), which is slightly longer than the maximum depth range theoretically predicted for our system (depth range limited by the LCFG dispersion rate to ≈14.3 mm, according to Eq. (17)). The mirror travel-range limit of 18 mm corresponds to a modulation frequency of up to 7.8 GHz at the detector (according to the stretched pulse chirp). Fig. 9(a) shows the A-line profiles with respect to the depth. The resolution varied from 87 μm (at z = 0mm) to 106 μm (at z = 18mm) but it is still within the theoretically estimated resolution assuming a Gaussian spectrum. The dynamic range, which is defined as the ratio of the peak reflectivity to the primary side-lobe peak , was reduced from 25 dB (at z = 0mm) to 15 dB (at z = 18mm), as shown in Fig. 9(b). This degradation is also visible in the A-line consecutive profiles shown in Fig. 9(a). Sensitivity variation ranged from -57 dB to -60 dB. These measurements confirmed that the axial resolution and system sensitivity exhibited a fairly uniform performance over the whole available depth range. We emphasize that this experimentally demonstrated uniformity in the behavior of our OFDR system is mostly associated with the fact that the LCFG group-delay slope is nearly constant over the whole operation bandwidth; this explains in part the performance uniformity degradation observed in an OFDR-RTFT system based on the use of conventional optical fiber .
A recently introduced technique for ultrahigh-speed OFDR, namely OFDR based on real-time Fourier transformation (OFDR-RTFT), has been theoretically and experimentally investigated in detail. This method is based on frequency-to-time conversion of the spectral interferograms using a passive, linear optical pulse stretcher (highly-dispersive medium). In this way, each frequency-domain interference pattern, from which the desired axial depth profile is reconstructed, can be captured directly in the time-domain over the duration of a single stretched pulse; as a result the A-line acquisition rate can be as high as the pulse repetition rate from the input pulsed source. An unprecedented A-line acquisition speed of 5-MHz has been previously demonstrated using this technique . Here, we have provided, for the first time to our knowledge, a rigorous theoretical analysis of OFDR-RTFT. The main design equations of OFDR-RTFT, specifically in terms of its key performance parameters at the system level (A-line rate, axial resolution, depth range and sensitivity), have been derived, evaluated and experimentally confirmed. Moreover, a simple Hilbert-transform based method has been demonstrated for compensating for the detrimental phase variations in the stretched pulse (associated with the nonlinear phase of the input pulse before stretching as well as with phase deviations in the linear dispersive medium) and those induced by dispersion imbalance in the fiber optics interferometer system. The theory introduced here is essential to be able to fully optimize the performance of an OFDR-RTFT system and also provides a complete understanding of the ultimate limitations and capabilities of this novel OFDR approach.
Based on our theoretical analysis, we have identified an optimal linear optical pulse stretcher for OFDR-RTFT, namely a compact LCFG. In the experiments reported here, single reflection depth profiles with optimal (transform-limited) 92.8 μm (average) spatial resolutions over a remarkably long 18 mm depth range have been obtained from OFDR-RTFT interferograms, each one measured directly in the time domain over the duration of a single stretched pulse (≈ 50ns). Here, the interferograms were captured using a single-shot acquisition in a sampling oscilloscope, which is not actually a real-time measurement. The demonstrated OFDR schematic offers however the potential for an axial-line rate up to 20-MHz (depending on the available sampling electronics, e.g. using a so-called real-time oscilloscope). Moreover, an improved sensitivity has been reported compared to the previously demonstrated OFDR-RTFT systems (SP-OCT), in part due to the use of an LCFG stretcher (instead of a long optical fiber).. We experimentally obtained a sensitivity up to -61 dB without using any balanced detection scheme. System performance variation over the A-line depth range in terms of resolution, dynamic range, and sensitivity has been also investigated and a fairly good performance uniformity has been demonstrated.
The unique features of LCFG-based OFDR-RTFT make this approach potentially attractive for imaging applications where ultrahigh acquisition speeds, high resolutions, long depth ranges and/or high sensitivities are required. Thus, we believe that this novel OFDR technique should prove very useful for ultrahigh-speed 2-D and 3-D biomedical image reconstruction using OCT, phase-contrast microscopy, and optical device characterization.
The authors would like to thank the anonymous reviewers for their invaluable comments and suggestions which have been extremely useful to improve the quality and presentation of the reported work. This work was supported in part by the Canadian Institute for Photonic Innovations (CIPI), by the Fonds Québécois de la Recherche sur la Nature et des Technologies (FQRNT), and by the Ministry of Education and Human Resources Development (MOEHRD), Korea through the Korea Research Foundation Grant, (KRF-2006-C00026).
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