## Abstract

The inclusion of a linear chirped fiber Bragg grating for short pulse dispersion is shown to enhance the time domain realization of optical frequency-domain reflectometry. A low resolution demonstrator is constructed with single surface scans containing 140 resolvable spots. The system dynamic range meets that shown in earlier demonstrations without digital post-processing for signal linearization. Using a conjugate pair of chirped pulses created by the fiber grating, ranging is performed with position and velocity information decoupled. Additionally, by probing the target with short pulses and introducing grating dispersion just before photodetection, velocity immune ranging is demonstrated.

©2007 Optical Society of America

## 1. Introduction

The need for high-speed, high-depth-resolution imaging motivates interest in swept-frequency optical ranging [1–7]. The method borrows heavily from mature radar principles, which emphasize the importance of distributing signal energy over extended time apertures and reducing time domain sampling rates through intermediate frequency generation [8, 9]. The range to target is deduced by simple measurement of the beat frequency between interfering, identically-chirped target and reference signals after the target signal is delayed by a round trip to the target. All range information is contained in the time aperture of the frequency sweep. Thus this time domain realization of spectral interferometry or optical frequency-domain reflectometry (here referred to as time domain OFDR) has the potential for fast data acquisition rates in a scan-free approach [3–5]. Within the last year, pulsed supercontinuum sources chirped in chromatically dispersive media have been used to demonstrate fine depth resolution with sub-microsecond sweep durations [6, 7]. In these works record speeds for optical coherence tomography (OCT) A-scans were achieved using high-speed temporal sampling. Fiber-based, passive dispersion technologies, including single mode fiber (SMF) and chirped fiber Bragg gratings (CFBG), offer large accumulated chromatic dispersion in small volumes with low loss. Wideband pulsed sources chirped in such dispersive media promise to enable time domain OFDR systems with micron scale resolution over centimeter ranges.

Applications utilizing large accumulated chromatic dispersion include, among others: optical pulse shaping [10–12], microwave spectrum analysis [13], microwave signal generation [14–16], temporal signal magnification [17, 18], and for optical communications: coherent code division multiple access encoding/decoding [19], and delay buffering [20,21]. All systems listed exhibit aberrations in the presence of significant higher order dispersion (see for instance: [14] or [22]), which causes a departure from the first order mapping of optical frequency to time delay. A dispersive element with an accurate linear relationship between frequency and time delay is therefore important for the applications. A technological solution is the linear CFBG, designed specifically to disperse a pulse with all higher order terms set to zero such that a constant chirp rate emerges. In this work, we analyze and demonstrate several key processing advantages afforded by the use of linear CFBGs in low resolution time domain OFDR. While our approaches derive from RADAR practices, the passive dispersion characteristics of the linear CFBG allow for uniquely optical implementations.

The paper is organized as follows: in section 2 we introduce a time domain OFDR system utilizing the linear CFBG and compare the realization to earlier work and to a traditional OFDR implementation, in section 3 we demonstrate an effective and straightforward method for Doppler-range decoupling through the use of a two pulse sequence from the linear CFBG, in section 4 we introduce dispersion-last time domain OFDR and demonstrate Doppler-immune ranging, and in section 5 we conclude the paper.

## 2. Linear CFBG OFDR system

Time domain OFDR deduces target range from the beat frequency between two identically chirped target and reference pulses, one delayed relative to the other. A thorough mathematical treatment of the dispersion-based technique can be found in Ref. [7]; a time-frequency visualization plot is presented in Fig. 1. Briefly, the spectral phase of a chromatically dispersed pulse is Taylor expanded around a carrier frequency, ω_{c}, as

where ω=Ω-ω_{c} if Ω is absolute frequency. Terms of orders higher than the dominant aberration, Φ_{3}, are ignored here but can be taken into account if needed. Arrival times of the frequency components of the pulse are given by the negative derivative of the spectral phase:

If time is referenced to the arrival of the carrier, we can work with relative time, *t*' = *t* - Φ_{1}. Equation (2) can also be inverted to find the instantaneous frequency arriving at a given time, *t*'. After reflection off a linear CFBG, where Φ_{3} and higher terms equal zero, the arriving instantaneous frequency is chirped linearly with time under the pulse envelope according to

A delay, τ, between pulse copies generates a difference (beat) frequency in the interference term after square-law detection,

at all points in time under the envelope, see the left side of Fig. 1. Given the known dispersion parameter, Φ_{2}, the captured beat frequency may be converted to delay, τ, and hence target range, with resolution limited by the bandwidth of the system. In contrast, when a dispersion element with a third order response is employed, the arriving instantaneous optical frequency in the dispersed pulse is chirped nonlinearly as

This situation is visualized in the right side of Fig. 1, where it is clear that a time-variant beat frequency,

forms between detected pulse copies under delay. (The approximation in Eq. (6) assumes *t*' ≫ *τ*, valid through most of the detection time when the delay, τ, is much shorter than the pulse duration.) When converting the beat frequency of Eq (6) to target range, the apparent line broadening drastically reduces measurement resolution at modest to large ranges or delays.

The optical chirp characteristics of passively dispersed short pulses have major ramifications for the implementation of time domain OFDR systems. Both prior demonstrations of dispersion-based time domain OFDR employ dispersive sources with significant higher order dispersion [6,7]. The measured beat signals in these experiments are time-variant, as in Eq. (6), and digital post-processing is required to compensate for higher order phase distortions. From a physical perspective the ranging technique is a time domain realization of OFDR. Therefore, to maximize resolution, a defined time/frequency map must instruct a data interpolation algorithm to sample the signal at times corresponding to the temporal arrival of regularly-spaced optical frequency components. As an example, one could resample a signal with spectral arrival times given by Eq. (5) at times *t*″ such that

in order to create regularly spaced frequency samples. The process effectively converts the right side of Fig. 1 to the idealized picture on the left side. If the interpolated samples are placed on a regular grid, a simple fast Fourier transform (FFT) provides accurate, bandwidth-limited depth resolution at all ranges. Inherent to the approach is the assumption that the beat frequency is very small compared to the bandwidth.

Time domain OFDR employing higher order chirped sources requires both the creation of an accurate time/frequency map and the continual triggering of the interpolation algorithm at an appropriate start time corresponding to the arrival of the carrier, ω_{c}, around which the Taylor expansion of Eq. (1) is constructed. Failure to satisfy either requirement leads to loss of both relative and absolute range accuracy. In practice the time/frequency map can be extracted using a standard dispersion measurement as in Ref. [6] or through an analysis of the interference phase of a target and reference pulse under known delay as in Ref. [7]. The latter method is particularly appealing for its ability to include phase distortions from many potential sources. However, like its spectral domain counterpart [23], using a time domain spectral-shearing interferometer for pulse phase extraction requires that the delay between pulse copies used for calibration be known with high accuracy. Simple analysis shows that a relative error, R, in pulse-to-pulse delay during calibration introduces a scaling factor (1+R) into all subsequent delay measurements to which the calibration is applied. As this ranging technique aspires to achieve and exceed 10^{4} resolvable spots, the 1+R factor can lead to appreciable depth-dependent inaccuracies. Thus the reference arm requires a translation stage with a large number of resolvable spots and knowledge of the zero delay offset point. In the case of a linear chirp, on the other hand, where only relative delays between targets in the image are relevant, such precision is unnecessary. With respect to triggering, system accuracy critically depends on applying the time/frequency map at the correct time (Φ_{1}) relative to pulse arrival. In real time operation, an external trigger derived from the pulsed laser source should be synchronized with the interpolation algorithm. This approach allows the map to be anchored at Φ_{1} in all pulses within the picosecond trigger jitter of commercial digitizers. Signal inspection to identify Φ_{1}, as in Ref. [6], using the real time Fourier transformation principle [24, 25], is limited by the temporal resolution of the detector. However, for the distortion-canceling, irregular sampling methods using an external trigger, whenever the trigger is lost the time/frequency map needs recreating before re-application.

To simplify system implementation, while addressing the potential range inaccuracies stemming from both time/frequency map generation and misidentification of Φ_{1}, we demonstrate a low resolution time domain OFDR system driven by sub-picosecond pulses dispersed in a linear CFBG. All calculations follow from the simplified analysis of Eqs. (3) and (4). As the beat signal frequency is time-invariant, locating Φ_{1} is not experimentally critical. The beat signal linewidth also does not broaden at extended ranges, thus irregular sampling is not required to provide accurate ranging results over a large number of resolvable spots. Our demonstration in this work is practically limited to moderate/low resolution OFDR. For a system employing optical bandwidths sufficient for high resolution OFDR, phase distortions both common and unshared between reference and target pulses quickly degrade resolution despite the use of linear CFBGs. Potential distortion sources include the use of non-transform limited launch pulses, group delay phase ripple within the CFBG, and dispersion of the target sample. The ability to compensate these effects in the physical domain is strained. For high resolution OFDR interpolative post processing is likely required in such a system. A number of works have shown the capabilities of post-processing to compensate for dispersive effects in OFDR [26,27]. These techniques are all relevant for application to time domain OFDR based on dispersed pulses. However, noting the loss of resolution at extended depth ranges in Ref. [6] despite interpolative data re-sampling, it is clear that high resolution time domain OFDR benefits from the minimization of total accumulated higher order phase in the physical layer before photo-detection. For short pulse signals, such dispersion is offered by linear CFBGs.

Our experimental demonstrator is depicted in Fig. 2. The source is a Ti:Sapphire pumped OPO creating 150 fs pulses at wavelengths in the conventional telecommunications band around 1550nm. No nonlinear spectral broadening is employed. The pulses are fiber launched with powers below the threshold for optical nonlinearities and are dispersed in a 3M Corp. CFBG offering Φ_{2}=-431.6ps^{2} (2711 ps/THz) of dispersion over a 35nm bandwidth centered at 1550nm. A loss of 4dB is incurred in the grating and required circulator, a dispersion to loss ratio quite similar to that of SMF. The tailored CFBG provides pure chromatic dispersion with all higher order dispersion terms approximately zero. Additional fiber patch cables adjust the measured dispersion value to Φ_{2}=-432.4ps^{2}. Due to system bandwidth, the higher order dispersion in these meters of cabling is assumed negligible. After dispersing, the pulse time aperture approaches 12 ns: a conservative match to the pulse repetition period and data collection window of 13 ns. As the beat signal synchronously repeats at the pulse repetition frequency but is not restricted to harmonics of the repetition rate, data taken over a time aperture covering multiple pulses in the train can produce destructive interference for the signal beat and should be avoided. The dispersed pulses are introduced into an interferometer for processing. A Michelson arrangement is used to demonstrate system performance. The target arm contains focusing optics and a translating reflective sample. The staircase-like sample target assembled either from 1mm thick gold coated microscope slides or from 150 μm thick gold coated cover glass is stepped through an 80-point one dimensional scan. The reference arm is biased to ensure all delays are positive. Ninety percent of signal power is directed to a 15GHz amplified photodiode (Agilent 11982A). The 15 GHz 3dB roll-off point corresponds to a range of 6.1 mm. The signals digitize in an Agilent sampling oscilloscope with an effective interleaved sample spacing of 3.2ps. This detection method exceeds by nearly an order of magnitude the current state of the art in real-time sampling rates but is used here to enable the demonstration of other system capabilities. The remaining ten percent of signal power goes to a commercial fiber coupled optical spectrum analyzer (OSA) (Ando 6317B) with .01 nm resolution. Assuming Nyquist sampling, the resolution leads to a range limit of 60 mm. A personal computer controls sample translation and data processing.

An examination of data captured from a single representative reflection point gives insight into system performance. A time domain waveform captured by the sampling scope is shown as inset in Fig. 2. An FFT of this waveform, Fig 3(a) red line, presents the information in the beat-frequency domain. The frequency axis is scaled to target range using the factor Φ_{2}c*π* where c is the speed of light and Φ_{2}=-432.4 ps^{2}. Corresponding to the single reflection surface at a range of 3.22 mm, a single spike is present in Fig. 3. The raw data is plotted against a linear amplitude scale and is normalized to the peak of the point spread function (PSF). Resolution is measured at 52 microns, full width at half maximum (FWHM), for the PSF. This resolution is consistent with the bandwidth of the source (<3.5 THz FWHM) and matches well with the 51 μm FWHM of the DC component. Note that the DC and harmonic components would have identical width were the beat signal an ideal sine wave. The axial resolution offered is sufficient for low resolution ultrafast metrology but falls short of the resolutions required for biomedical OCT applications. A broader-band source is expected to bring the resolution down to the 35 μm transform limit of the CFBG bandwidth with the penalty of increased side lobes. To cancel deterministic amplitude noise, the signal is divided by a time averaged copy of the reference pulse and re-windowed with a Hamming function (blue curve in Fig. 3(a)). The procedure strongly reduces much of the short range noise under 1000 microns at the expense of broadening the linewidth to 58 microns. It is important to note that higher frequency (longer range) deterministic spurs, arising from amplitude ripples in the CFBG etc., can not be fully cancelled in this manner. These frequency components in the signal slip out of phase with their counterparts measured from the reference arm alone. This effect accounts for the inability of the reshaping approach to completely remove the two spurious reflections around 1100 and 1600 microns. The issue may subside in practice when target signals are much weaker than reference signals. In Fig. 3(b) the power spectral density of the reshaped signal of Fig. 3(a) is displayed on logarithmic scale, again in blue. The response is normalized to the peak reflection signal. The dynamic range (DR) of the system is limited by spurs to roughly 26 dB. This DR is significantly impacted by spurious reflections in the source and optical system and from group delay ripple in the linear CFBG, whose physical manifestation can be treated like an accumulation of many weak reflected signals [28]. Each spurious reflection contributes small spikes, distributed in range, to the plots of Fig. 3. Apart from the spurs, much of the high frequency (ranges > 10000 microns) noise can be attributed to source jitter incurred over the extended time of interleaved detection. In a single pulse, real time implementation, jitter should only impact the ability of post processing to remove deterministic aberrations. However, for this multi-pulse demonstrator, 4x averaging can be seen to lower the background signal by 4 dB, shown inset to Fig. 3(b).

System sensitivity is gauged by the noise equivalent response. The magenta line of Fig. 3(b) shows the output with the sample arm blocked. The reference pulse alone is detected as before, including the amplitude reshaping process. The sensitivity value is -58 dB in Fig. 3(b) while typical values lie between -55 and -56 dB. These values are made by comparing the peak signal response to the level of the noise one standard deviation above the mean for range values of interest, as done in Ref. [7]. Switching from a Hamming window to a narrower Blackman filter improves the sensitivity values by 2 dB at the expense of losing 2–3 μm of resolution in the signal peak (see the noise floor reduction in the green curve of Fig. 3(b)). Reference measurements employing SMF as the dispersion source showed sensitivities of comparable value to those included here. The comparison result suggests that sensitivity is dominated by source and detector characteristics. Time domain OFDR systems are typically limited in sensitivity by laser source relative intensity noise (RIN) requiring the use of dual balanced detection [4, 5]. The RIN fluctuations manifests themselves as spurs in range measurements, like Fig. 3(b), thus elevating noise levels. However, to contribute negatively the periods of RIN fluctuations must be shorter than the observation window; otherwise they affect only “DC” values. For the slow, actively-swept sources used commonly in time domain OFDR, the degrading RIN is that of a typical continuous-wave laser source. With time domain OFDR based on dispersed ultrashort pulses, single pulse acquisition over a short observation window is the desired operating approach. The mapping of the modelocked spectrum to time implies that RIN will arise from mode-partition noise as was recently studied in Ref. [29]. A formal introduction of RIN treatment based on this new theory is compounded by the multi-pulse, interleaved acquisition in use in our demonstrator. While it is permissible to state that observed DR values are comparable to Refs. [6] and [7] where interpolation post-processing is utilized, the lack of formalism including all system parameters prevents comparison of sensitivity values.

Representative one dimensional surface scans are shown in Fig 4. In each of these 80-point scans, data taken using time domain OFDR is plotted in green while conventional OFDR data is plotted in black. The conventional OFDR data is acquired by scanning spectrometer and then interpolated using a wavelength to frequency mapping [30]. Both the time domain OFDR and conventional OFDR data were zero-padded by a factor of 32 for range-domain interpolation at the single micron scale. Figure 4(a) shows two overlaid scans across the sample made of 150 micron cover glass plates in a staircase arrangement. In the upper trace the reference mirror was biased to demonstrate precision at larger ranges. Figure 4(b) shows a single scan made over the larger sample of millimeter size microscope slides. Note that the time domain acquisition surpassed the 3dB roll-off point, 6.1 mm, of the high speed detector considerably with minimal penalty observed within the measurement. Zoom-in insets show that noise variance does not increase over the range. The slight upward trend of measured height with surface position noticeable in each inset graph is attributed to surface slope and is not present in (a). In Fig. 4 very good agreement is seen between time domain and conventional OFDR despite the dramatic difference in data acquisition time windows of the two OFDR techniques.

Although Fig. 4 shows a noise floor on a single flat surface lower than the FWHM of the PSF in Fig. 3, the latter constitutes a fairer metric of system resolution. The single scan in Fig. 4(b) provides ~50 micron resolution over 7 millimeters, thus demonstrating 140 resolvable range spots for the time domain OFDR system. In accordance with the analysis of Ref. [7], the number of resolvable spots (e.g. system time-bandwidth product) of this time domain OFDR systems is fundamentally limited to the product of time aperture and detector bandwidth. Here, a 13ns time window and 15 GHz detector imply an upper limit of 195 resolvable spots. The discrepancy between our 140 points and the theoretical limit can be attributed to our underutilization of available reflection bandwidth (3.5 THz of 4.3 THz max) and system time window (12ns of 13ns max). Demonstrated results show that time domain OFDR realized with the linear CFBG can deliver accurate ranging values over extended total ranges on the sub-microsecond time scale. Moderate to low resolution systems can exploit the linear CFBG technology to possibly forego interpolative data post-processing.

## 3. Doppler-range decoupling

A practical problem facing conventional ranging systems employing frequency swept signals is the coupling of range and Doppler information in the return target signal. Figure 5 gives a time/frequency visualization showing how an unexpected Doppler upshift, ω_{Doppler}, to a target pulse will be recognized by the system as an additional apparent delay. The problem has been long known in Radar signal processing [31] and has the potential to cause ambiguous results in a time domain OFDR system [32]. A common approach to Doppler-range decoupling is to increase the number of degrees of freedom in the system by utilizing a pulse sequence of non-identical chirps. The linear CFBG technology is well suited for a two pulse measurement in which the first pulse is dispersed with the blue side of the CFBG and the second pulse with the red side of the same grating. Figure 5(b) gives a visualization of the two pulse sequence. The passive dispersion provided by the two-port linear CFBG allows for near perfect signal conjugation or chirp sign flip between the two pulses. It can be seen that this sign flip will allow for maximum contrast between Doppler and range information.

To understand the utility of the two pulse sequence it helps to theoretically derive the instantaneous frequency of a dispersed and Doppler shifted pulse. Due to the time-variance of the Doppler phenomenon, standard frequency-domain transfer function techniques cannot be employed (a Doppler element generates frequencies at the output that are not present at the input). However, the complex field of a chirped Gaussian pulse reflected from a linear CFBG can still be written as a superposition of time exponents:

Again ω_{c} is the carrier frequency, and ∆ω is the 1/e point spectral half-width. E_{0} is included to scale the field amplitude. In keeping with the formalism for linear CFBG dispersion, the
pulse travels in the moving reference frame, *t*' , and the second order dispersion (quadratic phase) term is present while higher order terms are absent. When the Doppler shift acts on the chirped pulse, it stretches or compresses *t*' through a factor $\alpha =\left(1-2\frac{v}{c}\right)$ , where v is the velocity of the target projected onto the illumination direction and c is the speed of light [31]. The Doppler-shifted dispersed pulse is therefore given by:

The term, α, also scales the amplitude to include the increase or decrease in photon energy accompanying reflection from the moving target. Completing the square and taking the integral over the Gaussian, the output signal in time is:

Simplifications in the exponents are valid when $\frac{1}{\Delta {\omega}^{2}}\text{}\ll {\Phi}_{2}$, which is easily satisfied for the bandwidths and dispersion values used for time domain OFDR. Taking a derivative of the phase in Eq. (10), the instantaneous frequency under the pulse envelope is given by:

As expected, Doppler processing up- or downshifts the chirped signal carrier by the factor α relative to the dispersed-only pulse of Eq. (3). The appearance of α_{2} in the linear temporal sweep term is due to the combined effect of the frequency dependence of Doppler shifts and the variation in return time from the moving target. Following the measurement approach of time domain OFDR, the difference frequency (under the interaction envelope) between a dispersion-only frequency sweep of Eq. (3) and a delayed version of the Doppler shifted swept signal of Eq. (11) is

Comparison with Eq. (4) reveals the impact of target motion on the beat signal. The α^{2} stretch factor in the first term, within the sensitivity of a typical measurement, is indistinguishable from unity. The second term imparts the expected Doppler shift to the measured beat frequency. The last term leads to a time variant beat frequency and must be considered more closely. Its consequences are easier to observe when the beat frequency is converted to target range using the scale factor νcΦ_{2}. We denote by R_{1} and R_{2} the two apparent range values obtained with the two probe pulses. Let the first pulse reflect from the blue side of the CFBG, Φ_{2}<0, and the second from the red side, Φ_{2}>0, then the range values are:

where ω_{Doppler}=-ω_{c}2v/c. The potential loss of resolution from the line broadening as a result of the 2*t*' v term equals twice the product of the time aperture and the target velocity. If this product is much less than the range resolution, it can be ignored. Alternatively, resolution may be recovered with an interpolative post processing approach as discussed earlier. For a time domain OFDR system with supercontinuum bandwidths, as in Ref. [6], the time variance of the beat frequency due to significant Doppler shifts cannot be ignored if micron-scale resolution is to be preserved. Whether or not the 2*t*' v term is included, the range and Doppler information are readily separated when both R_{1} and R_{2} are available. Averaging the two range values delivers a true Doppler-free range, while subtracting R_{2} from R_{1} and dividing by 2cΦ_{2}
*π* yields the Doppler shift (in frequency units). To avoid measurement ambiguity, the setup must be configured so that *τ* > |Φ_{2}|ω_{Doppler}. A simple biasing of the reference delay allows this restriction to be easily met in practice. Two oppositely chirped, phase conjugate, pulses provide the best contrast between range values, R_{1} and R_{2}, for a given dispersion strength and are easy to obtain by reflecting off the two sides of the linear CFBG. Thus the isolation of true Doppler shift and range values is made in a straightforward and accurate manner by incorporation of linear CFBG technology.

In the demonstrator introduced in Section 2, the coupling factor, Φ_{2}cπ, corresponds to roughly 0.4 μm of range shift per MHz of Doppler shift, or equivalently, to ~0.5 μm of range shift per m/s of parallel target velocity. Coupling would be greater, however, in a system utilizing stronger dispersion, while tighter error tolerance would be imposed by bandwidth-driven resolution improvements. Time domain OFDR with 10^{4} resolvable spots over centimeter-scale ranges requires, as compared to our experimental setup, a ten fold increase in system bandwidth, a further five fold increase in dispersion strength of the CFBG, and a corresponding decrease in pulse repetition rate. A system with these specifications should be concerned with high-speed blood flow coupling to range values in a biomedical application [33, 34]. Additionally, as the utility of the OFDR technique extends beyond biological imaging to targets with larger depth features and potentially faster flow/travel rates, separation of Doppler and range information becomes increasingly important.

To demonstrate the ability of the two pulse sequence to decouple Doppler and range values, the experiment described in the preceding section is modified according to the schematic shown in Fig. 6. The fiber coupled pulse enters a manual switch box that passes the pulse to the blue or red port of the CFBG. Reflections from either side are switched back to the remainder of the system. The 50 micron resolution limit puts an initial restriction on observable Doppler shifts. For our demonstrator we use a +152 MHz acousto-optic modulator (AOM) from IntraAction Corp. to provide a Doppler upshift to the signal pulse, equivalent to target velocity ~-120m/s. The frequency shift corresponds to an apparent range shift of ±60 microns depending on chirp sign. The fiber coupled AOM is incorporated into the target arm of a Mach Zehnder interferometer. The reference arm contains a fiber coupled delay line to demonstrate the functionality of the system at multiple ranges. The time variant term in Eq. (13) is here neglected because it yields only a 3 micron variation over the entire 13 ns obsevation time window. The AOM is driven by a filtered and amplified signal derived from the 2^{nd} harmonic of the laser repetition rate. The RF phase is transferred to the optical signal in the AOM and thus to the photo-detected beat wave. Phase locking the AOM drive to the laser repetition rate enables the sampling oscilloscope, triggered on the repetition rate, to capture each interleaved pulse with an identical beat signal phase relative to the pulse envelope. This step prevents fringe washout in the recorded waveform. The locking would not be necessary if a real-time, non-interleaved digitizer were used, and a single pulse pair were acquired. As before, the signal and reference pulses are routed to both the fast photodiode and the OSA for comparison of time domain OFDR against conventional OFDR.

Typical data demonstrates the decoupling approach. To improve visual clarity, detected waveforms with low beat frequency are shown in Fig. 7. Pulses reflected from the blue and the red sides of the grating are drawn in blue, Figs. 7(a) and 7(c), and red, 7(b) and 7(d), respectively. In the time domain data the pulse reflected from the blue side of the CFBG, Fig. 7(a), shows a Doppler-induced increase in beat frequency relative to the pulse reflected from the red side, Fig. 7(b). The waveforms are also time reversed with respect to one another confirming the arrival of either the blue or red ends of the spectrum first. As chromatic dispersion is seen here to map frequency information to the time domain, it is not surprising that the same Doppler-Range phenomenon is observed in the frequency-domain data captured by the OSA and reproduced in Figs. 7 (c) and 7(d). Doppler-Range decoupling is a potentially compelling reason to utilize chirped femtosecond sources, as opposed to broadband continuous wave sources, in conventional OFDR. However, as noted above, strict phasing requirements for the beat signal were imposed in this demonstration due to multi-pulse acquisition in time and frequency domains. To apply this technique generally to conventional OFDR, the spectrometer should acquire only one pulse per integration window.

Figure 8 depicts a 20 point scan over a centimeter worth of delays with constant 60 micron up and downshifts in the observed data from the blue and red ports, see upper inset figure. The green line represents the measured true range obtained by averaging the two measurements, R_{1} and R_{2}, described in Eq. (13). The lower inset plot (obtained in another 55 point scan over centimeter range) shows the Doppler shift computed by subtracting R_{2} from R_{1}. As expected the value is nearly constant at +152 MHz. As was the case in the earlier demonstration, chirp rates differ slightly from those of the CFBG due to fiber patch cables used in the setup. For the purpose of calculating true range (green curve in Fig. 8) and the Doppler shift, a chirp rate of -432.6 ps^{2} was assumed for the blue port and 430.8 ps^{2} for the red port. The raw readings of R_{1} and R_{2} were weighted accordingly. Good agreement is seen between measured data and the specifications of the demonstrator setup.

## 4. Doppler immunity

While Doppler information is often desirable, as in functional OCT [34, 35], and was shown separable in the previous section, some range-sensitive imaging systems may wish to simply remove the effect to enhance image clarity. Earlier work in OCT showed that limiting the temporal duration of broadband signals or simply limiting the instantaneous duration of each frequency component within a swept source can greatly reduce Doppler blurring [36]. The former method physically limits the interaction time with a moving sample while the second utilizes post-processing to recreate a short interaction. A hybrid of these two solutions for Doppler immunity is easily implemented directly in the physical domain of a dispersion based time domain OFDR signal processor. A system variant in which the target pulse is dispersed after, rather than before, bouncing off the target proves Doppler-immune. In this case, the target pulse interacts with the moving sample over a limited time aperture but, due to the use of passive dispersion before photo-detection, can be ultimately received as a much longer frequency swept signal suitable for time domain OFDR. We compare the time-frequency mapping of pulses dispersed before and after a Doppler shift to clarify the difference between the two system configurations, and we demonstrate high-speed, Doppler-immune ranging.

Despite the linearity of the dispersion based time domain OFDR system, a pulse Doppler shifted first and subsequently dispersed has a distinctly different instantaneous frequency profile than one first dispersed and then Doppler shifted. To derive the former, a Doppler shift is introduced through the factor, α, as in Eq. (9), to a Gaussian pulse. We write:

where a change of variables, Ω = *α*Ω, was used on the right side. If chromatic dispersion is introduced as a quadratic spectral phase as in Eq. (8), a change of time variable,
*t*" = *t*'- Φ_{2} (*α* - 1) ω_{c}, facilitates completing the square and integrating to yield:

Here again we utilize the approximation, $\frac{1}{\Delta {\omega}^{2}}\ll {\Phi}_{2}$ to simplify the exponentials. A derivative of the temporal phase produces an instantaneous frequency,

identical to that for the dispersed only (i.e. non-Doppler) pulse treatment of Eq. (3). It should be noted, however, that the Doppler terms do not entirely drop out without the approximation $\frac{1}{\Delta {\omega}^{2}}\text{}\ll {\Phi}_{2}$.

An intuitive explanation for the Doppler immunity may be offered as follows. When a transform limited broadband pulse is subjected to a small Doppler shift, it merely changes the upper and lower limits of the pulse spectrum. The central components shift as well but are replaced in their former position by a neighboring component. Due to the linearity of the spectral phase of the pulse, all spectral components remain collocated at the temporal position, Φ_{1}, of the original carrier frequency. Consequently, except near the margins of the spectrum, the Doppler shifted pulse performs in subsequent processing identically to a pulse that has not been Doppler shifted. When the beat frequency signal forms between a dispersed reference pulse and a delayed copy of the target pulse of Eq. (14), it reflects only range information.

The treatment leading to Eq. (14) assumed an interaction of a transform-limited pulse with the target. In some applications this approach may be power limited by the maximum permissible exposure of the sample before ensuing damage occurs. For metrology with these target samples, a pre-stretching scheme would allow for greater average powers and/or reduced peak powers. The lack of transform limited incidence for the target pulse, however, implies that some Doppler-range coupling will occur. The factor introduced in Section 3, Φ_{2}cπ, describes the ratio of range adjustment to Doppler frequency. If a pre-stretch technique were employed, the net suppression of Doppler-range coupling would be limited to
}a factor of Φ_{2_CFBG}/Φ_{2_Pre-stretch}. As an example, tens of meters of patch cables will pre-stretch the 150fs pulses used here to the ten picosecond scale. The peak power drops by a factor of 100 while Doppler suppression remains greater than 20dB as Φ_{Pre-stretch} ≈ 1ps^{2}.

To confirm the Doppler immunity of the dispersion-last approach, our experimental system in Fig. 6 is modified by exchanging the order of the switch box, circulator, and CFBG with the Mach-Zehnder interferometer. All other system parameters are unchanged. Figure 9 shows sample waveforms, organized akin to Fig. 7, taken by the oscilloscope and OSA under short relative delay and +152 MHz Doppler shift. Signals dispersed from either side of the CFBG exhibit the same beat frequency. A slight discrepancy between visualized beat frequency for the blue and red conventional OFDR waveforms is attributable to phase drift of the interferometer (and the beat signal) during the several-second duration OSA scans. Note again the time reversal of the envelopes captured by the oscilloscope. Figure 10 shows a 55 point range scan demonstrating the overlap of data taken from the blue and red side of the CFBG. The difference between R_{1} (blue line) and R_{2} (red line) range values is on the order of 1μm, compared to ~120μm for the Doppler-first configuration, Fig. 8. Due to the strong agreement, a weighted average (measured true range) is not presented. If a Doppler shift were to be computed from this difference, as in Fig. 8 lower inset, the apparent Doppler frequency would be close to zero. This effect is illustrated inset in Fig. 10, where all apparent Doppler values are under 4 MHz with an average of 1.3 MHz, two orders of magnitude below the 152 MHz actual Doppler shift present. Comparing the results of Figs. 8 and 10, good agreement exists between theory and observation. While the data of Fig. 10 is all based on time domain detection, similar results were observed using conventional OFDR by OSA detection. A ranging system could clearly be designed with Doppler immunity if it were tolerable to transmit and receive short pulses, withholding most dispersion until the return of the target and reference pulses. The approach highlights the unique capability that chromatic dispersion offers for passive generation of high-speed, broadband frequency sweeps with low loss at any point of the processing chain. This approach to chirp signal generation is more flexible than actively swept source technology, which must drive the system from the launch.

## 5. Conclusion

We demonstrate several enhancements to the time domain realization of OFDR through the introduction of a linear CFBG as means for high-speed chirp generation. The utility of the linear CFBG is shown in the realization of a low resolution surface profile scan with a near-theoretically-limited number of resolvable spots. The idealized signal dispersion offered by the linear CFBG gives single point DR matching that of earlier demonstrated systems based on dispersed short pulses, despite the removal of potentially accuracy spoiling interpolative post-processing. A novel system arrangement is implemented with a two pulse sequence generated by dispersing the launch pulse from both ends of the same linear CFBG in succession. This approach permits the decoupling and isolation of Doppler and range information with maximum contrast. A further alteration leads to system operation in a dispersion-last approach providing Doppler-immune range values.

While time domain OFDR based on dispersed wideband pulses has disruptive potential, certain key technical challenges lie in its path. The use of linear CFBGs is restricted to systems with moderate numbers of resolvable spots until critical improvements to grating fabrication are implemented. Techniques to reduce amplitude and phase ripple, which impact system DR, and methods to further lengthen CFBGs to increase dispersion values and operating bandwidths are essential. Of equal importance, fast A/D converters are a gateway to extending the number of resolvable spots in real time operation. Current technological limits will restrict dispersion based time domain OFDR to mm ranges until A/D converters at several GHz can offer sufficient DR. The creation of highly stable broadband sources through supercontinuum generation or other means centered at wavelengths appropriate to targets of interest and where low loss dispersive fiber technologies exist is a last technical step. As enabling technologies improve, a time domain OFDR system based on dispersion of short pulses with micron scale resolution over a centimeter range utilizing a sub microsecond time aperture should emerge.

## Acknowledgments

This work was supported by the Defense Advanced Research Projects Agency, the UC Discovery Grants Program, the National Science Foundation, and the U.S. Air Force Office of Scientific Research.

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