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• Abstract

SECTION 1

SECTION 2

## PRINCIPLE OF OPERATION

An LFM waveform of duration $T$, bandwidth $B$, and central radio frequency $f_{\rm RF}$ can be expressed as TeX Source $$A_{\rm LFM}(t) = A_{0}\cos\left(2\pi f_{\rm RF}t + \pi{B \over T}t^{2}\right){\rm rect}\left({t \over T}\right)\eqno{\hbox{(1)}}$$ where $A_{0}$ is a constant magnitude, $t$ denotes time, and ${\rm rect}(\xi) = 1$ for $\vert\xi\vert\leq 0.5$ and equals zero elsewhere. The instantaneous frequency $f(t)$ of the waveform is linearly sweeping between $f_{\rm RF}\pm (1/2) B$ along the waveform duration $T$. The same waveform delayed by $\tau$ is given by TeX Source $$A_{\rm LFM}(t - \tau) = A_{0}\cos\left[-2\pi f_{\rm RF}\tau + \pi{B \over T}\tau^{2} + 2\pi\left(f_{\rm RF} - {B \over T}\tau\right)t + \pi{B \over T}t^{2}\right]{\rm rect}\left({t - \tau \over T}\right).\eqno{\hbox{(2)}}$$

The delayed waveform is approximated by the introduction of a phase offset $\Delta\varphi$ and a frequency offset $\Delta f$ to (1) so that [11] TeX Source $$A_{\rm LFM}^{\rm offset}(t) = A_{0}\cos\left[\Delta\varphi + 2\pi \left(f_{\rm RF} + \Delta f\right)t + \pi{B \over T}t^{2}\right]{\rm rect}\left({t \over T}\right).\eqno{\hbox{(3)}}$$

Provided that TeX Source \eqalignno{\Delta f = &\, -{B\tau \over T}&\hbox{(4)}\cr \Delta\varphi = &\, -2\pi f_{\rm RF}\tau + \pi {B\tau^{2} \over T}.&\hbox{(5)}}

When (4) and (5) are met, differences between (3) and (2) are confined to the edges of the rectangular temporal window, which is not delayed. As long as $\tau\ll T$ (or, equivalently, $\Delta f\ll B$), the differences between the offset waveform and the “truly delayed” one are negligible. Since $T$ typically equals many microseconds, substantial delays to the impulse response are possible without excessive distortion. The impulse response function of the offset waveform is calculated as its cross-correlation with a replica of the original signal [1] TeX Source $$h(\xi) = \int\limits_{- \infty}^{\infty}A_{\rm LFM}(t)A_{\rm LFM}^{\rm offset}(t + \xi)\, {\rm d}t.\eqno{\hbox{(6)}}$$

Both waveforms are often windowed in order to suppress the correlation sidelobes.

In MWP implementations, the RF LFM waveform is used to modulate an optical carrier, and is later recovered through the beating of the optical carrier and the modulation sidebands on a photodetector. The photonic realization of our proposed TTD method therefore requires that relative $\Delta\varphi$ and $\Delta f$ offsets are generated between carrier and sidebands. To that end, the optical carrier of frequency $f_{\rm opt}$ is split in two paths. Light in one path is modulated in suppressed-carrier single sideband (SC-SSB) manner to retain only one modulation sideband which carries the LFM waveform and is subsequently offset in phase and frequency. The original, unmodulated carrier is retained in the other path, and the two are recombined prior to detection. The processing of a single optical sideband was previously employed in several MWP TTD demonstrations [2], [8], [12], [13]. The delay of single-sideband analog waveforms benefits from piece-wise treatment [2]: Since $f_{\rm RF}\gg B$, it is sufficient for a MWP setup to provide an appropriate spectral phase across the window $[f_{\rm opt} + f_{\rm RF} - (1/2)B, \, f_{\rm opt} + f_{\rm RF} + (1/2)B]$, and at the carrier frequency $f_{\rm opt}$ itself, and the spectral phase at all other optical frequencies need not be specified. The above technique for the TTD of LFM waveforms may be regarded as a specific case of this principle, which takes advantage of the particular attributes of the waveform.

In our earlier demonstration [11], the frequency offset was implemented via an electrooptic phase modulator that was driven by a ramp waveform of magnitude $2V_{\pi}$ and period $1/ \Delta f$. Discontinuities and deviations from the ideal ramp shape generated high-order harmonics of the phase modulation, which resulted in multiple, periodic secondary correlation peaks. Although 500-MHz-wide LFM signals were delayed by as much as 250 ns [11], the ISLR levels were unacceptable: as low as only 7 dB for a 100-ns delay. Here, frequency offsets are directly implemented using an acousto-optic modulator (AOM) that is driven by a sine wave of frequency $f_{\rm AOM}$. In an AOM, a piezoelectric transducer is used to drive an ultrasonic acoustic wave through an optical medium. Part of the power of an incident optical beam is deflected by the acoustic wave via spontaneous Brillouin scattering, and collected into an output fiber port [14]. The deflected beam is offset in frequency by $f_{\rm AOM}$. The AOM is designed to operate at a nominal drive frequency of $f_{{\rm AOM}, 0}$, and we consider the timing of the impulse response that is obtained using that nominal drive frequency as our zero delay reference. Relative delay or advancement of the impulse response are obtained through deviations of the driving frequency: $\Delta f = f_{\rm AOM} - f_{{\rm AOM}, 0}$. Since the angle of deflection varies with the driving frequency [14], the usable range of $\Delta f$ is limited to deviations of only a few Megahertz from $f_{{\rm AOM}, 0}$. Nevertheless, this range of frequencies is sufficient to delay or advance the impulse response of LFM waveforms by tens of nanoseconds, as expressed in (4). As described next, the use of an AOM reduces the impulse response sidelobes and drastically improves its ISLR. In addition, the AOM requires a simpler waveform generator. The introduction of a phase bias according to (5) would require an electrooptic phase modulator in series with the AOM. Unlike the previous demonstration [11], phase modulation would only need to compensate for slow environmental drifts.

SECTION 3

## EXPERIMENTAL SETUP AND RESULTS

Fig. 1 shows the experimental setup for the variable delay of the impulse response of LFM waveforms with suppressed sidelobes. The output of a continuous wave (CW) laser diode is split in two paths. Light in the upper path is amplified and then modulated by an electrooptic amplitude modulator, driven by an LFM waveform $(T = 5 \ \mu\hbox{s}, B = 500 \ \hbox{MHz})$ from the output of an arbitrary waveform generator (AWG). The LFM's central frequency is up-converted to $f_{\rm RF} = 7.5 \ \hbox{GHz}$ using an RF mixer. The modulated optical waveform is brought into SC-SSB format using a narrow fiber Bragg grating (FBG), which reflects only a single optical sideband back into the fiber path and removes the optical carrier and the complementary sideband. The remaining sideband is frequency-offset by an AOM, that is driven by a sine wave of magnitude $V$ and frequency $f_{\rm AOM}$. The nominal drive frequency of the AOM used in the experiment is 40 MHz. The processed sideband is combined with the original optical carrier, which is retained in the lower path (see Fig. 1). A modified electrical LFM waveform is reconstructed through the beating of the carrier and sideband on a broadband detector. The detected waveform is down-converted to an intermediate central frequency of 1 GHz and sampled by a real-time oscilloscope of 6 GHz bandwidth. The impulse response of the waveform is calculated by cross-correlating the sampled signal against a reference waveform, which was acquired at the AWG output. Both waveforms were filtered by an optimized Hanning window for sidelobe suppression.

Fig. 1. Experimental setup for the variable delay of the impulse response of linear frequency modulated waveforms. CW: continuous wave. EDFA: erbium-doped fiber amplifier. PC: polarization controller. IF: intermediate frequency ($\sim$1 GHz). RF: radio frequency ($\sim$7.5 GHz). LO: radio-frequency local oscillator (6.5 GHz).

Fig. 2 shows the experimental impulse response functions, which had been obtained for five values of $f_{\rm AOM}$ in the range of 35–45 MHz. A relative delay of the impulse response by as much as ±50 ns is evident, in agreement with (4). The delay was restricted by the reduced efficiency of the AOM at drive frequencies that are far detuned from its predesigned value of 40 MHz. For all values of $\Delta f$, the full-width at half-maximum (FWHM) of the impulse response main lobe, signifying resolution, was 2 ns. The PSLR of all impulse response measurements was higher than 18 dB, and the ISLR was better than 20 dB. Our earlier work, which employed ramp phase modulation, had reached longer delay variations of 0–250 ns with equal resolution and PSLR [11]. However, the ISLR for delay variations larger than 20 ns were unacceptably low, between 7–9 dB [11]. The application of an AOM eliminated the multiple periodic correlation sidelobes that hindered the performance of the earlier setup. The remaining sidelobes stem from a residual incident beam that is not shifted, as well as from anti-Stokes and higher order scattering in the AOM. The drive voltage $V$ was adjusted to minimize the sidelobes. The range of frequency offset values $\Delta f$ may be increased, in principle, with the construction of double-pass AOM in which the angle of beam deflection remains fixed. Note that the bias phase $\Delta\varphi$ was not controlled in the experiment. Bias phase adjustments would be mandatory in beam steering experiments involving multiple elements to avoid spatial distortion [2].

Fig. 2. Measured, normalized impulse responses of 500-MHz-wide, 5- $\mu\hbox{s}$-long LFM waveforms. The frequency offsets $f_{\rm AOM}$ (in MHz) are 35 (blue), 38 (red), 40 (green), 42 (black), and 45 (magenta). A magnified view of the primary correlation peaks is shown in the inset.
SECTION 4

## CONCLUSION

The variable delay and advancement of the impulse response of LFM waveforms by ±50 ns was demonstrated experimentally. The delay-bandwidth product of the experimental demonstration is 50. For the first time, such long MWP TTDs of real-world radar waveforms are achieved with the high fidelity that is required in system applications. The method is suitable for providing the delays required in large phased-array antennas. The technique is applicable to the processing of LFM waveforms with arbitrary bandwidths and central RFs. Ongoing work is dedicated to the extension of the technique to the processing of additional waveforms, such as nonlinear frequency modulated (NLFM) radar signals.

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