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We present the results of an experiment designed to verify the results of a previously published theoretical model that predicts the range resolution and peak-to-side lobe ratio of sparse frequency linearly frequency modulated (SF-LFM) ladar signals. We use two ultra stable diode lasers which are frequency locked and can be current tuned in order to adjust the difference frequency between the two lasers. The results of the experiment verify the previously developed model proving that SF-LFM ladar signals have the ability to increase the range resolution of a ladar system without the need for larger bandwidth modulators. Finally we simulate a target at a range of approximately 150 meters through the use of a fiber optic delay line, and demonstrate the ability of SF-LFM ladar signals to detect a target at range.
©2010 Optical Society of America
1. Introduction
Coherent LFM radar pulse trains are common signal waveforms used by the radar community [1], but while these signals are fairly easy to generate in the radio frequency band they are much more challenging to produce in the optical bands. There are several different methods of producing frequency modulated signals in the optical domain, but it is extremely difficult to maintain linearity over large modulation bandwidths while maintaining the pulse energy and pulse repetition rates needed for airborne synthetic aperture ladar. Two examples of solutions to this problem are mentioned. Karlsson and Olsson utilized a rather intricate process which used an arbitrary waveform generator to linearize chirps on the order of a gigahertz [2,3]. Nordin and Hyyppa utilized complex thermal modeling to produce LFM output from distributed feedback diode lasers [4]. We overcome the difficulty of achieving large modulator bandwidths by taking a cues from sparse aperture imaging [5,6], chirped synthetic-wavelength interferometry [7], and sparse frequency radar [8,9] communities. By using two frequency-shifted coherent sources, each having a continuous wave LFM (CW-LFM) imposed on them [10], we can obtain a LFM signal with an extended spectrum. We proposed the superposition of frequency detuned laser sources, which are then linearly frequency chirped using conventional modulators, i.e. 30MHz – 100MHz, producing a signal with an effective bandwidth larger than the modulator bandwidth. This method allows for the generation of large effective bandwidths without the need for large modulator bandwidths, therefore eliminating the need for complex modulation techniques.
Our prior publications illustrated that sparse frequency linearly frequency modulation (SF-LFM) laser radar (ladar) signals have the ability to improve the range resolution of a ladar system without the need for larger modulator bandwidth [10–12] through the use of segmented bandwidth signals. Here, several narrow linewidth laser lines separated by a fixed frequency are sent through an acousto-optic modulator, which imposes the same frequency chirp on all laser lines. In the current paper we experimentally verify the theoretical framework using two laser lines. Because of the frequency offset of the two lasers, we have twice the amount of modulated bandwidth and we can vary the separation of the two frequency bands. Our previous publications present analytical and numerical models that are developed to characterize the range resolution of the signal as a function of difference frequency as well as the peak-to-side lobe ratio (PSLR) of the matched filter output of the signal.
In this paper we set out to verify the results of the model for the case consisting of the superposition of two chirped laser waveforms. The approximated analytic expression for the matched filter output (autocorrelation) of this signal is given by,
where I is the intensity of each of the superimposed chirped lasers, which were assumed to be equal in the model, ILO is the intensity of the local oscillator which is assumed to be much greater than I, B is the chirp bandwidth of the modulator, T is the pulse duration, fo is the optical carrier frequency, df is the difference frequency between the laser lines, and τ is the signal time delay [10]. Equation (1) shows that the matched filter output consists of the superposition of three sinc functions. The first term in Eq. (1) represents the central lobe of the autocorrelation, which narrows as the difference frequency is increased. The second term shows two symmetric sinc functions located at with amplitude decreasing as the difference frequency is increased. These symmetric sinc functions can cause ghosting when the modulated frequency bands overlap; however the ghosts vanish when the difference frequency is greater than the modulator bandwidth.The numerical model developed previously [10,11] allows for the calculation of the full width half max (FWHM) of the central lobe (δτ) which was measured at the −3dB point. The numerical model also calculates the PSRL of the matched filter output, determined by the ratio of the value of the largest side lobe divided by the central peak value. The range resolution δR, determined by the relationship,
was calculated as the model steps through difference frequencies in 1MHz increments [10]. In the following sections we discuss the experimental setup used to verify the results of the modeling. We present the resultant data and compare it to the numerical model.2. Experimental setup
For simplicity the experiment was conducted in polarization maintaining (PM) fiber optics with no free space components. To generate the signal we used an ultra stable diode laser (USDL) system from Innovative Photonic Solutions for the experiment. The USDL system consists of two external cavity diode lasers that are isolated in individual micro-Kelvin ovens allowing them to be frequency locked to each other with a controllable frequency separation. The individual lasers have independent battery operated current supplies, which allow for extremely stable operation and independent fine frequency tuning. By using the fine frequency adjustment (of current) and the coarse frequency adjustment (of temperature) the frequency difference between the two lasers is controllable to less than one megahertz and can be continuously tuned over several hundred megahertz. A fiber pigtailed acousto-optic frequency shifter from Brimrose Corporation powered by a variable frequency driver was used to linearly chirp the two laser lines.
The SF-LFM signal produced from the acusto-optic modulator (AOM) was mixed with an unchirped local oscillator and recorded on a fiber coupled high-speed photodiode. The output of the photodiode was coupled to an Acqiris DC252 two channel digitizer allowing for the signal to be digitally recorded and filtered. Unlike in our past publications an I/Q detection assembly was not used, but the signal was digitally post-processed to recreate the complex envelope of the signal. This was due to the relative complexity of implementing an I/Q detection system (i.e. a common path interferometer) and the relative simplicity of post-processing the data in MatLab.
In the set up shown in Fig. 1 , the two outputs from the USLD system are split by 50/50 fiber splitters. One output from each is then recombined (homodyne) and coupled to a photodiode (PD), digitized and then fast Fourier transformed (FFT), which allows the difference frequency (df) to be monitored in real time. One of the outputs from the bottom splitter (the stationary laser line) is split again so one leg can be used as the local oscillator (LO). This leaves one fiber with each laser line propagating in it; these fibers are then coupled together and sent through the AOM, which has a linear frequency ramp applied to it, producing the desired dual chirp SF-LFM signal. The signal is then coupled with the LO allowing for the heterodyne mixing to occur, and then the heterodyned signal is coupled to the other photodiode. Just as with the other photodiode the output is coupled to the digitizer. The output of the digitizer is recorded and processed to determine the range resolution and the PSLR.
Fig. 1 A schematic of the experimental setup (no delay). The system components: splitter (Spl), Coupler (Cpl), photodiode (PD), the ultra stable laser diodes (USLD) and other components are discussed in the text.
The heterodyned signal was processed by taking the FFT and multiplying it by a narrow band filter, customized to pass only the positive frequency content of the two chirps, and then taking the inverse FFT (IFFT). This process not only eliminates any unwanted noise in the signal it also digitally I/Q’s the signal resulting in the generation of a complex signal. The resultant complex signal is then autocorrelated to generate the matched filter output. Finally a simple algorithm was developed to measure the range resolution (δR) and PSLR from the matched filter output.
3. Verification of range resolution and PSLR modeling
The following results were measured for a SF-LFM ladar signal with a modulator bandwidth of 37MHz and pulse duration of 4µs. The measured range resolution and PSLR for various difference frequencies between zero and twice the modulator bandwidth are shown in Table 1 . Figure 2 compares the data from Table 1 with the theoretical curves produced from the algorithms described above. From Fig. 2a it is clear that the experimental data matches the curves extremely well in the region where df > 23MHz, but there are significant discrepancies in the region where df < 23MHz. Unlike in Fig. 2a, Fig. 2b shows that the data matches the theory extremely well regardless of the difference frequency.
Table 1. Range resolution and PSLR for a SF-LFM ladar signal with a modulator bandwidth of 37MHz and a pulse duration of 4µs.
Fig. 2 The experimental data from Table 1 plotted on top of the modeled results for a 37MHz SF-LFM chirp ladar signal (a) PSLR verse difference frequency (b) range resolution verse difference frequency.
The reason for the discrepancy in Fig. 2a is that our theory assumes an ideal AOM that is able to apply a linear sawtooth frequency ramp. In fact the AOM we used in the experiment was not capable of producing a sawtooth waveform without a loss of linearity at pulse durations greater than 4μs. When our AOM operates in this regime, it obtains a triangle-waveform instead of the sawtooth, since we are forcing the AOM to ramp for time durations near the driver’s slew rate. Furthermore, the triangular wave is slightly asymmetric in that the positive slope is slightly different in magnitude from the negative slope. We are able to accommodate for this waveform by modifying our numerical model from the ideal case to match the actual experimental conditions. For a detailed description of how the experimental waveform was characterized and determined to be different from the theoretical expectations see Ref. (13). The modified numerical model shows that when the chirped bandwidths significantly overlap (df < 23MHz) that the PSLR has a large oscillatory component, See Fig. 3a . The reason for the oscillation is that the symmetric delta-functions are the dominant side lobe peaks and they have amplitudes that rapidly change in this region. When the chirped bandwidths do not overlap (df > 23MHz) the symmetric delta function amplitudes decrease below the amplitudes of the second peak of the central sinc function, which is a smoothly varying function of difference frequency. Since the symmetric delta functions in Eq. (1) have no effect on the FWHM of the central sinc function, it is no surprise that the range resolution shown in Fig. 3b was unchanged by the modifications to the model.
Fig. 3 Comparison of experimental data and modeling with the waveform modified to fit the experimental chirp function for the range resolution (a) and PSLR (b) for a 4µs LFM chirp with a 37MHz bandwidth.
Our experiment does not have sufficient frequency resolution to resolve the high frequency oscillation, so in order to more accurately compare the data to the modified model a moving average was taken over the region of rapid oscillation. Figure 4 shows that the data points all fall within one standard deviation of the moving average. We believe that if the AOM could maintain linearity over longer pulse durations, or had a shorter slew rate that the data points in that region would match the theory. For more details on how the slew rate of the AOM affects the PSLR of the signal see Ref [13].
Fig. 4 Zoomed in view of the data points from Fig. 4.3a showing the data follows the moving average of the theory and is within one standard deviation.
4. Target Simulation (~1µs Delay)
In order to simulate a target at 150m, a 200m fiber-optic delay line was introduced into the setup which would match the round trip time of flight for a target at 150m (τ ≈1µs). Since the phase of a LFM signal cannot be controlled to an accurate enough degree for digital matched filtering, it was necessary to use coherent on receive processing to analyze the return signal [1]. Coherent on receive processing was implemented by splitting a portion of the signal before it entered the delay line, and coupling it to a photodetector. The resultant signal was then digitized and utilized as the matched filter, which was cross-correlated with the delayed signal. Figure 5 shows a diagram of the modified experimental setup.
Fig. 5 Schematic of the new experimental setup to test a time delay in the system. The signal was delayed by ~1µs. Refer to Fig. 1 for the component legend.
Figure 6 shows the results of the target simulation experiment using two 37MHz chirps with a 50.1281MHz difference frequency. Figure 6a is the entire matched filter output of the signal which shows a main lobe shifted to the right as well as a smaller lobe on the left separated by a distance of T/2 from the main lobe. This second lobe is due to the fact that we are operating near the slew rate of the AOM, and would not show up if we had an AOM with a shorter slew rate. Figure 6b shows the matched filter output of the signal zoomed in to show that the main lobe was shifted to τ/T = 0.252, which corresponds to a time delay of 1.008µs.
Fig. 6 Matched filter output of a SF-LFM Signal with T = 4µs, B = 37 MHz, df = 50.1281 MHz, and τ ~1µs. (a) Full matched filter output. (b) Matched filter output zoomed in about τ/T = 0.25.
In free space range is defined as. Therefore, using the measured time delay (τ) it can be shown that a target was simulated at a distance (R) of 151.2 meters. By measuring the −3dB bandwidth of the central lobe in Fig. 6b and using Eq. (2) it can also be shown that the range resolution (δR) is 1.44 meters, which is in agreement with the theoretical models presented in the previous section and Ref. (2).
5. Conclusion and Outlook
The experimental results verify all the predictions of our previous models showing the viability of this signal for real world applications. This provides the necessary proof that these signals are not only mathematically interesting but also physically realizable. Forcing the AOM into such a short pulse duration further highlights the need for SF-LFM ladar signals that can increase the range resolution of a ladar system without the need to increase the modulator bandwidth. In the future we would like to investigate methods to reduce or eliminate the effects of the ghosting without sacrificing range resolution by adding difference frequency jitter, for applications where it is advantageous to have the chirped bandwidths overlapped. We would also like to attempt to use a second AOM to generate several lines from a single laser in order to experimentally verify the results of the multiple chirp modeling [11]. Finally now that the proof of principle experiment has been performed we would like to take the experiment out of fiber and verify the ability of this signal to detect actual targets at range.
Acknowledgments
This effort was supported in part by the U.S. Air Force through contract number FA8650-06- 2-1081, and the University of Dayton Ladar and Optical Communications Institute (LOCI). The authors would like to extend special thanks to Nicholas Miller, Dr. Bradley Duncan, Dr. Igor Anisimov, and everyone else at LOCI for their help and support. The views expressed in this article are those of the authors and do not reflect on the official policy of the Air Force, Department of Defense or the U.S. Government.
References and links
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