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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 15 — Jul. 19, 2010
  • pp: 15400–15407
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Experimental verification of sparse frequency linearly frequency modulated ladar signals modeling

Robert V. Chimenti, Matthew P. Dierking, Peter E. Powers, Joseph W. Haus, and Eric S. Bailey  »View Author Affiliations


Optics Express, Vol. 18, Issue 15, pp. 15400-15407 (2010)
http://dx.doi.org/10.1364/OE.18.015400


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Abstract

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 OSA

1. Introduction

Coherent LFM radar pulse trains are common signal waveforms used by the radar community [1

1. N. Levenon, and E. Mozeson, Radar Signals, (Wiley-Interscience, 2004).

], 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

2. C. J. Karlsson and F. Å. A. Olsson, “Linearization of the frequency sweep of a frequency-modulated continuous-wave semiconductor laser radar and the resulting ranging performance,” Appl. Opt. 38(15), 3376–3386 (1999), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-15-3376. [CrossRef]

,3

3. C. J. Karlsson, F. Å. A. Olsson, D. Letalick, and M. Harris, “All-Fiber Multifunction Continuous-Wave Coherent Laser Radar at 1.55 num for Range, Speed, Vibration, and Wind Measurements,” Appl. Opt. 39(21), 3716–3726 (2000), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-39-21-3716. [CrossRef]

]. Nordin and Hyyppa utilized complex thermal modeling to produce LFM output from distributed feedback diode lasers [4

4. D. Nordin and K. Hyyppa, “Using a discrete thermal model to obtain a linear frequency ramping in a FMCW system,” Opt. Eng. 44(7), 74202–74205 (2005). [CrossRef]

]. We overcome the difficulty of achieving large modulator bandwidths by taking a cues from sparse aperture imaging [5

5. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–5943 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-23-5933. [CrossRef] [PubMed]

,6

6. R. L. Lucke, “Fundamentals of Wide-Field Sparse-Aperture Imaging,” in 2001 IEEE Aerospace Conference Proceedings (Institute of Electrical and Electronics Engineers, Big Sky,” Montana (March): 10–17 (2001).

], chirped synthetic-wavelength interferometry [7

7. P. de Groot and J. McGarvey, “Chirped synthetic-wavelength interferometry,” Opt. Lett. 17(22), 1626–1628 (1992), http://www.opticsinfobase.org/abstract.cfm?URI=ol-17-22-1626. [CrossRef] [PubMed]

], and sparse frequency radar [8

8. W. X. Liu, M. Lesturgie, and Y. L. Lu, “Real-time sparse frequency waveform design for HFSWR system,” Electron. Lett. 43(24), 1387–1389 (2007). [CrossRef]

,9

9. M. J. Lindenfeld, “Sparse Frequency Transmit and Receive Waveform Design,” IEEE Trans. Aerosp. Electron. Syst. 40(3), 851–861 (2004). [CrossRef]

] communities. By using two frequency-shifted coherent sources, each having a continuous wave LFM (CW-LFM) imposed on them [10

10. R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Sparse frequency LFM ladar signals,” Opt. Express 17(10), 8302–8309 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8302. [CrossRef] [PubMed]

], 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.

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,
|χ(τ,0)|=I×ILO|sinc(Bτ)ei2π(fo+B2)τ(1+ei2πdfτ)+BdfB[δ(τ+TdfB)+δ(τTdfB)]ei2π2dfB(df+2fo)τ{sinc((Bdf)τ)ei2πi(fo+B+df2)τ,    if dfB0,    if df>B|,
(1)
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

10. R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Sparse frequency LFM ladar signals,” Opt. Express 17(10), 8302–8309 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8302. [CrossRef] [PubMed]

]. 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 τ=±Tdf/Bwith 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

10. R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Sparse frequency LFM ladar signals,” Opt. Express 17(10), 8302–8309 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8302. [CrossRef] [PubMed]

,11

11. R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Multiple chirp sparse frequency LFM ladar signals,” Proc. SPIE 7323, 73230N (2009). [CrossRef]

] 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,
δR=δτ×c2,
(2)
was calculated as the model steps through difference frequencies in 1MHz increments [10

10. R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Sparse frequency LFM ladar signals,” Opt. Express 17(10), 8302–8309 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8302. [CrossRef] [PubMed]

]. 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
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 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.

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

Table 1. Range resolution and PSLR for a SF-LFM ladar signal with a modulator bandwidth of 37MHz and a pulse duration of 4µs.

table-icon
View This Table
. Figure 2
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.
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.

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
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.
. 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.

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
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.
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

13. R. V. Chimenti, “Sparse Frequency Linear Frequency Modulated Laser Radar Signal Generation, Detection, and Processing,” M. S. Thesis (University of Dayton, Dayton, OH, 2009).

].

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

1. N. Levenon, and E. Mozeson, Radar Signals, (Wiley-Interscience, 2004).

]. 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
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.
shows a diagram of the modified experimental setup.

Figure 6
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.
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.

5. Conclusion and Outlook

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

1.

N. Levenon, and E. Mozeson, Radar Signals, (Wiley-Interscience, 2004).

2.

C. J. Karlsson and F. Å. A. Olsson, “Linearization of the frequency sweep of a frequency-modulated continuous-wave semiconductor laser radar and the resulting ranging performance,” Appl. Opt. 38(15), 3376–3386 (1999), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-15-3376. [CrossRef]

3.

C. J. Karlsson, F. Å. A. Olsson, D. Letalick, and M. Harris, “All-Fiber Multifunction Continuous-Wave Coherent Laser Radar at 1.55 num for Range, Speed, Vibration, and Wind Measurements,” Appl. Opt. 39(21), 3716–3726 (2000), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-39-21-3716. [CrossRef]

4.

D. Nordin and K. Hyyppa, “Using a discrete thermal model to obtain a linear frequency ramping in a FMCW system,” Opt. Eng. 44(7), 74202–74205 (2005). [CrossRef]

5.

N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–5943 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-23-5933. [CrossRef] [PubMed]

6.

R. L. Lucke, “Fundamentals of Wide-Field Sparse-Aperture Imaging,” in 2001 IEEE Aerospace Conference Proceedings (Institute of Electrical and Electronics Engineers, Big Sky,” Montana (March): 10–17 (2001).

7.

P. de Groot and J. McGarvey, “Chirped synthetic-wavelength interferometry,” Opt. Lett. 17(22), 1626–1628 (1992), http://www.opticsinfobase.org/abstract.cfm?URI=ol-17-22-1626. [CrossRef] [PubMed]

8.

W. X. Liu, M. Lesturgie, and Y. L. Lu, “Real-time sparse frequency waveform design for HFSWR system,” Electron. Lett. 43(24), 1387–1389 (2007). [CrossRef]

9.

M. J. Lindenfeld, “Sparse Frequency Transmit and Receive Waveform Design,” IEEE Trans. Aerosp. Electron. Syst. 40(3), 851–861 (2004). [CrossRef]

10.

R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Sparse frequency LFM ladar signals,” Opt. Express 17(10), 8302–8309 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8302. [CrossRef] [PubMed]

11.

R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Multiple chirp sparse frequency LFM ladar signals,” Proc. SPIE 7323, 73230N (2009). [CrossRef]

12.

R. V. Chimenti, E. S. Bailey, R. V. Dierking, M. P. Powers, P. E. Haus, and J. W. Haus, “A review of sparse frequency linearly frequency modulated (SF-LFM) laser radar signal modeling with preliminary experimental results,” 15th Coherent Laser Radar Conference (2009).

13.

R. V. Chimenti, “Sparse Frequency Linear Frequency Modulated Laser Radar Signal Generation, Detection, and Processing,” M. S. Thesis (University of Dayton, Dayton, OH, 2009).

OCIS Codes
(040.2840) Detectors : Heterodyne
(280.3400) Remote sensing and sensors : Laser range finder

ToC Category:
Remote Sensing

History
Original Manuscript: February 8, 2010
Revised Manuscript: June 17, 2010
Manuscript Accepted: June 23, 2010
Published: July 6, 2010

Citation
Robert V. Chimenti, Matthew P. Dierking, Peter E. Powers, Joseph W. Haus, and Eric S. Bailey, "Experimental verification of sparse frequency linearly frequency modulated ladar signals modeling," Opt. Express 18, 15400-15407 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15400


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References

  1. N. Levenon and E. Mozeson, Radar Signals, (Wiley-Interscience, 2004).
  2. C. J. Karlsson and F. Å. A. Olsson, “Linearization of the frequency sweep of a frequency-modulated continuous-wave semiconductor laser radar and the resulting ranging performance,” Appl. Opt. 38(15), 3376–3386 (1999), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-38-15-3376 . [CrossRef]
  3. C. J. Karlsson, F. Å. A. Olsson, D. Letalick, and M. Harris, “All-Fiber Multifunction Continuous-Wave Coherent Laser Radar at 1.55 num for Range, Speed, Vibration, and Wind Measurements,” Appl. Opt. 39(21), 3716–3726 (2000), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-39-21-3716 . [CrossRef]
  4. D. Nordin and K. Hyyppa, “Using a discrete thermal model to obtain a linear frequency ramping in a FMCW system,” Opt. Eng. 44(7), 74202–74205 (2005). [CrossRef]
  5. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–5943 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=ao-46-23-5933 . [CrossRef] [PubMed]
  6. R. L. Lucke, “Fundamentals of Wide-Field Sparse-Aperture Imaging,” in 2001 IEEE Aerospace Conference Proceedings (Institute of Electrical and Electronics Engineers, Big Sky,” Montana (March): 10–17 (2001).
  7. P. de Groot and J. McGarvey, “Chirped synthetic-wavelength interferometry,” Opt. Lett. 17(22), 1626–1628 (1992), http://www.opticsinfobase.org/abstract.cfm?URI=ol-17-22-1626 . [CrossRef] [PubMed]
  8. W. X. Liu, M. Lesturgie, and Y. L. Lu, “Real-time sparse frequency waveform design for HFSWR system,” Electron. Lett. 43(24), 1387–1389 (2007). [CrossRef]
  9. M. J. Lindenfeld, “Sparse Frequency Transmit and Receive Waveform Design,” IEEE Trans. Aerosp. Electron. Syst. 40(3), 851–861 (2004). [CrossRef]
  10. R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Sparse frequency LFM ladar signals,” Opt. Express 17(10), 8302–8309 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8302 . [CrossRef] [PubMed]
  11. R. V. Chimenti, M. P. Dierking, P. E. Powers, and J. W. Haus, “Multiple chirp sparse frequency LFM ladar signals,” Proc. SPIE 7323, 73230N (2009). [CrossRef]
  12. R. V. Chimenti, E. S. Bailey, R. V. Dierking, M. P. Powers, P. E. Haus, and J. W. Haus, “A review of sparse frequency linearly frequency modulated (SF-LFM) laser radar signal modeling with preliminary experimental results,” 15th Coherent Laser Radar Conference (2009).
  13. R. V. Chimenti, “Sparse Frequency Linear Frequency Modulated Laser Radar Signal Generation, Detection, and Processing,” M. S. Thesis (University of Dayton, Dayton, OH, 2009).

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