TIME REVERSED PHOTONIC BEAMFORMING OF ARBITRARY WAVEFORM LADAR ARRAYS

Joseph L. Cox U. S. Air Force

Space and Missile Systems Center Los Angeles, CA

Henry Zmuda

Department of Electrical and Computer Engineering University of Florida

Gainesville, FL

Rebecca J. Bussjaeger Reinhard K. Erdmann

Michael L. Fanto Michael J. Hayduk John E. Malowicki Sensors Directorate

Air Force Research Laboratories Rome, NY

ABSTRACT

Herein is described a novel approach of performing adaptive photonic beam forming of an array of optical fibers with the expressed purpose of performing laser ranging. The beam forming technique leverages the concepts of time reversal, previously implemented in the sonar community, and wherein photonic implementation has recently been described for use by beamforming of ultra-wideband radar arrays. Photonic beam forming is also capable of combining the optical output of several fiber lasers into a coherent source, exactly phase matched on a pre-determined target. By implementing electro-optically modulated pulses from frequency chirped femtosecond-scale laser pulses, ladar waveforms can be generated with arbitrary spectral and temporal characteristics within the limitations of the wide-band system. Also described is a means of generating angle/angle/range measurements of illuminated targets.

KEYWORDS: Photonics, Adaptive Beamforming, Phased Array Antennas, Time Reversal, Ladar, Arbitrary Waveform Generation

1 INTRODUCTION 1.1 Photonic Beam-Forming For Array Antennas A ladar designer may wish to increase the useable range of a ladar system or increase its probability of detection, decrease the false alarm rate or otherwise capture some new phenomenology with the system. Other than advancements in detectors or increasing the size of the ladar optics, the ladar designer will typically require higher optical output power from a laser source while maintaining the desired spectral and temporal characteristics. Superposing the output from several sources into one is an advantage of beam forming, yet in laser sources this is a difficult enterprise due to rapid decoherence of the laser pulses or the inability to adequately phase match them. Beam-forming of radio

Invited Paper

Enabling Photonics Technologies for Defense, Security, and Aerospace Applications III, edited by M.J. Hayduk, A.R. Pirich,P.J. Delfyett Jr., E.J. Donkor, J.P. Barrios, R.J. Bussjager, M.L. Fanto, R.L. Kaminski, G.Li, H.Mohseni, E.W. Taylor,

Proc. of SPIE Vol. 6572, 657209, (2007) · 0277-786X/07/$18 · doi: 10.1117/12.723749

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frequency arrays by means of photonics has been widely described in the literature [1-4]. Further, recent publication by the authors asserts the use of photonic beam forming for adaptive RF phased arrays is possible using time reversal [5]. The theoretical ladar system described in this paper utilizes a broadband supercontinuum source, whereby its frequency is linearly chirped in time, as an initial laser probe pulse illuminating the target scene as in Figure 1(a). The energy from the target and clutter is received by each element of the array, described in Figure 1(b), amplified, photonically time-reversed, and retransmitted toward the reflecting object, Figure 1(c). An important application of phased array antennas is that of adaptation of the arrays. This is where the control signals used to create a particular radiation pattern are generated in a manner so as to optimize some aspect of antenna performance such as its ability to automatically track one or more targets, to place nulls strategically, or some other user-defined function [4]. Thus, this ladar system generates adaptive beamforming by shaping the laser pulse beam-front using the time reversal concept. In addition, the system is able to make use of the superposition of several independent laser pulses from small lightweight fiber optic lasers to generate high peak powers at the reflecting object on the return pulse.

Figure 1: Sequence of events to implement time reversal. (a) An interrogation signal is transmitted from one antenna element. (b) The interrogation signal is scattered from the target(s) and received by all the array elements. (c) A time reversal processor is used to time gate the received signal, time reverse them, and amplify then retransmit these signals resulting in maximum target fluence. 1.2 Time Lensing Time Reversal, a process well-known in the sonar community and explored in the literature under the title of “time lens” [6], is an adaptive array process based on the premise that if one can reverse a signal ( )f t , that is generate

( )f t− , then the signal can in principle be traced back to its source [7]. This is particularly useful in laser propagation applications since the information contained in the time reversed signal includes any irregularities in the path of propagation, i.e., index variations in a random atmosphere, beam diffraction due to propagation in inhomogeneous (e.g., layered) media, multiple scattering due to clutter or additional targets, and so forth. This information in turn allows for the automatic determination of the array element time delay needed to steer the aperture back on to the target from which the signal originated. Normally the required time reversal signal processing in a corresponding RF phased array would be performed by a digital signal processing system (DSP) [8, 9]. However, for those instances where the operational bandwidth is so large that a suitably fast analog-to-digital converter (ADC) does not exist, a photonic can be used to perform the time reversal in the optical domain. The applicability of time reversal to the present discussion is based on the reciprocal nature of the wave equation. In essence, a signal ( )f t is passed through a system with an output of ( )f t T− + , the delay T being necessary to ensure that the signal remains causal. This effect can be realized using a photonic system, as will be explained in Section 2.1. Supposing that the signal ( )f t is the retro-reflected signal from a distant target, the time-

reversed ( )f t T− + signal, when re-transmitted through the ladar system, will serve as a precise true-time delay beam forming system. Thus the re-transmitted energy is focused by means of time reversal and a “time lens” is achieved. This concept has been employed with great success for the treatment of cancer using microwave hyperthermia with array apertures and time reversed signal processing, but at frequencies and bandwidths significantly lower than those discussed in this paper [10].

(a) (b) (c)

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1.3 Supercontinuum Sources The use of supercontinuum sources for laser radar is covered in the literature [11]. Previous work utilized femtosecond and shorter pulsed lasers for the purpose of generating white light sources extent in space for further illumination of the scene. The white light was generated in the atmosphere due to the ionization of the air by extreme electric field intensities. Like the system that will soon be described in this paper, the supercontinuum ladar uses dispersion to focus energy temporally and spatially. There were suggestions in the literature that the supercontinuum ladar be used for fluorescence imaging of extended targets. However, use of the supercontinuum for arbitrary waveform generation or manipulation of the spectrum by modulation of the pulses, as will be described in this paper, has not been shown. 1.4 Arbitrary Waveform Generation For maximum flexibility in a fielded ladar, it is necessary for the ladar to project and process a wide-range of waveforms as required by the operational circumstance. To address this need, the U.S. Air Force is currently investing in ladar technologies that accept, process and transmit any arbitrary waveform within the optical and electrical limitations of the system. A key advantage of using a photonic based time reversed ladar is the ability to manipulate the spectrum of the ladar pulses in an unprecedented way. This paper presents the methods that could be used to select many wavelengths among a wide spectral region, or modulate ultra-short pulses with radiofrequency signals, and time stretch or compress the resultant pulses. Thus, this ladar could produce an unprecedented capability to tune a ladar source spectrally and temporally. 1.5 Comparison to Conventional Laser Radar 1.5.1 Scene Illumination Type. Conventional ladar is usually broadly categorized as either scanning or scannerless [12]. Scanning ladar is accomplished by push-broom or rastor scanning methods that systematically illuminate the entire field of view, one or more IFOV “voxels” at a time. Scannerless ladar illuminates the entire FOV simultaneously, and typically receives the reflected energy from each IFOV voxel onto an independent element of a focal plane array. The system described in this paper performs an initial scene illumination via a laser probe pulse, and develops a priori knowledge of the target scene for the subsequent time reversed ladar beam forming. The resultant time reversed pulses are directed to specific targets or clutter within the scene. Therefore, the photonically based time reversed ladar is a hybrid of scannerless and scanning ladars with the added efficiency of only illuminating on the return pulse voxels that were illuminated during the laser probe pulse. 1.5.2 Coherency. Conventional ladars are further characterized by their use of the phase envelope in detection of the target and classified as either coherent or direct detect [12]. Coherent ladars rely upon the phase envelope of the received pulse, generated by a stable local oscillator, to perform heterodyne mixing to retrieve range and velocity information about the target, whereby direct detect methods do not require phase information but instead use the intensity of the received pulse to determine range to target and only by multiple range reports to develop target velocity information. The system characterized herein has the potential to perform coherent beam combining, and within acceptable coherent lengths produce coherent imaging of the target without the utilization of a local oscillator. The analysis for performance estimation of the system assumes the implementation of femtosecond pulses, hence very short temporal coherent lengths, and is assumed to be operating as a direct detect ladar. 1.5.3 Temporal Characteristics. The temporal characteristics, pulse widths, of laser pulses in laser radars are generally restricted to a narrow region available to the system designer [13]. Laser sources, particularly solid state, are typically operated at a few nanoseconds or more pulse width. For direct detect applications, use of pulse widths smaller than a few nanoseconds is rare as detectors and the associated electronics typically are not fast enough to accommodate the speed of the pulse. Coherent applications suffer similarly since the shorter the laser pulse, the shorter the coherence length and, therefore, the shorter the operational range of the ladar [12]. Utilization of dispersive pulse shaping could enable a designer to select a wide range of temporal profiles from femtoseconds or less to several nanoseconds or more. These pulse widths could be generated while still maintaining the desired spectral characteristics of the original pulse. Furthermore, with photonic time-stretching and time compression techniques, the received pulse can be further manipulated in the time domain so as to be as acceptable as possible to the detector and associated electronics. 1.5.4 Spectral Characteristics. Desired spectral characteristics within the supercontinuum band can be produced using the arbitrary waveform generation techniques previously described. This contrasts sharply with the narrow spectral line widths used in both coherent and direct detect laser radars, restricted to specific wavelengths as allowed by the solid state gain medium selected or by the availability of non-linear optics for frequency conversion [12].

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Many systems under development have co-axially combined pulse trains of different wavelengths, yet the choice of wavelengths were still severely limited by the available laser sources. Supercontinuum ladar has been previously reported in the literature with ultra-wide spectral width [11]. Furthermore; some systems have employed wide spectral reception in order to detect fluorescence from the illuminated target. Certainly, the ability to select certain wavelengths separately from pulse to pulse, as will be shown in this paper, is unique to laser radars and is fundamental to arbitrary waveform generation. 1.5.5 Beamforming. Most conventional ladar is predicated on a divergent laser pulse from an optic train designed such that the waist of the beam is at or near the laser aperture [12]. As a result, nearly all of objects within range of the ladar lie in the Fraunhofer region. The mathematics of size determination is quite simple with angular range measurements followed by the derived range information. Many laser radar systems tend to use Gaussian or super-Gaussian laser modes due to the predictable propagation characteristics throughout the operating ranges of the ladars. For direct detect systems that require illumination of only one voxel at a time, the Gaussian mode can overfill the voxel resulting in partial illumination of neighboring voxels and hampering the cross-range resolution, or under fill resulting in decreased signal from the target voxel. Scannerless scene illumination systems typically desire to use square focal plane arrays due to the ease of FFTs in image processing. These systems work best with near uniform illumination, a characteristic that is only obtained near the peak of the Gaussian profile. Implementation of the Gaussian mode on an FPA based ladar results in non-uniform illumination and decreased probability of detection about the edges of the FPA or in systemic waste of available fluence outside of the ladar FOV. A successful approach to mitigate these issues is to produce square “top hat” beam profiles for the propagating beams by implementation of various diffractive optics. Unfortunately, these methods tend to be useful for only a specified narrow range as the diffractive effects become unbearable too close or too far away from the allowable range. The photonic based time reversed ladar is assumed to use Gaussian propagation modes emanating from the fiber outputs. There is no intentional focus or collimation of the superimposed phase fronts before or after the target. The beamforming effect of the time lensing is only relevant exactly at the target of interest. Therefore, the beamforming result will be as large as the target reflected from the initial probe pulse but as minimized as is possible by the combination of time reversed pulses. Maximum possible uniform fluence on the interrogated target will be generated. Furthermore, the technique is applicable whether the target is within the Fraunhofer or Fresnel region of the ladar. The use of adaptive optic beamforming has been suggested for implementation on ladar. Adaptive optic beamforming requires the a priori knowledge of the atmospheric characteristics in order to induce compensation and are limited by their size, weight, complexity and expense. The adaptive beam forming generated by the photonic time reversed ladar could provide similar results in a simple low-cost and straightforward approach. 1.5.6. Depth of Field. Historically, ladars have only produced one reported range per voxel, known as single pulse detection. Lately, multiple pulse detection has been implemented to produce more detailed imagery of semi-transparent scenes, such as camouflage or foliage [14]. The depth of the field of view is a measurement of the number of possible range reports within each IFOV. For the case of a time reversed ladar, the system could detect as many voxels as are illuminated by the return pulse. Because the time reversed beam forming does not propagate in a strictly constant divergence as do other ladars, it is possible to have much higher intensities behind an initial range report than would be possible with a conventional ladar. 1.5.7 Clutter Rejection. As the rejection of clutter is a critical function in ladar systems design, most systems depend upon a few popular methods to do so: spectro-polarimetry [15] or multiple pulse detection [14]. This system would be capable of performing clutter rejection at the voxel level by implementation of spectral discrimination. Polarization, however, is neglected throughout the paper for the purpose of simplification of the concepts. With appropriate modulation of the ultra-short pulses, it is possible to produce vibrometry of the target scene and therefore eliminate all objects within the scene that do not represent vibrations (such as engine noise, etc.). Range gating is another popular tool in conventional ladars if there is enough knowledge of the target space to narrow the scene by use of range gates. Similar to the range gate approach, a time reversed ladar would use time gating to “clock out” areas of the target scene.

2 PHOTONIC IMPLEMENTATION 2.1 Photonic Based Time Reversal The fundamental element of photonic based time reversal is the use of dispersion within certain fibers to manipulate the optical waveform in the temporal dimension. Central to the photonic time reversed ladar system is the chirped fiber Bragg grating, Figure 2. Chirped fiber Bragg gratings are typically used to negate the effects of dispersion.

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Figure 2: Chirped fiber Bragg grating. A pulse impending upon the fiber Bragg grating is reflected such that the longer wavelength is reflected first and the shorter wavelength is reflected last. in optical channels [16], but in our application the dispersive effects are made use of by inverting a chirped pulse in time. If a pulse chirped with the shortest wavelength first is sent to a properly constructed fiber Bragg grating, it will be inverted in time so that the longest wavelength is sent first and the shortest wavelength last. Therefore, the signal would be time reversed. 2.2 Fiber Optic Transmit and Receive Element In photonic based implementation of UWB radar beamforming a small 16 nm spectral band of an available 274 nm band was recommended as this band was the only band considered flat enough for modulation of an RF signal. In this ladar application, spectral flatness is of less of a concern as the characteristic spectra of the source is assumed to be well known and accounted for in the generation of the arbitrary ladar waveform. The electro-optical modulator (see Figure 3) may be able to increase the flatness of the amplitude of the spectral band if desired. Thus, a spectral band of 274nm, centered at 1550nm, is readily available in commercial supercontinua sources [5]. Dispersive fiber may be used to increase the chirp from the laser source, if the chirp is insufficient to begin with. The chirped pulses are stretched in time to an acceptable pulse width and then modulated by the use of an electro-optical modulator. Characteristics of the modulation and requirements for the pulse stretch are described later in the section on arbitrary waveform generation. The transmission of the pulse from the system is assumed to be through SMF-28 fiber terminating on air. The core diameter of SMF-28 fiber, where upon the optical energy is carried within the fiber, is 8.25µm [17] and the coating diameter is 245 µm. For the purposes of modeling the far field effects, the propagation is assumed to be Hermite-Gaussian, TEM0,0, diffracted by the circular aperture that is defined by the coating. Using this dimension instead of the fiber core is more realistic in the context of manufacturability of the array. This far field diffraction can be obtained easily by the use of mode matching optics transforming the Bessel nature of the field propagating within the fiber, so the realism of this assumption is assured. This component is described in Figure 3 as a beam expander.

Figure 3: Generation of interrogation pulse. A pulsed laser is used to generate a comb of fsec class pulses which are passed through dispersive fiber to increase the temporal width of the frequency chirp. An electro-optical modulator is activated on each pulse, altering the spectra as desired. Beam shaping optics may or may not be used to output the interrogation pulse and receive energy from the target. At the bottom of the diagram are the chirped Bragg grating and optical amplifier used during receive mode. The spectral diagrams on the right demonstrate the output pulse, top, and energy reflected off of the target, bottom.

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The system only requires one transmission module and it may be separate optically from any of the receive modules. In the example provided in Figure 4, the transmitter is assumed to be co-boresighted with a receiver whereby optical separation is provided with a circulator. A telescope or “beam expander” may be attached to the end of the receive module to decrease the FOV and increase signal received. Each receiver module consists of two components, an optical amplifier and a chirped Bragg grating. The chirped Bragg grating is positioned between the receiver and the chirped Bragg grating so that the received, and therefore re-transmitted signal, will benefit from two-fold gain. Gain from an Erbium doped fiber amplifier is typically 30dB in the spectral range of interest [18]. The chirped Bragg grating reverses the optical signal in time as described in section 1.2. The time-reversed, amplified signal is re-transmitted to free space and back to the original target by the process of time lensing. As was described in previous publication [5], dispersive effects on a Gaussian pulse maintain the Gaussian characteristics of the pulse.

Figure 4: The time reversal module consists of a chirped Bragg grating and an optical amplifier. The spectral characteristics of the chirped input and output pulses are described on the right. 2.3 Beamforming Array An array of receive elements can be constructed of any convenient number of elements and are naturally conformal to any desired shape thanks to the time reversal process. Each receive element operates independently from the other elements and the transmitter. Described in Figure 5 is a notional linear array of a single transmitter and four receive modules. The far field pattern of an independent element transmitting a time-reversed pulse is no different than if it where transmitting any other pulse. As in phased array radars, the constructive and destructive interference of the radiating elements is what constructs the unique far field pattern and increases the overall system gain in the direction of the target.

Figure 5: A small linear array of four receive elements and one transmit element (in green). The time reversed output from the receive modules is described on the right. 2.4 Arbitrary Waveform Generation By implementation of photonic time stretch techniques, the pulse envelope can be expanded to much greater pulse widths than will be generated by the supercontinuum generator. Pulse widths of several nanoseconds can be obtained with suitable lengths of dispersive fiber. However, since this technique is not readily adaptable from pulse to pulse as it requires a structural implementation as opposed to digitization, it is not as flexible operationally as are the digital techniques that will be soon described. Nevertheless, for the purpose of arbitrary waveform generation, a pulse width of arbitrary lengths can be selected over a very broad range, from a few picoseconds depending on the source to several nanoseconds.

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2.4.1 Spectral Modification. An example of spectral modification of the laser pulse is described in Figure 6. Field data obtained on Bradford pear bark [15] is shown in Figure 6(a). The data covers the spectral region of the chirped pulse, displayed with respect to time in Figure 6(b). The necessary waveform for the EOM to produce this pulse, assuming flat spectra, is shown in Figure 6(c). The experimentally derived data in Figure 6(a) spans 150 points between 1413nm and 1675nm, the spectral range of this theoretical ladar, with an average of 1.86nm between points. Time stretching of the supercontinuum pulse by use of additional dispersive fiber could easily increase the pulse length to 1 µsec [19, 20] making it easier to digitize a representative signal for the EOM. The spectra are well defined with 12 bits of resolution and the DAC would need to operate at 150MHz, well within the range of commercially available DACs. An additional application of spectral modification is to compensate for loss mechanisms in the atmosphere. For example, if spectral transmissivity through the atmosphere is known and a uniform spectral illumination is desired at the target, compensation can be performed simply by applying the inverse of the known atmospheric transmissivity to the EOM waveform. Incidentally, with a properly arranged detector, the transmissivity through the atmosphere can be determined on the first pulse from the interrogator such that subsequent pulses have corrected for the atmospheric aberrations.

Figure 6: The optical signature of Bradford pear bark [15], (a), can be transmitted from the ladar by modulating the chirped pulse, (b), with the RF signal in (c). Another application of the spectral modification technique is shown in Figure 7. Figure 7(a) is spectral reflectance data obtained of soil from Eglin Air Force Base in the identical spectral band and resolution as the Bradford pear bark in Figure 6(a) [15]. Identical Bradford pear bark data is shown next to Eglin soil as Figure 7(b). If the interrogation pulses were incident on Eglin soil and modulated such that the spectral output was as shown in Figure 7(c), the Eglin soil would irradiate with an optical signature identical to Bradford pear bark. The laser intensity plot of Figure 7(c) was obtained by dividing the reflectance of Bradford pear bark at each data point in the spectral band by the corresponding reflectance of Eglin soil.

Figure 7: The optical signature of Bradford pear bark [15], (b), can be generated on reflection from Eglin soil [15], (a), by modulating the interrogation pulses to have the output in (c).

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2.4.2 Spectral Discrimination. If a flat spectrum is transmitted to the target, its return pulse will be altered spectrally by the atmosphere and most notably by the spectral reflectance of the target. If the spectra can be adequately resolved and compared quickly with a database of spectra types the ladar could discriminate on the basis of the spectra. To implement this process at each element of the array the target return pulse would be stretched within a dispersive fiber to an acceptable pulse length such that when the pulse is detected commercially available ADCs can resolve the signal and recover the target spectra with enough resolution to be analyzed. The spectra would be compared digitally to a database of desired target spectra and if a match is discovered, the system would recognize the returned pulse as having originated from a genuine target. Alternatively, the analog to digital code detection described could be placed just after the optical amplifier at each element and use only a small portion of the signal by means of a fiber coupler. The remaining signal would be passed through a suitable delay line and connected to an optical switch (or shutter) that is activated by the digital comparator. Therefore, only targets (or clutter) that have the appropriate spectra will receive focused energy during the return pulse. 2.4.3 Pulse Doppler and Chirped Waveforms. A variety of radio-frequency waveforms can also be modulated onto the spectrum. Depicted in Figure 8(a) is a Doppler waveform, and in Figure 8(b) a chirped waveform. A sinusoid modulated over a 1 µsec pulse and compressed to 10 fsec would have a pulse compression ratio of 108. The chirped waveform would experience similar compression, but would be most useful in absolute range determination after heterodyning the received pulse (down-converted to RF by means of a detector) with the transmitted waveform [21]. Shown in Figure 8(a) is an arbitrary 5MHz CW sinusoid modulating the optically chirped pulse spectrum. In Figure 8(b) a linearly chirped RF waveform from 1 MHz to 10 MHz has modulated the 1 µsec long optically chirped pulse. 2.4.4 Code Modulation of the Waveform. In a spectrally rich environment whereby several sensors are operating simultaneously, it may be useful for the ladar array to act only upon signals that are authenticated as originating from the array. A possible solution is to generate a pseudo-random binary code known only to the ladar system and use the code to modulate the spectrum. Figure 8(c) shows a representation of a code modulated waveform, with 4 bits of code used for each of 150 points in the spectral band (a total of 600 bits for the code). The modulation would be carried out by an EOM, as before, with the RF signal input to the EOM generated by a DAC having digital input from the pseudo-random code generator. The ladar system can be arranged such that it will only recognize as a genuine target return pulses that match the original code. Alternatively, as described in the spectral discrimination discussion, the code detection can be placed in the front end of each receive module aft of the optical amplifier such that only pulses that match the original code will receive a return pulse from the array.

Figure 8: Three types of RF waveforms are shown modulated onto a time-stretched (to 1 µsec) optically chirped pulse: (a) a sinusoid, (b) an RF chirp, and (c) a code modulated waveform. 2.5 Target Detection and Signal Processing The array used for beamforming can also be used for target detection, ranging, and angle/angle measurements. Within each receive module is generated the time reversed signal of the return pulse. This is mathematically the same as phase conjugation [22]. Combination of the phase conjugated pulse with the original pulse creates a brief spike in optical intensity. Figure 9 shows the combination of a target return pulse with its phase conjugate by coupling some of the energy from the return pulse into a fiber which is then sent to a short delay line, and coupling a similar amount of energy from the time reversed pulse into a second fiber. The two pulses are sent into a head on collision down a length of fiber generating the optical spike. Only a small portion of the energy is removed from the system for the purpose of detection, the rest is allowed to propagate to the target. A significant point is that detection of the target can be performed

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Figure 9: A portion of the signal returned from the target is removed from the optical train and combined with a similar intensity of signal from the chirped Bragg grating. The collision of the return pulse with its phase conjugated counterpart generates a brief spike in optical intensity. after the interrogation pulse that is prior to the time reversed pulse reaching the target. Because the intensity of the time reversed pulse at the target is expected to be much greater than the intensity of the interrogation pulse, the probability of detection of the target should be much greater after the return from the time reversed pulse is received. This process can be extended to mixing of signals within a subsection of the array. Figure 10 is a depiction of nine elements of an array cross-mixing the conjugated pulses with the original receive pulses. For each array element a pulse is returned from the target and energy deposited down a length of fiber. Residual energy from the return pulse is time-reversed by the receive module and some of the reversed energy is propagated down the fiber of a neighboring receive element. As the pulses received from the target are identical in phase, the conjugation of a pulse with its neighbor is identical to the same process with only one receive element. The choice of which elements to cross-mix is made such that the propagation distances between each pair of elements is identical compared to the center of the array subsection. In Figure 10 the top-most element is cross-mixed with the bottom-most element, the second to top element is cross mixed with the second from the bottom, and so on. Further description of this pairing is shown in Figure 11(a), showing the “focal plane” of a single quad cell whereby each element is numbered with respect to its pair. Each of the pairs are equidistant from the center of the subarray labeled “A”. The center receive element would have its conjugated pulse mixed with itself and is represented in Figure 10 by the middle element whose phase conjugated pulse does not cross over to any neighboring element.

Figure 10: Cross mixing of the return pulses with the phase conjugated pulses of equidistant neighboring receive modules will generate optical intensity spikes simultaneously across the subarray.

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D

Figure 11: Cross mixing of conjugated pulses is shown within the subarray labeled “single quad cell” on the left (a) representing one fourth of the larger detector on the right (c). The simultaneous optical spike corresponds with the distance from the center of the cell to the target. This characteristic enables angle/angle calculations (b) to be performed at the quad cell detector. Because each of the elements is cross-mixed with a pair that is equidistant from the center of the array subsection, regardless of the position of the target with respect to the array, all of the signals generated by combination of conjugated with original pulses will occur at the same time. The timing of this signal “spike” is dependent upon the distance of the center of the subarray, designated by the letter A in Figure 11(a), to the target. By concentrating these fibers in such away that all pulses from a subarray are equidistant from a detector, the detector will receive all of the energy from each paired element simultaneously. A device suggested to accomplish this is to place a detector within as small as possible transparent cylinder, and wrap the fibers around the cylinder. This way the signals, though potentially generated at any location along the fiber, will be equidistant from the detector at the center. The signal received by a detector at a single quad cell is enough for the system to measure range to target. But by gathering the signal from four separate quad cells aligned as shown in Figure 11(c), the timing offset from the received signals can be interpreted into angle/angle measurements [23]. This is typically an untenable process as the time differential between the cells of a quad cell detector is very short compared to the pulse width of the reflected pulse. But in a quad cell detector as shown if Figure 11 (c) of SMF-28 fibers (245 µm thick coating), the distance between the center of the cells is on the order of a picosecond (10-12) which is still far larger than the femtosecond (10-15) class pulses generated by the interrogation source. As a stressing case, assuming the quad cell consisted of only 10 by 10 array of fiber elements, 245 µm thick, and a relatively long 20 fsec pulse is used by the interrogator, the quad cell should be able to distinguish angles of at least 60 mrad (3.4 deg). The authors anticipate fielding arrays of many more elements and use of much shorter pulses for interrogation. The equations to derive angle/angle measurements are well known and provided in Figure 11(b).

3 ANTICIPATED PERFORMANCE

3.1 Signal to Noise Slight modifications of the standard ladar equations [23] are necessary to estimate performance of this ladar against a target. The common expression for signal received by a scene illumination ladar is:

2202

2

22 21

4

44

GANR

DRPS

INT

TR τπ

σρλππ

π⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

= (1)

The first part of the equation in brackets represents the illumination pulse in its entirety; P represents the power of the source laser, R for range, DINT for the diameter of the interrogation beam optic, and λ for wavelength. The remainder of the equation is the reception of the signal, transmission through N detector elements of area A, receiving gain through the optical amplifiers, G, twice, while suffering optical transmissivity τ0 twice, and transmitting back towards the target

(a)

(b)

(c)

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of optical cross section σ and reflectivity ρ. In this case, STR represents the signal received, processed, and retransmitted by the time reversal unit and is substituted for the source power in the next modified ladar equation:

2430

228λπσρτ

RNAGSS TR= (2)

Substituting for STR:

485

2330

223316λπτρσ

RDGNPA

S INT= (3)

For simplification, and to estimate performance for a worst case design, we assume no significant beam expansion on the interrogation pulse or the receive elements. But instead of using the beam core as the waist diameter (8.25 µm), calculations will assume that the coating diameter (245 µm) [17] would represent the beam waist (2ω0) and detector diameter. The target is assumed to be a Lambertian surface with a reflectivity of 0.5 and a cross section of 1 m2. The transmissivity through the system is assumed to be 0.8, higher than normal optical systems due to complete processing within photonic domains, but debited by 20% for the purpose of angle/angle calculations. The gain from the optical amplifiers for the spectral region of interest is typically 30dB [18]. Over 200W peak pulse power have been demonstrated in supercontinuum sources with spectra of more than 200nm [24]. Therefore, a reasonable assumption is that the interrogation pulse will have a maximum peak power of 100W (50% optical efficiency). As beam divergence is chromatic, the longest wavelength within the spectral band, 1675 nm, will be used for SNR calculations. Solving for S, N, and R:

8

3

037.11RNS = (4)

The result implies that as range is doubled the number of elements must increase by over 6-fold to maintain the same SNR. These calculations assumed that the ladar would be performing incoherent beam combining as the pulse widths and range to target were such that pulses propagated far beyond their coherence length. By lowering the range or by increasing the pulse width of the interrogation source, the coherence could be preserved, the beam forming conducted coherently, and the intensity on target greatly increased. In spite of the tremendous opportunities supercontinuum sources provide for ladar, the small pulse width of the ladar allows for the reception of a large amount of noise. Detector bandwidth is usually expressed as the inverse of pulse width [23]. For a 20 fsec source the bandwidth would be 5x1013 Hz, on the order of optical frequencies, with an approximate background emittance noise level of -48.5dB. A detector quantum efficiency of 50% is assumed for this estimation. To arrive at acceptable SNR levels of 20dB or more, at a range to target of 100m, an array of nearly 11,000 fibers (a square array of ~105 elements by ~105 elements) would be required. If stacked in a square array coating to coating, the array of fibers would measure 2.55 cm on each side. Because of the microscopic scale of fibers and their ease of manufacture into photonic components, an array of such large numbers is possible and commercially viable. As noted previously, the performance estimation for the time reversed ladar array is based upon several assumptions that must be carefully considered when making a detailed design and thorough analysis. The worst case scenarios of beam widths and system gains were used to determine if the system were viable. 3.2 Multiple Time Reversals An expected yet unstudied result of this ladar would be the development of multiple time reversals. After an interrogation pulse is received from a target and a time reversed signal is propagated to the target, the array could remain in a mode of receiving energy, re-amplifying, and transmitting back to the target. Since the time reversal process is indiscriminate to the nature of the original signal, it will perform the same time lensing function on the time-reversal of a time-reversed signal with only minor modification of the ladar. If energy propagated to the target is greater than the energy originally transmitted to it, gain would be achieved and the signal strength would be expected to grow until system saturation or the relative motion of the system causes a “break-lock” condition. In order to achieve maximum detection of targets in a wide field of view, it may be necessary to increase the gain on the optical amplifiers such that they are vulnerable to developing false reports due to spurious signals. By using the concept of multiple time reversals, false reports in the system will be systematically eliminated by nature of not receiving a re-transmitted pulse off of the target, and by increasing the system signal to noise ratio repetitively.

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3.3 Jitter and Motion Compensation Though a time-reversed ladar system is predicated on a priori knowledge of a target in the field of view, and that the nature of time lensing is acutely dependent upon position of transmitter, target, and the timing of pulses, a time-reversed ladar system is robust against most aspects of jitter or relative motion. Any relative target motion (to include jitter) after the initial interrogation pulse but prior to the receipt of the reflection off of the target would not affect time reversal or the time lensing process so long as the target remains within the field of view of the array elements. Similarly, once the target is irradiated with a time-reversed pulse, the relative motion between the array and the target are inconsequential so long as the target remains within the broad field of view. This is true with any form of jitter or relative motion. The only time when array jitter or relative motion may be cause for concern is during the time-reversal process itself from the reception of the energy onto a receive element, to the re-transmission of its time-reversed pulse. The timeline for this function will vary depending upon the types of delays that are necessary to process temporal or spectral waveforms. For arrays that are arranged such that their detector elements are remoted with fiber optics to another location where the photonic processing will take place, the effects of jitter and relative motion will be much more serious. For more modest applications of time-reversal arrays, adverse results on the system are anticipated for target motion or jitter greater than the optical propagation length of the femtosecond class source (6 nm for a 20 fsec pulse) within time reversal processing timelines on the order of a few picoseconds. This translates into velocities on the order of 1 Mm/s or ~1/100 speed of light. It is unlikely that relative motion or jitter will be as extreme for the applications for which this technology is proposed. The time-reversed ladar array will re-transmit time-reversed pulses that will focus with time lensing onto the point of the original transmission. If the target has moved away from its original location when it was interrogated it will have degraded intensity or may move beyond the field of view. Therefore, the primary concern within the realm of jitter and motion compensation would be the absolute motion of the target between interrogation and completion of time lensing of the time reversed pulse. When implementing multiple time reversals on a target, motion compensation is no more difficult than with the initial interrogation pulse and first time reversal. With each subsequent pulse against a target the time reversal operation is “reset” to the new target location and time reversal is conducted with the same motion resiliency as with the initial pulse. The time-reversed array will, therefore, automatically track targets within its field of view so long as the motion of the targets does not exceed the range or cross-range illumination provided by the array.

3.4 Grating Lobes The coating diameter of SMF-28 fiber is 245±5µm [17]. The least costly technique to manufacture the fiber array would be to join each element at the outer coating layer of the fibers, resulting in a closest spacing between elements equal to the coating diameter. Sophisticated manufacturing may attempt to strip the coating from the fiber elements exposing the cladding prior to joining the elements of the array. If so, the minimum separation between elements would be the cladding diameter of 125±0.3µm. Either way, the separation between elements would be far greater than one half of a wavelength over the entire spectral band and grating lobes would appear [21]. Because the location of the grating lobes is dependent on wavelength, the lobes will suffer from severe chromatic aberration and are not likely to focus energy onto clutter or targets to any intensity similar to the primary propagation beam. 3.5 Ambiguous Range As with most ladar systems, range ambiguity is a function of the pulse repetition frequency (PRF) [12]. The maximum PRF of a ladar or radar is typically chosen as the inverse of the time of flight from the sensor to the target and back to the sensor. For a time-reversed ladar, the PRF should be halved as the system requires two illuminations of the target to effectively range, the first interrogation pulse and a second time-reversed pulse. Unique to the time-reversal concept is that the PRF of the system, during multiple time reversal events, is an automatic photonic-based process and will depend solely on the range to target. Hence, targets at different ranges from the ladar will experience different PRFs, each PRF optimized, automatically, to be at the maximum rate acceptable to negate range ambiguity. For a nominal distance of 100 meters, the PRF of the time reversed ladar should be no greater than 750 kHz. If spectral discrimination were implemented, and the pulse to pulse spectra were modified, the PRF would be a limiting factor to the repetition of an identical spectrum and the system overall would have a nearly unlimited range of PRFs available. 3.6 Range Resolution Range resolution as defined in the literature [23], for a 20 fsec source will be 3 µm. Typical ladar systems of 10 nsec pulse widths, for example, have range resolutions of 1.5 m. Thus this system represents a potential 500,000 fold

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improvement in range resolution. But, as conventional ladar range resolution is limited to the laser pulse widths, a ladar array will most likely be limited in range resolution by the ability of the system to detect the timing of received pulses accurately. This will be particularly true with the quad cell detector herein described. Errors in manufacture, alignment, or mechanical aberrations could easily spread in time the energies of multiple pulses, greatly increasing the actual range resolution. 3.7 Future Considerations Aspects of polarization were ignored for this initial study. The benefits of polarization for target discrimination are well known, and laser sources are selected for ladar due to their polarizing capabilities. With the integration of polarization preserving fiber optics, there is good potential for modification of this system for beamforming of polarized pulses, as well as detection of the polarization state of the energy reflecting off the target.

4 CONCLUSIONS A novel method has been presented that extends the concepts of time reversal as applied to ultra-wideband radar to laser ranging and imaging ladar. This paper addressed the concept of beam forming for power combining and beam steering of a laser to providing time-of-flight ranging to a target as well as angle/angle determination. It did not address the added complexity of multiple targets within a field of view. There are many advantages to using a system based upon supercontinua than conventional laser sources. Within the paper are described the utility of added spectral information within a pulse and the discrimination of targets based upon their perceived spectra. Included is a unique ability to correct for spectrally resolved atmospheric attenuation. Also, the ability to modulate many varieties of waveforms onto the interrogation pulse and receive these pulses by the array provides an unprecedented operational flexibility to ladar. Because time reversal process is independent of the distance of any single element to the target, it is inherently capable of application to conformal arrays and the beamforming will be independent of any thermal, vibrational, or alignment considerations. In a leap beyond application to conformal arrays, the use of photonic fibers readily lends this type of ladar to construction in spaces and structure previously not considered for ladar systems. Upon placing the front end “beam shaping” component of the detectors on the surface of a structure, the fibers, time reversal units, and interrogation system can all be placed remotely from the surface to any place within the body of the structure. In addition to the compact size, the lightweight characteristic of fibers lends this system to implementation on micro and nano-structures. The ability for quad cell detection to accurately determine the angle/angle measurements to a target may be adversely affected by the mechanical environment and would have to be studied by a system designer. The time reversal process is independent of the index of refraction of the medium through which the ladar is propagating. Therefore, it is well suited for use in inhomogeneous media: large temperature gradients, smoke, and battlefield induced contaminants. From large turbulent eddies in the atmosphere that create beam wander in the propagation of a laser pulse to small turbulent eddies that create distorted wavefronts [23], all effects are negated in the time reversal process. Unlike conventional ladar systems, the time-reversed ladar will not suffer from variation in beam sizing, beam breathing and scintillation effects. Applications of this system are possible in fire, rescue, and combat operations where conventional ladar is unusable today.

5 REFERENCES

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5. J. Cox, H. Zmuda, J. Li, P. Sforza, “Ultra wideband photonic control of an adaptive phased array antenna” Proceedings of SPIE, Vol 6243, May 2006, pp 62430V1-9.

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24. G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng and M. Kaivola “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,”Optics Express, Vol. 10, No. 20, pp.1083-1098, Oct 2002.

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