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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 52, Iss. 4 — Feb. 1, 2013
  • pp: 579–599
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Coherent and incoherent synthetic-aperture imaging ladars and laboratory-space experimental demonstrations [Invited]

Liren Liu  »View Author Affiliations


Applied Optics, Vol. 52, Issue 4, pp. 579-599 (2013)
http://dx.doi.org/10.1364/AO.52.000579


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Abstract

This paper reviews our studies on coherent and incoherent synthetic-aperture imaging ladars (SAILs). Using optical diffraction, a systematic theory of side-looking SAIL was mathematically formulated and the necessary conditions for assuring a correct phase history are established. Based on optical transformation and regulation of wavefront, a down-looking SAIL of two distinctive architectures was invented and the basic principle, systematic theory, design equations, and necessary conditions are presented. An incoherent spotlight-mode SAIL was proposed, and detailed mathematically. To validate the concepts, laboratory experiments were conducted. The spatially and temporally dependent laser speckles are analyzed by applying the partial coherence theorem, and proposals to reduce their effect are given. Optical antennas and their components are discussed. It is shown that for down-looking SAIL the width of the scanning strip may be greatly increased without loss of high resolution, and the influences from atmospheric turbulence and unmodeled line-of-sight motion can be automatically compensated.

© 2013 Optical Society of America

1. Introduction

A synthetic-aperture imaging ladar (SAIL) can provide fine resolution, two-dimensional (2D) active imaging at long range with small-diameter optics. This technique was first demonstrated to full 2D strip-mode SAIL imaging at 1.55 μm in laboratory-scale experiments in 2002, with an innovative proposal of using an HCN spectral filter to synchronize the starting chirp frequency [1

1. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002). [CrossRef]

]. And a similar experiment, but of a fixed target with a moving aperture, was afterwards reported [2

2. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005). [CrossRef]

]. The setups for these two experiments were based on fiber optics. As an alternative, we designed a strip-mode SAIL setup consisting of free-space optics to perform laboratory-scale 2D imaging [3

3. Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009). [CrossRef]

]. Then we completed a strip-mode SAIL demonstrator of an antenna aperture of ϕ300mm and verified its performance in the near field of diffraction at a 14 m distance of laboratory space [4

4. L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011). [CrossRef]

]. Moreover, two airborne SAIL imagers of Raytheon 1.5 μm Fiber Laser System and Northrop Grumman 9.11 μm Ladar Transceiver were stated in 2006 [5

5. J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.

]. And in 2011 Lockheed Martin gave an experimental demonstration of SAIL imaging at 1.5 μm from an airborne platform to a cooperative target at a distance of 1.6 km [6

6. B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

], in which the oblique ranging was achieved with short pulsing and not with frequency sweeping as in the other experiments.

Typical implementation of strip-mode SAIL employs synthetic-aperture imaging by use of phase-history reconstruction in the azimuth direction and oblique ranging with frequency sweeping or short pulsing in the range direction for 2D imaging. As seen in the earliest synthetic-aperture one-dimensional (1D) experiments (summarized in [1

1. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002). [CrossRef]

,2

2. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005). [CrossRef]

]), a number of difficulties in both the time domain and the space domain prevented a straightforward duplication of synthetic-aperture radar (SAR) 2D imaging techniques in the optical field. Fortunately, as seen in the mentioned 2D imaging experiments, there were methods of mitigating these problems. In the time domain, the suggestions included the uses of an HCN filter to synchronize the starting chirp frequency [1

1. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002). [CrossRef]

], a reference channel or a vibrometer to measure and correct the environmentally induced phase errors [1

1. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002). [CrossRef]

6

6. B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

], a digital reference channel associated with an algorithm to match the local oscillator (LO) path length to the real range to target to eliminate the high-order phase errors in chirping [2

2. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005). [CrossRef]

], and a photon-limited phase-sensitive heterodyne detection [7

7. R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report NRL/FR/7218-02-10, 051 (2002).

]. In the space domain, the difficulties come from the fact that the size of optical antenna is up to six orders of magnitude larger than the wavelength used. Based on optical diffraction, the antenna systems that can generate a correct spatial quadratic phase history and the best imaging resolution was reported by us, and the system performance of SAILs for different operation modes was theoretically derived [8

8. L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

,9

9. L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009). [CrossRef]

].

These techniques represented an important step forward in developing a robust imaging system. But the phase errors caused from atmospheric turbulence and mechanical trembling in real-word environments are still the problems. The phase-stable delay fiber for matching the actual path length of the target and the array of heterodyne detectors necessary for large receiving aperture are the needs of engineering complexity. And a small footprint is inherent to SAIL. Obviously, there are many challenges that need to be overcome to move this 2D imaging technique into practical use.

It is known that the transformation and regulation of wavefronts in optics is more flexible and variable than in microwave, so that we may carry the SAIL investigation forward not from a direct duplication from the microwave SAR idea but on the basis of optical transformation and regulation. In this way, we proposed a concept of down-looking SAIL [10

10. L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).

,11

11. L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

], with a transmitter of two coaxial and deflected polarization-orthogonal beams of spatial spherical or parabolic phase difference and a receiver of self-heterodyne detection, so that the implementation employs synthetic-aperture imaging by use of quadratic phase-history reconstruction in the direction of travel and quadratic phase-history or linear phase-history reconstruction in the orthogonal direction of travel. Similar to the side-looking SAIL, the down-looking SAIL achieves the fine-resolution, long-distance, and 2D imaging with modest aperture diameters, but has its inherent feature that the size of the optical footprint together with the associated imaging resolution is controllable and changeable in a large scale. Consequently, the width of scanning strip may be greatly enhanced but still with a high resolution, and the influence from atmospheric turbulence and unmodeled line-of-sight motion can be automatically compensated. It is interesting to see that the algorithm for image reconstruction in the down-looking SAIL with 1D linear and 1D quadratic phase-history reconstruction is the same as that in the side-looking SAIL. The down-looking SAIL greatly relaxes the difficulties in the side-looking SAIL. The down-looking SAIL can be used not only to ground observation but also in a reverse-SAR mode to imaging of moving targets in space.

So far the discussed SAIL techniques belong to optical coherent processing; accordingly there are the rigorous constraints on the amplitude, polarization, frequency, temporal phase, and spatial phase of the optical signal both in the time domain and in the space domain. Based on computer tomography (CT) an incoherent spotlight-mode SAIL was proposed by us [12

12. L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010). [CrossRef]

], in which the data of range resolution or Doppler resolution are first collected in terms of the rotated oblique projections of a target illuminated with a laser beam, and the image is then reconstructed by virtue of CT algorithms. This incoherent SAIL supplies a variety of dimensional transformations for imaging, and has the features that the data collection and image reconstruction are easy and the system construction is simple.

A speckle pattern, with both random intensity and phase distributions, is formed at the receiving plane when the coherent light is reflected from the rough surface of target. The speckle effect degrades the image quality of coherent SAIL. Particularly, the speckles were found to be dependent spatially and even temporally if a chirping laser is used. We analyzed such a speckle effect in terms of partial coherence theory, and the proposals to reduce the speckle effect were given [13

13. L. Liu, “Synthetic aperture imaging ladar (VI): space-time speckle effect and heterodyne SNR,” Acta Opt. Sin. 29, 2326–2332 (2009). [CrossRef]

,14

14. L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011). [CrossRef]

].

For a better understanding, Fig. 1 depicts the scenarios of SAIL: (a) side-looking strip-map mode with which the transmitted beam is continuously scanned obliquely across a target; (b) down-looking strip-mode with which the transmitted beam is scanned vertically across a target; and (c) side-looking spotlight mode with which the transmitted beam is held obliquely at one position on target for a dwell period.

Fig. 1. SAIL with (a) strip-mode side-looking; (b) strip-mode down-looking; and (c) spotlight-mode side-looking.

This paper reviews our efforts in developing SAIL. As key and common elements, optical antennas and their accessories are summed up and uniformly analyzed in the following section. Section 3 gives a full mathematical formulation on side-looking strip-mode SAIL. The difficulties and corresponding resolutions are touched upon, and a comparison of required laser powers for different antennas and detectors is outlined. The basic principle, systematic theory, and design equations of the down-looking SAIL with 1D linear and 1D quadratic phase-history reconstruction are presented in Section 4. Section 5 presents those of the down-looking SAIL with 2D quadratic phase-history reconstruction. The theories of these coherent SAILs are reformulated in a unified mathematics in this paper. Section 6 describes the principle and mathematical theory of spotlight-mode incoherent SAIL. Section 7 gives a brief theoretical discussion on the spatial-temporal speckle effect and methods for speckle reduction. The laboratory-space experimental demonstrations are shown in Section 8.

2. Optical Antenna System

An optical antenna plays a key role for production of the phase history necessary for synthetic-aperture imaging. The optical antenna system includes a transmitting antenna, a receiving antenna, and their accessories [8

8. L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

11

11. L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

].

In analysis, an optical field is expressed as a complex amplitude of e(x,y,z)=a(x,y,z)exp(jφ(x,y,z)). And the optical intensity and optical power are defined, respectively, by I(x,y,z)=|e(x,y,z)|2 and p(z)=SI(x,y,z)dxdy/2, where S is the area at z. Note that the mathematical formulations in this section are the basis of the manipulations in the following sections.

Because a rectangular aperture provides a better imaging resolution than a circular aperture [9

9. L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009). [CrossRef]

], for simplification in this paper rectangular apertures are adopted.

A. Optical Transmitting Antenna

Three basic transmitting antennas used in SAIL are illustrated in Fig. 2—that is, telescope, lens imaging amplifier, and lens projector. For the former two antennas, the ladar and target should be located mutually under the far-field diffraction [9

9. L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009). [CrossRef]

], i.e., the Fraunhofer diffraction. For a clear expression, plane wavefronts are adopted in the analyses of this paper.

Fig. 2. Optical transmitting antennas based on (a) telescope; (b) imaging amplification; and (c) quasi-geometric projection.

1. Telescope Transmitting Antenna

For a rectangular aperture of et(x,y)=E0rect(x/Dx)rect(y/Dy), we have
eZ(x,y)=EejkZjλZexp(jπx2+y2λZ)DxDysinc(DxxλZ)sinc(DyyλZ).
(1b)

The divergence angle of transmitted beam at the full width (FW) at minimum is Δθx=2λ/Dx and Δθy=2λ/Dy. The included spatial quadratic phase is the origin of phase history for the side-looking SAIL imaging.

2. Lens for Imaging Amplification

As depicted in Fig. 2(b), an optical field on the entrance focal plane of a lens can be amplified onto the target plane far away from the lens by the Fraunhofer diffraction. The amplification factor is defined as M=Z/f1, and the diffracted field can thus be written as
eZ(x,y)=ejk(Z+2f1)Met(xM,yM)exp(jπx2+y2λZ).
(2)

3. Lens for Quasi-Geometric Projection

As seen in Fig. 2(c), a focused laser spot can be transformed again by a lens into an illumination spot in the front of a target to form a geometric projection of quadratic wavefront with a curvature radius of R=Zf12/Δf1. The amplification factor is M=R/(Z+f1R), and the optical field on the target plane is approximately
eZ(x,y)=ejk(Z+f1)Met(xM,yM)exp(jπx2+y2λR).
(3)

B. Optical Receiving Antenna

Three typical telescope designs for spatial heterodyne detection are illustrated in Fig. 3; they range from using the telescope for pupil plane heterodyne detection, applying a focusing lens for imaging plane heterodyne detection, and using the telescope for self-heterodyne.

Fig. 3. Optical receiving antennas of (a) telescope for heterodyne detection; (b) focusing lens for heterodyne detection; and (c) telescope for self-heterodyne detection.

For analysis, it is necessary to know the return field from a resolution element of the target. An element with the widths of lx and ly and the location of (xp,yp) is described as Etrect((xxp)/lx,(yyp)/ly); thus the return field at the receiving plane can be obtained by the Fraunhofer diffraction as
er(x,y)=EtejkZlxlyjλZsinclx(xxp)λZsincly(yyp)λZexp(jπx2+y2λZ)exp(j2πxxp+yypλZ).
(4a)

For a more general approximation, the return field can be modified to
er(x,y)=EtejkZlxlyjλZKr,x(xxp)Kr,y(yyp)exp(jπx2+y2λZ)exp(j2πxxp+yypλZ),
(4b)
where Kr,x(x) and Kr,x(x) represent the reflective directivities depending on the surface properties of the target. These functions will be further modified for the study of speckle effect. Two spatial phase terms are included in the right of the equation; the first term is a common quadratic phase from the Fraunhofer diffraction, and the second term is a linear phase that remains to further discuss because it has its effect on heterodyne detection.

1. Telescope Antenna for Heterodyne Detection

As shown in Fig. 3(a), the return field from a point target of (xp,yp) on the entrance pupil plane of the telescope can be simplified to er(x,y)=Erexp[j2π(xxp/λZ+yyp/λZ)+jφ(t)], where φ(t) contains the temporal signal and additional noise. The optical field of the LO beam on the detector plane is elo(x,y)=Elo, which can be translated to the same pupil plane as eloc(x,y)=Elo=Elo/M, where M=f2/f1. Therefore, the optical intensity on the receiving plane of telescope is given by
I(x,y:t)=Eloc2+Er2+2ElocErcos[2π(xθx+yθy)φ(t)].
(5)

The viewing angles are defined as θx=xp/Z, θy=yp/Z. And the received optical power can be further achieved by an integral over the pupil area S as
p(θx,θy:t)=(Eloc2+Er2)S+2ElocErSΘ(θx,θy)cosφ(t),
(6)

where the angular sensitivity or directivity function of heterodyne detection is characterized by [8

8. L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

]
Θ(θx,θy)=Scos[2π(xθx+yθy)]dxdy/S.
(7a)

For a rectangular aperture of Dx×Dy, we have
Θ(θx,θy)=sinc(Dxθxλ)sinc(Dyθyλ).
(7b)

The FWs at minimum of viewing angle are Δθx=2λ/Dx and Δθy=2λ/Dy.

2. Focusing Lens Antenna for Heterodyne Detection

As shown in Fig. 3(b), if the aperture on front focal plane is a rectangular function of rect(x/Dx)rect(y/Dy), the focused spot of return beam is thus sinc(xDx/λf1)sinc(xDy/λf1).

For a single photo-detector [type A detector in Fig. 3(b)] of a full sensitive area of rect(x/dx)rect(y/dy) and with a light spot of the same size of dx=λf1/Dx,dy=λf1/Dy, the FWs at minimum of viewing angle may nearly be Δθx=3dx/f1=3λ/Dx and Δθy=3dy/f1=3λ/Dy.

Type B is a single detector with the sensitive area bigger than the return spot, and type C is an array of (M+1)(M+1) detectors. The total width of these two detectors is Mdx and Mdy and the FWs at minimum are Δθx=2Mdx/f1 and Δθy=2Mdy/f1.

3. Telescope Antenna for Self-Heterodyne Detection

As shown in Fig. 3(c), two beams of er1(x,y:t) and er2(x,y:t) are coaxially incident at the telescope entrance pupil; the resulting intensity on the entrance aperture is thus
I(t)=Er12+Er22+2Er1Er2cos[φr1(t)φr2(t)].
(8)

It is seen that there is no receiving directivity.

4. Optical Circulator for Concurrent Transmission and Reception

If the same primary lens of the telescope is used for coaxial transmitting and reception, the other optical elements used in the transmitting path may be different from those in the reception path. Therefore, it is necessary to utilize an optical circulator to spatially separate the paths [8

8. L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

]. A conventional way is to use a polarization beam splitter (PBS) as a circulator with polarization multiplexing of beams.

C. Free-Space Optical Hybrid

By use of an optical 2×4 90° hybrid [15

15. R. Garreis and C. Zeiss, “90° optical hybrid for coherent receivers,” Proc. SPIE 1522, 210–219 (1991). [CrossRef]

,16

16. Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009). [CrossRef]

], the optical phase difference between two beams can be detected as the real and imaginary parts of a phase complex exponential. Figure 4 illustrates a tunable 2×4 90° free-space optical hybrid [16

16. Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009). [CrossRef]

]. The signal beam with a 45° polarization is split into two separate beams with orthogonal polarizations by PBS 1; the LO beam is first delayed in polarization by a λ/4 waveplate (WP) and subsequently spilt by PBS 1. In the in-phase channel the one combination of two beams of orthogonal polarizations from beam 1 and beam 2 is rotated to a 45° polarization by a λ/2 WP and then incident to PBS 2. They are split by PBS 2 into two separate combinations of beams of identical polarizations, which are photo-detected to a photocurrent output of real part of the tested phase by a balanced detection device. In the same way, the quadrature channel outputs a photocurrent of imaginary part. The optical fields of the incident signal beam and the LO beam are denoted as e1(t)=E1exp[jφ1(t)] and e2(t)=E2exp[jφ2(t)], and the temporal phase difference is Δφ(t)=φ1(t)φ2(t). The in-phase and quadrature signals are thus
iIP(t)=KdE1E2Ad2cos[Δφ(t)+φrPBSφtPBSφpWP],
(9a)
iQD(t)=KdE1E2Ad2cos[Δφ(t)φrPBS+φtPBSφvWP].
(9b)

Fig. 4. Component layout for a 2×4 90° free-space optical hybrid.

Here, φrPBS and φtPBS represent the phase changes of the vertical polarization reflected from PBS 1 and the phase change of the horizontal polarization transmitted through PBS 1, and φvWP and φpWP represent the phase changes of the vertical polarization and of the horizontal polarization as the LO beam passes through the λ/4 WP; Kd is the opto-to-electronic conversion factor. Thus, φIP(t)φQD(t)=2(φrPBSφtPBS)(φpWPφvWP). We have two possibilities to rotate the λ/4 WP to finely tune as depicted on the upper left in the figure. For the first polarization arrangement, we have φvWPφpWP=π/2+2θ, and if θ=(φrPBSφtPBS), then φIP(t)φQD(t)=π/2. For the second arrangement, we have φvWPφpWP=π/2(2θ)2, and if θ2=(φrPBSφtPBS)/2, then φIP(t)φQD(t)=π/2.

In the following digital processing, the phase-complex exponential can be easily evaluated by
i(t)=iIP(t)+jiQD(t)=KdE1E2Ad2exp[jΔφ(t)].
(10)

D. Measurement of Wavefront

A large-aperture equal-path interferometer [17

17. Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008). [CrossRef]

] based on the double-shearing principle [18

18. Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004). [CrossRef]

] was developed by us to precisely measure the wavefronts of laser beams. And a quasi-interferometry using incoherent Fourier filtering was before suggested to test strongly varied phase objects [19

19. L. Liu, “Quasi-interferometry with coded correlation filtering,” Appl. Opt. 21, 2817–2826 (1982). [CrossRef]

]. Therefore, the quality and the curvature of the wavefront from a SAIL can be measured.

3. Side-Looking Synthetic-Aperture Image Ladar

Here we concentrate [8

8. L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

,9

9. L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009). [CrossRef]

] on the systematic theory, design equations and necessary conditions of the strip-mode side-looking SAIL as shown in Fig. 1(a), and give a comparison of the required laser powers for different antennas and detectors.

A. Construction of Side-Looking SAIL

Figure 5 illustrates a typical construction of strip-mode side-looking SAIL with key functional modules. The illumination source consists of an FM-chirped laser and a laser amplifier, which outputs a linearly polarized beam. The beam is then divided into three beams, and the one with the most energy is transmitted via an optical circulator and a telescope to a target. The second beam is used to trigger the HCN filter to synchronize the starting frequency for each laser pulse, and the third beam will be delayed to a 2×4 90° free-space optical hybrid as an LO beam for deramping detection. The return signal beam from the target is fed back to the hybrid via the circulator based on polarization multiplexing. The photo-current is decomposed into a real and imaginary parts to form a phase-complex exponential between the return signal and the LO from two balanced detection devices. These are the in-phase channel and quadrature channels of the hybrid system. Then the two parts are digitally composed into a phase-complex exponential for further storing and processing.

Fig. 5. Schematic of a typical side-looking SAIL with key functional modules.

B. 2D Data Collection of Strip-Mode Side-Looking SAIL

Fig. 6. Geometry for strip-mode side-looking SAIL imaging.

A target point (αp,βp) at the slant plane has thus the principal coordinates of (xp=αpsinφ,yp=βp,zp=αpcosφ). Slow time (ts) is defined as the time duration in the azimuth direction. Because the SAIL operates in a repeated pulsing, we have a discrete slow time of ts=nTs, where Ts is the period between laser pulses. The fast time along the range direction at the nth pulse is defined as tn,f. An ideal FM-chirped waveform is generated by linearly sweeping the frequency of the laser during the fast time, which at the nth pulse can be expressed as φ(tn,f)=(f0+Δfn)tn,f+f˙tn,f2/2+φn, where f0 is the starting frequency, Δfn is the deviation of the starting frequency at the nth pulse, f˙ is the chirping rate, and φn is the initial phase at starting. It was known that the use of the HCN channel leads to Δfn being nearly a constant, and thus it can be neglected.

The optical field of complex amplitude on the pupil of the transmitting telescope with a rectangular aperture of Dxt×Dyt can be described by
e0(x,y:tn,f)=E0rectxDxtrectyDyt,
(12)
and with a temporal phase signal of
φFM,n(tn,f)=2πf0tn,f+f˙2tn,f2+φn.
(13)

This beam moved with a velocity of v along the azimuth direction and is transmitted onto the target plane with an average wavelength of λ. The diffracted field of complex amplitude on the principal target plane is thus given by
et(x,y:tn,f,nTs)=E0DxtDytjλZKt,x(x)Kt,y(y:nTs)exp[jπλZ(x2+(yvnTs)2)]×exp{j[φFM,n(tn,f+τp2)φFM,n(tn,f)]},
(14)
and
Kt,x(x)=sincDxtxλZ,Kt,y(y:nTs)=sincDyt(yvnTs)λZ,
(15)
where τp=2(Z+zp)/c is the round-trip delay, and the Kt functions represent the directivity of the transmitted beam on the target plane.

And the reflected field from a target element at (xp,yp) with a size of lx×ly and a reflectivity of ρp at the same pupil plane, i.e., the receiving plane, can be deduced as
er(x,y:xp,yp:tn,f,nTs)=E0DxtDytjλZρplxlyjλZKr,x(x:xp)Kr,y(y:yp:nTs)exp{jπλZ[xp2+(ypvnTs)2]}exp(jπλZ{(xxp)2+[y(ypvnTs)]2})exp{j[φn,FM(tn,f+τp)φn,FM(tn,f)]},
(16)
where Kr,x(x:xp)=Kr,x(xxp) and Kr,y(y:yp:nTs)=Kr,y[y(ypvnTs)].

The field of the LO beam equivalently on the receiving plane can be written by a field of complex amplitude as
elo(x,y:)=EloMexp{j[φn,FM(tn,f+τlo)φn,FM(tn,f)]}.
(17)

With the help that bbcos[2πXx+φ(t)]dx=cosφ(t)bbcos(2πXx)dx, the AC component of optical intensity can thus be obtained as
IAC(x,y:xp,yp:tn,f,nTs)=2E0EloMDxtDytjλZρplxlyjλZη×Kr,x(0:xp)Kr,y(0:yp:nTs)Kt,x(xp)Kt,y(yp:nTs)×cos({πλZ/2[xp2+(ypvts)2]2πλZ[xxp+y(ypvts)]}+2πf˙Δτptn,f),
(18)
where the LO beam delay is τlo, and the relative delay of the signal to the LO is Δτp=τpτlo. It was known that the nonlinear chirping effect can be greatly reduced as τloτp. For simplicity all the losses during the signal transmission from the laser source to the detector are included by η.

The received optical power is the integration of the intensity over a receiving aperture of Dxr×Dyr (which matches the used aperture of the detector) as
p(xp,yp:tn,f,nTs)=E0EloMDxtDytjλZρplxlyjλZDxrDyrη×Kr,x(0:xp)Kr,y(0:yp:nTs)Kt,x(xp)Kt,y(yp:nTs)Θx(xp)Θy(ypvnTs)cos{πλZ/2[xp2+(ypvnTs)2]+2πf˙Δτptn,f}.
(19)

Similarly, we have the angular sensitivity of heterodyne as
Θ(xp,ypvnTs)=Θx(xp)Θy(ypvnTs)=sinc(DxrxpλZ)sinc[Dyr(ypvnTs)λZ].
(20)

Note here that the common regime of the illuminated area and the detectable area at the target plane is defined as a ground footprint.

A photo-detector converts optical power to photo-current. And there is a sampling interval for photo-current, denoted as Tf, which should be less than the pulse duration. Thus the sampling can be described by a temporal window of rect{[tn,f(τp+Tf/2)]/Tf}.

Finally, we have a 2D data-collection equation of strip-mode side-looking SAIL for a target point of (xp,yp) as
i2D(xp,yp:tn,f,nTs)=ni1D(xp,yp:tn,f,nTs).
(23)

It is clear that the adjustment of sampling interval Tf is important for reducing the influences caused by the nonlinear chirping as well as temporally varied speckles from the target.

C. Imaging Processing of Strip-Mode Side-Looking SAIL

In the image processing, the 2D data are first compressed by the Fourier transform in the range direction and then compressed by the match filtering with a conjugate quadratic phase to the phase history, exp{j2π[(mn)vTs]/λZ}, in the azimuth direction. That is
I(ξ,mTs)=KdE0EloKsn(Kr,y(0:yp:nTs)Kt,y(yp:nTs)Θy(ypvnTs)×exp{j[πλZ/2(ypvnTs)2]}exp{jπλZ/2[(mn)vTs]}×Kr,x(0:xp)Kt,x(xp)Θx(xp)exp(j2πf˙Δτptn,f)rect[tn,f(τp+Tf/2)Tf]exp(j2πξtn,f)dtn,f).
(24a)

Approximately, we have the image of the target point as
I(ξ,mTs)=K[SR(ξ)*δ(ξf˙Δτp)][SA(m)*δ(mvTsyp)],
(24b)
where SR(ξ) is the impulse response in the range direction, SA(m) is the impulse response in the azimuth direction, and the star denotes a convolution. The coordinates of image plane are given by x=Δz=(c/2f˙)ξ and y=(vTs)m.

D. Detection Abilities of Different Receiving Schemes

For the side-looking SAIL, to provide a large illumination spot, the transmitting aperture must be small, and to match such a spot by heterodyne detection, the receiving aperture must have the identical size. In a conventional design, the received power from the target may be too weak. So, it is necessary to increase the receiver aperture but keep the receiver viewing angle the same as the transmitted beam divergence. We define a detectable receiving intensity to compare the reception abilities among a heterodyne detector with a telescope antenna, a heterodyne detector with an antenna of focusing lens, and an array of heterodyne detectors with an antenna of focusing lens.

The heterodyne detection of SAIL works in the photon-limited regime [7

7. R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report NRL/FR/7218-02-10, 051 (2002).

]; the dominant source of noise is shot noise from the photon number of LO. The root-mean-square (rms) shot-noise power was evaluated as Pσ=(hνPol/2τpus)1/2, where τpul is the pulse duration. We assume that the return power has a signal-to-noise ratio of Sshot to the rms noise power. In addition, the total noise (Pn) of the detector will limit the detection of the signal; we further assume that the AC component has a signal-to-noise ratio of Snos to the detector noise power.

1. Heterodyne Receiver Using a Telescope and a Detector

The detector has a sensing area of d×d matched to the telescope aperture of D×D. The FW at minimum of the heterodyne directivity is Δθtel=2λ/D, which is defined as the same as the divergence angle of transmitted beam.

The desired optical intensity at the incident plane of antenna can be arrived at
Irtel(d:D)=(hv2τpul)1/3[Pn(d×d)2]2/3(SnoiseSshot)2/32D2.
(26)

2. Heterodyne Receiver Using a Big Lens and a Detector

The receiver uses a detector and a lens of the same size as in the case of the telescope; to reach the identical receiving angle, its focal length is f=dD/2λ.

If the lens aperture increases by a factor of N, correspondingly, the spot shrinks by an identical factor, and the viewing angle keeps just Δθ=d/f=Δθtel. The equivalent noise of detector is Pn(d×d). The desired optical intensity at the incident plane of lens is given by
Ir(d:N)=Irtel(d:D)/N1/3.
(27)

3. Heterodyne Receiver Using a Big Lens and an Array of Detectors

Assume that the detector array has a total size of d×d; the aperture increases by a multiple of N, and the number of detectors in the array is N. The equivalent noise of each detector is Pn(1:N)=Pn(d×d)/N; thus the required optical intensity at the incident plane of the lens is given by
Ir(1:N)=Irtel(d:N)N.
(28)

It is concluded that, in comparison with the telescope receiver, the receiver with a lens bigger by N times in aperture and an array of detectors can reduce the laser transmitting power by N times, and the receiver with a lens N times bigger and a single detector has only a power-reducing factor of N1/3. In an earlier paper [20

20. A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 54, 1350–1356 (1966). [CrossRef]

], from a linear antenna theorem, the latter receiver has no effective improvement.

4. Down-Looking SAIL with 1D Linear and 1D Quadratic Phase-History Reconstruction

The down-looking SAIL consists of a transmitter of two coaxial and deflected polarization-orthogonal beams; the measurement uses optical components to perform wavefront transformation on separated polarizations to produce beams with spatial parabolic or spherical phase difference. The beams are coaxially recombined before transmission. The receiver is designed for self-heterodyne detection of the two coaxial beams [10

10. L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).

,11

11. L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

]. The necessary spatial phase histories are not extracted alone from the natural Fraunhofer diffraction of the beams; this is the inherent difference to the side-looking SAIL.

The suggested down-looking SAIL works in strip mode and has two basic operational architectures, i.e., with 1D linear phase and 1D quadratic phase-history reconstruction [10

10. L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).

] and with 2D quadratic phase reconstruction [11

11. L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

]. This section deals with the former, and the latter will be discussed in the following section.

A. General Construction of Down-Looking SAIL

The two kinds of down-looking SAILs have the same construction, and Fig. 7 illustrates a general scheme with the key functional modules. The beam from the source laser is equally divided by a PBS into two beams with orthogonal polarizations. The horizontally polarized beam and the vertically polarized beam pass through their own channel of deflector and reshaping lens and are then joined into a combination of coaxial beams by the second PBS. The two beams are first regulated independently by the H-reshaping lens and the V-reshaping lens, and then continued to shape together by the transformation lens. In terms of imaging amplification of the main lens, the resulting H-polarization and V-polarization wavefronts on its focal plane are transmitted onto the target.

Fig. 7. General construction of down-looking SAIL with key functional modules.

The receiving telescope is used to collect the return signals from the target. The received beams are first separated to an H-polarization beam and a V-polarization beam by a PBS, and then enter a 2×4 90° optical hybrid and divided into an in-phase channel and a quadrature channel. The beams in each channel are detected in self-heterodyne by a balanced photo-detector device. Finally, the collected data are stored and processed by a computer.

It should be noted here that in the down-looking SAIL with 1D linear and 1D quadratic phase-history reconstruction described here, the two beams are deflected at the opposite directions, and in the one with 2D quadratic phase-history reconstruction to be discussed in Section 5, the two beams are deflected at the same direction.

B. Formation of Wavefronts of Paraboloidal Phase Difference

To produce two beams with a spatial parabolic phase difference on the focal plane of the main lens is a key problem. A typical design is given in Fig. 8, in which the only reshaping element is a cylinder lens in the V-polarization channel. The upper two parts illustrate the structure of the horizontal-polarization channel. By a simple defocusing from the focus of the main lens with a distance of R1in, the incident plane wavefront at the plane of the H-deflector can be transformed into a 2D divergent spherical wavefront with a radius of R1in at the focal plane of main lens. The lower two parts illustrate the structure of the vertical-polarization channel. In the (x,z) plane, a 1D divergent spherical wavefront with a radius of R1in at the main lens focal plane is generated by a simple defocusing from the transformation lens. And in the (y,z) plane, a negative cylinder lens is added to diverge the incident plane wavefront to form a 1D convergent spherical wavefront with a radius of R2in at the focal plane. Therefore, the incident plane wavefront at the plane of deflectors is transformed into a 2-D paraboloidal wavefront at the focal plane of th main lens on two independent principal planes. If the cylinder lens is located on the focal plane of the transformation lens, we have its focal length as
fx=f22R1in+R2in.
(29)

Fig. 8. Configuration (upper two) to produce a 2-D spherical wavefront, and configuration (lower two) to produce a 2-D paraboloidal wavefront.

In both channels, a deviation of the deflector to the focal plane of the transformation lens causes an additional phase of exp(jπΔf2θ2/λ) with respect to the deflection angle of θ.

On this progress, we have the inner optical fields at the focal plane of main lens as
eHin(x,y:tn,f,nTs)=E0rect(x+vxintn,fLxin)rect(yLyin)exp[jπΔf2λf2(vxintn,f)2]×exp{jπ[(x+vxintn,f)2λR1in+y2λR1in]},
(30a)
eVin(x,y:tn,f,nTs)=E0rect(xvxintn,fLxin)rect(yLyin)exp[jπΔf2λf2(vxintn,f)2]×exp{jπ[(xvxintn,f)2λR1iny2λR2in]},
(30b)
where rect(x/Lxin)rect(y/Lyin) is the designed amplitude distributions of the two inner fields, and their widths are Lxin and Lyin; the center scan is given by vintf=f2θ˙xtn,f, θ˙x is the angular deflection rate, and the scanning period is Tf beginning from Tf/2 and ending at Tf/2.

C. 2D Data Collection

Figure 9 illustrates the geometry for down-looking SAIL imaging. By an amplified transmitting, we have the H-polarization and V-polarization fields on the target plane as
eHT(x,y:tn,f,nTs)=E0ejkZMrect(xvxtn,fLx)rect(yvynTsLy)exp[jπΔf2λf22(vxtn,fM)2]×exp{jπλ[(xvxtn,f)2R1+(yvynTs)2R1]}exp{jπλZ[x2+(yvynTs)2]},
(31a)
eVT(x,y:tn,f,nTs)=E0ejkZMrect(x+vxtn,fLx)rect(yvyTsLy)exp[jπΔf2λf22(vxtn,fM)2]×exp{jπλ[(x+vxtn,f)2R1(yvynTs)2R2]}exp{jπλZ[x2+(yvynTs)2]},
(31b)
where R1=M2R1in, R2=M2R2in, vx=Mvxin, ts=nTs, and vy is the motion velocity of the centers of the illumination spots along the travel direction. The size of the illumination spot is Lx=MLxin and Ly=MLyin.

Fig. 9. Geometry for down-looking SAIL imaging.

The reflected field from a target element at (xp,yp) with a size of lx×ly and a reflectivity of ρp at the entrance pupil of the receiving telescope, which is with a distance of ΔY to the transmitting lens along the travel direction, is thus obtained as
eHTR(x,y:xp,yp:tn,f,nTs)=E0Mρpej2kZjλZlxlyηKr,x(x:xp)Kr,y(y:yp:nTs)rect(xpvxtn,fLx)rect(ypvynTsLy)×exp{jπλ[(xpvxtn,f)2R1+(ypvynTs)2R1]}exp[jπΔf2λf22(vxtn,fM)2]exp(jφH)×exp{jπλZ[xp2+(ypvynTs)2]}exp(jπλZ{(xxp)2+[yΔY(ypvynTs)]2}),
(32a)
eVTR(x,y:xp,yp:tn,f,nTs)=E0Mρpej2kZjλZlxlyηKr,x(x:xp)Kr,y(y:yp:nTs)rect(xp+vxtn,fLx)rect(ypvynTsLy)×exp{jπλ[(xp+vxtn,f)2R1(ypvynTs)2R2]}exp[jπΔf2λf22(vxtn,fM)2]exp(jφV)×exp{jπλZ[xp2+(ypvynTs)2]}exp(jπλZ{(xxp)2+[yΔY(ypvynTs)]2}).
(32b)
where Kr,x(x:xp)=Kr,x(xxp) and Kr,y(y:yp:nTs)=Kr,y[yΔY(ypvynTs)]. And the phase difference between the two return fields is seen as a spatial paraboloidal phase of
ΔφTR(xp,yp:tn,f,nTs)=πλ[2xpvxtn,fR1/2+(ypvynTs)2R3]+φHφV,
(33)
where 1/R3=1/R1+1/R2; usually R1=R2 is designed and thus R3=R1/2. And φH and φV represent the phase variations caused by the atmospheric turbulence and other interferences in the H-channel and V-channel. Under the coaxial heterodyne detection, we have φHφV0; this means that the phase variations are compensated. It is also seen that the self-heterodyne detection does not introduce the directivity for reception. From the equations, it is indicated that in the orthogonal direction of travel a target point yields a linear phase-modulation history with respect to the target location of xp, and in the travel direction a quadratic phase history with respect to the target location of yp.

Thus the AC component of optical intensity from the self-heterodyne can tbe obtained as
IAC(x,y:xp,yp:tn,f,nTs)=2E02(ρplxlyηMλZ)Kr,x2(x:xp)Kr,y2(x:yp:nTs)Kt,x(xp:tn,f)Kt,y2(yp:nTs)cos[2πλR1/2xp(vxtn,f)+πλR3(ypvynTs)2]
(34)
and
Ks=DxrDyr(ρplxlyηMλZ)2,
(35)
Kt,x(xp:tn,f)=rect(xpvxtn,fLx)rect(xp+vxtn,fLx)Kt,y2(yp:nTs)=rect(ypvynTsLy).
(36)

The optical power is simply the integration of the intensity over the reception aperture. And the resuling photocurrents from the 2×4 90° optical hybrid are then processed into the complex exponential, which can be expressed by a point-target radar equation as
i1D(xp,yp:tn,f,nTs)=I(xp,yp:tn,f,nTs)exp[j2πλR1/2xp(vxtn,f)+jπλR3(ypvynTs)2],
(37a)
I(xp,yp:tn,f,nTs)=KdE02KsKr,x2(0:xp)Kr,y2(0:yp:nTs)Kt,x(xp:tn,f)Kt,y2(yp:nTs).
(37b)

Finally, we have a complete 2D data-collection equation for a target point of (xp,yp) as
i2D(xp,yp:tn,f,nTs)=ni1D(xp,yp:tn,f,nTs).
(38)

D. Image Processing and Imaging Resolution

In the image processing, the 2D data are first compressed by the Fourier transform in the orthogonal direction of travel and then compressed by the match filtering with a conjugate quadratic phase of the phase history in the travel direction. The algorithm is the same as that in side-looking SAIL. We have
I(ξ,mTs)=KdE02Ksn(Kr,y2(0:yp:nTs)Kt,y2(yp:nTs)×Kr,x2(0:xp)Kt,x(xp:tn,f)exp[j2πλR1/2xp(vxtn,f)]exp(j2πξtn,f)d(vxtn,f)×exp[jπλR3(ypvynTs)2]exp{jπλR3[vy(mn)Ts]}).
(39a)

Approximately, we have the image of the target point as
I(ξ,mTs)=K[Sot(ξ)*δ(ξ+xpλR1/2)][Stv(m)*δ(mvTsyp)],
(39b)
where Sot(ξ) is the impulse response in the orthogonal direction of travel, and Stv(m) is the impulse response in the travel direction. The coordinates of the image plane are x=λR1ξ/2 and y=(vTs)m.

From the definition of Kt,x(xp:tn,f), it is found that the scan range of the laser spot on the target plane along the orthogonal direction is between kLx and kLx, where k0.5 is a designed scanning factor. Thus the resulting effective width of the strip is (12k)Lx, and within this width the sampling interval of linear phase modulation maintains the maximum of 2kLx. The impulse response along the orthogonal direction is ideally a function of sinc[ξ(2kLx)]; thus the FWHM of theoretical range imaging resolution is given by
dx=λR14kLx=λMR1in4kLxin.
(40a)

The impulse response along the travel direction is ideally a function of sinc(yLy/λR3); the FWHM of theoretical imaging resolution in the travel direction is given by
dy=λR3Ly=MλR3inLyin.
(40b)

5. Down-Looking Synthetic-Aperture Imaging Ladar with 2D Quadratic Phase-History Reconstruction

In this down-looking SAIL with 2D quadratic phase-history reconstruction, the two beams in the transmitter are deflected at the same direction and the spatial phase difference between the two wavefronts is thus a spatial quadratic phase.

A. Formation of Wavefronts of Quadratic Phase Difference

A typical design is shown in Fig. 10 to produce two beams with a quadratic phase difference on the focal plane of the main lens. The upper part illustrates the configuration of the horizontal-polarization channel. And by a simple defocusing from the focus of the transformation lens with a distance of R1in, the incident plane wavefront at the plane of the H-deflector can be transformed into a 2D divergent spherical wavefront with a radius of R1in at the focal plane of the main lens. The lower part is the configuration of the vertical-polarization channel to transform the incident plane wavefront at the plane of the V-deflector into a 2D convergent wavefront. This is done by adding a negative lens to further diverge the incident plane wavefront to form a 2D convergent spherical wavefront with a radius of R2in at the focal plane of the main lens. If the negative lens is located on the focal plane of the transformation lens, we have its focal length as
fx=f22R1in+R2in.
(41)

Fig. 10. Upper configuration to produce a 2-D divergent spherical wavefront, and lower configuration to produce a 2-D convergent spherical wavefront.

In both channels, a deviation of the deflector to the focal plane of the transformation lens yields an additional phase of exp(jπΔf2θ2/λ).

It is recognized that a deflection rate θ˙H in the H-channel generates a center displacement of spherical wavefront of f2θ˙H=vHin, and in the V-channel a center displacement of (R1in+R2in)θ˙V=vVin. The two channels must have the same displacement of centers during the scanning time of tn,f; thus vHin=vVin is desired and the deflection rates must have the relation of
θ˙V=f2R1in+R2inθ˙H.
(42)

If R1in=R2in=f2/2 we have simply θ˙V=θ˙H.

On this basis, we have that the inner optical fields to be transmitted at the focal plane of the main lens are given by
eHin(x,y:tn,f,nTs)=E0rect(x+vxintn,fLxin)rect(yLyin)exp[jπΔf2λf2(vxintn,f)2]×exp{jπ[(x+vxintn,f)2λR1in+y2λR1in]},
(43a)
eVin(x,y:tn,f,nTs)=E0rect(x+vxintn,fLxin)rect(yLyin)exp[jπΔf2λf2(vxintn,f)2]×exp{jπ[(x+vxintn,f)2λR2in+y2λR2in]}.
(43b)

B. 2D Data Collection

By the same manipulation, we have the received photocurrent as
i1D(xp,yp:tn,f,nTs)=I(xp,yp:tn,f,nTs)exp{jπλR3[(xpvxtn,f)2+(ypvynTs)2]},
(44a)
I(xp,yp:tn,f,nTs)=KdE02KsKr,x2(x:xp)Kr,y2(y:yp:nTs)Kt,x2(xp:tn,f)Kt,y2(yp:nTs).
(44b)

Finally, the complete 2D data-collection equation for a point target of (xp,yp) is achieved as
i2D(xp,yp:tn,f,nTs)=ni1D(xp,yp:tn,f,nTs),
(45)
where
Kt,x(xp:tn,f)=rect(xpvxtn,fLx),Kt,y(yp:nTs)=rect(ypvynTsLy).
(46)

It is seen that the spatial phase difference between the two returned fields has a spatial quadratic phase of
ΔφTR(xp,yp:tn,f,nTs)=πλR3[(xpvxtn,f)2+(ypvynTs)2]+φHφV.
(47)

As seen before, we have the automatic compensation of φHφV0.

C. Image Processing and Imaging Resolution

This SAIL has the quadratic phase history in both directions. In the image processing, the 2D data are first compressed by the match filtering with a conjugate quadratic phase of the phase history in the orthogonal direction of travel and then compressed by the match filtering with a conjugate quadratic phase of the phase history in the travel direction. That is,
I(ξ,mTs)=KdE02Ksn(Kr,y2(0:yp:nTs)Kt,y2(yp:nTs)×Kr,x2(0:xp)Kt,x2(xp:tn,f)exp[jπλR3(xpvxtn,f)2]exp[jπλR3(ξ+vxtn,f)2]d(vxtn,f)×exp[jπλR3(ypvynTs)2]exp{jπλR3[vy(mn)Ts]2}).
(48a)

Approximately, we have the image of the target point as
I(ξ,mTs)=K[Sot(ξ)*δ(ξ+xp)][Stv(m)*δ(mvTsyp)].
(48b)

The coordinates of the image plane are obviously x=ξ, y=(vTs)m.

From the definition of Kt,x(xp:tn,f), it is seen that the scan range of the laser spot center on the target plane along the orthogonal direction is between kLx and kLx, where k0.5. Thus the resulting effective width of the strip is (12k)Lx, and within this width the sampling interval of the quadratic phase is 2kLX. The impulse response along the orthogonal direction is ideally a function of sinc[x(2kLy)/λR3]; thus the FWHM of theoretical imaging resolution along the orthogonal direction is given by
dx=λR12kLx=λMR1in2kLxin.
(49a)

The impulse response along the travel direction is ideally a function of sinc(yLy/λR3); the FWHM of theoretical imaging resolution along the travel direction is given by
dy=λR3Ly=MλR3inLyin.
(49b)

D. Discussion on Down-Looking SAIL

It is interesting to see that a down-looking SAIL with 1D linear and 1D quadratic phase-history reconstruction (Section 4) produces in the orthogonal direction of travel a linear phase modulation proportional to the lateral distance xp of the target point to the coordinate original, so that two target points of (xp,yp) and (xp,yp) opposite each other to the original can be distinguished by the Fourier transform for focusing. And a down-looking SAIL with 2D quadratic phase reconstruction (Section 5) produces a 2D quadratic phase centered at the location (xp,yp) of the target point, so that it can be recognized from (xp,yp) by the matched filtering. In the both cases, a vertical observation (down-looking) is possible. On the contrast, in side-looking SAIL a linear phase modulation proportional to the relative range zp of the target point is yielded and two points opposite to the original cannot be distinguished. Thus oblique observation is necessary.

Because the illumination spot size and the phase history are not from the straight transmitter aperture diffraction, the imaging resolution, illumination size, and equivalent curvature radii of the illumination wavefronts can be separately designed and controlled. This provides a great flexibility in designing the system requirements. Consequently, the width of the scanning strip may be greatly enhanced but still retain high resolution. Furthermore, by using self-heterodyne, the influence from atmospheric turbulence and unmodeled line-of-sight motion can be automatically compensated. We note that the FWHM imaging resolution for the down-looking SAILs is dependent on the target distance, which is similar to the imaging by a lens.

For the self-heterodyne, there exists interference between any two target points. However, it has been proven that due to the heterodyne angular directivity, nearly all of the interference noises can be naturally suppressed [11

11. L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

].

With self-heterodyne detection by a single detector, the AC component is given by Pac=P21+P22. In a similar way, we need Pac=SnosPn. If the aperture increases a multiple of N, the total required reception intensity for the self-heterodyne on the receiving aperture can be reached as
Ir(d:Kd)=2SnosPn(d×d)ND2.
(50)

It is thus seen that by the use of a quite big receiving lens, the required transmitter power for the down-looking SAIL might be a little more than that for the side-looking SAIL with a detector array but significantly less than that for the side-looking SAIL with a single detector.

It is found that down-looking SAIL can be immediately operated in an inverse-SAIL mode because of vertical observation. In addition, a scanning mode can be achieved where the target is relatively stationary, but the SAIL scans to acquire the target data. This is a new application of aperture synthesizing.

6. Incoherent Spotlight-Mode SAIL

In fact, the suggested incoherent SAIL [12

12. L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010). [CrossRef]

] can perform four operation modes of spotlight mode, inverse spotlight mode, CT spotlight modes, and circular spotlight-mode, which includes two sensing techniques of range resolution and Doppler resolution, and provides a variety of dimensional transformations for imaging that are not only for 2D range or Doppler-resolution imaging of 2D targets but also for three-dimensional (3D) range resolution as well as in the depth compressed 2D range resolution and 2D Doppler-resolution imaging of 3D targets. In this paper, we concentrate on the incoherent SAIL with range-resolved slant projections, and the ranging is achieved with an incoherent laser short-pulse waveform.

A. Construction of Incoherent Spotlight-Mode SAIL

Figure 11 illustrates a typical configuration of incoherent spotlight-mode SAIL with key functional modules. The beam from a pulsed laser is transmitted to a target by a telescope and the return signal from the target is collected by a telescope and then detected by a photo-detector. The optical spot is deflected to be held at the target during the SAIL motion by an optical deflector. The range-time intensity-profile data of different deflecting angles are stored in a computer and then processed in terms of the CT algorithms for image reconstruction.

Fig. 11. Schematic of typical spotlight-mode incoherent SAIL with key functional modules.

B. 2D Data Collection from Range-Resolved Oblique Projections

The relative rotation between the target and the SAIL can be described by the coordinate systems as depicted in Fig. 12. The basic reference coordinates perpendicular to the beam are (x,y), the object reference coordinates are (α,β), and the in-between coordinates are (x,y). The cross-over point of the three coordinates to the original position of SAIL is the principal target range Z. Define φ(0|φ|<π/2) as the horizontal tilt, and θ(ts) as the rotating angle, which is given by tanθ(ts)=vtscosφ/Z, where vts is the moving distance of SAIL at a time of ts. Furthermore, we have the θ-dependent base distance as Z(θ(ts))=[Z2+(vts)2]1/2. The SAIL observes the ranging-time intensity profile from the target by a ray-sum f[z+Z(θ),θ(ts)] of object with respect to the rotated angle, where the range is defined as an increment z to its base distance Z(θ).

Fig. 12. Coordinate systems for relative rotation between target and incoherent SAIL.

To study the ray-sum, we first establish the relations among the different coordinates systems, which are
x=x,y=ycosθ,
(51a)
α=xsinφ,β=ycosθxctanφtanθ.
(51b)

The data from the object are collected by the oblique projections. From Fig. 12 it is seen that the range from a target point (α,β) to the basic reference plane is given by
z=αcosφcosθβsinθ.
(52)

Therefore, the contour equation of the additional range z of the target on the target plane to the basic plane can be achieved as
α=z/cosφcosθ+βtanθ/cosφ.
(53)

By using the coordinate transformation for a clockwise rotation as
a=αcosγβsinγ,b=αsinγβcosγ,
(54)
it is found that if the rotated angle is reached at
tanγ=tanθcosφ.
(55)

The b-axis will be parallel to the contour lines and the range of the target point (α,β) to the basic reference plane is determined to locate along the a-axis at
a=αcosφcosθβsinθcosθtan2θ+cos2φ=z(α,β)cosθtan2θ+cos2φ.
(56)

Denote the object function as g(α,β) of the intensity reflective distribution. The collected profile data from the object in the b-direction under the rotated angle of γ is generated by a 1D integration along the a-axis of range contour lines as
GR(a:γ)=g(asinγ+bcosγ,acosγbsinγ)db.
(57)

It is obvious that GR(a:γ) is just the wanted ray-sum of f[z+Z(θ),θ(ts)], but neglecting the base range Z(θ).

C. Tomographic Reconstruction of Range-Resolved Images

The reconstruction of a range-resolved image is performed with the standard CT demodulation algorithms such as the backprojection, filtered backprojection, and Radon-Fourier transform methods [12

12. L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010). [CrossRef]

].

By using the backprojection method, the estimated image is given by
gB(α,β)=iGR(αcosγiβsinγi)Δγ,
(58)
where γi is the rotated angle of the ith projection, and Δγ is the angular spacing between the neighboring projections.

D. Resolution in Range-Resolved Imaging

As the width of the laser pulse for range-resolved imaging is Δτ, the resolution of range resolution is thus Δz=cΔτ/2. Therefore, the imaging resolution Δa(θ) for an individual projection can be obtained as
Δa(θ)=Δz(α,β)sin2θ+cos2θcos2φ.
(59a)

If there is a number of N inverse projections, the diameter of the image point at the maximum intensity from the N-level threshold of the overlapping of these projections is given by
dI=Δa(θ=0)=Δzcosφ.
(59b)

It is thus seen that the less the tilting angle φ of the object plane, the less the diameter of the imaging point, and a larger angular interval of rotation will lead to a smaller imaging spread.

In order to achieve a qualified reconstruction of image, the collected projections must be backprojected accurately. Otherwise, the unknown location of the rotation axis and random translation of the target will bring image blur and geometric distortion in the reconstruction. Projections registration is thus an important technique. Feature tracking and phase-retrieval methods can improve imaging quality.

If the tilt of object plane is π/2—i.e., the object plane is vertical to the incidence of laser beam—the spotlight-mode SAIL is simplified to a kind of reflective tomography laser radar [22

22. R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

,23

23. A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001). [CrossRef]

]. Therefore, the image becomes only the outline of opaque object.

7. Speckle Effect in Coherent SAIL

The laser waves scattered from a diffused target will cause the speckles on the receiving plane. Particularly, in a coherent SAIL using FM chirp, the speckles are spatially and temporally varied [13

13. L. Liu, “Synthetic aperture imaging ladar (VI): space-time speckle effect and heterodyne SNR,” Acta Opt. Sin. 29, 2326–2332 (2009). [CrossRef]

]. In coherent synthetic-aperture imaging, the speckle effect not only decreases the signal-to-noise ratio in heterodyne detection but also degrades the imaging quality. A complex coherence function of integrated speckle over the antenna aperture is thus defined, and its width is the possible length for the aperture synthesizing. Therefore, a design principle for the transmitter aperture, receiver aperture, and available length for synthesizing is given, and the methods to reduce the speckle effect are suggested [14

14. L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011). [CrossRef]

].

A. Speckle Characteristics in Coherent SAIL

The speckle effect is characterized here by a resolution element of the target.

1. Average Size of Speckle

The statistically averaged size of the speckle can be given by its intensity self-correlation function. When the resolution element of the diffused target is a rectangle of lx×ly, the averaged size of the speckle on the receiving plane is given by
Sx=λZlxcosφ;Sy=λZly.
(60)

2. Complex Coherence Function of Integrated Speckles Over Receiving Aperture

The received speckle field is an integration of the speckles over the receiving aperture. The motion distance during which the amplitude and phase of the received speckle field maintain relatively stable as a SAIL travels in the azimuth direction is the length capable for the aperture synthesizing. A complex coherence function of the integrated received speckles with respect to the SAIL travel is thus defined.

If the receiving aperture is a rectangle of B(x,y)=rect(x/Dxr)rect(y/Dyr), then the complex coherence function of integrated speckles over the receiving aperture [integrated complex coherence function (ICCF)] has been obtained as [14

14. L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011). [CrossRef]

]
μ(Δx,Δy)=[sinc(ΔxSx)*tri(ΔxDxt)][sinc(ΔySy)*tri(ΔyDyt)]sinc(αSx)tri(αDxt)dαsinc(βSy)tri(βDyt)dβ.
(61)

It has been indicated that the speckle sinc function represents the speckle complex coherence factor (SCCF) with the width of FW at minimum of 2Sx or 2Sy, and the antenna triangle function stands for the autocorrelation function (AACF) of the antenna aperture with the width of FW at minimum of 2Dxr or 2Dyr.

For a better understanding, Fig. 13 depicts normalized 1D complex coherence functions under the different ratios of the width of the antenna autocorrelation function to the width of speckle complex coherence factor. It is demonstrated that (1) if the antenna AACF width is larger than the speckle SCCF width, the width of the ICCF is determined by the antenna aperture and is given by 2Dt; (2) if the SCCF width is larger than the AACF width, the ICCF width is determined by the speckle size and is given by 2S; and (3) if the AACF width is near to SCCF width, the ICCF width is determined by the antenna aperture or speckle size, and is given by 2Dt or 2S. In the cases (1) and (2), the speckle effect is relatively weak.

Fig. 13. Normalized 1D integrated complex coherence functions at normalized coordinates, ratio of antenna width ACF to speckle width CCF: (a) 5:1; (b) 1:1, and (c) 1:5.

B. Synthetic-Aperture Imaging Estimation Using Complex Coherence Function

We can thus evaluate the speckle effect for the side-looking SAIL in terms of the ICCF. The estimation in the azimuth direction is straightforward in that it is preferred to design a large averaged size of speckle, which can yield a large synthesizing length. In the range direction, frequency chirping will result in a spatial and temporal variation of speckles. Due to the target tilt, typically of 45°, for example, the additional phase shift reaches a wave number of 2 and equivalently a linear phase term to the maximum of 2π(2λ/Lx)x, where Lx is the width of the illumination spot. This linear phase generates in turn a 2Sx deviation of speckles in the range direction. From the ICCF function it can be found that to design a small averaged size of speckle to reduce the chirping effect is thus preferred. In conclusion, it is contradictory to design an ICCF to satisfy the requirements in both the range and azimuth directions. It needs a carefully balanced design.

For the down-looking SAILs, the two returned speckle patterns from two coaxial transmitted beams may be identical in an ideal situation. Thus the random phases within speckles may be counteracted from the self-heterodyne detection, so that only the randomly varied amplitudes have the effect.

C. Methods to Reduce Speckle Effect

The first possible method to reduce the speckle effect is to use a sliding spotlight mode that has a magnification from the SAIL movement to the beam scan, thus being capable of effectively utilizing the synthesizing length limited from the speckle effect.

Multichannel arrangement is a traditional way to reduce the speckle effect. The possible designs include a receiver with separated multiple transmitters or a transmitter with separated multiple receivers.

As usual, the speckled image can be improved by using digital processing in terms of, for example, the nearest-neighbor operations.

8. Experimental Demonstrations in Laboratory Space

A demonstrator of side-looking SAIL, a setup of down-looking SAIL with 2D quadratic phase-history reconstruction, and a setup of circular spotlight-mode incoherent SAIL were experimentally demonstrated in the laboratory space.

A. Demonstrator of Side-Looking SAIL and its Verification in Laboratory Space

A side-looking SAIL demonstrator with an optical antenna aperture of φ300mm was designed, and to verify the performance in the laboratory space a secondary transmitting telescope was included in [4

4. L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011). [CrossRef]

].

As illustrated in Fig. 14, the illumination source consisted of a 1.55 μm FM-chirped laser and a fiber laser amplifier. A beam divider split the source beam into three beams: the one of the most energy was the beam transmitting to the target, and the other two were used, respectively, as the LO beam for heterodyne detection and the trigging beam of the HCN gas filter for synchronizing the starting frequency for every chirping. The magnification of main telescope was 10×, and its objective had a diameter of φ300mm and a focal length of 1200 mm. The magnification of the secondary telescope was 10×, and its objective had a diameter of φ35mm and a focal length of 140 mm. The source beam was first collimated to a diameter of φ2.9mm, and furthermore via the secondary telescope expanded and converted into the main telescope, which had a defocusing of 110.8 mm to the focal plane of the main telescope objective. This design resulted in a focusing of the transmitting beam in the front of the target plane, which was equivalent to a curvature radius of R=3.3m at an interval of 14 m between the SAIL setup and the target. The illumination size on the target plane was 32mm×32mm as the exit pupil of the main telescope had a stop of 100mm×100mm. The target plane was tilted at a 45 deg angle to the optical axis of system. The light backscattered from the target was collected by the same main telescope. It should be noted that the transmitting beam was arranged to be horizontally polarized, and due to the λ/4 waveplate the return signal beam became vertically polarized in the front of PBS 1. As consequence, PBS 1 acted as an optical circulator to extract out the return signal beam, which was then transferred to PBS 2 through a 4f imaging system. PBS 2 combined the LO beam and return signal beam, which then passed through a λ/2 WP to rotate the polarizations by 45 deg. They were detected by the following 2×2 optical hybrid and balanced heterodyne photo-detectors. The photo-detection areas were both limited by the receiving stops of a size of 0.2mm×0.2mm, which was equivalent to a size of 2mm×2mm on the main telescope pupil. Therefore, the detectable area for heterodyne detection on target plane was 22mm×22mm, and the footprint on the target plane had a similar size. The FW at minimum of theoretical imaging resolution in the azimuth direction was thus 0.38 mm.

Fig. 14. Design of a side-looking SAIL demonstrator and experimental arrangement for near-distance verification.

The average transmitting laser power was approximately 500 mW, and with a repetition rate of 2.5pulses/s. To match the spectrum of HCN filter, the chirp range of the laser was regulated from 1549 nm to 1553 nm with a span of 4 nm, and the chirp rate is 1.2674×1013Hz/s. The FW at minimum of theoretical imaging resolution in the range direction was thus 0.84 mm, i.e., 0.6 mm in the depth. So, the designed full imaging resolution was 0.38mm(azimuth)×0.84mm(range). The LO delay was realized by using either a fiber as depicted in the figure or a multireflection device in the free space. The range difference between the round trip of the signal and the LO path was 60 mm; thus the heterodyne beat frequency was about 2.5 kHz.

Fig. 15. Reconstructed image from a side-looking SAIL experiment.

The full imaging resolution of the SAIL setup was also measured by using a circular target of φ1mm. The resulting image was nearly a rectangle of FW at minimum of 1.4mm(azimuth)×1.2mm(range), which was in good agreement with the theoretical design.

In the experiment, the random phase fluctuations environmentally induced by air current and mechanical interference affected the imaging quality. A stationary mirror was added to locate near the target, which as a reference measured such phase errors in real time during the data collection, and then they were used simultaneously to compensate in the image processing.

B. Laboratory Demonstration of Down-Looking SAIL with 2D Quadratic Phase-History Reconstruction

Figure 16 illustrates the schematic of the experimental setup of a down-looking SAIL with 2D quadratic phase-history reconstruction [24

24. E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012). [CrossRef]

]. For convenience, the experiments were so arranged that the target was stationary and the beams were 2D scanned by a scanning mirror.

Fig. 16. Experimental setup of down-looking SAIL.

The laser was with 300 mW at 532 nm. The source beam was split by PBS 1 into two beams of orthogonal polarizations, and then combined by PBS 2 into a combination of two cocentric coaxial beams of orthogonal polarizations. Lens 1 and lens 3 jointly produced a beam projection on the target plane with a curvature radius of R1=2m. And Lens 2 and lens 3 jointly formed a projection with a radius of R2=3m. Correspondingly, the equivalent radius of the spatial phase difference was R3=6m. The common illumination spot was approximately φ30mm. The scanning mirror had a scanning frequency of 20 Hz along the horizontal, and 10 mHz along the vertical.

The light backscattered from the target was collected by a separated telescope, and its objective aperture was φ50mm. PBS 3 separated the return signal beams into two beams according to their states of polarization. These two beams were then incident to a 2×4 90° optical hybrid followed with two balanced detection devices, which as indicated in Fig. 7 output the photocurrents of the in-phase and quadrature channels. The collected data were stored and processed in a computer.

The target tested was a Chinese character “ao-52-4-579-i001” (light) of the size 5mm×5mm, cut from a reflective tape. The reconstructed image is depicted in Fig. 17. The speckle effect on the reconstruction was obvious. A point target of 0.3mm×0.3mm was also tested to evaluate the imaging resolution. The resulting image had the FW at minimum of 0.4mm×0.35mm, which was reasonable in comparison with the theoretical FW at minimum of 0.26mm×0.26mm.

Fig. 17. Down-looking SAIL experiment: (a) amplitude distribution from in-phase channel; (b) amplitude distribution from quadrature channel and (c) reconstructed image.

C. Laboratory Demonstration of Circular Spotlight-Mode Incoherent SAIL

The experimental arrangement of a laboratory circular spotlight-mode incoherent SAIL [25

25. Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012). [CrossRef]

] is shown in Fig. 18. The illumination source was a pulsed 1064 nm laser with a pulsewidth of 8 ps (equivalently a range of 2.4 mm), and with pulse energies of 0.2–0.55 mJ. The beam from the source laser was linearly polarized and first extended to a diameter of 15 mm as its output. The extended beam passed through a PBS and a λ/4 WP, and then transmitted to the target via a telescope, which had a magnification of 20 and a primary lens aperture of ϕ300mm. The target used was a letter “E” of 295×230mm made from a reflective tap, and tilted 62 deg to the horizontal. The distance between the primary lens and the target was about 10 m. The target was rotated by a rotation stage. The range-time intensity-profile signals from the target were received by the same telescope. The return beam was first changed in polarization by 90° with the λ/4 WP, and then extracted by the PBS to the signal analyzer of a Tektronix Oscilloscope of bandwidth 20Gs/s (equivalently a range resolution of 7.5 mm). The sample length was 25 ns, equivalent to an interval of round-trip range of 3.75 m. In the experiment, the projections were sampled with an increment of 1°.

Fig. 18. Experimental setup of circular spotlight-mode incoherent SAIL.

The reconstructed images from the rotation coverage of 360°, 180°, 90°, and 45° by using the filtered backprojection algorithm are shown in Fig. 19. As predicted, the image reconstructed from 360° projections was well rebuilt with high resolution, the quality of the image from 180° projections was satisfactory, the image from 90° projections was distorted, and the image from 45° projections was hardly recognized.

Fig. 19. Reconstructed images from a circular incoherent SAIL experiment with rotation coverage of (a) 360°, (b) 180°, (c) 90°, and (d) 45°.

9. Conclusion

In this paper we presented a review of our studies on the antenna system, the side-looking SAIL, the down-looking SAIL with 1D linear and quadratic phase-history reconstruction, the down-looking SAIL with 2D quadratic phase-history reconstruction, the incoherent spotlight-mode SAIL, the speckle effect and reduction methods, and the experimental demonstrations.

The down-looking SAIL is based on a key idea of wavefront transformation and regulation by optical techniques. It is understood that to exactly transform or regulate a wavefront into a new one in a limited space is difficult in the microwave domain. In other words, the suggested down-looking SAIL is not a duplication of SAR, but a construction developed from optics and suitable only for optical implementation. The down-looking SAIL has its inherent features; it is thus seen that the down-looking SAIL really relaxes the difficulties in side-looking SAIL and has great potential for outside application.

Various engineering difficulties for the side-looking SAIL were discussed in the course of this paper as well as in the references. For the down-looking SAIL, most obvious are the laser technology issues of developing high-power, high-pulse-repetition-rate, single-mode, and single-frequency lasers, and the optical technique for the wavefront transformation can be further improved. Another major engineering problem is providing line-of-sight pointing control capable of executing multiple scans of the scene to further increase area coverage.

References

1.

M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002). [CrossRef]

2.

S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005). [CrossRef]

3.

Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009). [CrossRef]

4.

L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011). [CrossRef]

5.

J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.

6.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.

7.

R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report NRL/FR/7218-02-10, 051 (2002).

8.

L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).

9.

L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009). [CrossRef]

10.

L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).

11.

L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).

12.

L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010). [CrossRef]

13.

L. Liu, “Synthetic aperture imaging ladar (VI): space-time speckle effect and heterodyne SNR,” Acta Opt. Sin. 29, 2326–2332 (2009). [CrossRef]

14.

L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011). [CrossRef]

15.

R. Garreis and C. Zeiss, “90° optical hybrid for coherent receivers,” Proc. SPIE 1522, 210–219 (1991). [CrossRef]

16.

Y. Zhou, L. Wang, Y. Zhi, Z. Luan, J. Sun, and L. Liu, “Polarization-splitting 2×4 90 free-space optical hybrid with phase compensation,” Acta Opt. Sin. 29, 3291–3294 (2009). [CrossRef]

17.

Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008). [CrossRef]

18.

Z. Luan, L. Liu, S. Teng, and D. Liu, “Jamin double-shearing interferometer for diffraction limited wavefront test,” Appl. Opt. 43, 1819–1824 (2004). [CrossRef]

19.

L. Liu, “Quasi-interferometry with coded correlation filtering,” Appl. Opt. 21, 2817–2826 (1982). [CrossRef]

20.

A. E. Siegman, “The antenna properties of optical heterodyne receivers,” Proc. IEEE 54, 1350–1356 (1966). [CrossRef]

21.

T.-C. Poon, M. Wu, K. Shinoda, and Y. Suzuki, “Optical scanning holography,” Proc. IEEE 84, 753–764 (1996). [CrossRef]

22.

R. M. Marino, R. N. Capes, W. E. Keicher, S. R. Kulkarni, J. K. Parker, L. W. Swezey, J. R. Senning, M. F. Reiley, and E. B. Craig, “Tomographic image reconstruction from laser radar reflective projections,” Proc. SPIE 999, 248–263 (1988).

23.

A. S. Hanesa, V. N. Benhamb, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micron reflection tomography imaging system,” Proc. SPIE 4167, 230–241 (2001). [CrossRef]

24.

E. Dai, J. Sun, A. Yan, Y. Zhi, Y. Zhou, Y. Wu, and L. Liu, “Demonstration of a laboratory Fresnel telescope synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0528003 (2012). [CrossRef]

25.

Y. Yan, X. Jin, J. Sun, Y. Zhou, and L. Liu, “Research of spotlight mode incoherently synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0211003 (2012). [CrossRef]

OCIS Codes
(100.2000) Image processing : Digital image processing
(100.3010) Image processing : Image reconstruction techniques
(110.0110) Imaging systems : Imaging systems
(280.3640) Remote sensing and sensors : Lidar
(280.6730) Remote sensing and sensors : Synthetic aperture radar
(110.3175) Imaging systems : Interferometric imaging

ToC Category:
Remote Sensing and Sensors

History
Original Manuscript: August 22, 2012
Manuscript Accepted: November 26, 2012
Published: January 24, 2013

Virtual Issues
(2013) Advances in Optics and Photonics

Citation
Liren Liu, "Coherent and incoherent synthetic-aperture imaging ladars and laboratory-space experimental demonstrations [Invited]," Appl. Opt. 52, 579-599 (2013)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-52-4-579


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References

  1. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985. (2002). [CrossRef]
  2. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging ladar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005). [CrossRef]
  3. Y. Zhou, N. Xu, Z. Luan, A. Yan, L. Wang, J. Sun, and L. Liu, “2D imaging experiment of a 2D target in a laboratory-scale synthetic aperture imaging ladar,” Acta Opt. Sin. 29, 2030–2032 (2009). [CrossRef]
  4. L. Liu, Y. Zhou, Y. Zhi, J. Sun, Y. Wu, Z. Luan, A. Yan, L. Wang, E. Dai, and W. Lu, “A large-aperture synthetic aperture imaging ladar demonstrator and its verification in laboratory space,” Acta Opt. Sin. 31, 0900112 (2011). [CrossRef]
  5. J. Ricklin, M. Dierking, S. Fuhrer, B. Schumm, and D. Tomlison, “Synthetic aperture ladar for tactical imaging (SALTI) flight test results and path forward,” presented at the Coherent Laser Radar Conferences, Snowmass, Colorado, USA, 9–13 July 2007.
  6. B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic Aperture Ladar Flight Demonstration,” in CLEO: 2011– Laser Applications to Photonic Applications, Technical Digest (CD) (Optical Society of America, 2011), paper PDPB7.
  7. R. L. Lucke, L. J. Rickard, M. Bashkansky, J. Reintjes, and E. Funk, “Synthetic aperture ladar (SAL): fundamental theory, design equations for a satellite system, and laboratory demonstration,” Naval Research Laboratory Report , 051 (2002).
  8. L. Liu, “Optical antenna of telescope for synthetic aperture ladar,” Proc. SPIE 7094, 70940F (2008).
  9. L. Liu, “Antenna aperture and imaging resolution of synthetic aperture imaging ladar,” Proc. SPIE 7468, 74680R (2009). [CrossRef]
  10. L. Liu, “Fresnel telescope full-aperture synthesized imaging ladar: principle,” Acta Opt. Sin. 31, 0128001 (2011).
  11. L. Liu, “Principle of down-looking synthetic aperture imaging ladar,” Acta Opt. Sin. 32, 0920002 (2012).
  12. L. Liu, “Spotlight-mode incoherently-synthetic aperture imaging ladar: fundamentals,” Proc. SPIE 7818, 78180U (2010). [CrossRef]
  13. L. Liu, “Synthetic aperture imaging ladar (VI): space-time speckle effect and heterodyne SNR,” Acta Opt. Sin. 29, 2326–2332 (2009). [CrossRef]
  14. L. Liu, “Structure and operating mode of synthetic aperture imaging ladar for speckle reduction,” Acta Opt. Sin. 31, 1028001 (2011). [CrossRef]
  15. R. Garreis and C. Zeiss, “90° optical hybrid for coherent receivers,” Proc. SPIE 1522, 210–219 (1991). [CrossRef]
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  17. Z. Luan, L. Liu, L. Wang, and D. Liu, “Large-optics white light interferometer for laser wavefront test: apparatus and application,” Proc. SPIE 7091, 70910Q (2008). [CrossRef]
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