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Dutton et al. Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B A63

LADAR resolution improvement using receiversenhanced with squeezed-vacuum injection and

phase-sensitive amplification

Zachary Dutton,1,* Jeffrey H. Shapiro,2 and Saikat Guha1

1Raytheon BBN Technologies, 10 Moulton St., Cambridge, Massachusetts 02138, USA2Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

*Corresponding author: [email protected]

Received November 30, 2009; revised February 3, 2010; accepted February 4, 2010;posted February 12, 2010 (Doc. ID 120564); published March 18, 2010

The use of quantum resources—squeezed-vacuum injection (SVI) and noise-free phase-sensitive amplification(PSA)—at the receiver of a soft-aperture homodyne-detection LAser Detection And Ranging (LADAR) systemis shown to afford significant improvement in the receiver’s spatial resolution. This improvement originatesfrom the potential for SVI to ameliorate the loss of high-spatial-frequency information about a target or targetcomplex that is due to soft-aperture attenuation in the LADAR’s entrance pupil, and the value of PSA in re-alizing that potential despite inefficiency in the LADAR’s homodyne detection system. We show this improve-ment quantitatively by calculating lower error rates—in comparison with those of a standard homodyne de-tection system—for a one-target versus two-target hypothesis test. We also exhibit the effective signal-to-noiseratio (SNR) improvement provided by SVI and PSA in simulated imagery. © 2010 Optical Society of America

OCIS codes: 270.0270, 280.3420, 270.6570.

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. INTRODUCTIONkey feature of a remote sensing system is its ability to

btain detailed spatial information about targets of inter-st, in both transverse and longitudinal (range) dimen-ions. High-resolution spatial information is essential foruch tasks as target classification, image processing, andracking of multiple closely spaced targets [1]. Forodest-range �1–100 km� terrestrial applications under

lear-weather conditions, LAser Detection And RangingLADAR) systems [2,3] offer superior spatial resolution,hen compared to microwave radars, owing to their usef much shorter wavelengths. When atmospheric turbu-ence can be neglected, the spatial resolution of such aystem is generally limited by the Rayleigh resolution ofts receiving optics (1.22� /D, where � is the LADARavelength and D is the aperture diameter for an unob-

cured circular entrance pupil) and the signal-to-noise ra-io (SNR).

It is known [4–6] that quantum entangled states offerotential for improved resolution in low-loss interferom-try, time-of-flight ranging, and other parameter estima-ion problems. It is an open question, however, whetheruantum effects can be used to improve remote sensing,n which high loss (�100 dB for standoff sensing of auasi-Lambertian reflector) is unavoidable. For example,00N states [7], which achieve Heisenberg-limited phase

ensing in a lossless scenario, have been shown to be of noalue in high-loss environments [8]. Although the poten-ial advantages afforded by the quantum properties ofight are indeed destroyed by even modest losses, theossibility remains that an optical receiver could utilizeonclassical effects—in processing the LADAR return

0740-3224/10/060A63-10/$15.00 © 2

btained from conventional laser illumination of aarget—to achieve better performance than the standardeception techniques, i.e., direct, homodyne, and hetero-yne detection.In this paper, we analyze two ways in which quantum

ffects can be used to improve a LADAR receiver that isquipped with a soft-aperture entrance pupil. First we in-estigate squeezed vacuum injection (SVI), as proposed in9], to reduce the vacuum noise incurred on the high-patial-frequency target information that has been at-enuated by the soft aperture. SVI requires the use of ho-odyne detection, so the LADAR receiver is only

ensitive to the quadrature in which the noise reductionas occurred. However, the effectiveness of this noise re-uction is severely impacted by inefficiency in that homo-yne measurement. Thus, for our second use of nonclas-ical effects, we analyze phase-sensitive amplificationPSA) [9–11], which enables noise-free amplification of aingle field quadrature, to overcome any homodyne ineffi-iency. We perform a quantitative analysis of these two ef-ects and show that they each improve the SNR of the re-eived LADAR signal, which translates directly intomproved spatial resolution in the acquired image.

The main thrust of this paper will be to quantify theesolution improvement alluded to above. However, be-ause the presence of a soft-aperture entrance pupil isecessary to realize the gains due to SVI, it is worth giv-

ng a brief appraisal of why such a soft-aperture systemould be of interest in a conventional LADAR system.uppose we have a LADAR receiver whose optics containn unobscured circular entrance pupil, so that itsiffraction-limited point-spread function (PSF) is the

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A64 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Dutton et al.

amiliar Airy disk pattern, with its quasi-periodic sidelobeings. By incorporating a soft aperture inside this en-rance pupil, i.e., an apodizing mask that gradually rollsff in transmission from unity at its center to zero or nearero at radius D /2, a somewhat broadened but smoothed,idelobe-free PSF can be obtained. This new PSF willvoid image contamination from off-axis bright specularshat could preclude resolution of weak on-axis objectshen side-lobe returns from the former compete withain-lobe returns from the latter within an image pixel.The remainder of this paper is organized as follows. In

ection 2 we describe the LADAR configuration that weill consider, including both its baseline operation, and

ts quantum enhanced operation with SVI and PSA. Inection 3 we establish a rigorous quantitative methodol-gy for LADAR spatial resolution by defining that reso-ution to be the ability to reliably distinguish one pointarget from two closely spaced point targets based on theinimum error-probability decision rule from binary hy-

othesis testing. We then perform this decision-theorynalysis for soft-aperture LADAR receivers that employither standard homodyne reception or quantum en-ancement with SVI and/or PSA. We assume quasi-ambertian reflection at rough-surfaced targets, so thathe target returns exhibit fully developed speckle statis-ics, as is typically the case for the �1 �m wavelengthsmployed in LADAR systems. We find that PSA can sig-ificantly improve spatial resolution for the case of ineffi-ient homodyne detection even without SVI. SVI alonean also improve spatial resolution, but only for highly ef-cient homodyne reception. The SVI advantage is re-ained, in the presence of homodyne inefficiency, whenufficient PSA is also employed. The combination of SVIlus PSA will be shown to afford a substantial improve-ent in spatial resolution as a consequence of the SNR

ain they provide. In Section 4 we use a modulationransfer function (MTF) analysis to simulate and comparemagery corresponding to conventional homodyne recep-ion and quantum enhancement with SVI and/or PSA.his simulated imagery shows spatial resolution improve-ent consistent with what was obtained from the

ecision-theory analysis. Indeed, in both cases the perfor-ance improvement can be directly ascribed to an effec-

ive increase in SNR provided by the nonclassical effectsf SVI and PSA. In Section 5 we close with some conclud-ng remarks that draw together the key lessons of ournalysis and indicate areas for future research.

. DESCRIPTION OF THEUANTUM-ENHANCED LADAR RECEIVERsimple schematic of our quantum-enhanced LADAR re-

eiver concept is shown in Fig. 1. Not shown is the LA-AR transmitter that floodlights the target region with aulse of laser light. This illumination results in a target-eturn pulse from a planar object at range L that, becausehe transmitter emits coherent-state light, can be takeno be a classical field. The photon-units complex envelopef the classical target-return field that arrives at the re-eiver’s entrance pupil is then given by

ER��,t� =� d��IT�p

��

s�t − 2L/c�T��e−ik��·��/L+ik��2/2L

i�L,

�1�

here ��= �x� ,y��, ��= �x� ,y�� are, respectively, the trans-erse spatial coordinates in the target and pupil planes.n Eq. (1), IT is the target-plane intensity (in W/m2) pro-uced by the transmitter pulse; �� is the photon energyt the LADAR wavelength; �p is the duration of thatulse; s�t� is the normalized—squared magnitude inte-rates to one—baseband shape of the transmitter pulse;�� is the target’s field-reflection coefficient; and the ex-onential represents the Fraunhofer diffraction from po-ition �� in the target plane to position �� in the receiver’sntrance pupil, with k=2� /� being the wave number athe LADAR wavelength. Note that we have neglected at-ospheric extinction, although that is easily added to our

ormulation, and ignored the effects of atmospheric turbu-ence, although they can also be incorporated into ournalysis [12]. We have also neglected background light,ecause it is typically negligible in homodyne detectionystems [13].

We assume that the receiver’s entrance pupil containsspatially dependent (soft-aperture) transmission maskith field transmittance A��=e−2��2/R2

[14]. Equation1) shows that the field entering this aperture is propor-ional to the spatial Fourier transform of the target’seld-reflection coefficient T��. Thus the soft aperture at-enuates high-spatial-frequency target information,ence limiting the LADAR’s spatial resolution. Let usrst use classical electromagnetics and semiclassical pho-odetection theory to characterize the statistics of theaseline sensor, after which we will change to quantumheory for the SVI plus PSA case.

The classical field emerging from the soft aperture is

ER� ��,t� = A��ER��,t�, �2�

hich is then focused onto a continuum array where ho-odyne detection is performed. Assuming that we mea-

ure the real quadrature of the target return in the imagelane—after suppressing some inessential absolute and

ig. 1. (Color online) Diagram of our quantum-enhancedADAR receiver. At the far left we show the targets (at range L

rom the receiver) considered in the resolution analysis—oneoint target on axis and two point targets symmetrically dis-osed at ±�0 in angle about the optical axis—that will be pre-ented in Section 3. The baseband photon-units field operator ERf the target return is transmitted through a spatially dependentsoft) aperture A�� and a squeezed-vacuum field operator ES isnjected according to Eq. (5). The resulting field operator ER� thenndergoes PSA before being mixed with a LO and homodyne de-ected. We assume a continuum homodyne-detection array. Thisuantum-enhanced receiver will be compared with a classicalaseline LADAR system in which ES is in its vacuum state ando PSA is employed.

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uadratic phase factors—followed by time-domainatched-filtering for a pulse arriving from range L, we

btain the following noisy spatially dependent outputrom the matched filter,

yd�� =��IT�p

��

Re�� d��T��m�� − �� + nd��,

�3�

here the first term is the target return, under the as-umptions of one-to-one [15] imaging and measurement ofhe real-part quadrature, and the second term is the localscillator (LO) shot noise. In Eq. (3), � is the homodynefficiency—the product of the mixing efficiency and quan-um efficiency of the continuum array—and

m�� =� df A��Lf�ei2�f·� =�

2 R

�L2

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s the PSF associated with the Gaussian soft aperture.he noise term nd�� is a real-valued, zero-mean, whiteaussian noise process whose spectral density, with ourormalization of the homodyne output, is Snd,nd

�f�=1/4,here f= �fx , fy� is the the spatial frequency vector associ-ted with �= �x ,y�.We now assume that our sensor is augmented with two

istinct quantum effects. In order to accurately describeheir impact on the detected return, the above expres-ions must be rewritten using quantum field operators inlace of classical stochastic fields. First we assume thatVI is employed at the soft aperture, so that the photon-nits baseband field operator emerging from this aper-ure satisfies

ER� ��,t� = A��ER��,t� + �1 − A2��ES��,t�. �5�

n this expression the field operator ER is in the coherenttate whose eigenfunction is given by Eq. (1), conditionedn knowledge of T��, and ES is in a squeezed-vacuumtate arranged so that its low-noise quadrature will beeasured by the image-plane homodyne detector. In par-

icular, the squeezed vacuum field ES is generated as fol-ows:

ES��,t� = cosh�r�Ein��,t� − sinh�r�Ein† �− ��,t�, �6�

here r0 and Ein�� , t� is a vacuum-state field, so thathe first quadrature of this operator’s contribution to themage-plane field—where the homodyne detection willake place—will be squeezed by a factor of e−2r. Withoutnjection of this squeezed vacuum, ES�� , t� would be in itsacuum state. Because the pupil plane is a Fourier-ransform plane relative to the image formed on the de-ector array, squeezed-vacuum injection reduces whatould otherwise be a higher vacuum-noise contribution atigh spatial frequencies in the field quadrature that theomodyne setup measures.For our second quantum enhancement over baseline

ADAR operation, we utilize PSA to noiselessly amplifyhe quadrature of ER� �� , t� that will be measured by theontinuum homodyne array. PSA can be realized with anptical parametric amplifier (OPA) [11], and can amplify a

ingle field quadrature with unity noise figure. Further-ore, this noiseless amplification can occur over a wide

patial bandwidth and thus cover many transverse spa-ial modes. The baseband field operator after PSA thenakes the form

E��,t� = �GER� ��,t� + �G − 1ER�†��,t�, �7�

here G1 is the OPA gain, and amplification of the real-art quadrature—to match the low-noise quadrature ofhe SVI—has been assumed.

The final step in our quantum-enhanced LADAR is theame as in the baseline system, viz., continuum homo-yne detection of the real-part quadrature followed byime-domain matched filtering for the return from a tar-et at range L. The noisy output from the matched filter isgain classical, as in the baseline case from Eq. (3), but itsarget-return component has been amplified by the PSA’suadrature gain Geff��G+�G−1�2,

yd�� =�Geff�IT�p

��

Re�� d�� T��m�� − �� + nd��,

�8�

nd the spectral density of the zero-mean additive Gauss-an noise has been altered by SVI and PSA so that it isow given by

Snd,nd�f� =

�Geff

4�A2��Lf� + �1 − A2��Lf� e−2r� +

1 − �

4.

�9�

he terms in this spectral density have the following im-ortant physical interpretations. From left to right weave contributions from (i) the target return’s coherent-tate quadrature quantum noise after PSA and inefficientetection, �GeffA2��Lf� /4; (ii) the quadrature quantumoise from squeezed-vacuum injection after PSA and inef-cient detection, �Geff�1−A2��Lf� e−2r /4; and (iii) theuadrature quantum noise from inefficient detection,1−�� /4.

In the limit of no amplification �Geff=1� and no squeez-ng �r=0�, the noise spectrum expression reduces to thelassical result, Snd,nd

�f�=1/4, and Eq. (8) becomes Eq.3), confirming the validity of semiclassical theory fornalyzing the baseline LADAR [16]. In this case, thequared strength of the target-related component in yd��s attenuated by the homodyne efficiency �, whereas the

ean-squared noise strength is unaffected, degrading theomodyne SNR in the baseline LADAR by a factor �.owever, when PSA gain Geff�1 is introduced, theomodyne-inefficiency term in Eq. (9) becomes negligible.n this case the squared target-return strength and theoise spectrum are both proportional to �, and we recoverhe SNR corresponding to unity homodyne efficiency,=1. When strong squeezing is introduced, the spatial

requencies f for which target information is most attenu-ted by the soft aperture A��Lf� have their noise greatlyeduced, further improving their SNR. Taken together,hese two effects act to eliminate SNR degradation that isue to both soft-aperture attenuation and inefficient de-ection. Note that in this analysis we have assumed that

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he spatial bandwidths of the SVI and the PSA are suffi-iently large to have negligible variation over the spatialrequencies captured by the soft aperture, viz., SVI andSA extend at least to spatial frequencies satisfying

�Lf � �R.For the hypothesis testing problem to be considered in

ection 3, as well as for gaining deeper insights into theehavior of our SVI plus PSA enhanced receiver, it is con-enient to convert the continuum photodetector returnsyd�� :−�x ,y�� into Fourier space by defining

Yd�f� �� d� yd��e−i2�f·�, for − � fx, fy � ,

�10�

here yd�� is given by the quantum-enhanced expressionq. (8). We can recover the baseline case by setting r=0nd Geff=1, so the quantum theory provides a convenientnified starting point.It is easily verified that

Re�Yd�f� =�Geff�IT�p

��

Ev�Tr�f� A��Lf� + nr�f�, �11�

nd

Im�Yd�f� = −�Geff�IT�p

��

Od�Tr�f� A��Lf� + ni�f�,

�12�

here

Tr�f� �� df Re�T�� e−i2�f·� �13�

s the Fourier transform of the real part of the target’seld-reflection coefficient, Ev�Tr�f� and Od�Tr�f� denotehe even and odd parts of that Fourier transform,nd �nr�f� ,ni�f�� is a set of independent, identicallyistributed, real-valued, zero-mean, nonstationary whiteaussian noise processes whose common correlation

unction is

�nj�f1�nj�f2�� =Sndnd

�f1�

2��f1 − f2�, for j = r,i. �14�

In Section 3 we utilize the Fourier description to derivehe structure and performance of the minimum error-robability receiver for deciding between equally likelyypotheses (corresponding to one and two targets). Inection 4 we utilize this description to understand the be-avior of our simulated imagery.

. RESOLUTION BEHAVIORquations (8) and (9) imply that our quantum-enhanced

eceiver concept will improve the SNR of a homodyne LA-AR whose homodyne efficiency is appreciably lower

han unity and increase the SNR of the high spatial-requency information when that LADAR employs a soft-perture entrance pupil. To rigorously quantify how thisNR improvement translates into improved spatial reso-

ution, we consider a fundamental binary hypothesis-

esting scenario in which the target is assumed to be ei-her a single on-axis point object or two closely spacedoint objects that are symmetrically disposed at angles�0 about the optical axis, as shown in Fig. 1. Under bothypotheses we will assume that the targets give fully de-eloped speckle statistics with the same average photonumber being received over the LADAR receiver’s en-rance pupil [17]. Because the error probabilities dependn the magnitude of the angular separation between thewo targets, we find it sufficient and convenient to per-orm our calculations for a one-dimensional (1D)ersion—spatial coordinate x—of the Section 2 setup.

The target characteristics for hypotheses H1 (one point-arget present) and H2 (two point-targets present) can beaken to be [18]

�x��

=��dTv0��x��, under H1

��dT/2�v+��x� − �0L� + v−��x� + �0L� , under H2�

�15�

here dT is the (1D) LADAR cross-section of the singleoint target; �� · � is the unit-impulse function; andv0 ,v+,v−� is a set of independent, identically distributed,ero-mean, unity-variance, isotropic complex-Gaussianandom variables representing the speckle behavior ofach point target.

It turns out to be easiest to treat the preceding hypoth-sis testing problem in the Fourier space, by usingYd�fx� :−� fx�� as the data from which a decision is toe made. Hypothesis-testing problems involving station-ry white Gaussian noise are easily solved [19], so wehall preprocess Yd�fx� to remove its nonstationarity, viz.,e apply a whitening filter and base our one-target ver-

us two-target decision on

Yd��fx� �� 2

Sndnd�fx�

Yd�fx�, �16�

here Yd�fx� and Sndnd�fx� are the 1D analogs of the 2D

ases from Eqs. (11) and (9). The real and imaginary partsf Yd��fx� are then

Re�Yd��fx� =� 2Geff�IT�p

��Sndnd�fx�

Ev�Tr�fx� A��Lfx� + nr��fx�,

�17�

nd

Im�Yd��fx� = −� 2Geff�IT�p

��Sndnd�fx�

Od�Tr�fx� A��Lfx� + ni��fx�,

�18�

here nr��fx� and ni��fx� are independent, identicallyistributed, real-valued, zero-mean, stationary whiteaussian noise processes, each with unity spectral den-

ity. Here A�x�=e−2x2/R2is the 1D soft-aperture function

nd IT is now the 1D target-plane irradiance (in W/mnits).

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Evaluating the target-return contributions to these ex-ressions for our two target hypotheses we find that

Ev�Tr�fx� =��dT Re�v0�, under H1

��dT/2 Re�v+ + v−�cos�2��0Lfx�, under H2�

�19�

nd

d�Tr�fx�

= �0, under H1

��dT/2 Im�v+ − v−�sin�2��0Lfx�, under H2.� �20�

tandard results in binary hypothesis testing for Gauss-an random processes [19] now allow us to state that theandom vector r��rc r rs T, with

rc �� dfx Re�Yd��fx� A��Lfx�

�2Sndnd�fx�

4��L�2

�R2 1/4

cos�2��0Lfx�,

�21�

r �� dfx Re�Yd��fx� A��Lfx�

4S �f �4��L�2

�R2 1/4

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sin�2��0Lfx�,

�23�

s a sufficient statistic for the minimum error-probabilityeceiver. When only one target is present (hypothesis H1),e find r is a zero-mean Gaussian random vector whose

ovariance matrix is

1

= �8Geff�nC1

2 + C0 + C2 �2�4Geff�nC0 + 1�C1 0

�2�4Geff�nC0 + 1�C1 �4Geff�nC0 + 1�C0 0

0 0 C0 − C2�

�24�

ikewise, when two targets are present (hypothesis H2), rs again a zero-mean Gaussian random vector, this timeith a covariance matrix given by

�2 = �� 0

0T �2Geff�n�C0 − C2� + 1 �C0 − C2�� , �25�

T
here 0 = �0 0 and
� = ��2Geff�n�C0 + C2� + 1 �C0 + C2� �2�2Geff�n�C0 + C2� + 1 C1

�2�2Geff�n�C0 + C2� + 1 C1 4Geff�nC12 + C0

� . �26�

he constants appearing in these covariance matrices are

C0 =� dfx

A2��Lfx�

4Sndnd�fx��4��L�2

�R2 , �27�

C1 =� dfx

A2��Lfx�


�R2 cos�2��0Lfx�, �28�

C2 =� dfx

A2��Lfx�


�R2 cos�4��0Lfx�, �29�

nd

n =ITdT�p

��L��R2

4. �30�

or the baseline sensor there are simple closed-form ex-ressions available for the �Ci�, viz., C0=1, C1=e−�k�0R / 4�2,nd C2=e−�k�0R / 2�2. More important, however, is the physi-al interpretation of n. This quantity is the average pho-on flux—under either hypothesis—that passes throughhe LADAR receiver’s soft aperture, so that 2�n is theNR for conventional homodyne reception of this light atixing efficiency �.

For equally likely hypotheses, the minimum error-robability decision rule, based on the sufficient statistic, is [19]

rT��1�−1 − ��2�−1 r

decide H2

�

decide H1

2 ln ��2�1/2

��1�1/2 , �31�

here ��j� denotes the determinant of �j. We computedhe error probability, PE, for this decision rule by runningonte-Carlo simulations of the sufficient statistic r, gen-

rated in accordance with zero-mean Gaussian statisticssing the covariance matrices �1 and �2, and applyinghe decision rule from Eq. (31). We did this for a variety nalues and two-target angle offsets �0. For each case wean 10,000 trials of hypotheses H1 and H2. This simula-ion procedure was performed for baseline sensor opera-ion, for enhancement with SVI only, for enhancementith PSA only, and for enhancement with both SVI andSA. In what follows we present a summary of our re-ults. We also computed the Chernoff bound [20] on PE,ut we found that this upper bound was not sufficientlyight in the PE regime of interest.

Figure 2(a) plots PE versus �0 for the baseline sensorr=0 and Geff=1) at unity homodyne efficiency ��=1� foreveral different n values. As expected, PE decreases withncreasing target separation � and with increasing n. In-
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erestingly, the error probabilities have nonzero asymp-otes in the limit �0→ [21]. Physically, this is due topeckle precluding perfect distinguishability between onend two point targets at finite SNR, regardless of theirngular separation.To assess the role of SNR on resolution, we define tar-

ets to be resolved when the minimum error-probabilityecision between equally likely one-target and two-targetypotheses satisfies PE�0.03, as shown by the dashed

ine in Fig. 2(a). In that figure we are using the Rayleigh

(b)

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ig. 2. (Color online) (a) Error probability, PE, for minimumrror-probability decisions between equally likely one-target andwo-target hypotheses at three different n values plotted versushe angular separation �0: n=21 dB (circles), n=16 dB (squares),=11 dB (diamonds). The angular separations are normalized to

he Rayleigh resolution �0�Ray��0.32� /R. The error probabilities

ere found via 10,000 Monte-Carlo trials of the sufficient statis-ic vector r under each hypothesis. The LADAR’s resolution, de-ned to be the angle �0

�res� at which PE=0.03 occurs, are the pointshere the curves cross the dashed line. All the PE curves asymp-

ote to nonzero constants with increasing �0, and at n=17 dB thisalue is above the PE=0.03 resolution threshold. (b) Resolutionrelative to the Rayleigh resolution) plotted versus n for the clas-ical baseline (open circles) and SVI-enhanced systems (filledircles) both with unity homodyne efficiency �=1 and with 15 dBf SVI, i.e., e2r=101.5. There is an 8 dB SNR shift between theaseline and SVI curves at high n values. (c) Classical baselineesolution (open circles) for an inefficient detection system�=0.25�, for that system enhanced with 15 dB SVI but no PSAfilled circles), and for it enhanced with 15 dB of SVI plus Geff15 dB of PSA (filled squares). This latter case shows an 11 dBNR shift at high n values. Also shown is the case �=1 with SVI

open squares), which corresponds to the performance obtainedn the limit Geff→.

esolution, �0�Ray�=0.32� /R, for our A�x�=e−2x�2/R2

soft ap-rture, as a normalization constant for �0, so that a sys-em that can resolve angles smaller than �0

�Ray� haschieved sub-Rayleigh performance [22]. From Fig. 2(a)e see that �0

�res�=1.6�0�Ray� is realized when n=16 dB, and

0�res�=0.9�0

�Ray� is achieved when n=21 dB. For n=11 dBhe targets cannot be resolved, owing to the �0→symptotic behavior noted above. The open (black) circlesn Fig. 2(b) denote the baseline sensor’s resolution as aunction of n. We note the resolution diverges, i.e., �0

�res�

, for n�14 dB. Using the Chernoff bound mentionedbove, we have been able to estimate that the resolutioncales as n−3/10 in the limit of large n. At n values nearero resolution divergence, only numerical results are avail-ble.Now let us consider the resolution possible when 15 dB

VI—corresponding to e2r=101.5—is employed at the softperture, without PSA, when the LADAR still has unityomodyne efficiency. The resolution versus n behavior ofhis case is given by the filled (red) circles in Fig. 2(b),hich show that there is approximately a factor of two

mprovement in resolution at high n values. In this re-ime the SVI resolution curve is, in essence, the baselineesolution curve shifted to the left by �8 dB. We also seehat SVI has lowered the n value at which resolution di-erges to n=10 dB. Because we have assumed �=1, nourther performance improvement can be obtainedhrough the addition of a PSA stage.

To quantify the importance of PSA when the LADAReceiver has an inefficient homodyne setup, we now as-ume �=0.25. Figure 2(c) shows the resolution versus nehavior for (i) the classical baseline sensor with this in-fficiency; (ii) the ideal ��=1� case with 15 dB of SVI ando PSA; and (iii) the inefficient ��=0.25� case with 15 dBf SVI and Geff=15 dB of PSA. Here we see that the clas-ical curve gets shifted to the right by 6 dB=1/� relativeo its unity efficiency performance from Fig. 2(b). Thishift is directly ascribable to the factor of � reduction inhe SNR of conventional homodyne operation. The SVI-nhanced sensor offers virtually no improvement in thisnefficient case because the resulting injection of ��1acuum-state quantum noise renders the squeezing inef-ectual. However, utilizing sufficient PSA in conjunctionith SVI recovers almost all of the SNR shift from the �1 case, i.e., the filled (blue) squares for SVI plus PSA at=0.25 come close to the �=1 SVI-enhanced SNR shifthown in Fig. 2(c) by the open (green) squares. Overall,he SVI and PSA values employed in Fig. 2(c) at �=0.25rovide approximately a three-fold improvement in reso-ution and approximately an 8 dB shift in the divergenceoint.Having established the performance gain possible by

tilizing PSA and SVI quantum enhancements to a homo-yne LADAR receiver, we gain a more systematic under-tanding of these improvements by plotting the SNRhift—observed to be 8 dB and 11 dB in Figs. 2(b) and(c), respectively—for a variety of detector efficiencies �nd gains Geff. Figure 3(a) plots this shift versus homo-yne efficiency when only PSA has been added to theaseline configuration. PSA with sufficient gain restoreshe system SNR to its �=1 value, and so does not offer

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mprovement at high values of the homodyne efficiency.he top curve (black circles) shows the theoretical maxi-um �Geff→� SNR shift 1/�. The other curves show the

hift for various finite Geff. All these curves approach Geffn the limit �→0. Figure 3(b) shows the shifts with 15 dBf SVI employed in addition to PSA. Here the SNR shift isdB when �=1, cf. Fig. 2(b). For smaller � this shift be-

omes degraded when no PSA is employed (Geff=1, bottomurve). As Geff is increased, the SNR shift is regained, ap-roaching the theoretical limit 8 dB+1/� for sufficientlyigh Geff. For very low � values, PSA is responsible for allhe SNR advantage, i.e., the SVI advantage disappears,s implied by the fact that the curves in Figs. 3(a) and(b) to converge to the same values as �→0.

. SIMULATED IMAGERYhe simple binary hypothesis-testing problem from thereceding section afforded a detailed quantitative studyf the resolution improvement offered by SVI and PSA en-ancements. It is also of interest, however, to provide aualitative indication of how these quantum enhance-ents lead to improvements in image quality for more

omplicated scenes. In this section we will provide suchndication by simulating imagery from the 2D setup de-cribed in Section 2. The system we analyze is like thatiagrammed in Fig. 1, but with an extended 2D target in-tead of either one or two point targets, and with augmen-ation, described below, to image both quadratures of thearget’s field-reflection coefficient.

SNR

shift

SNR

shift

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(a)

(b)

15

10

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0.2 0.4 0.6 0.8 1.00.0

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ig. 3. (Color online) (a) SNR shift versus homodyne efficiencyfor several different PSA gain values (from bottom to top)

eff=4.8 dB (open squares), 7.0 dB (open circles), 10 dBfilled diamonds), and 15 dB ( filled squares). The top curve (filledircles) is 1/�, the limit for Geff→. (b) The same curves with5 dB of SVI in addition to PSA. Here we also plot the case witho PSA Geff=0 dB ( open diamonds). Note that for �=1 all curvesonverge to the 8 dB SVI SNR shift. For low values of � and Geff,he vacuum noise introduced by inefficient detection destroys theVI advantage. For higher Geff values, PSA preserves the SVIdvantage. At low � values, the SVI plus PSA advantage is pri-arily due to the PSA stage, as the curves converge to the same

alues seen in (a) for the non-SVI case.

Suppose that the LADAR receiver’s entrance pupil isreceded by a 50–50 beam splitter whose two outputs arendividually processed—in the manner described in Sec-ion 2—to obtain SVI plus PSA enhanced imagery of theeal and imaginary quadratures of the target’s field-eflection coefficient [23]. Thus we obtain a return similaro Eq. (8) only with half the signal power in each quadra-ure rather than all of it in one of them:

y1�� =�Geff�IT�p

2��

Re�� d��T��m�� − �� + n1��,

�32�

nd

y2�� =�Geff�IT�p

2��

Im�� d��T��m�� − �� + n2��.

�33�

n the following analysis we also assume that theaussian-attenuation soft aperture is embedded in a cir-

ular hard-aperture pupil of diameter D so that

A�� =�e−2��2/R2, for �� D/2

0, otherwise. � �34�

he noises n1�� and n2�� are independent, identicallyistributed, zero-mean Gaussian random processes, eachith the spectral density given in Eq. (9). Once again, we

an revert to baseline operation by setting r=0 and Geff1. In that case the dual-homodyne receiver has statistics

hat are equivalent to those of a heterodyne-detection la-er radar.

Assembling the real and imaginary quadrature imagesnto a single complex-valued entity gives us

y�� = y1�� + iy2��

=�Geff�IT�p

2�� d��T��m�� − �� + n��

= s�� + n��, �35�

here s�� is the signal (target-return) component of themage and n��n1��+ in2�� is the noise component.vidently, from Eq. (4), the MTF for the target-returnomponent of y�� is A��Lf�. Combined with the spatial-requency content of the noise terms n1�� and n2�� weave a complete understanding of the spatial-frequencyehavior of the dual-homodyne image, y��, of an arbi-rary field-reflection coefficient T��. However, becausehere is no target information at spatial frequencies above/2�L in magnitude, whereas there is noise at these spa-

ial frequencies, we will apply an ideal low-pass spatiallter to y�� to eliminate the pure-noise spatial frequen-ies. Thus we shall display simulated intensity imagesy��2, where

y�� =� d��y��hD�� − �� = s�� + n�� 36�

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hD�� f��D/2�L

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At this point, all that is needed to obtain a simulatedntensity image based on the preceding MTF is to supply

test object and its statistics. We will assume that anyarget of interest will produce fully developed speckle sta-istics. Hence, we will use the idealized model for such anbject that has frequently been employed in LADARheory [12], namely, T�� is a zero-mean, complex-valuedaussian random process whose only nonzero second mo-ent is [24]

�T*��1��T��2�� = �2T��1��1� − �2��, �38�

here T�� is the average target intensity-reflectance atransverse coordinate ��.

In Fig. 4 we show results of such simulations when�� is taken to be a United States Air Force (USAF) res-lution chart. The original T�� [Fig. 4(a)] is blurred byhe Gaussian soft-aperture entrance pupil with R=D /2Fig. 4(b)]. This blurred image represents the LADAR in-ensity image obtained with that soft aperture in the limitf high SNR and full speckle suppression by multi-imageveraging. Owing to the speckle statistics, a single LA-AR intensity image formed with dual-homodyne detec-

ion will have very strong spatial variations, even for fea-ureless targets. To see that this is so, we first define the

(a)

(d)

(b) (c)

(e) (f)

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Apertureblurred image

Homodyne

SVI only PSA only PSA+SVI

ig. 4. (Color online) Simulated intensity images for ouruantum-enhanced LADAR receiver when the planar target ishe USAF resolution chart, shown in (a), that gives rise to fullyeveloped laser speckle. We have assumed a target range=1 km, a 15 m�15 m square target region, �=1.55 �m, an=4 mm soft-aperture receive pupil inside a D=8 mm diameterard aperture imaged onto a continuum-detector homodyne ar-ay. (b) Image of the resolution chart after blurring by transmis-ion through the soft aperture. This corresponds to the image inhe limit of high SNR and averaging a large number of intensitymages with statistically independent speckle. Images (c)–(f)how detected images averaging over M=100 intensity imagesssuming independent speckle fluctuations and homodyne effi-iency �=0.25. (c) Image obtained by the baseline homodyneADAR. (d) Image obtained by the SVI-enhanced LADAR with5 dB squeezing. Little image improvement is seen in compari-on with the baseline case owing to homodyne inefficiency. (e) Im-ge obtained by the PSA-enhanced LADAR with Geff=15 dB.ome image improvement is seen in comparison with theaseline case. (f) Image obtained using SVI plus PSA enhancedADAR. Substantial image improvement over the baseline case

n (c) is seen.

mage signal-to-noise ratio, SNRI��, at a single point in aADAR intensity image to be the square of its average

arget-return component, ��s��2�, divided by the vari-nce of the intensity image at that point, Var��y��2��.e then find that

SNRI�� =SNR��/2

1 + SNR��/2 + 1/2SNR��→ 1, �39�

s SNR��→, with SNR��s��2� / ��n��2� being theatio of the average intensity of the target-return compo-ent to the average intensity of the noise component athat point in the image [12]. The high-SNR asymptote forNRI represents speckle-limited operation. Of course, ifll points in the image suffer identical speckle fluctua-ions, then speckle just induces a random variation ofverall image contrast. Such is not the case, however, be-ause the �-correlation property of target’s field-reflectionoefficient makes each diffraction-limited field of view inhe intensity image have an independent (or nearly inde-endent) speckle fluctuation. For this reason, averaging

LADAR intensity images—taken in such a mannerhat the speckle decorrelates between acquisition of eachmage—is often employed, resulting in a factor-of-M in-rease in SNRI. The image SNR for this M-image averages SNRI

�M��=MSNRI��, where SNRI�� is given by Eq.39). Note that soft-aperture attenuation of high-spatial-requency target information implies that those spatialrequencies will not reach the speckle-limited SNRI valuen a single intensity image. So, although averaging M in-ensity images will increase SNRI for the high-spatial-requency target information from our baseline LADAR,erformance can be—and, as we will now show, will be—ignificantly improved by means or our quantum en-ancements, SVI and PSA. To provide a single-numberssessment of image SNR, in what follows, we will employNRI

�M� defined to be the spatial average of SNRI�M��

ver the entire image region.Figure 4(c) shows the M=100 case for the baseline

ADAR’s intensity image of the USAF resolution charthen �=0.25. In the remaining panels of Fig. 4 we show=100 image averages with SVI utilized [Fig. 4(d)], with

SA utilized [Fig. 4(e)], and with both SVI and PSA em-loyed [Fig. 4(f)], all with �=0.25 homodyne efficiency.ecause of the low homodyne efficiency, SVI alone offers

ittle improvement over the baseline image, but some im-ge improvement is obtained with PSA alone. However, asan be seen by comparing Figs. 4(c) and 4(f), the combina-ion of SVI and PSA provides the greatest improvement inmage quality in comparison with the baseline case. Themage SNR averaged over the entire image is SNRI

�M�

6 dB for the classical �=0.25 baseline case (Fig. 4(c)),hereas the SNRI=13 dB when SVI and PSA are both ap-lied (Fig. 4(f)). This 7 dB enhancement in average imageNR translates to the observed improvement in imageuality.It is also instructive to use data derived from the USAF

esolution chart to visualize the signal and noise spatial-requency behavior of our quantum-enhanced receiver. Inig. 5(a) we show a cut in one dimension of the spectralensities of the signal and noise components of Eq. (36)or the baseline sensor with �=1. These spectra were ob-

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ained from simulation data, rather than theory, leadingo the jagged nature seen in the figure. The signal spec-rum is shaped by the soft aperture whereas the noise ist the white (vacuum) noise level at all spatial frequen-ies out to the cutoff imposed by hD��. Figure 5(b) pre-ents similar baseline-sensor results when the homodynefficiency is reduced to �=0.25. Here we see that the sig-al spectrum is scaled down by a factor of � while theoise level is unchanged, leading to a much reduced SNR.hen SVI is employed with �=0.25 and no PSA, the

queezing has only marginal effect on the noise spectrumnd the performance is very similar to the classical casen Fig. 5(b). In Fig. 5(c) we show the �=0.25 case withSA enhancement of Geff=10 dB. The PSA amplifies bothhe signal and noise before the detector, leading to anNR comparable to the �=1 case in Fig. 5(a). Finally, inig. 5(d), we show the case with SVI and PSA both uti-

ized. Here we find that SVI suppresses the noise level inhe high-frequency components, while PSA overcomes theomodyne inefficiency, leading to a noise spectrum whosehape is matched to the soft aperture, further improvinghe SNR beyond what we had for PSA-only operation.

. DISCUSSIONe have shown that two complementary quantum effects,SA and SVI, can be used to overcome performance limi-

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ig. 5. (Color online) 1D slices showing the spatial-frequencyontent (magnitude-squared Fourier transforms) of the signalnd noise from an M=100 simulated intensity-image average us-ng the USAF resolution chart from Fig. 4(a) as the target. Inach plot the signal contribution is the solid curve and the noiseomponent is the dashed curve. (a) Spatial-frequency content forhe baseline sensor (no quantum enhancement) with perfect ho-odyne efficiency, �=1. (b) Spatial-frequency content for the

aseline sensor with �=0.25; the signal is reduced but the noiseevel is unaffected. (c) Spatial-frequency content with �=0.25nd PSA with Geff=10 dB employed; the SNR level is restored tohat obtained with perfect homodyne efficiency. (d) Spatial-requency content when �=0.25 and 15 dB of SVI employed in toddition Geff=10 dB of PSA; the noise is suppressed at spatialrequencies that are attenuated by the soft aperture, giving anmproved SNR relative to that in (c) and (a).

ations in LADAR receivers that employ soft-aperture en-rance pupils and homodyne detection, thus improving re-eiver SNR and hence spatial resolution. PSA recoversNR that is lost to inefficient homodyne detection, i.e., in-fficiency in homodyne mode-matching and sub-unity de-ector quantum efficiency. Although we have studied itsse only in homodyne detection, PSA can provide similarenefits when used in conjunction with direct detection.eanwhile, the complimentary technique of SVI reduces

he noise on a single quadrature of the high-spatial-requency target information that is attenuated by theoft aperture, again leading to an improvement in SNRnd hence spatial resolution. SVI must be accompaniedy PSA, when homodyne efficiency is not close to one, toreserve the SNR of the high-spatial-frequency compo-ents upon detection. Also, unlike PSA, SVI relies on a co-erent imaging of the squeezed quadrature, so it cannote used with direct detection.It is important to note that SVI and PSA do not rely on

ransmission of fragile quantum states over the LADAR-o-target-to-LADAR path through the atmosphere,ecause our quantum-enhanced LADAR employs alassical-state (laser) transmitter with its SVI and PSAffects being confined to the LADAR receiver, whereinosses can be minimized. Thus our system is capable ofelivering its quantum-enhanced performance despite the100 dB of LADAR-to-target-to-LADAR transmission

oss encountered with typical quasi-Lambertian reflec-ors.

We have analyzed the benefits of our quantum en-ancements in two ways. We used a one-target versuswo-target hypothesis test to rigorously establish the im-roved spatial resolution—both in SNR shift and diver-ence shift—that SVI and PSA enable. We also exhibitedhe qualitative improvement our enhancements providesing simulated imagery of a USAF resolution chart andlots of the resulting signal and noise spectra.In the future, more detailed implementation consider-

tions need to be done for our quantum-enhanced LADARoncept. For example, we have assumed spatially whiteSA gain and spatially white SVI, whereas in reality theonlinear crystals used to do squeezed-state generationnd PSA only afford finite spatial bandwidths [25]. Fur-hermore, we have assumed that the squeezing, thehase-sensitive gain, and the homodyne detection areerfectly aligned to the same field quadrature. Finding re-eiver configurations that can approach these ideal condi-ions is a nontrivial task, but is one that must be com-leted if our LADAR concept is to be brought to fruition.inally, we have assumed that our targets are stationary,lanar objects at a known range. Real LADAR systemsust confront moving targets whose range—prior to in-

errogation by the LADAR transmitter—will not benown. Additional refinements to our LADAR receiverrchitecture—accompanied by their quantitative perfor-ance assessment—will be needed to accommodate such

cenarios.

CKNOWLEDGMENThis work was supported by the Defence Advancedesearch Projects Agency (DARPA) Quantum Sensors

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rogram. We thank Prem Kumar, Michael Vasilyev,orace Yuen, Geoff Burdge, Jon Habif, John Myers, and

. Bidigare for useful discussions.

EFERENCES1. B. Bhanu, D. E. Dudgeon, E. G. Zelnio, A. Rosenfeld, D.

Casasent, and I. S. Reed, eds., Special Issue on AutomaticTarget Recognition, IEEE Trans. Image Process. 6 (1997).

2. G. W. Kamerman, ed., Selected Papers on LADAR, SPIEMilestone Series, Vol. MS133 (SPIE, 1997).

3. G. R. Osche, Optical Detection Theory for Laser Applica-tions (Wiley-Interscience, 2002).

4. C. M. Caves, “Quantum mechanical noise in an interferom-eter,” Phys. Rev. D 23, 1693–1708 (1981).

5. D. J. Wineland, J. J. Bollinger, W. M. Itano, and F. L.Moore, “Spin squeezing and reduced quantum noise in spec-troscopy,” Phys. Rev. A 46, R6797–R6800 (1994).

6. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum me-trology,” Phys. Rev. Lett. 96, 010401 (2006).

7. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P.Williams, and J. P. Dowling, “Quantum interferometric op-tical lithography: exploiting entanglement to beat the dif-fraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000).

8. G. Gilbert, M. Hamrick, and Y. S. Weinstein, “On the use ofphotonic N00N states for practical quantum interferom-etry,” J. Opt. Soc. Am. B 25, 1336–1340 (2008).

9. P. Kumar, V. Grigoryan, and M. Vasilyev, “Noise-free ampli-fication: towards quantum laser radar,” presented at the14th Coherent Laser Radar Conference, Snowmass, Colo-rado, 9–13 July 2007.

0. M. I. Kolobov and L. A. Lugiato, “Noiseless amplification ofoptical images,” Phys. Rev. A 52, 4930–4940 (1995).

1. S.-K. Choi, M. Vasilyev, and P. Kumar, “Noiseless opticalamplification of images,” Phys. Rev. Lett. 83, 1938–1941(1999).

2. J. H. Shapiro, B. A. Capron, and R. C. Harney, “Imagingand target detection with a heterodyne-reception optical ra-dar,” Appl. Opt. 20, 3292–3313 (1981).

3. J. H. Shapiro, S. Guha, and B. I. Erkmen, “Ultimate chan-nel capacity of free-space optical communications,” J. Opt.Networking 4, 501–516 (2005).

4. Strictly speaking, the soft aperture must be embedded in-side a hard-aperture pupil, e.g., A��=e−2��2/R2

for ��D /2 and A��=0 otherwise. Without appreciable loss ofgenerality we shall ignore that constraint here and in Sec-tion 3 because: (1) we are interested in soft apertures whosetransmission at the hard-aperture limit is �1%; and (2) we
will not be assuming so much SVI and PSA that the hard-
aperture limit will constrain the soft-aperture resolutionimprovement afforded by these quantum enhancements.We shall, however, impose the hard-aperture limit in Sec-tion 4, when we employ MTF analysis to generate simu-lated baseline and quantum-enhanced imagery.

5. One-to-one imaging corresponds to setting F=L in Fig. 1, achoice that simplifies the notation. In reality, of course, thetarget range will satisfy L�F, however, this merely intro-duces a minification factor into the analysis.

6. J. H. Shapiro, “The quantum theory of optical communica-tions,” IEEE J. Sel. Top. Quantum Electron. 15, 1547–1569(2009).

7. This equal-strength assumption makes the LADAR’s taskof distinguishing between the one-target and two-target hy-potheses entirely a matter of the spatial pattern in the im-age plane rather than detected target-return strength.

8. Strictly speaking, �T�x��1 is required. However, becauseof the quasi-Lambertian nature of the target reflection, thesimple expressions we have provided lead to appropriatestatistics for the classical target-return field arriving at theLADAR receiver’s entrance pupil.

9. H. L. Van Trees, Detection, Estimation, and ModulationTheory, Part I (Wiley, 1968).

0. H. Chernoff, “A measure of asymptotic efficiency for tests ofa hypothesis based on the sum of observations,” Ann. Math.Stat. 23, 493–507 (1952).

1. Strictly speaking � /2 is the maximum value of �0. It is onlybecause we are employing paraxial optics that �0→ ap-pears to be possible. In practice, however, we will never beconcerned with angular separations that take us outsidethe realm of paraxial optics.

2. Our Gaussian soft-aperture definition for Rayleigh reso-lution is chosen so that the depth of the trough between theaverage photon-flux density of the two point targets on thedetector array is equal to the trough depth present for thesame two targets when they are imaged through an unob-scured hard pupil of length D and they are separated by thehard-aperture Rayleigh angle 2�0=� /D.

3. Alternatively, for stationary targets, we can use a two-pulseillumination sequence, with the first pulse employed to im-age the real quadrature and the second pulse employed toimage the imaginary quadrature, to achieve similar results.

4. Once again we are violating �T��1, and once again ourfield-reflection model does not pose a problem in that itgives physically reasonable statistics for the Fraunhofer-diffracted target return that is collected by the LADAR’s en-trance pupil.

5. M. Vasilyev, N. Stelmakh, and P. Kumar, “Estimation of thespatial bandwidth of an optical parametric amplifier with
plane-wave pump,” J. Mod. Opt. 56, 2029–2033 (2009).

ladar resolution improvement using receivers enhanced with squeezed-vacuum injection and...

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