T
pw
E
B(
vm
a
At3vcmvas(
ct[
Dutton et al. Vol. 27, No. 10 /October 2010 /J. Opt. Soc. Am. B 2007
LADAR resolution improvement using receiversenhanced with squeezed-vacuum injectionand phase-sensitive amplification: erratum
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 August 12, 2010 (Doc. ID 133183);posted August 12, 2010 (Doc. ID 133183); published September 15, 2010
We correct errors that were made in “LADAR resolution improvement using receivers with squeezed-vacuuminjection and phase-sensitive amplification,” J. Opt. Soc. Am. B 27, A63–A72 (2010). © 2010 Optical Society ofAmerica
OCIS codes: 270.0270, 280.3420, 270.6570.
twapvt
i
w
ivf
v
here were three minor typographical errors in [1]:Equation (7) is a pupil-plane representation of the
hase-sensitive amplification (PSA) and hence should beritten as
E���,t� = �GER� ���,t� + �G − 1ER�†�− ��,t�. �1�
quation (13) should be
Tr�f� �� d� Re�T����e−i2�f·�. �2�
elow Eq. (16) the references should be to Eqs. (10) and9).
More importantly, there is a serious error in the deri-ation of the one-versus-two spatial resolution perfor-ance. In particular, Eq. (14) should be replaced with
�nr�f1�nr�f2� =Sndnd
�f1�
2���f1 − f2� + ��f1 + f2��, �3�
nd
�ni�f1�ni�f2� =Sndnd
�f1�
2���f1 − f2� − ��f1 + f2��. �4�
s a result, the sufficient-statistic vector and its condi-ional covariance matrices that were developed in Sectionare not correct. Here we will summarize a proper deri-
ation of the sufficient-statistic vector and its conditionalovariance matrices. In terms of these new quantities theinimum error-probability decision rule from Eq. (31) is
alid. As we shall show, these corrections do not negateny of the conclusions drawn about the improvement inpatial resolution afforded by squeezed-vacuum injectionSVI) and PSA.
Equations (8) and (9) of [1] provide the correct statisti-al characterization of the quantum-enhanced LADAR. Inhe Section 3 treatment of resolution behavior, Eq. (15) of1] continues to provide the target characterizations for
0740-3224/10/102007-2/$15.00 © 2
he one-versus-two hypothesis test. Now, however, thehitening filter should be applied as follows to generaten appropriate sufficient statistic for minimum error-robability discrimination. With Yd�fx� defined as the 1Dersion of Eq. (10) from [1], we will use a whitening filtero obtain
Yd��fx� =Yd�fx�
�Sndnd�fx�
, �5�
nstead of Eq. (16) from [1]. In the space domain this gives
yd��x� =� dfxYd��fx�ei2�fxx
=�Geff�IT�p
���� dx� Re�T�x���m��x − x�� + nd��x�,
�6�
here
m��x� =� dfx
A��Lfx�
�Sndnd�fx�
ei2�fxx �7�
s a real, even function, and nd��x� is a zero-mean, real-alued, white Gaussian noise process with correlationunction
�nd��x1�nd��x2� = ��x1 − x2�. �8�
In terms of yd��x� we construct a 3D sufficient-statisticector, rT= �rc r rs�, as follows:
rc =� dx4��L�2
�R2 �1/4m��x − �0L� + m��x + �0L�
2�2yd��x�,
�9�
010 Optical Society of America
Gddec
�
,
wa
w
wplToasorcsccnttve
2008 J. Opt. Soc. Am. B/Vol. 27, No. 10 /October 2010 Dutton et al.
r =� dx4��L�2
�R2 �1/4m��x�
2yd��x�, �10�
rs =� dx4��L�2
�R2 �1/4m��x − �0L� − m��x + �0L�
2�2yd��x�.
�11�
iven the true hypothesis, r is a zero-mean Gaussian ran-om vector and thus completely characterized by its con-itional covariance matrices, �1 and �2, under hypoth-ses H1 and H2, respectively. Moreover, straightforwardalculations yield
cbsfv
DfEhd
R
1
= �4Geff�nC1
2 + C0 + C2 �2�2Geff�nC0 + 1�C1 0
�2�2Geff�nC0 + 1�C1 �2Geff�nC0 + 1�C0 0
0 0 C0 − C2
�12�
here C0, C1, C2, and n are given by Eqs. (27)–(30) of [1],nd
�2 = �� 0
0T �Geff�n�C0 − C2� + 1��C0 − C2�� �13�
ith
� = ��Geff�n�C0 + C2� + 1��C0 + C2� �2�Geff�n�C0 + C2� + 1�C1
�2�Geff�n�C0 + C2� + 1�C1 2Geff�nC12 + C0
� . �14�
Because r is zero-mean and conditionally Gaussianith covariance matrices �1 and �2, the minimum error-robability decision rule for choosing between equallyikely hypotheses H1 and H2 is given by Eq. (31) of [1].hus, to correct our resolution behavior work from [1], wenly need account for any differences between the covari-nce matrices given above and the corresponding expres-ions from [1], viz., Eqs. (24)–(26). It turns out that thenly change in going from Eqs. (24)–(26) of [1] to the cor-ect results presented above is to replace n by n /2. Physi-ally, this means that to fix the incorrect one-versus-twopatial resolution performance reported in [1] we just in-rease n by a factor of two �3 dB� in Fig. 2 and the asso-iated discussion. In particular, because this 3 dB shift inapplies to both baseline operation and all forms of quan-
um enhancement—SVI only, PSA only, SVI plus PSA—he quantitative statements made in [1] about the one-ersus-two resolution improvement afforded by quantumnhancement are unaffected. Finally, we note that be-
ause Eqs. (8) and (9) from [1] did not need correction, andecause these equations provided the basis for the imageimulations presented in [1], those simulations are unaf-ected by the corrections described herein for the one-ersus-two spatial resolution problem.
The research reported here was supported by theARPA Quantum Sensors Program. The authors grate-
ully acknowledge B. J. Yen for pointing out the error inq. (14) of [1] and its consequences. They also appreciateis confirming the correctness of the new one-versus-twoerivation presented above.
EFERENCES1. Z. Dutton, J. H. Shapiro, and S. Guha, “LADAR resolution
improvement using receivers enhanced with squeezed-vacuum injection and phase-sensitive amplification,” J.Opt. Soc. Am. B 27, A63–A72 (2010).
