Fundamental limit of resolving two point sources limited by an arbitrary point spread function

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by  Ronan Kerviche

Fundamental limit of resolving two point sources
limited by an arbitrary point spread function
Ronan Kerviche Saikat Guha Amit Ashok
University of Arizona Raytheon BBN Technologies University of Arizona
Email: [email protected] Email: [email protected] Email: [email protected]
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Abstract—Estimating the angular separation between two time (hence N) held fixed. In a recent breakthrough result,
1
0 incoherently radiating monochromatic point sources is a canon- Tsang et al. showed that, assuming a Gaussian ASF, intensity
ical toy problem to quantify spatial resolution in imaging. In
2 measurement in the infinite Hermite-Gauss (HG) basis in the
recent work, Tsang et al. showed, using a Fisher Information
image-plane coordinates attains a Fisher Information I (θ)
n analysis, that Rayleigh’s resolution limit is just an artifact of HG
a the conventional wisdom of intensity measurement in the image that is independent of θ no matter how small is θ, and equals
J plane.Theyshowedthattheoptimalsensitivityofestimatingthe the high-θ MSE attained by conventional image-plane direct

8 angle is only a function of the total photons collected during detection [5]. They also showed that the quantum Fisher
1 the camera’s integration time but entirely independent of the Information (QFI) I (θ) for estimating θ—which makes no
Q
angular separation itself no matter how small it is, and found
assumptions on how the optical field collected by the aperture
] theinformation-optimalmodebasis,intensitydetectioninwhich
s achievestheaforesaidperformance.Weextendtheaboveanalysis, gets pre-processed and detected—equals IHG(θ), establishing
c
which was done for a Gaussian point spread function (PSF) that a linear spatial mode sorting prior to detection, which
i
t to a hard-aperture pupil proving the information optimality of separates mutually-orthogonal HG modes and detects each
p
image-plane sinc-Bessel modes, and generalize the result further
o with individual detector pixels, is an optimal detector for this
to an arbitrary PSF. We obtain new counterintuitive insights
. problem. This showed that Rayleigh’s criterion is an artifact
s on energy vs. information content in spatial modes, and extend
c theFisherInformationanalysistoexactcalculationsofminimum of the conventional philosophy of intensity measurement in
si meansquarederror,bothforGaussianandhardaperturepupils. the image plane. There is rich information content in the
y phase of the image to extract, which optimally using a shot-
h I. INTRODUCTIONANDBACKGROUND noise-limited intensity measurement, one must use a non-
p
Consider estimating the angular separation 2θ between trivial spatial-mode transformation to the aperture field prior
[
twoincoherently-radiatingλ-wavelengthquasimonochromatic to detection to manipulate the post-detection shot noise so

1 point sources in the far field that are symmetrically disposed as to maximize the information content about θ in the noisy
v
aboutthelineofsight.Theapertureofthecamerahasdiameter detection outcomes.
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1 D, and during the integration time the total mean photon Thisresultopensupavarietyofinterestingquestions,some
9 number collected is denoted N. A conventional camera uses important ones being: (a) what is the information-optimal
4 a lens in the plane of the aperture pupil to focus the image in mode basis for the two-point-source problem with a hard-
0 an image plane, and detects the image-plane intensity pattern aperture pupil (sinc ASF) and for other general ASFs, (b)
.
1 using a detector pixel array. The field amplitude in the image what is the right minimal set of modes that carry almost
0 plane is an aperture-blurred version of the true object profile, all the relevant information about a passive imaging problem,
7 i.e., a scaled version of convolution of the object-plane field (c) what is the actual advantage in MSE (Fisher Information
1
(two independent delta functions for the above problem) with provides a lower bound on the MMSE, via the Cramer Rao
:
v the amplitude spread function (ASF) A(x) of the camera’s lower bound (CRLB), which is not always achievable), and
Xi aperture. It is well known that no matter what θ is, the (d) how does this theory generalize to more complex imaging
minimummeansquarederror(MMSE)ofestimatingθ canbe problems, and to broadband light.
r
a made arbitrarily small by using a long enough exposure, i.e., In this paper, we focus on the two-point-source problem
by taking N →∞. Rayleigh showed that, for a conventional described above. Our contributions are summarized below:
camera that measures the intensity in the image plane, even 1. With a rectangular hard aperture, i.e. sinc ASF, we show
if that intensity measurement is done with infinitely many thatmeasuringinthesinc-Bessel(SB)modebasisachievesthe
infinitesimally-tiny shot-noise-limited detector pixels, when θ QFIandtheFisherInformationisindependentofθ,analogous
decreasesbelow∼λ/D [1],themeansquarederror(MSE)of to measuring in the HG mode basis with a Gaussian aperture.
estimatingθdrasticallydegrades(increases)fortheintegration 2. We evaluate the exact MMSE of estimating θ with the
optimal mode basis and compare with the CRLB.
ThisworkwassupportedbytheDARPAREVEALprogramundercontract 3. We illustrate a counterintuitive distribution of energy vs.
numberHR0011-16-C-0026.Whilepreparingthispaperwebecameawareof
informationintheindividualmodesoftheoptimalmodebasis,
relatedwork[9],whichhassomeoverlapwithresultspresentedinthispaper.
AllthedetailedproofsarerelegatedtoanAppendixattheendofthepaper. which provide insights on efficient measurement design. In
particular we find that if one were to extract and detect a number collected over the camera’s integration time, the
single mode and its orthogonal complement (e.g., the binary number of photons in the qth SB mode is Nm (θ).
q
SPADE of [5]), the 1st mode is optimal rather than the 0th EachofseparatedSBmodesisdetectedusingashot-noise-
mode, in the deep sub-Rayleigh limit. We discuss information limited detector. The total number of orthogonal temporal
loss due to leaky mode separation. modes in the collected field M ≈ T(∆ν), where T is the
4. We provide the optimal binary SPADE measurement for a integrationtimeand∆νisthebandwidthofthelightaroundits
generalASFwhichattainstheQFIinthelowθ limit,andnew center frequency. The number of photons per temporal mode
insights into constructing information optimal mode bases. N (cid:28) 1 at optical frequencies. N = MN . It is simple to
0 0
show that with N (cid:28) 1 and M (cid:29) 1, and with the photon
II. OPTIMALMODESFORHARDAPERTURE 0
statistics in the individual modes being distributed with the
Let A(x) be the (generally complex-valued) energy- thermal (geometric, or Bose Einstein) distribution, that the
normalizedAmplitudeSpreadFunction(ASF)oftheaperture, total number of photons in the qth spatial mode is Poisson
i.e., (cid:82)+∞|A(x)|2 dx = 1. The image plane field is an distributedwithmeanNm (θ).InordertocalculatetheFisher
−∞ q
incoherent sum of two symmetrically-shifted copies of the Information of θ in the qth SB mode, we will rely on the
ASF at ±θ, each of which is perfectly self coherent. If one following result:
projectsthe imageplanefieldonto thecomplex-valuedspatial Lemma 1 (Fisher Information For A Poisson Corrupted Process):
mode f(x), (cid:82)+∞|f(x)|2 dx=1, the fraction of the intensity Letf beaC functionwithvaluesinR+ mappingthevariable
−∞ 1
intheimage-planefieldthatappearsinthef(x)modeisgiven of interest θ to a measurement Y ∼ P(y|f(θ)) where P
by: is the Poisson distribution. Then the Fisher Information
1(cid:12)(cid:12)(cid:90) +∞ (cid:12)(cid:12)2 associated to the process can be written as:
mf(θ)= 2(cid:12)(cid:12) −∞ f(x)A(x+θ) dx(cid:12)(cid:12) I(θ)= f(cid:48)(θ)2. (4)
1(cid:12)(cid:12)(cid:90) +∞ (cid:12)(cid:12)2 f(θ)
+ (cid:12) f(x)A(x−θ) dx(cid:12) , (1) This expression is similar to that obtained for a Gaussian-
2(cid:12) (cid:12)
−∞ corrupted measurement process, I(θ) = f(cid:48)(θ)2/η2, where
where f(x) denotes complex conjugate. We will call m (θ) f(cid:48)(θ)isthesensitivityandη2 isthenoisevariance.TheFisher
f
the measurement function for mode f. In [5], the authors Information in the qth SB mode evaluates to:
showed that if A(x) is Gaussian, then projecting the image- m(cid:48)(θ)2
plane field simultaneously onto the infinite Hermite Gauss Iq(θ)=N mq(θ) (5)
q
(HG) orthonormal mode basis functions f (x), q = 0,1,...,
q 4π2N (cid:18)qσ (cid:18)πθ(cid:19) (cid:18)πθ(cid:19)(cid:19)2
attains a vector of measurements whose Fisher Information
= (1+2q) j −j .
content on θ is quantum optimal, and independent of θ. σ2 πθ q σ q+1 σ
We will develop a similar strategy but for the practically (6)
relevantcaseofaspace-limited(hard)aperture.Thisproduces Lemma 2 (Series of Spherical Bessel Function Of The First Kind):
a cardinal sine (sinc) ASF, i.e., A(x)=sinc(x). We have the following result on a series of Spherical Bessel
In order to match the energy distribution of the PSF, we Function of the First Kind on any finite interval I of R and
will choose an orthonormal basis for which the ASF is the containing 0 (see Appendix for proof):
fiFrusntcbtiaosnissoffunthcetioFnir.stTKhiisndis,Mthe(xca)s=e o√f1th+e2Sqpjhe(rπicxa)l,qBe∈ssNel, ∀x∈I, (cid:88)∞ (1+2q)q2j (x)2+(2+4q)qj (x)j (x)
q q x2 q x q q+1
which are all either even or odd. Note that j0(πx)=sinc(x). q=0
It is simple to verify that (cid:82) M (x)sinc(x(cid:48)−x) dx=M (x(cid:48)). 1
q q +(1+2q)j (x)2 = . (7)
We will introduce a spatial scale factor σ of dimensions of q+1 3
length in the image plane coordinate (x) to capture the actual The measurement outputs on any orthogonal mode set are
‘length’oftheASFintheimageplane.Itwilldependuponthe statistically independent Poisson random variables. So, the
diameter of the aperture and the focal length of the imaging total Fisher Information for the vector-parametrized estimator
system. The fraction of the total energy collected in the qth isequaltothesumoftheindividualFisherInformationsfrom
mode is given by: eachmode.UsingLemma2,wededucethatmeasuringallthe
(cid:18)θ(cid:19)2 SB modes leads to a θ-independent Fisher Information, i.e.,
mq(θ)=Mq σ . (2) (cid:88)∞ 4π2N
∀θ ∈I, I(θ)= I (θ)= . (8)
q 3σ2
From [8], equation 1.10.50 :
q=0
∞ ∞ In [5], it was shown that I(θ) in (8) is the QFI for estimating
(cid:88) (cid:88)
∀x∈R, (1+2q)j (x)2 =1⇒ m (x)=1, (3)
q q θ with a hard aperture. Hence, we now have a proof that a
q=0 q=0 specificSBmodesortingpriortodirectdetectionachievesthe
shows that the sinc Bessel (SB) modes capture all the energy QFI, in the sense that the classical Fisher Information of the
in the image-plane field. With N being the total mean photon SB-mode measurement exactly matches the QFI.
III. ENERGYVS.INFORMATIONCONTENTINMODES Proposition 4 (Insensitivity Property): Given any properly
With the Gaussian ASF A(x)=(2πσ2)−14 exp(−x2/4σ2), normalized and continuous mode g(x) over R+ and any
the measurement function and Fisher information in the qth ASF A(x), the first order derivative m(cid:48)g(θ) of the associated
image-plane HG mode, q ∈N, are respectively given by [5]: measurement function mg(θ) goes to 0 as θ →0+.
Hence, in order for the information content in mode g,
m (θ) = 1 (cid:18) θ2 (cid:19)qe−θ2/4σ2, and (9) Ig(θ) = m(cid:48)g(θ)2/mg(θ) to go to a constant as θ → 0+, the
q q! 4σ2 following equivalence relation must be satisfied:
Iq(θ) = σN2q!(cid:18)q− 4θσ22(cid:19)2(cid:18)4θσ22(cid:19)q−1e−θ2/4σ2.(10) ∃c>0,s.t. cmg(θ)θ→∼0+ m(cid:48)g(θ)2. (14)
For the above to hold, it is necessary (but not sufficient in
ThetotalFisherinformationfrommeasuringallthe(infinitely
general) for m →0 as θ →0+. Thus, m (θ) should neither
many) HG modes equals the QFI bound for any θ, i.e. [5], g g
have sensitivity nor should it capture any energy at θ = 0.
∞
(cid:88) N Yet, if (14) is satisfied, measuring g will produce non zero
I(θ)= I (θ)= . (11)
q σ2 information for θ → 0+. Note that the insensitivity property
q=0 applies to I(cid:48)(x,θ) in (12) for the direct measurement as well:
The Fisher Information attained by infinite-spatial-resolution it equals 0 regardless of the ASF. So an image-plane direct
image-plane direct detection is given by, measurement provides no information about θ when θ →0+.
(cid:90) ∞ I(cid:48)(x,θ)2 ∂I(x,θ) A binary SPADE (Bin-SPADE) receiver measures a single
I = dx, with I(cid:48)(x,θ)= , (12) mode g(x) and its orthogonal component (i.e., the leftover
Direct I(x,θ) ∂θ
−∞ energy) leading to a simple implementation [5]. For both
where I(x,θ)=|A(x,θ)|2 is the normalized spatial distribu- Gaussianandhardapertures,BinSPADEreceiversconstructed
tion of energy in the image plane, also the probability density forg(x)beingtherespectiveq =0mode(theASFmode)and
function of measuring a photon at spatial position x, condi- onewiththeq =1mode(q =1HGorSBmode,respectively)
tioned on θ. I approaches the QFI in (11) for θ → ∞, attain a non-zero information at θ → 0+, which equals the
Direct
but goes to zero as θ → 0. So, the information advantage of respective QFI limit. We will refer to these two receivers
the HG mode measurement over image-plane direct detection as 0-BinSPADE and 1-BinSPADE, respectively. The Fisher
is maximum at small θ (sub Rayleigh regime) [5]. InformationforthesetwomeasurementsfortheGaussianASF
Comparing (9) and (10) with (2) and (6), we see several are given by:
analogous trends. First, measuring all the modes (HG or SB,
N θ2
respectively)ineithercasecapturesalltheimage-planeenergy I (θ)= 4σ2 , and (15)
for any θ. Hence, any spatial mode orthogonal to the span of 0-BinSPADE σ2exp(cid:0) θ2 (cid:1)−1
4σ2
the respective mode sets would neither have any energy nor (cid:16) (cid:17)2
1− θ2
anyinformationcontent.Further,inbothcasestheq =0mode N 4σ2
I (θ)= , (16)
capturesalloftheenergyatθ =0.Finally,asnotedin[6],only 1-BinSPADE σ2exp(cid:0) θ2 (cid:1)− θ2
4σ2 4σ2
theq =1HGmodecontributestothetotalFisherinformation
whereas, for sinc ASF (and SB modes), the respective Fisher
I(θ) at low θ for the Gaussian aperture case. With a hard
Informations are given by:
aperture, the q =1 SB mode has that exact same property.
Inwhatfollows, wewillconsiderinformationcontributions (cid:16)σ cos(cid:0)πθ(cid:1)− σ2 sin(cid:0)πθ(cid:1)(cid:17)2
fromindividualmodesandinsetsofmodes(withoutresolving 4N θ σ πθ2 σ
I (θ)= , (17)
modes in the set). We first consider the following lemma: 0-BinSPADE σ2 1−j (cid:0)πθ(cid:1)2
0 σ
Lemma 3 (Fisher Information Inequality On Aggregated Measurements): 12N σ2(cid:2)sin(cid:0)πθ(cid:1)−2j (cid:0)πθ(cid:1)(cid:3)2
Canonosritdheornomremaaslurfeammeinlyt foufncmtioodness.{mLeqt(θu)s} scaoyrrewsepomndaikneg aton I1-BinSPADE(θ)= σ2 θ2 1σ−3j (cid:0)π1θ(cid:1)2σ . (18)
1 σ
aggregated measurement where we project the image plane
Even though both BinSPADE receivers attain the QFI for
field on to a collection of modes S, i.e., an effective
θ → 0, the former significantly outperforms the latter for
(cid:80)
measurement function m (θ) = m (θ). This
S q∈S q higher θ (see Fig. 1). However 1-BinSPADE is much more
measurement cannot give us more information than the
robust to imperfect implementation. Let us say (cid:15) > 0 is a
sum of the information in the individual modes in the set,
leakage parameter such that the BinSPADE receiver measures
i.e.,
(1−(cid:15))m (θ)anditsorthogonalcomplement.Then,absolutely
g
θ ∈R, I (θ)= m(cid:48)S(θ)2 ≤(cid:88)I (θ)=(cid:88)m(cid:48)q(θ)2. (13) any (cid:15) > 0 results in I0-BinSPADE(θ) to collapse to 0 at θ = 0.
S m (θ) q m (θ) On the contrary, the performance of 1-BinSPADE at small
S q
q∈S q∈S θ remains fairly stable and leakage tolerant, with the loss in
We now want to find a single mode g whose information Fisherinformationbeingproportionaltotheenergylossinthe
contentdoesnotgotozero(i.e.,goestoaconstantc)asθ →0. detected mode, thus continuing to satisfy the condition stated
To see the requirement on g, let us consider the following: above and hence retaining a non-zero information at θ →0+.
1 For a Gaussian Aperture Point Sources, Gaussian PSF, 0−BinSPADE Point Sources, Gaussian PSF, 1−BinSPADE
Fisher Information2σNormalized to N/0000....024680 0.5 1 For a Rectanθ1g/.uσ5lar Aperture 2 QD01W−−iuriBBteahiicn nnltteSS uMaPPmkeAAa aIDDgnsefEEuo2,r r.efm5rmoametino 1tns% B toou 5n%d 3 θσ/ Normalized RMSE, MLE111000−−021 θσ/ Normalized RMSE, MLE111000−−021 NCD131000orira 00erPmm 0cPh0tae oh MlrPtoi−zoethRenoaodsnast sooRun rBMesomSuEenndt
1
Fisher Information22πσNormalized to 4N/30000....2468 QD01W−−iuriBBteahiicn nnltteSS uMaPPmkeAAa aIDDgnsefEEuo,r refmrmoametino 1tns% B toou 5n%d Fthiagn.102−63.0 5,0N0o00r.5msaamlizpθe1/lσeds.R(Ml1e.S5ftE) 0p-lBo2tisnS1f0rP−o3A0 mDEMo0f.o5nrteGCaθau1/σrslsoiaEns1.t5aipmearttiuo2rne.w(ritihghmt)o1re-
0 BinSPADE for Gaussian aperture. The estimators for both can be written
0 0.5 1 1.5 2 2.5
θ/σ analytically (with Lambert-W function for the 1-BinSPADE) over the range
Fig. 1. Performance comparison of Direct Measurements versus 0 and 1- ofθ shown,wheretheassociatedmeasurementfunctionisinvertible.
BinSPADEreceiversforGaussian(top)andhard(bottom)apertures.They-
axesarenormalizedtotherespectiveQuantumFisherInformationbounds. Point Sources, Sinc2 PSF, 0−BinSPADE Point Sources, Sinc2 PSF, 1−BinSPADE

100 100
IV. MEANSQUAREDERROR(MSE)ANALYSIS
Let us recall the following inequality between the Mean
SFqisuhaerreIEnfrorormrMa(SMtiEoSθ(cid:98)nE(θIθ(cid:98)))(θ≥o)f:(a1n−eIsBt(iθθ(cid:98)m(cid:48))(aθt)o)r2θ(cid:98)+, iBtsθ(cid:98)(bθia)2s,Bθ(cid:98), and(1th9e) θσ/ Normalized RMSE, MLE1100−−21 θσ/ Normalized RMSE, MLE1100−−21 NCD131000orira 00erPmm 0cPh0tae oh MlrPtoi−zoethRenoaodsnast sooRun rBMesomSuEenndt
where B(cid:48)(θ) is the first derivative of the bias function with
θ(cid:98)
respecttoθ.ForcomparingtheMSEattainedbymode-sorting
10−3 10−3
receivers and image-plane detection, we will use the Normal- 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
θ/σ θ/σ
izedRootMeanSquareError,NRMSE =(cid:112)MSE (θ)/θ and Fig. 3. Normalized RMSE plots from Monte Carlo Estimation. (left) 0-
θ(cid:98) (cid:112) θ(cid:98) BinSPADE for rectangular aperture. (right) 1-BinSPADE for hard aperture.
the adapted Cramer-Rao bound defined as 1/θ I(θ) [3], [4]. Note that the θ/σ range restriction for 1-BinSPADE corresponds to the
For a set of independent and identically distributed (i.i.d.) presenceofthefirstzerointheFisherInformation.
measurements{y },q =0,...,Qdrawnfromtheconditional
q
distributionP(y |θ)(aswillbethecasewhenasetoforthog-
q at high SNR and generally follows the same trend across all
onal image-plane spatial modes are detected simultaneously),
θ. As noted by Tsang et al. in [5], the estimator can become
the Maximum Likelihood Estimator (MLE) is given by:
super efficient when it is biased, which happens at very low
(cid:40) Q (cid:41) and large θ. In both graphs, we also plot the numerically-
(cid:89)
θ(cid:98)(y0,...,yQ)=argmax P(yq|θ) . (20) evaluatedCRBforimage-planedirectmeasurementusing(12).
q=0 The MSE attained by the BinSPADEs are seen to outperform
image-plane detection by a clear margin for θ much smaller
Assuming the prior knowledge of the total photon number
than the Rayleigh separation (θ/σ = 1). Despite the superior
N collected during the integration time, the MLE for a
Fisher information of the BinSPADE receivers, the NRMSE
measurement that just measures the q(x) mode, is given by:
highlights the unavoidable precision decay at low θ. Both
θ(cid:98)(yq)=m−q1(yq/N). (21) the BinSPADEs and direct-measurement receivers can reach
arbitrary precision in estimating θ provided they can collect
For the q-BinSPADE receiver (i.e., one that measures mode q
a large enough photon flux (from increased exposure time).
and its orthogonal complement), the total photon number can
But the BinSPADEs remain more efficient than the direct
be inferred from the sum count over the two measurements:
measurement strategy in the sub Rayleigh regime.
θ(cid:98)(yq,yq,r)=m−q1(yq/(yq+yq,r)). (22)
V. EFFICIENTBINARYSPADEFORARBITRARYASF
When N is high, yq +yq,r becomes a good estimator for N √
Consider a general (complex-valued) ASF, A(x/σ)/ σ,
and the two previous expressions become similar.
(cid:82)∞ |A(x)|2 dx=1, in C∞, and its autocorrelation function,
In Figs. 2 and 3, we show the performance of MLE for −∞
the 0 and 1-BinSPADE receivers for the Gaussian and hard
(cid:90) +∞
apertures, respectively, for three choices of the total number Γ (x(cid:48))= A(x)A(x+x(cid:48)) dx. (23)
A
ofdetectedphotons.TheMLEperformsveryclosetotheCRB −∞
Using (1) the measurement functions associated with mea- A(x)) and hence the first mode (M ) simply equals the ASF.
0
suring the A(x) mode and its orthogonal complement are: The measurement functions for a simultaneous measurement
(cid:12)(cid:12) (cid:18)θ(cid:19)(cid:12)(cid:12)2 of the Mq modes can be expressed as:
ThmeAF,0i(sθh)er=I(cid:12)(cid:12)nΓfAormaσtio(cid:12)(cid:12)n ,formthAi,s0,rg(eθn)e=ral1iz−edm0A-,B0(inθS).PA(D24E) mA,q(θ)= 12(cid:12)(cid:12)(cid:12)(cid:12)(cid:88)q ωσkk,qΓ(Ak)(cid:18)σθ(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)2+ 12(cid:12)(cid:12)(cid:12)(cid:12)(cid:88)q ωσkk,qΓ(Ak)(cid:18)−σθ(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)2
receiver evaluates to: (cid:12)k=0 (cid:12) (cid:12)k=0 (cid:12)
Re(cid:104)Γ(1)(cid:0)θ(cid:1)Γ (cid:0)θ(cid:1)(cid:105)2 (30)
4N A σ A σ
I (θ)= , (25) Weconsider ageneralized 1-BinSPADEreceiver associated
0-BinSPADE σ2 (cid:12)(cid:12)ΓA(cid:0)σθ(cid:1)(cid:12)(cid:12)2(cid:16)1−(cid:12)(cid:12)ΓA(cid:0)σθ(cid:1)(cid:12)(cid:12)2(cid:17) with the M1 mode. The corresponding measurement function
is given by:
w(GhaeuressiΓa(An1)anisdthhearfidrsatpedretruirvea)tisvteudoifedΓAb.efAorse,fotrhitshegetwneoraclaizseeds m (θ)= (cid:12)(cid:12)(cid:12)Γ(A1)(cid:0)σθ(cid:1)−Γ(A1)(0)ΓA(cid:0)σθ(cid:1)(cid:12)(cid:12)(cid:12)2, (31)
A,1 (cid:12) (cid:12)2
0-BinSPADE receiver collects all the image-plane energy in −Γ(2)(0)−(cid:12)Γ(1)(0)(cid:12)
the 0 (ASF) mode for θ → 0+. This follows from the peak A (cid:12) A (cid:12)
property of the autocorrelation function and the normalization and the associated Fisher Information is given by:
onfottehethaAtSIF0,-BwinhSPiAchDEg(iθv)eissuinsdmepAe,n0d(0en)t=of1a.nIytpishainseteprerestsienngtitno 4N Re(cid:104)f(1)(cid:0)σθ(cid:1)f(cid:0)σθ(cid:1)(cid:105)2
I (θ)= , (32)
theaperturefunctionandissolelybasedonitsintensityprofile. 1-BinSPADE σ2 (cid:12)(cid:12)f(cid:0)θ(cid:1)(cid:12)(cid:12)2(cid:16)−f(1)(0)−(cid:12)(cid:12)f(cid:0)θ(cid:1)(cid:12)(cid:12)2(cid:17)
Thus, most optical aberrations such as defocusing, spherical σ σ
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
aberration or coma, among others, if taken into account into with: f θ =Γ(1) θ −Γ(1)(0)Γ θ . (33)
the projection mode, do not degrade the information, unlike σ A σ A A σ
the image-plane direct measurement.
Assuming Γ (x) can be expanded as in (26), we find that
AssumingΓ (x)admitsasecondorderexpansionnearθ = A
A the generalized 1-BinSPADE receiver attains the same Fisher
0, one can write a Taylor series as follows:
Information as that of the generalized 0-BinSPADE receiver:
α
Γ (x) = 1+iβx− x2+O(x3), (26) 4N/σ2(α−β2),atθ →0+.Asdiscussedbeforeinthecontext
A θ→0 2 of Gaussian and hard rectangular apertures, a similar robust-
with α≥0, and β ∈R. One can further show that: ness advantage (for 1 over 0-BinSPADE) exists in the general
I0-BinSPADE(θ)θ→→0 4σN2 (α−β2), (27) cfoarsea.Wgehneetrhaelrcoormnpolte(cid:80)x-v∞qa=l0ueIdq(Aθ)(x=)4wNit/hσo2u(αr −coβn2st)r,u∀cθtiohnoldosf
the measurement modes, is left open. In future work, it will
which is a θ-independent constant as before. Note that a real
also be interesting to investigate a systematic generalization
ASFbenefitsfromthefactthatβ =0.Also,fromtheWiener-
of the optimal modes for a pre-detection mode-sorting based
Khinchintheorem,increasingαisequivalenttogreaterspatial
receiverformorecompleximagingproblems,andtoprovethat
variations in the ASF profile, i.e., sharper or numerous edges.
the quantum Fisher Information limit for any incoherent-light
It is simple to verify that (27) reduces to the corresponding
imaging problem can be attained by an appropriate receiver
formulas for the QFI for the Gaussian (α=1/4, β =0) and
that applies a pre-detection linear mode transformation.
hard rectangular apertures (α=π2/3, β =0), attained by the
respective 0-BinSPADE receivers as discussed above. REFERENCES
LetusnowconstructanorthonormalbasisforageneralASF
[1] Lord Rayleigh F.R.S., “Investigations in optics, with special reference
by using derivatives of A(x). Let us first note the following tothespectroscope”,PhilosophicalMagazineSeries5,8,49(1879).
identity relating the qth derivative of A(x) to that of Γ : [2] C.E.Shannon,“Amathematicaltheoryofcommunication,”BellSyst.
A
Tech.J.,27,pp.379–423and623–656(1948).
(cid:90) +∞ [3] Rao, Calyampudi Radakrishna (1945). ”Information and the accuracy
A(q)(x)A(x+x(cid:48)) dx=(−1)qΓ(Aq)(x(cid:48)) (28) attainable in the estimation of statistical parameters”. Bulletin of the
−∞ CalcuttaMathematicalSociety.37:81–89.MR0015748.
[4] Cramer,Harald(1946).MathematicalMethodsofStatistics.Princeton,
Clearly, the functions (A(q)) need not be orthogonal, and
q NJ:PrincetonUniv.Press.ISBN0-691-08004-6.OCLC185436716
hence cannot be used for parallel measurements. We do a [5] M.Tsang,R.Nair,andX.M.Lu,“QuantumTheoryofSuperresolution
Gram-Schmidt orthogonalization by selecting weights ω ∈ for Two Incoherent Optical Point Sources,” Phys. Rev. X 6, 031033
k,q
CsuchthateachmeasurementmodeM (x)canbewrittenas (2016)
q [6] R.NairandM.Tsang,“Far-FieldSuperresolutionofThermalElectro-
a linear combination of the (A(q))q functions, magneticSourcesattheQuantumLimit,”Phys.Rev.Lett.117,190801
(2016).
q
√1σMq(cid:16)σx(cid:17)=(cid:88)(−1)kσωqk+,q12A(q)(cid:16)σx(cid:17), (29) [[87]] MCSu..bLAwubaprvoaemlaeonnwdgtiShtz.,IPmI.iraSagtneidngogul,n”a,,P““hHUyalsnt.idmRbaeotvoe.kPLorefetctM.is1aio1thn7e,mB1oa9tu0inc8ad0l2oFf(u2nQ0cu1tia6on)n.tus,m”Ua.nSd.
k=0
NationalBureauofStandards,Washington,DC,(1964).
such that {Mq} forms an orthonormal basis (after removing [9] J. Rehacek, M. Pau´r, B. Stoklasa, Z. Hradil and L. L. Sa´nchez-Soto,
some possibly identically null functions). It is easy to verify “Optimal measurements for resolution beyond the Rayleigh limit,” To
bepublishedinOSAOpticsLetters(2016)
that ω = 1 (due to the energy normalization property of
0,0
APPENDIX that, for the first series (40) we have the convergence bound:
∀P,Q∈N,Q>P, |A (x)−A (x)|
Fisher Information for a Poisson corrupted process: Let b,Q b,P
fa vbaeriaabfulencθtitohnatRis→coRrr+upmteoddbeylinPgotihsseoonuntpouisteo:f a process on =(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88)Q 2qbx+1jq(x)2−2qbjq(x)jq+1(x)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (44)
(cid:12)q=P+1 (cid:12)
f(θ)yexp(−f(θ))
pf(y|θ)= y! , (34) ≤π(1+X2)exp(cid:18)X2(cid:19) (cid:88)∞ qb+1X2q−1. (45)
2 2qq!
q=P+1
such that ln(p (y|θ)) is twice differentiable with respect to θ.
f As the underlying series is positive, monotonic and con-
Wenotef(cid:48)andf(cid:48)(cid:48),respectivelyitsfirstandsecondderivatives.
verges (by the ratio test), we can choose a P large enough
Then we can write the corresponding Fisher Information as:
to reduce the remainder, and subsequently the bound, to any
(cid:20) ∂2 (cid:12)(cid:12) (cid:21) small (cid:15) we wish. In other words:
I(θ)=−Ey ∂θ2 ln(pf(y|θ))(cid:12)(cid:12)θ (35) ∀(cid:15)>0,∃P ∈N,∀p,q ≥P,x∈I,|A (x)−A (x)|<(cid:15).
b,p b,q
E [y|θ](cid:18) f(cid:48)(θ)2(cid:19) (46)
=− y f(cid:48)(cid:48)(θ)− +f(cid:48)(cid:48)(θ) (36)
f(θ) f(θ)
We can thus conclude that the function series is uniformly
f(cid:48)(θ)2 Cauchy and converges uniformly over I.
= (37)
f(θ) For the second series (41) we have:
∀P,Q∈N,Q>P, |B (x)−B (x)|
b,Q b,P
(cid:12) (cid:12)
Fisher Information For Multiple Independent Measure- (cid:12) Q (cid:12)
ments: Ifweconsidertheoutputofmultipleindependentmea- =(cid:12)(cid:12)(cid:12) (cid:88) qbjq(x)2− x2qbjq(x)jq+1(x)−qbjq+1(x)2(cid:12)(cid:12)(cid:12) (47)
surementfunctions(f0(θ),...,fQ(θ)),alltwicedifferentiable (cid:12)q=P+1 (cid:12)
with respect to θ, and their respective outputs y0,...,yQ, we (cid:18) X2(cid:19) (cid:18)X2(cid:19) (cid:88)∞ qb
have for the Fisher Information: ≤π 1+ exp X2q, (48)
2 2 2qq!
q=P+1
I(θ)=−E (cid:34) ∂2 ln(cid:32)(cid:89)Q p (y |θ)(cid:33)(cid:12)(cid:12)(cid:12)θ(cid:35) (38) where the same conclusion applies.
y0,...,yQ ∂θ2 q q (cid:12)(cid:12) Finally, we consider the new set of series of functions,
q=0
related respectively to A and B :
=(cid:88)Q ffq(cid:48)((θθ))2 =(cid:88)Q Iq(θ) (39) b,Q (cid:88)bQ,Q
q=0 q q=0 ∀Q,b∈N,∀x∈I, Cb,Q(x)= qbjq(x)2 (49)
q=0
Q
Uniform Convergence Of Series Based On Spherical D (x)=(cid:88)qbj (x)j (x). (50)
b,Q q q+1
Bessel Functions Of The First Kind: We consider the two
q=0
following series of functions over a finite interval I of R
We note that their point-wise convergence can easily be
containing zero, for some fixed positive integer b :
established at x = 0, as ∀q > 0, j (0) = 0, j (x) = 1 and,
q 0
Q considering the previous results, we can apply the Differenti-
(cid:88) ∂
∀Q∈N, x∈I, A (x)= qb j (x)2 (40) ation Theorem to obtain their respective uniform convergence
b,Q ∂x q
as well as the relations:
q=0
Q ∂
(cid:88) ∂ C (x)= A (x), and (51)
Bb,Q(x)= qb∂xjq(x)jq+1(x) (41) b,Q ∂x b,Q
q=0 ∂
D (x)= B (x). (52)
b,Q ∂x b,Q
We have the following loose upper bound on the Spherical
Bessel Functions, from their series definition:
Series of Spherical Bessel Function Of The First Kind:
√π (cid:88)∞ (−1)k (cid:16)x(cid:17)2k We will use the following two results from [8] (Equations
∀q ∈N, x∈R, j (x)= xq 1.10.50 and 1.10.52), where Si(x) denotes the Sine Integral.
q 2q+1 k!Γ(cid:0)k+q+ 3(cid:1) 2
k=0 2 ∞
(42) (cid:88)j (x)2 = Si(2x), and (53)
√π (cid:18)|x|(cid:19)q (cid:18)x2(cid:19) q 2x
q=0
|j (x)|≤ exp (43)
q 2q! 2 4 ∞
(cid:88)
(1+2q)j (x)2 =1 (54)
q
WecanpickX suchthat,∀x∈I, |x|≤X andwecantest q=0
By combining them, we obtain: Finally, we can combine the results (57) to (64) to get:
(cid:88)∞ 1(cid:18) Si(2x)(cid:19) (cid:88)∞ q2 q
qj (x)2 = 1− (55) (1+2q) j (x)2−2(1+2q) j (x)j (x)
q 2 2x x2 q x q q+1
q=0 q=0
∞ 1
(cid:88)jq+1(x)2 = Si2(2xx) −j0(x)2 (56) +(1+2q)jq+1(x)2 = 3. (65)
q=0
∞
(cid:88) 1 3Si(2x)
qj (x)2 = − +j (x)2 (57) ASF Insensitivity Property: Considering any continuous
q+1 2 4x 0 and differentiable ASF A(x) as well as mode g(x), the
q=0
(cid:88)∞ Si(2x) measurement function mg for two point sources separated by
(1+2q)j (x)2 =1− +j (x)2 (58) 2θ can be written as:
q+1 x 0
q=0 1 (cid:12)(cid:12)(cid:90) +∞ (cid:16)x(cid:17) (cid:18)x+θ(cid:19) (cid:12)(cid:12)2
Thanks to the Differentiation results of the previous lemma mg(θ)= 2σ2 (cid:12)(cid:12) g σ A σ dx(cid:12)(cid:12)
−∞
(51), we can write from the derivative of (53):
1 (cid:12)(cid:12)(cid:90) +∞ (cid:16)x(cid:17) (cid:18)x−θ(cid:19) (cid:12)(cid:12)2
∞ + (cid:12) g A dx(cid:12) . (66)
(cid:88) 1 2σ2 (cid:12) σ σ (cid:12)
j (x)j (x)= (1−j (2x)). (59) −∞
q q+1 2x 0
q=0 We have for the limit of its first derivative:
We then find the following as the solution of a first order ∂
lim m (θ)= lim
linear differential equation involving (55) and (57) as well as θ→0∂θ g θ→0
(cid:34) (cid:35)
the property (52) and the constraint that the series is equal to 1 (cid:90) +∞ (cid:16)x(cid:17) (cid:16)x(cid:17) (cid:90) +∞ (cid:16)x(cid:17) (cid:16)x(cid:17)
0 at x=0: Re g A(cid:48) dx g A dx
σ3 σ σ σ σ
−∞ −∞
(cid:88)∞ qj (x)j (x)= Si(2x) + 3sin(2x) − 1 + cos(2x) 1 (cid:34)(cid:90) +∞ (cid:16)x(cid:17) (cid:16)x(cid:17) (cid:90) +∞ (cid:16)x(cid:17) (cid:16)x(cid:17) (cid:35)
q q+1 4 16x2 2x 8x − Re g A(cid:48) dx g A dx
q=0 σ3 −∞ σ σ −∞ σ σ
(60)
=0. (67)
With this result, we can compute the derivative of (55) to
We can proceed similarly for the Fisher Information in the
express the following series and its respective shifted version:
case of direct detection. We recall the expression:
∞ (cid:18) (cid:19)
(cid:88)q2j (x)2 = 1Si(2x) 1 +2x + cos(2x) (cid:90) ∞ I(cid:48)(x)2 ∂I(x,θ)
q 8 x 8 I (θ)= dx, with I(cid:48)(x,θ)= (68)
q=0 Direct I(x) ∂θ
−∞
j (2x) 1
+ 0 − (61) In the case of two point sources, the normalized intensity
8 2
∞ (cid:18) (cid:19) profile can be written with the previous ASF as : I(x) =
(cid:88) 1 9 cos(2x)
q2j (x)2 = Si(2x) +2x + (|A(x + θ)|2 + |A(x − θ)|2)/2. One can write for its first
q+1 8 x 8
q=0 derivative:
j (2x) 3
+ 08 −j0(x)2− 2 (62) ∀x∈R, limI(cid:48)(x,θ)=
θ→0
(cid:104) (cid:105)
The next series is also found as the solution of another limRe A(cid:48)(x+θ)A(x+θ)−A(cid:48)(x−θ)A(x−θ)
θ→0
differential equation involving (61) and (62), with the same
=0 (69)
value constraint at x=0 than previously:
And : lim I (θ)=0. (70)
Direct
(cid:88)∞ Si(2x) sin(2x) x θ→0
q2j (x)j (x)=− − +
q q+1 2 8x2 3
q=0 Fisher Information Inequality On Aggregated Measure-
1 cos(2x)
+ − (63) ments: Let mS be an aggregated measurement over a col-
2x 4x lection S of measurement functions from orthogonal modes,
(cid:80)
And after deriving (61) we obtain: i.e. mS(θ) = q∈Smq(θ), all corrupted by Poisson noise.
At a location θ where ∀q, m (θ)>0, we have the following
q
(cid:88)∞ 6x2+1 3sin(2x)+6xcos(2x) measurement function and Fisher Information:
q3j (x)2 =− Si(2x)−
q 16x 32x
(cid:16) (cid:17)2
q=0 (cid:80) m(cid:48)(θ)
x2 1 I (θ)= q∈S q (71)
+ + (64) S (cid:80) m (θ)
3 2 q∈S q
Wecansimplifythenotationsforthecurrentlocationintosets For a rectangular aperture:
(m ) and (m(cid:48)) and observe that:
q q (cid:16)σ cos(cid:0)πθ(cid:1)− σ2 sin(cid:0)πθ(cid:1)(cid:17)2
4Nρ θ σ πθ2 σ
(cid:88)mq(cid:88)mm(cid:48)qq2−(cid:88)m(cid:48)q2 I0-BinSPADE,ρ(θ)= 1σ2N2 ρσ2(cid:2)sin1(cid:0)−πθρ(cid:1)j−0(cid:0)2πjσθ(cid:1)(cid:0)2πθ(cid:1)(cid:3)2 (79)
q∈S q∈S q∈S I (θ)= σ 1 σ (80)
=(cid:88) (cid:88) mmrm(cid:48)q2+ mmqm(cid:48)r2−2m(cid:48)qm(cid:48)r (72) 1-BinSPADE,ρ σ2 θ2 1−3ρj1(cid:0)πσθ(cid:1)2
q r
q∈Sr∈S,r>q Finally, for a generic ASF A(x), properly normalized and
in C2:
Here, if m(cid:48) or m(cid:48) are equal to zero or their product is
negative, andqwith ther previous positivity constraint, the sum Re(cid:104)Γ(1)(cid:0)θ(cid:1)Γ (cid:0)θ(cid:1)(cid:105)2
4Nρ A σ A σ
is clearly positive. Otherwise, one can write: I (θ)= (81)
(cid:18) (cid:19) 0-BinSPADE,ρ σ2 (cid:12)(cid:12)ΓA(cid:0)σθ(cid:1)(cid:12)(cid:12)2(cid:16)1−ρ(cid:12)(cid:12)ΓA(cid:0)σθ(cid:1)(cid:12)(cid:12)2(cid:17)
1 m m
rm(cid:48)2+ qm(cid:48)2−2m(cid:48)m(cid:48)
m(cid:48)qm(cid:48)r mq q mr r q r Re(cid:104)f(1)(cid:0)θ(cid:1)f(cid:0)θ(cid:1)(cid:105)2
m m(cid:48) m m(cid:48) 4Nρ σ σ
= mrq m(cid:48)rq + mrq mr(cid:48)q −2≥0 (73) I1-BinSPADE,ρ(θ)= σ2 (cid:12)(cid:12)f(cid:0)σθ(cid:1)(cid:12)(cid:12)2(cid:16)−f(1)(0)−ρ(cid:12)(cid:12)f(cid:0)σθ(cid:1)(cid:12)(cid:12)2(cid:17)
(82)
As we have, m m(cid:48)/m m(cid:48) >0 and ∀x>0, x+1/x≥2.
r q q r (cid:18)θ(cid:19) (cid:18)θ(cid:19) (cid:18)θ(cid:19)
Thus, (72) is always positive and we can conclude with the with: f =Γ(1) −Γ(1)(0)Γ . (83)
σ A σ A A σ
inequality:
OnecannoticefromthecommonnormalizationoftheASF
(cid:16) (cid:17)2
(cid:80)q∈Sm(cid:48)q(θ) =I (θ)≤(cid:88)m(cid:48)q(θ)2 =(cid:88)I (θ) (74) aAn(dx)asthiattitshealasoutohceormrreitliaatnio,nthΓaAt (Γθ()1)i(s0e)qiusaplutoreolyneimatagθin=ar0y;.
(cid:80) m (θ) S m (θ) q A
q∈S q q∈S q q∈S Hence, in (81) the numerator always tend to zero for θ →
0, but the denominator only does so if the efficiency ρ is
exactly equal to one. Thus, for any ρ<1, the generalized 0-
Remark 5 (Generalized projection modes for a BinSPADE receiver):
BinSPADEdoesnotcollectinformationfornarrowseparation
One can expand the generalized mode expression (29) in the
angles : I →0 near 0.
0-BinSPADE,ρ<1
case of the two first modes for a normalized ASF A(x) into :
Ontheotherhand,withtheexpansion(26)onecandevelop
the limit of (82) as :
1 (cid:16)x(cid:17) 1 (cid:16)x(cid:17)
√ M = √ A (75)
σ 0 σ σ σ limI (θ)= 4Nρ(α−β2). (84)
1 (cid:16)x(cid:17) 1 −A(1)(cid:0)x(cid:1)+Γ(1)(0)A(cid:0)x(cid:1) θ→0 1-BinSPADE,ρ σ2
√σM1 σ = √σ (cid:114) σ A(cid:12) (cid:12)σ2 , (76) Thus, the generalized 1-BinSPADE is resilient to leakage in
−Γ(2)(0)−(cid:12)Γ(1)(0)(cid:12) the deep sub-Rayleigh limit.
A (cid:12) A (cid:12)
where A(1)(x) is the first derivative of the ASF, Γ is the
A
autocorrelationofAandΓ(1),Γ(2) arerespectivelyitsfirstand
A A
second derivatives. Here, one can note, as the autocorrelation
ishermitian,thatΓ(1)(0)ispurelyimaginary.Thusinthecase
A
of a purely real ASF it is equal to zero and the first mode can
be constructed with only A(1)(x).
Remark 6 (Fisher Information of a leaky BinSPADE receiver):
Considering a leakage factor 0 < (cid:15) < 1, we define the
efficiency of a BinSPADE receiver as ρ = 1−(cid:15). We have
the following expressions of the Fisher Information for 0 and
1-BinSPADE respectively; first, for a Gaussian aperture:
Nρ θ2
I (θ)= 4σ2 (77)
0-BinSPADE,ρ σ2 exp(cid:0) θ2 (cid:1)−ρ
4σ2
(cid:104) (cid:105)2
1− θ2
Nρ 4σ2
I (θ)= (78)
1-BinSPADE,ρ σ2 exp(cid:0) θ2 (cid:1)−ρ θ2
4σ2 4σ2