Sub-Rayleigh-diffraction-bound quantum imaging
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Giovannetti, Vittorio et al. “Sub-Rayleigh-diffraction-bound
quantum imaging.” Physical Review A 79.1 (2009): 013827. ©
2009 The American Physical Society
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http://dx.doi.org/10.1103/PhysRevA.79.013827
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PHYSICAL REVIEW A 79, 013827 共2009兲
Sub-Rayleigh-diffraction-bound quantum imaging
1
Vittorio Giovannetti,1 Seth Lloyd,2 Lorenzo Maccone,3 and Jeffrey H. Shapiro2
NEST-CNR-INFM and Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126, Pisa, Italy
MIT, Research Laboratory of Electronics, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
3
QUIT, Dipartimento di Fisica “A. Volta,” Universitá Pavia, via Bassi 6, I-27100 Pavia, Italy
and Institute for Scientific Interchange, I-10133 Torino, Italy
共Received 5 May 2008; published 28 January 2009; publisher error corrected 3 March 2009兲
2
The spatial resolution of an imaging apparatus is limited by the Rayleigh diffraction bound, a consequence
of the imager’s finite spatial extent. We show some N-photon strategies that permit resolution of details that are
smaller than this bound, attaining either a 1 / 冑N enhancement 共standard quantum limit兲 or a 1 / N enhancement
共Heisenberg-like scaling兲 over standard techniques. In the incoherent imaging regime, the methods presented
are loss resistant, since classical light sources suffice. Our results may be of importance in many applications:
microscopy, telescopy, lithography, metrology, etc.
DOI: 10.1103/PhysRevA.79.013827
PACS number共s兲: 42.50.Dv, 03.67.⫺a, 42.30.Va
Quantum effects have been used successfully to provide
resolution enhancement in imaging procedures. Among the
many proposals that have been made 关1兴, arguably the most
famous is the quantum lithography procedure 关2兴. These
methods take advantage of the fact that the effective wavelength of a multiphoton light state is shorter than its electromagnetic field wavelength: the light generation, propagation,
and detection can be performed at optical wavelengths,
where it is simple to manipulate, whereas the quantum correlations in the employed states allow one to perform imaging at the shorter multiphoton wavelength. Such proposals
are then based on entangled or squeezed light sources, as
entanglement or squeezing are necessary for efficient quantum enhancements 关3兴. If, however, efficiency considerations
are dropped, it is also possible to employ classical-state light
sources and post-selection at the detection stage to filter desirable quantum states from the classical light 关4兴. In fact, in
many practical situations efficiency considerations do not
play any role, as the quantum enhancement is typically of the
order of the square root of the number of entangled systems
关3兴, whereas in practical situations the complexity of generating the required quantum states has a much worse scaling.
Many post-selection imaging procedures employing only
classical light sources have been proposed and analyzed
关5–17兴, and cover a wide range of interesting situations.
Analogous methods have been employed successfully also in
fields not directly related to imaging 关18兴.
Here we show how one can achieve a resolution enhancement beyond what the apparatus’ structural limits impose for
conventional imaging, i.e., the Rayleigh diffraction bound
xR. In particular we show that employing appropriate light
sources together with N-photon coincidence photodetection
yields a resolution ⬃xR / 冑N. A resolution ⬃xR / N can also be
obtained by introducing, at the lens, a device that is opaque
when it is illuminated by fewer than N photons. The first
type of enhancement—a 1 / 冑N standard quantum limit 共SQL兲
for imaging—is an N-photon quantum process, but it is
roughly equivalent to the classical procedure of averaging
the arrival positions of N photons that originate from
the same point on the object. The second type of
1050-2947/2009/79共1兲/013827共4兲
enhancement—a 1 / N Heisenberg-like scaling for imaging—
is a quantum phenomenon that derives from treating the N
photons as a single field of N-times higher frequency. In the
incoherent imaging regime, both methods can tolerate arbitrary amounts of loss at the expense of reduced efficiency but
without sacrificing resolution. To compare the performance
of our technique with conventional imaging methods, an explicit setup that uses visible light will be analyzed in detail,
and shows that the decrease in efficiency is manageable for
N ⱗ 5. It is worth stressing that, via proper scaling of the
system parameters, our analysis can be applied to a broader
context 共not necessarily involving optical frequencies兲.
We start by reviewing some basics of conventional imaging. Then we discuss the SQL coherent and incoherent subRayleigh imaging procedures. We conclude with a procedure
achieving the 1 / N Heisenberg scaling.
Rayleigh bound. Consider monochromatic imaging using
a circular-pupil thin lens of radius R and focal length f that is
placed at a distance Do from an object of surface area A, and
at a distance Di from the image plane, where 1 / D0 + 1 / Di
= 1 / f. In conventional imaging, the object is illuminated by
an appropriate 共spatially coherent or incoherent兲 source and
the image plane distribution of the light intensity, corresponding to the probability of detecting a photon at each
image-plane point rជi, is recorded. For photodetectors whose
spatial-resolution area S and temporal-resolution time ⌬t are
sufficiently small, the preceding probability is P1共rជi兲
ជi , t兲E共+兲
ជi , t兲典, where angular brackets de⯝ 共␩Sc⌬t兲具E共−兲
i 共r
i 共r
note ensemble average over the illumination’s state, ␩ is the
detector quantum efficiency, and E共+兲 = 关E共−兲兴† is the positivefrequency component of the electric field. This field compoជi , t兲 = 兰d3kជ E共+兲
ជi ; kជ 兲e−ikt/ca共kជ 兲, where a共kជ 兲 is
nent obeys E共+兲
i 共r
i 共r
the field annihilation operator for the optical mode with wave
vector kជ and E共+兲
i is the solution to the associated Helmholtz
equation at the image plane. The latter can be written in
terms
of
the
corresponding
object-plane
field
ikជ t·rជo
ជ
ជ
ជ
共r
,
k
兲
=
e
,
where
k
is
the
transverse
component
of
kជ ,
E共+兲
o
t
o
using classical imaging equations. For monochromatic light
in the paraxial regime kt Ⰶ k, it follows that 关19,20兴
013827-1
©2009 The American Physical Society
PHYSICAL REVIEW A 79, 013827 共2009兲
GIOVANNETTI et al.
a
b
a
b
c
d
c
d
f
e
FIG. 1. Coherent imaging. 共a兲 Object to be imaged. 共b兲 Conventional coherent image 共3兲 computed for Do / R = 200 共e.g., corresponding to Do = 10 cm and R = 0.5 mm兲, k = 8.57⫻ 105 m−1 共red
light兲, and m = 1 共note that, modulo a rescaling, the same images
would be found for any value of m兲. Owing to diffraction, no object
details below the Rayleigh bound xR = 0.89 mm 共segment in the top
right corner of the pictures兲 are discernible in the image. 共c兲 N-fold
coincidence detection 共N = 5兲 through Eq. 共6兲: resolution is not enhanced over the previous image despite the Nth-order compression
of the point-spread function. 共d兲 SQL reconstruction with illumination by the superposition of Fock states from Eq. 共7兲 with N = 3 and
⌬kt / k = 0.1. Sub-Rayleigh resolution is present. 共e兲 The same as 共d兲
for N = 5; more resolution enhancement occurs. 共f兲 Heisenberg-like
coherent reconstruction from Eq. 共11兲 with N = 5; still further enhancement is evident. All plots are normalized so that the minimal
and maximal intensities are 0 and 1, respectively.
ជi ;kជ 兲 =
E共+兲
i 共r
冕
d2rជo
ជo ;kជ 兲,
A共rជo兲h共rជi,rជo兲E共+兲
o 共r
A
共1兲
where rជi and rជo are two-dimensional vectors in the image and
object planes, A共rជo兲 is the object aperture function 关22兴, and
h共rជi , rជo兲 is the point-spread function of the imaging apparatus
given by 关12,19,20兴
h共rជi,rជo兲 =
R 2k 2A
4 ␲ D oD i
ei␽somb共Rk兩rជo + rជi/m兩/Do兲,
共2兲
somb共x兲 ⬅ 2J1共x兲 / x being the Airy function and m = Di / Do
the image magnification factor. The phase ␽ in Eq. 共2兲 plays
no role in the protocol, and can be neglected or compensated
using standard imaging techniques 关19兴.
Coherent imaging prevails when collimated coherent-state
illumination is employed, i.e. 关see Figs. 1共a兲 and 1共b兲兴,
ជi兲 = ␩Sc⌬tIo
P共c兲
1 共r
冏冕
d2rជo
A共rជo兲h共rជi,rជo兲
A
冏
2
,
共3兲
共+兲
with Io ⬅ 具E共−兲
o Eo 典 being the field intensity on the object
plane. Incoherent imaging occurs when the object is illuminated by independent 共monochromatic兲 beams propagating
from all directions 关see Figs. 2共a兲 and 2共b兲兴 whence
FIG. 2. Incoherent imaging of the object shown in Fig. 1共a兲 with
the same parameters. 共a兲, 共b兲 Conventional incoherent images from
Eq. 共6兲 with N = 1 and 5, respectively. The images are featureless
blurs, because the Rayleigh bound xR 共segment in top right corner兲
is too large to resolve the details. 共c兲, 共d兲 Reconstruction via
Eq. 共10兲 using the mixture ␴ of coherent states for N = 3 and 5,
respectively. The price one has to pay for the increase in resolution
of 共c兲 and 共d兲 is a reduction in intensity of the images by a factor
103 and 105, respectively 共for the parameters given in the text兲.
ជ i兲 =
P共inc兲
1 共r
␩Sc⌬t
Io
2 ␲ k 2A
冕
d2rជo
兩A共rជo兲h共rជi,rជo兲兩2 .
A
共4兲
When the lens radius R is sufficiently large, Eqs. 共4兲 and 共3兲
produce inverted, magnified, perfect images of the object,
because R2somb共R兩xជ 兩兲 → 4␲␦共2兲共xជ 兲 for R → ⬁ 关19兴. For small
R, the convolution integrals in Eqs. 共4兲 and 共3兲 produce
blurred images. The amount of blurring can be gauged
through the Rayleigh bound: for a point source at rជo in the
object plane, the resulting image-plane intensity is proportional to somb2共Rk兩rជo + rជi / m兩 / Do兲, which comprises a pattern
of circular fringes in rជi that are centered on −mrជo. The radius
of the first fringe, xR ⯝ 0.61⫻ 2␲mDo / 共kR兲, about −mrជo encloses ⬃84% of the light falling on the image plane 关20兴.
Intuitively, the image of an extended object is then a
weighted superposition of radius-xR circles off centered
about each −mrជo. This is the Rayleigh bound; using conventional imaging techniques, one cannot resolve details smaller
than xR.
SQL. The main idea of SQL sub-Rayleigh imaging is to
use an appropriate light source and to replace intensity measurement with spatially resolving N-fold coincidence detection strategies. The probability of detecting N photons at position rជi on the image plane is 关21兴
PN共rជi兲 ⯝
共␩Sc⌬t兲N
N!
ជi,t兲兴N典,
ជi,t兲兴N关E共+兲
具关E共−兲
i 共r
i 共r
共5兲
which can be accomplished by means of doppleron absorbers
关23兴, photon-number resolving detectors, or N-fold coincidence counting 共which exploits the full photon statistics: the
N value needs not be predetermined兲. Note that multiphoton
detection alone does not guarantee sub-Rayleigh performance. In fact, for the coherent imaging of Eq. 共3兲, N-photon
detection gives
013827-2
PHYSICAL REVIEW A 79, 013827 共2009兲
SUB-RAYLEIGH-DIFFRACTION-BOUND QUANTUM IMAGING
ជi兲兴N/N!.
PN共rជi兲 = 关P共c兲
1 共r
共6兲
The N in the exponent gives an N-fold compression of the
fringes in the point-spread function. It amounts to taking the
Nth power of the light intensity, which is formally equivalent
to a classical post-processing of the signal of Eq. 共3兲. As
shown in Fig. 1共c兲 no resolution enhancement is obtained
using only N-photon detection: an appropriate light source is
also needed.
A first example of such an input source is a superposition
of N-photon Fock states, focused in a small area s
⬅ 共␲⌬k2t 兲−1 Ⰶ A centered at positions rជo on the object
兩␺典 ⬅
1
冑M
冕
d2rជo兩N典rជo,
兩N典rជo ⬅
1
ជ
冑N! 关b 共ro兲兴
†
N
兩0典, 共7兲
where b共rជo兲 is the annihilator of the associated localized spa4A
tial mode 关24兴 and M ⯝ ⌬k
2 is the normalization. Inserting
t
this state into Eq. 共5兲, we find
PN共rជi兲 ⯝
Q共rជi,rជo兲 ⬅
⌬k2t A
4␲
冕
␰N
冏冕
d2rជo N
Q 共rជi,rជo兲
A
冏
2
,
d2rជ
A共rជ兲h共rជi,rជ兲F⌬kt共兩rជo − rជ兩兲,
A
共8兲
⌬␻⌬t
where F⌬kt共x兲 ⬅ ␲⌬k2t A somb共⌬ktx兲, and ␰ ⬅ ␩ ␲⌬k2A AS is a
t
dimensionless quantity that is typically very small because of
the monochromatic 共⌬␻⌬t Ⰶ 1兲 and focusing assumptions
共␲⌬k2t A Ⰷ 1兲. Equation 共8兲 can be simplified by assuming
Do / R Ⰷ k / ⌬kt, which implies that each number state in the
superposition is focused to a spot much smaller than the
object-plane Rayleigh limit of the lens. In this case, h can be
extracted from the integral yielding Q共rជi , rជo兲 ⯝ h共rជi , rជo兲Ã共rជo兲
d2rជo
with Ã共rជo兲 ⬅ 兰 A A共rជ兲F⌬kt共兩rជo − rជ兩兲. Now Eq. 共8兲 becomes
PN共c兲共rជi兲 ⯝
⌬k2t A
4␲
␰N
冏冕
d2rជo
关Ã共rជo兲h共rជi,rជo兲兴N
A
冏
2
,
共9兲
which, contrary to Eq. 共6兲, cannot be obtained through classical post-processing of P1. It generalizes coherent imaging
共3兲 to N-photon detection. The point-spread function that
governs spatial resolution is now hN—which is narrower
than h—so that when A共rជo兲 ⯝ Ã共rជo兲 there is an enhancement
in resolution, see Figs. 1共d兲 and 1共e兲.
An analogous generalization for incoherent imaging is obtained by replacing the state Eq. 共7兲 with an incoherent mixd2rជo
ture of focused Fock states, i.e., ␳ = 兰 A 兩N典rជo具N兩. In this case,
Eq. 共9兲 becomes
PN共inc兲共rជi兲 ⯝ ␰N
冕
d2rជo
兩Ã共rជo兲兩2N兩h共rជi,rជo兲兩2N ,
A
共10兲
which generalizes Eq. 共4兲 to N-photon detection. The attainable resolution is shown in Figs. 2共c兲 and 2共d兲.
The Fock states employed in Eqs. 共7兲 and 共10兲 are highly
sensitive to loss and are quite difficult to create. Nevertheless, N-fold incoherent imaging can be realized with lossresistant and easy to create light sources, which are prefer-
able in most situations. One can use an incoherent mixture of
coherent states that randomly illuminate all points on the
d2rជ
object:
␴ = 兰 A o 兩␣典rជo具␣兩,
where
兩␣典rជo ⬅ exp关␣b†共rជo兲
*
− ␣ b共rជo兲兴兩0典. Equation 共10兲 still applies with an extra multiplicative factor of 兩␣兩2N / 共N!兲. The state ␴ can be easily
prepared by shining a highly focused laser beam on the object, one point at the time. This state is robust to loss, as the
loss parameter ␩ just takes 兩␣典 into 兩冑␩␣典. An arbitrary
amount of loss can be tolerated—without sacrificing
resolution—simply by increasing 兩␣兩. In the same way, we
can overcome the reduction in efficiency 共exponential in N兲
associated with the low intensities of the signals that come
from the factor ␰N. To show that this procedure is practical,
we calculate the reduction in image intensity for realistic
values of the parameters ␩ = 10−3, ⌬␻⌬t ⬃ 10−1, S = A
⫻ 10−6 共and the values given in the figure captions for the
remaining ones兲. Then, the ratio between the maximum intensities of our method and the classical procedure can be
straightforwardly evaluated from the coefficients in front of
the function h of Eq. 共2兲 and in front of the probability 共10兲.
It follows that for a pulsed laser with an easily attainable
兩␣兩2 ⬃ 5 ⫻ 1011, the light intensity reaching the image plane is
⬃105 times lower for N = 5 than a classical procedure with
the same illumination: the resolution increase of Fig. 2共d兲
comes at the cost of a 105 reduction in the absolute intensity.
Nonetheless, a good quality image can easily be obtained:
the ⬃107 N-photon events necessary for imaging can be obtained in ⬃105 laser pulses, i.e., typically in a small fraction
of a second.
The amount of resolution gain from the procedures detailed above can be roughly estimated by gauging by narrowing of the point-spread function h that results from taking its
Nth power. For instance, one can evaluate the radius xR共N兲
that contains 84% of the area under somb2N. Numerical
analysis shows that xR共N兲 / xR ⬃ 1 / 冑N, which suggests an
SQL 关3兴 for imaging. This should be taken only as a rough
estimate, as xR共N兲 is also the radial dimension of a pointlike
object imaged using the post-processing strategy of Eq. 共6兲.
For more extended objects, the resolution enhancement depends also on ⌬kt. The 1 / 冑N scaling exposes the classical
nature of this enhancement: the same effect can be attained
by averaging the arrival positions of N photons at the image
plane. This is surely advantageous over N-photon detection
in many situations, but it is impractical for lithography or
film photography, and it cannot reproduce the coherent imaging case of Eq. 共9兲. Moreover, from general principles
关1,3兴 one would expect that also a 1 / N Heisenberg-like scaling is achievable, i.e., a resolution ⬃xR / N, not achievable
with classical strategies.
Heisenberg-like scaling. The 1 / N scaling can be obtained
by treating the N photons as a single entity of N-times higher
frequency. This situation can be simulated, at least in principle, by inserting immediately in front of the lens a screen
divided into small sections each of area sF such that, if less
than N photons reach one section, they are absorbed, otherwise they are coherently transmitted. Such a screen does not
currently exist but, in principle, one could be built, e.g., by
using doppleron materials 关23兴. Then, if the object is illuminated by the focused coherent states described above, only N
013827-3
PHYSICAL REVIEW A 79, 013827 共2009兲
GIOVANNETTI et al.
photons that originate at rជo, successfully transit the screen
within one of its area-sF segments, and get detected at rជi can
contribute to the image at that point. Again loss just reduces
the efficiency, but not the resolution. In this case, the operaជi , t兲兴N of the N-photon absorption probability 共9兲
tors 关E共+兲
i 共r
共or N-fold coincidence detection兲 are approximately given by
关25兴
ជi,t兲兴N ⯝ ␥
关E共+兲
i 共r
冕
d2rជo
hN共rជi,rជo兲关Ã共rជo兲b共rជo兲兴N .
A
共11兲
s
order 共 ␲RF 2 兲N. Equation 共11兲 describes the absorption of N
frequency-␻ photons that originated near rជo and then propagated through the imaging apparatus as if they were a single
frequency-N␻ photon. It gives rise to coherent and incoherent images that are formally equivalent to those of Eqs. 共3兲
and 共4兲 for a light beam of wave number Nk, thus realizing
the Heisenberg-like scaling of an N-fold resolution improvement over the Rayleigh bound, albeit with an even worse
efficiency than the N-photon detection methods given
above—see Fig. 1共f兲.
Here hN is obtained from Eq. 共2兲 by replacing k with Nk, i.e.,
hN is the point-spread function for photons having N-times
higher frequency than the illumination; and ␥ accounts for
the spatial resolution of the doppleron screen, i.e., it is of
We thank F.N.C. Wong for comments. This research was
supported in part by the W. M. Keck Foundation for Extreme
Quantum Information Theory and by the DARPA Quantum
Sensors Program.
关1兴 E.g., see M. I. Kolobov, Rev. Mod. Phys. 71, 1539 共1999兲.
关2兴 A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P.
Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 共2000兲.
关3兴 V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330
共2004兲; Phys. Rev. Lett. 96, 010401 共2006兲; S. L. Braunstein,
Nature 共London兲 440, 617 共2006兲.
关4兴 K. L. Pregnell and D. T. Pegg, J. Mod. Opt. 51, 1613 共2004兲.
关5兴 H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory 24, 657
共1978兲.
关6兴 P. R. Hemmer, A. Muthukrishnan, M. O. Scully, and M. S.
Zubairy, Phys. Rev. Lett. 96, 163603 共2006兲; A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, J. Opt. B: Quantum
Semiclassical Opt. 6, S575 共2004兲.
关7兴 D. Korobkin, and E. Yablonovitch, Opt. Eng. 41, 1729 共2002兲.
关8兴 C. Thiel et al., e-print arXiv:quant-ph/070.1024.
关9兴 S. J. Bentley and R. W. Boyd, Opt. Express 12, 5735 共2004兲.
关10兴 M. Zhang et al., e-print arXiv:quant-ph/0612060.
关11兴 K. Wang and D.-Z. Cao, Phys. Rev. A 70, 041801共R兲 共2004兲.
关12兴 Y. Shih, IEEE J. Sel. Top. Quantum Electron. 13, 1016 共2007兲.
关13兴 E. Yablonovitch and R. B. Vrijen, Opt. Eng. 38, 334 共1999兲.
关14兴 A. Pe’er et al., Opt. Express 12, 6600 共2004兲.
关15兴 B. I. Erkmen and J. H. Shapiro, Phys. Rev. A 77, 043809
共2008兲.
关16兴 Quantum Imaging, edited by M. I. Kolobov 共Springer, New
York, 2006兲.
关17兴 L. A. Lugiato, A. Gatti and E. Brambilla, J. Opt. B: Quantum
Semiclassical Opt. 4, S176 共2002兲.
关18兴 F. S. Cataliotti, R. Scheunemann, T. W. Hansch, and M. Weitz,
Phys. Rev. Lett. 87, 113601 共2001兲; C. Skornia, J. von
Zanthier, G. S. Agarwal, E. Werner, and H. Walther, Phys. Rev.
A 64, 063801 共2001兲; M. W. Mitchell, J. S. Lundeen, and A.
M. Steinberg, Nature 共London兲 429, 161 共2004兲; K. J. Resch,
K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L.
OBrien, and A. G. White, Phys. Rev. Lett. 98, 223601 共2007兲.
J. W. Goodman, Introduction to Fourier Optics 共McGraw-Hill,
New York, 1988兲.
M. Born and E. Wolf, Principles of Optics 共Cambridge Univ.
Press, Cambridge, 1999兲.
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics 共Cambridge Univ. Press, Cambridge, 1995兲.
The function A measures the attenuation of the light due to the
object. We assume A to be independent of kជ ; the generalization
to a kជ -dependent A is straightforward.
J. J. Tollett, J. Chen, J. G. Story, N. W. M. Ritchie, C. C.
Bradley, and R. G. Hulet, Phys. Rev. Lett. 65, 559 共1990兲; N.
P. Bigelow and M. G. Prentiss, ibid. 65, 555 共1990兲.
d2kជ t
ជ
d␻
It is given by b共rជo兲 ⬅ 兰 冑 兰
eikt·rជoa共kជ 兲, where the fre2
关19兴
关20兴
关21兴
关22兴
关23兴
关24兴
⌬␻ 冑␲⌬kt
quency integration is performed over a small bandwidth ⌬␻
around the mean frequency ␻ = kc and the wave-vector integration is performed over 兩kជ t兩 艋 ⌬kt. The exact expression for M
N
is 兰d2rជo兰d2rជ⬘oF⌬k2共兩rជo − rជ⬘o兩兲 / AN, which, for highly localized
t
2
modes ␲⌬kt A Ⰷ 1, gives M ⯝ 4A / ⌬k2t .
ជi , t兲兴N as
关25兴 Use the screen-to-lens transfer function to write 关E共+兲
i 共r
共+兲 共1兲
共+兲 共N兲
a convolution of the products El 共rជl , t兲 ¯ El 共rជl , t兲,
ជl , t兲 being the electric field on the screen. The screen
E共+兲
l 共r
removes all contributions but those for which the rជ共j兲
l fall in the
same area-sF region. Equation 共11兲 follows by writing the re共+兲 共N兲
ជ共1兲
ជo , t兲
sulting operator as a convolution of E共+兲
o 共r
o , t兲 ¯ Eo 共r
through the object-to-lens transfer function, and noting that for
N
focused sources it is approximated by 关b共rជ共1兲
o 兲兴 .
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