Sub-Rayleigh Quantum Imaging
No imaging apparatus can produce perfect images: spatial resolution is limited by the Rayleigh diffraction bound that is a consequence of the imager’s finite spatial extent. We show some -photon strategies that permit resolution of details that are smaller than this bound, attaining either a enhancement (standard quantum limit) or a enhancement (Heisenberg limit) over standard techniques. In the incoherent imaging regime, the methods presented are loss resistant, because they can be implemented with classical-state light sources. Our results may be of importance in many applications: microscopy, telescopy, lithography, metrology, etc.
Quantum effects have been used successfully to provide resolution enhancement in imaging procedures. Among the many proposals that have been made review (), arguably the most famous is the quantum lithography procedure litho (). All of these methods take advantage of the fact that the de Broglie wavelength of a multi-photon light state is much shorter than the photon’s electromagnetic field wavelength chuangyama (): 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 much shorter de Broglie wavelength. Such proposals are then based on light sources of highly entangled or squeezed states, as entanglement or squeezing are necessary to achieve efficient quantum enhancements metrology (). 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 postsel (). 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 metrology (), 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 scullylith (); korobkin (); agarwal (); boyd (); zhang (); wang (); yablo (); peer (); shih (); lugiato (); jeff (); jeffhorace (); libroqim (), and cover a wide range of interesting situations. Analogous methods have been employed successfully also in fields not directly related to imaging altri ().
This Letter discusses how one can achieve a resolution enhancement beyond what the apparatus’ structural limits impose for conventional imaging (i.e., the Rayleigh diffraction bound ). In particular we show that employing appropriate light sources together with -photon coincidence photodetection at the output yields a resolution . A resolution can also be obtained by introducing, at the lens plane, a device that is opaque when it is illuminated by fewer than photons. The first type of enhancement—a standard quantum limit for imaging—is an -photon quantum process, but it is roughly equivalent to the classical procedure of averaging the arrival positions of photons that originate from the same point on the object. The second type of enhancement—a Heisenberg bound for imaging—is a quantum phenomenon that derives from treating the photons as a single field of -times higher frequency. In the incoherent imaging regime, both methods presented here can tolerate arbitrary amounts of loss at the expense of reduced efficiency but without sacrificing resolution.
We start by reviewing some basics of conventional imaging. Then we discuss coherent and incoherent sub-Rayleigh imaging procedures that attain the standard-quantum limit, and finally we introduce our approach to realizing the Heisenberg limit for imaging.
Consider monochromatic imaging using a circular-pupil thin lens of radius and focal length that is placed at a distance from an object of surface area , and at a distance from the image plane, where . In conventional imaging, the object is illuminated by an appropriate (spatially coherent or incoherent) source and the image plane distribution of the light intensity, corresponding to the probability of detecting a photon at each image-plane point , is recorded. For photodetectors whose spatial-resolution area and temporal-resolution time are sufficiently small, the preceding probability satisfies , where angular brackets denote ensemble average over the illumination’s state, is the detector quantum efficiency, and is the positive-frequency component of the electric field. This field component obeys , where is the field annihilation operator for the optical mode with wave vector , and 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 , where is the transverse component of , using classical imaging equations. For monochromatic light in the paraxial regime , it follows that goodman (); born ()
where and are two-dimensional vectors in the image and object planes, is the object aperture function nota1 (), and is the point-spread function of the imaging apparatus given by shih (); goodman (); born ()
with being the well known Airy function, and being the image magnification factor. In Eq. (2), is a phase factor which can be generally neglected or compensated.
Incoherent imaging occurs when the object is illuminated by independent (monochromatic) beams propagating from all directions, whence
with being the field intensity on the object plane. Coherent imaging prevails when collimated coherent-state illumination is employed, giving rise to
When the lens radius is sufficiently large, Eqs. (3) and (4) produce inverted, magnified, perfect images of the object, because for goodman (). For insufficient to reach this asymptotic behavior, the convolution integrals in Eqs. (3) and (4) produce blurred images. The amount of blurring can be gauged through the Rayleigh diffraction bound: for a point source at in the object plane, the resulting image-plane intensity is proportional to , which comprises a pattern of circular fringes in that are centered on . The radius of the first fringe born (),
about encloses 84% of the light falling on the image plane. Intuitively, the image of an extended object is then a weighted superposition of radius- circles of centered about each . This is the Rayleigh diffraction bound; using conventional imaging techniques one cannot resolve details smaller than .
Standard quantum limit:–
The main idea of sub-Rayleigh imaging is to use an appropriate light source and to replace intensity measurement with spatially-resolving -fold coincidence detection strategies. Specifically we will focus on the probability of detecting photons at position on the image plane mandel (), i.e.,
which can be accomplished by means of doppleron absorbers doppleron (), photon-number resolving detectors, or -fold coincidence counting. The last two approaches are more convenient than the first, as they exploit the full photon statistics so that the value need not be predetermined. Note that multi-photon detection alone does not guarantee sub-Rayleigh performance. In fact, for the coherent imaging of Eq. (4), -photon detection gives
Here, the factor of in the exponent gives an -fold compression of the fringes in the point-spread function. This compression, however, is not intrinsically quantum. It amounts to taking the th power of the light intensity, which is simply a classical post-processing of the signal in Eq. (4). Thus no resolution enhancement is obtained through simple -photon detection, see Fig. 1(c).
As our first example of a source that can be used to beat the Rayleigh bound, consider an input state that is the superposition of -photon Fock states that have been focused to a small area centered at positions on the object plane, viz.,
where , and is a dimensionless quantity that is typically very small because of the monochromatic () and focusing assumptions (). Equation (9) can be simplified by assuming , 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 can be extracted from the integral yielding with . Now Eq. (9) becomes
which, contrary to Eq. (7), cannot be obtained through post-processing of , and which generalizes coherent imaging (4) to -photon detection. The point-spread function that governs spatial resolution is now —which is narrower than —so that when there is an enhancement in resolution over the Rayleigh bound. More generally, even if and differ significantly, one can still beat the Rayleigh bound if is sufficiently large and , see Figs. 1(d) and (e).
The states employed in Eqs. (8) and (11) are highly sensitive to loss. Nevertheless, -fold incoherent imaging can be realized with loss-resistant light sources. Suppose we use an incoherent mixture of coherent states that randomly illuminate all points on the object: , where . Equation (11) still applies with an extra multiplicative factor of to account for the Poissonian photodetection statistics of coherent states. The state can be prepared by shining a highly-focused laser beam on the object, one point at the time. This state is highly robust to loss, because loss parameter just takes into . Hence an arbitrary amount of loss can be tolerated—without sacrificing resolution—simply by increasing .
The improved resolution afforded by the procedures detailed above can be roughly estimated by gauging by narrowing of the point-spread function that results from taking its th power. This can be done, for instance, by evaluating the radius that contains 84 of the area under in the plane. Numerical analysis shows that , which suggests a standard quantum limit review () for imaging. This should be taken only as a rough estimate, as is also the radial dimension of a point-like object imaged using the post-processing strategy of Eq. (7). For more extended objects, the actual resolution enhancement will also depend on . The scaling exposes the classical nature of this enhancement: the same effect can be attained by averaging the arrival positions of photons at the image plane. This is surely advantageous over -photon detection in many situations, but it is impractical for lithography or film photography, and it seems impossible to classically reproduce the coherent imaging case of Eq. (10). In addition, from general principles review (); metrology () one would expect the ultimate bound (a Heisenberg limit for imaging) to have , scaling, i.e., a resolution , which is not achievable with classical strategies.
The Heisenberg scaling can be obtained by treating the photons as a single entity of -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 such that if less than 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 doppleron (). Then, if the object is illuminated by the focused coherent states described above, only photons that originate at , successfully transit the screen within one of its area- segments, and get detected at can can contribute to the image at that point. In this case, the operators of the -photon absorption probability (10) are approximately NOTENEW1 ()
Here: is obtained from Eq. (2) by replacing with , i.e., is the point-spread function for photons having -times higher frequency than the illumination; and accounts for the spatial resolution of the doppleron screen, i.e., it is of order . Equation (LABEL:enne) describes the absorption of frequency- photons that originated near and then propagated through the imaging apparatus as if they were a single frequency- photon. It gives rise to coherent and incoherent images that are formally equivalent to those of Eqs. (4) and (3) for a light beam of wave number , thus realizing the Heisenberg limit of an -fold resolution improvement over the Rayleigh bound, see Fig. 1(f). This method shows that Heisenberg-limited resolution can be obtained using classical light, albeit with an even worse efficiency than the -photon detection methods given above.
This research was supported in part by the W. M. Keck Foundation for Extreme Quantum Information Theory and by the DARPA Quantum Sensors Program.
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