We present the quantum theory of superresolution for discrete subwavelength structures. It allows to formulate, in particular, the standard quantum limit of superresolution achieved for illumination of the structure by light in coherent state. Our theory is based on discrete prolate spheroidal sequences and functions which are the proper basis set of the problem. We demonstrate that the superresolution factor is much higher for discrete structures than for continuous objects for the same signal-to-noise ratio. This result is a clear illustration of the crucial role of a priori information in superresolution problems.
©2008 Optical Society of America
Subwavelength optical structures and nanostructures are the subject of very active research in the last years. Enabled by nano-imprint lithography fabrication, these structures have numerous applications. A family of subwavelength structures known as subwavelength optical elements are considered as very promising for creation of new integrated optical components with very high level of integration and functionality.
Another area of application of subwavelength optical structures is in optical data storage. The ultimate storage density of optical discs presently is limited by the size of the laser spot used for readout. This size has its physical origin in the phenomenon of diffraction and is proportional to the wavelength of the light and inversely proportional to the numerical aperture of the optical imaging system. The increase in the storage density of the optical discs during the last years has been achieved by reducing the wavelength and increasing the numerical aperture. One of the possibilities for further increase of the storage density of optical discs is to use subwavelength structures engraved on the surface of the disc [1, 2] together with multilevel information encoding. Multilevel information encoding in several independent variables such as polarization of light, its phase, or optical angular momentum, is currently discussed in the literature . The task of the readout system is to reliably discriminate between several bits of information inside the diffraction-limited spot of the laser. This problem is closely related to superresolution, i. e. reconstruction of the original object from its diffraction-limited image with the resolution beyond the Rayleigh limit.
The question of quantum limits of optical superresolution has been addressed recently in a series of papers [4, 5, 6]. In particular, the theory developed in these papers has established the standard quantum limit of superresolution reached when the object is illuminated by a light wave in a multimode coherent quantum state. It was also shown that one can further improve the superresolution factor beyond the standard quantum limit using the illumination scheme with multimode squeezed light .
As in the case of classical superresolution by the data inversion , the quantum theory of superresolution uses a priori information about the object. In Ref. [4, 5, 6] this a priori information is in the assumption that the object has a finite support. In optical data storage the digital information is optically encoded in a relief pattern in form of pits or holes on the surface of a disc arranged along the tracks that are followed by the laser beam when the disc rotates. The significant information is encoded in the height or the depth of different pits along the track. Thus for the readout of digital information from the disc it is not necessary to resolve the exact relief of the structure, but is sufficient to recover a finite number of discrete values in order to completely characterize the input signal. This is only one example of many other possible applications when one is interested not in exact shape of the input signal but only in a discrete number of its samples. The quantum theory from Ref. [4, 5, 6] is not adapted for this situation. Let us mention that the question of superresolution of discrete optical images and sampled signals in general has been discussed in the literature (see, for example Refs. [9, 10]). However, to the best of our knowledge, the quantum theory of superresolution for imaging discrete structures has not been formulated until now.
In this paper we present the quantum theory of superresolution specially developed for imaging of discrete subwavelength structures. We show that the proper basis functions for description of this problem are the discrete prolate spheroidal sequences and functions introduced by Slepian  and some other authors . When the input optical signal, containing K distinct discrete values, is decomposed in this basis set, the problem becomes K-dimensional instead of infinitely dimensional as in the continuous case. We show that when quantum fluctuations of light are not taken into account these K discrete values can be reconstructed exactly for arbitrary K. Taking into account quantum fluctuations, the number K depends on the signal-to-noise ratio. We shall demonstrate that due to the discrete nature of this problem one can obtain significantly higher superresolution factor than in the continuous case for the same signal-to-noise ratio. This is the main result of the paper.
In Fig. 1 we have shown a simplified optical scheme for far-field imaging of a discrete subwavelength structure (not necessarily binary). We consider for simplicity a one-dimensional case but our results can be generalized into two dimensions. A discrete structure is modelled as a sequence of rectangular pixels of size Δ within a finite support area X. The number K of pixels is given by K=X/Δ. Choosing X of the order of the wavelength we obtain the situation of subwavelength structure. We restrict ourselves to the paraxial approximation. The question of exact diffraction theory of light by subwavelength structures is outside of the scope of this paper.
A thin lens L placed as indicated in Fig. 1, performs the spatial Fourier transform of the object into the Fourier plane with a diaphragm of finite size d. Only the spatial Fourier components within the diaphragm are transmitted trough the system and can be detected, while those outside are absorbed.We shall assume that the transmitted Fourier components are measured by means of a homodyne detection scheme with a local oscillator and a sensitive CCD camera.
Let us introduce the dimensionless spatial coordinates in the object plane as s=2x/X, and in the Fourier plane as ξ=2y/d (see Fig. 1). The dimensionless photon annihilation operators â(s) in the object plane and f̂(ξ) in the Fourier plane obey the standard commutation relations,
They are normalized so that 〈â †(s)â(s)〉 gives the mean photon number per unit dimensionless length in the object plane and 〈f̂ †(ξ)f̂(ξ)〉 - in the Fourier plane. The spatial Fourier transform (Tâ)(ξ) performed by the Fourier lens L, in terms of these dimensionless variables reads
where is the space-bandwidth product of the imaging system.
In the dimensional coordinates s the size of a pixel becomes Δ=2/K. For symmetry reasons we will consider the case of odd number K=2M+1. Let us introduce the “pixellized” annihilation operators
Here m=0,±1,±2,… is the position of the center of m-th pixel. The support area corresponds to m=0,±1,…±M.
The operators â(m) satisfy the commutation relations
of the annihilation and creation operators of the discrete modes corresponding to individual pixels. Using these discrete operators we can introduce the “coarse-grained” field operator Â(s) as
where p(s) is the frame function of the pixel equal to 1 for |s|≤Δ/2 and zero for |s|>Δ/2. Substituting the field operator Â(s) into Eq. (2) we obtain the Fourier spectrum F̂(ξ)
where P(ξ) is the Fourier spectrum of the pixel frame function. It does not contain any information about the coefficients â(m) of the discrete input structure and can be eliminated from the measured spectrum ̂F(ξ). Therefore, in what follows we will consider the corrected Fourier spectrum f̂(ξ)=F̂(ξ)/P(ξ) given by
where It is easy to check using Eq. (4) that the operators f̂(ξ) satisfy the commutation relations (1).
One may naturally ask whether division of the Fourier spectrum F̂(ξ) by the Fourier spectrum of the pixel frame function P(ξ) is correctly defined operation and will not encounter division by zero, for example. The answer to this question is straightforward. Indeed, it is very easy to see that P(ξ)=Δsinc(cξΔ/2) and, therefore, its first zero is outside the area |ξ|<1/2W of the object spectrum F̂(ξ).
The discrete prolate spheroidal sequences (DPSS) ψµ (m),µ=0,1…2M, are the K-dimensional eigenvectors satisfying the following equation :
where λµ are the nondegenerate real eigenvalues such that 1>λ 0>…>λ 2M>0. The DPSS have the following properties:
Slepian  has shown that DPSS form an orthonormal basis in the subspace of bandlimited discrete sequences, i. e. such sequences that their spectrum is identically zero outside the area |ξ|<1/2W. It is worth noting that our dimensionless Fourier frequencies ξ differ from the Fourier frequencies mostly used in the literature on the DPSS by a factor 1/W. This factor appears also in the support area for the Fourier spectrum of the DPSS. Let us underline that contrary to the continuous case [4, 5, 6] there are only K=2M+1 basis sequences ψµ(m).
Since the Fourier spectrum given by Eq. (7) is a continuous function, we need another basis set in the Fourier plane. This set is given by the discrete prolate spheroidal functions (DPSF) ψµ(ξ), the eigenfunctions of the following equation:
with the same eigenvalues λµ,µ=0,1…2M as in Eq. (8). The DPSF are doubly orthogonal:
The two sets DPSS and DPSF are related by the following Fourier transform
Using the DPSS ψµ(m) as basis functions in the object plane and the DPSF ψµ (ξ) in the Fourier plane, we can write the Fourier transform (7) as a unitary transformation of the photon annihilation operators, associated with the discrete prolate modes. In order to obtain this unitary transformation we shall split the object plane into the “core” region |m|≤M, corresponding to the object support area, and the “wings” region |m|>M, where the classical amplitude of the object is identically zero. The orthonormal basis sequences in these two regions are given by
Similarly, we shall split the Fourier plane into the core region |ξ|≤1 corresponding to the transmission area of the diaphragm, and the wings region |ξ|>1 of the absorption part. The orthonormal basis functions in these regions are
Using the basis sequences (13) and (14) we can write the operators â(m) in the object plane as
and the operators f̂(ξ) in the Fourier plane as
In these expressions the coefficients âµ and f̂µ are the annihilation operators of the discrete prolate modes in the core regions of the object and the Fourier plane, and the coefficients b̂µ and ĝµ are the corresponding annihilation operators on the wings regions of these planes. These operators obey the standard commutation relations of the annihilation and creation operators of the discrete modes. The last terms â ⊥(m) and f̂ ⊥(ξ) are added in order to satisfy the commutation relations (4) and (1). However, these terms are orthogonal to the decomposition over the discrete prolate modes and will not appear in the unitary transformation.
Substituting Eqs. (15) and (16) into the Fourier transform (7) and using Eqs. (12)–(14) we arrive at the following unitary transformation of the annihilation operators of the discrete prolate modes from the object plane into the Fourier plane:
Let us assume that we can measure the spatial Fourier amplitudes f̂(ξ) within the transmission area of the diaphragm using a homodyne detection scheme with a sensitive CCD camera. Then we can calculate the coefficients â (r) μ of the reconstructed object as
where the superscript (r) stands for “reconstructed”. As follows from Eq. (18), the reconstruction of the input object is not exact because of the second term with the annihilation operators b̂μ describing the vacuum fluctuations in the wings area. These vacuum fluctuations prevent from reconstruction of higher modes âμ, because the eigenvalues λμ become rapidly very small after the index μ has attained some critical value. The case of coherent input state in the core region and the vacuum state in the wings region determines the standard quantum limit of superresolution .
The physical meaning of superresolution for discrete objects is different from the continuous ones. Indeed, in the discrete case we have an object which is completely characterized by K components and, therefore, superresolution cannot be interpreted as an attempt to resolve fine details of the object as in the continuous case. However, both situations become closely related in the language of the spatial Fourier spectra. In the discrete case of a object with K components its spectrum is a periodic function which in the dimensional coordinates ξ is determined within |ξ|≤1/2W. Therefore, if W=1/2 all the spatial Fourier components are transmitted through the diaphragm and can be perfectly reconstructed from the measurement in the Fourier plane. However, if W<1/2 part of the spatial spectrum is absorbed by the diaphragm and lost. Superresolution is an attempt to recover the absorbed Fourier components both in the continuous and the discrete case.
The superresolution factor for imaging discrete structures can be evaluated in terms of the point-spread function (PSF) similar to the continuous case . In Ref.  the superresolution factor for continuous objects reconstructed from diffraction-limited images, was defined as the ratio of the widths of two point-spread functions: imaging and reconstruction. Similar approach can be used for the discrete case.
In order to introduce the imaging PSF in the discrete case we have to establish the relationship between the discretized field operator â(m) in the object plane and an equivalent operator ê(m) in the image plane,
with h(m,m′) being the imaging PSF. There are two approaches for obtaining this relationship. In the first one we can use the relation between the continuous field operators ê(s) in the image plane and â(s) in the object plane, and then discretize ê(s) similar to Eq. (3). This approach corresponds to an optical formation of the image by means of two Fourier lenses as in Refs. [4, 5].The resulting relation between ê(m) and â(m) will be complicated because one has to take into account the Fourier spectrum P(ξ) of the pixel frame function.
In order to obtain more transparent results we shall chose another approach. Namely, we shall assume that the relation between ê(m) and f̂(ξ) is given by the Fourier transform inverse to Eq. (7), i. e. that we are using the corrected Fourier spectrum f̂(ξ) instead of F̂(ξ) from Eq. (6). The resulting relationship between ê(m) and â(m) will not correspond to an optical formation of the continuous image ê(s) because we have corrected the Fourier spectrum. Nevertheless, we shall chose this approach due to the transparency of final results and will loosely speak about the “imaging” PSF keeping in mind the spectrum correction. The relationship between â(m) and ê(m) is easily obtained as
The reconstructed discrete field operator â (r)(m) in the core region |m|≤M can be written as the following decomposition over the DPSS φμ (m):
is the reconstruction point-spread function. Since the DPSS φμ (m) are complete over the core interval |m|≤M they satisfy the following relation
for |m|≤M, |m′|≤M, which means that the reconstruction PSF is equal to
and Eq. (23) can be written as
As in Eq. (18) the last term sets the quantum limit of reconstruction of the input field operatorâ(m).
Using the imaging PSF h(m,m′) and the reconstruction PSF h (r)(m,m′) we can define the superresolution factor (SF) as the ratio of the widths of these two functions. It follows from Eqs. (21) and (26) that SF=1/2W. The same result can be obtained comparing the spectral widths of the diffraction-limited image and that of the reconstructed one. Indeed, the size of the diaphragm in the Fourier plane, |ξ|<1, determines the spectral width of the diffraction-limited PSF of the imaging system. The spectral width of the reconstructed object is given by 1/2W. The ratio of two gives the same result SF=1/2 W.
The condition of superresolution W<1/2 in the discrete case can be formulated in terms of the number of pixels K in the input signal. Indeed, since , the condition of superresolution W<1/2 is equivalent to K>S, where S=2c/π is the Shannon number of the system. For example, for c=1, we have S=0.64 and thus already with one pixel, K=1, we achieve superresolution. In Fig. 2 we have shown the superresolution factor SF as a function of the signal-to-noise ratio for discrete structures (bold line) and continuous objects (dotted line). The signal-to-noise ratio for the coherent input state is given by the total mean number of photons 〈N̂〉 in the object defined as 
The evaluation of the superresolution factor for discrete case has been done exactly as for the continuous case in Ref. .We refer the interested reader to this publication for further details. The curve for the continuous case is taken from Ref. . One can see that the superresolution factor for the discrete structures is much higher at the same signal-to-noise ratio. This result illustrates clearly the crucial role of a priori information in the ill-posed inverse problems like superresolution.
In conclusion, we have presented a quantum theory of superresolution for imaging discrete subwavelength structures. Our motivation is twofold: on the one hand, to find an ultimate quantum limit of superresoltion for such structures, imposed by the quantum nature of the light and, on the other hand, to demonstrate that the superresolution factor for discrete case is higher than that for continuous case because of additional a priori information.We have obtained the standard quantum limit of superresolution for imaging of discrete structures using the light in coherent state. We have evaluated the superresolution factor as a function of the signal-to-noise ratio for this case. Our results show clearly that the potential of superresolution for discrete structures is higher than for continuous objects. Let us finally note that illumination of discrete structure by properly designed multimode squeezed light as in Ref.  can further increase the superresolution factor beyond the standard quantum limit. This subject will be discussed elsewhere.
This work was partially supported by the project INTAS 05-1000008-7904.
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