Abstract
Wavefront sensing is a set of techniques providing efficient means to ascertain the shape of an optical wavefront or its deviation from an ideal reference. Owing to its wide dynamical range and high optical efficiency, the Shack–Hartmann wavefront sensor is nowadays the most widely used of these sensors. Here we show that it actually performs a simultaneous measurement of position and angular spectrum of the incident radiation and, therefore, when combined with tomographic techniques previously developed for quantum information processing, the Shack–Hartmann wavefront sensor can be instrumental in reconstructing the complete coherence properties of the signal. We confirm these predictions with an experimental characterization of partially coherent vortex beams, a case that cannot be treated with the standard tools. This seems to indicate that classical methods employed hitherto do not fully exploit the potential of the registered data.
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Introduction
Light is a major carrier of information about the universe around us, from the smallest to the largest scale. Threedimensional objects emit radiation that can be viewed as complex wavefronts shaped by diverse features, such as refractive index, density or temperature of the emitter. These wavefronts are specified by both their amplitude and phase; yet, as conventional optical detectors measure only (timeaveraged) intensity, information on the phase is discarded. This information turns out to be valuable for a variety of applications, such as optical testing^{1}, image recovery^{2}, displacement and position sensing^{3}, beam control and shaping^{4,5,6}, as well as active and adaptive control of optical systems^{7}, to mention but a few.
Actually, there exists a diversity of methods for wavefront reconstruction, each one with its own pros and cons^{8}. Such methods can be roughly classified into three categories: (a) interferometric methods based on the superposition of two beams with a welldefined relative phase; (b) methods based on the measurement of the wavefront slope or wavefront curvature and (c) methods based on the acquisition of images followed by the application of an iterative phaseretrieval algorithm^{9}. Notwithstanding the enormous progress that has already been made, practical and robust wavefront sensing still stands as an unresolved and demanding problem^{10}.
The timehonoured example of the Shack–Hartmann (S–H) wavefront sensor surely deserves a special mention^{11}: its wide dynamical range, high optical efficiency, white light capability and ability to use continuous or pulsed sources make of this setup an excellent solution in numerous applications.
The operation of the S–H sensor appeals to the intuition, giving the overall impression that the underlying theory is obvious^{12}. Indeed, it is often understood in an oversimplified geometricaloptics framework, which is much the same as assuming full coherence of the detected signal. By any means, this is not a complete picture: even in the simplest instance of beam propagation, the coherence features turn out to be indispensable^{13}.
It has been recently suggested^{14} that S–H sensing can be reformulated in a concise quantum notation. This is more than an academic curiosity, because it immediately calls for the application of the methods of quantum state reconstruction^{15}. Accordingly, one can verify right away that wavefront sensors may open the door to an assessment of the mutual coherence function, which conveys full information on the signal.
In this paper, we report the first experimental measurement of the coherence properties of an optical beam with a S–H sensor. To that end, we have prepared several coherent and incoherent superpositions of vortex beams. Our strategy can efficiently disclose that information, whereas the common S–H operation fails in the task.
Results
S–H wavefront sensing
The working principle of the S–H wavefront sensor can be elaborated with reference to Fig. 1. An incoming light field is divided into a number of subapertures by a microlens array that creates focal spots, registered in a camera, typically a chargecoupled device (CCD). The deviation of the spot pattern from a reference measurement allows the local direction angles to be derived, which in turn enables the reconstruction of the wavefront. In addition, the intensity distribution within the detector plane can be obtained by integration and interpolation between the foci.
Unfortunately, this naive picture breaks down when the light is partially coherent, because the very notion of a single wavefront becomes somewhat ambiguous: the signal has to be conceived as a statistical mixture of many wavefronts^{16}. To circumvent this difficulty, we observe that these sensors provide a simultaneous detection of position and angular spectrum (that is, directions) of the incident radiation. In other words, the S–H is a pertinent example of a simultaneous unsharp position and momentum measurement, a question of fundamental importance in quantum theory and about which much has been discussed^{17,18,19}.
Rephrasing the S–H operation in a quantum parlance will prove pivotal for the remaining discussion. Let ρ be the coherence matrix of the field to be analysed. Using an obvious Dirac notation, we can write G(x′,x′′)=‹x′ρx′′›=Tr(ρx′›‹x′′), where x› is a vector describing a pointlike source located at x and Tr is the matrix trace. Thereby, the mutual coherence function G(x′,x′′) appears as the position representation of the coherence matrix. As a special case, the intensity distribution across a transversal plane becomes I(x)=Tr(ρx›‹x). Moreover, a coherent beam of complex amplitude U(x), can be assigned to a ket U›, such that U(x)=‹xU›.
To simplify, we restrict the discussion to one dimension, denoted by x. If the setup is illuminated with a coherent signal U(x), and the ith microlens is Δx_{i} apart from the S–H axis, this microlens feels the field U(x−Δx_{i})=‹xexp(−iΔx_{i}P)U›, where P is the momentum operator. This field is truncated and filtered by the aperture (or pupil) function A(x)=‹xA› and Fourier transformed by the microlens before being detected by the CCD camera. All this can be accounted for in the form
where X is the position operator and we have assumed that the jth pixel is angularly displaced from the axis by Δp_{j}. The intensity measured at the jth pixel behind the ith lens is then governed by a Bornlike rule
with π_{ij}›=exp(iΔx_{i}P)exp(iΔp_{j}X)A›. As a result, each pixel performs a projection on the position and momentumdisplaced aperture state, as anticipated before.
Some special cases of those aperture states are particularly appealing. For pointlike microlenses, A(x)→δ(x) and π_{ij}›→x=Δx_{i}› (that is, a position eigenstate): they produce broad diffraction patterns and information about the transversal momentum is lost. Conversely, for very large microlenses, A(x)→1 and π_{ij}›→p=Δp_{j}› (that is, a momentum eigenstate): they provide a sharp momentum measurement with the corresponding loss of position sensitivity. A most interesting situation is when one uses a Gaussian approximation^{14}; now A(x)=exp(−x^{2}/2), which implies π_{ij}›→α_{ij}›, that is, a coherent state of amplitude α_{ij}=Δx_{i}+iΔp_{j}. This means that the measurement in this case projects the signal on a set of coherent states and hence yields a direct sampling of the Husimi distribution^{20} Q(α)=‹αρα›.
This quantum analogy provides quite a convenient description of the signal: different choices of CCD pixels and/or microlenses can be interpreted as particular phasespace operations^{21}.
S–H tomography
Unlike the Gaussian profiles discussed before, in a realistic setup the microlens apertures do not overlap. If we introduce the operators Π_{ij}=π_{ij}›‹π_{ij}, the measurements describing two pixels belonging to distinct apertures are compatible whenever [Π_{ij}, Π_{i′j}]=0, i≠i′, which renders the scheme informationally incomplete^{22}. Signal components passing through distinct apertures are never recombined and the mutual coherence of those components cannot be determined.
Put differently, the method cannot discriminate signals comprised of sharply localized nonoverlapping components. Nevertheless, these problematic modes do not set any practical restriction. As a matter of fact, spatially bounded modes (that is, with vanishing amplitude outside a finite area) have an unbounded Fourier spectrum and so, an unlimited range of transversal momenta. Such modes cannot thus be prepared with finite resources and they must be excluded from our considerations: for all practical purposes, the S–H performs an informationally complete measurement and any practically realizable signal can be characterized with the present approach.
To proceed further in this matter, we expand the signal as a finite superposition of a suitable spatially unbounded computational basis (depending on the actual experiment, one should use plane waves, Laguerre–Gauss beams and so on). If that basis is labelled by k› (k=1,…, d, with d being the dimension), the complex amplitudes are ‹xk›=ψ_{k}(x). Therefore, the coherence matrix ρ and the measurement operators Π_{ij} are given by d × d nonnegative matrices. A convenient representation of Π_{ij} can be obtained directly from equation (2), viz,
where ψ_{m,i}(x) is the complex amplitude at the CCD plane of the ith lens generated by the incident mth basis mode ψ_{m}.
This idea can be illustrated with the simple yet relevant example of square microlenses: A(x)=rect(x). We decompose the signal in a discrete set of plane waves ψ_{k}(x)=exp(−ip_{k}x), parameterized by the transverse momenta p_{k}. This is just the Fraunhofer diffraction on a slit, and the measurement matrix is
The smallest possible search space consists of two plane waves (which is equivalent to a singlequbit tomography). By considering different pixels j belonging to the same aperture i, linear combinations of only three out of the four Pauli matrices can be generated from equation (4). For example, a lens placed on the S–H axis (Δx_{i}=0) fails to generate σ_{y} and at least one more lens with a different Δx_{i} needs to be added to the setup to make the tomography complete.
This argument can be easily extended: the larger the search space, the more microlenses must be used. In this example, the maximum number of independent measurements generated by the S–H detection is (2M+1)d−3M, for M lenses. A ddimensional signal—a spatial qudit—can be characterized with about M~d/2 microlenses. This should be compared with the d quadratures required for the homodyne reconstruction of a photonic qudit^{23,24}.
Experiment
We have validated our method with vortex beams^{25,26}. Consider the oneparameter family of modes specified by the orbital angular momentum ℓ, V_{ℓ}=‹r, ϕV_{ℓ}›∝e^{iℓϕ}, where (r, ϕ) are cylindrical coordinates. In our experiment, the partially coherent signal
was created; that is, modes V_{−3} and V_{−6} are coherently superposed, while V_{3} is incoherently mixed. Figure 2 sketches the experimental layout used to generate equation (5). Imperfections of the setup and sensor noise make the actual state to differ from the true state. Calibration and signal intensity scans are presented in Fig. 3.
The coherence matrix of the true state is expanded in the sevendimensional (7D) space spanned by the modes V_{ℓ}, with ℓε{−9, −6, −3, 0, +3, +6, +9}. The resulting matrix elements are plotted in Fig. 4.
To reconstruct the state we use a maximum likelihood algorithm^{27,28}, whose results are summarized in Fig. 4. The main features of ρ_{true} are nicely displayed, which is also confirmed by the high fidelity of the reconstructed state F(ρ_{true}, ρ)=Tr=0.98. The offdiagonal elements detect the coherence between modes, whereas the diagonal ones give the amplitude ratios between them. The reconstruction errors are mainly due to the difference between the true and the actually generated state.
To our best knowledge, this is the first experimental measurement of the coherence properties with a wavefront sensor. The procedure outperforms the standard S–H operation, both in terms of dynamical range and resolution, even for fully coherent beams. For example, the highorder vortex beams with strongly helical wavefronts are very difficult to analyse with the standard wavefront sensors, while they pose no difficulty for our proposed approach.
The dynamical range and the resolution of the S–H tomography are delimited by the choice of the search space {k›} and can be quantified by the singular spectrum^{29} of the measurement matrix Π_{ij}. For the data in Fig. 4, the singular spectrum (which is the analogue of the modulation transfer function in wave optics) is shown in Fig. 5. Depending on the threshold, around 20 out of the total of 49 modes spanning the space of 7 × 7 coherence matrices can be discriminated. The modes outside this field of view are mainly those with significant intensity contributions out of the rectangular regions of the CCD sensor. Further improvements can be expected by exploiting the full CCD area and/or using a CCD camera with more resolution, at the expense of more computational resources for data postprocessing.
3D imaging
Once the feasibility of the S–H tomography has been proven, we illustrate its utility with an experimental demonstration of 3D imaging (or digital propagation) of partially coherent fields.
As it is well known^{16}, the knowledge of the transverse intensity distribution at an input plane is, in general, not sufficient for calculating the transverse profile at other output planes. Propagation requires the explicit form of the mutual coherence function G_{in} at the input to determine I_{out}:
Here x′ (x′′) and x are the coordinates parameterizing the input and output planes, respectively, and h(x, x′) the response function accounting for propagation.
The dependence of the farfield intensity on the beam coherence properties is evidenced in Fig. 6 for coherent, partially coherent and incoherent superpositions of vortex beams.
Once the coherence matrix is reconstructed, the forward/backward spatial propagation can be obtained using tools of diffraction theory and, consequently, the full 3D spatial intensity distribution can be computed. In particular, the intensity profile at the focal plane of an imaging system can be predicted from the S–H measurements. This has been experimentally confirmed, as sketched in Fig. 7. We prepared the partially coherent superposition V_{4}+V_{−4}›‹V_{4}+V_{−4}+kV_{0}›‹V_{0} and characterized it by the S–H tomography method. The reconstructed coherence function (upper left) was digitally propagated to the focal plane of a lens and the intensity distribution at this plane was calculated (upper right) and compared with the actual CCD scan in the same plane (lower right). Excellent agreement between the predicted and measured distributions was found.
We emphasize that the standard S–H operation fails in this kind of application^{30}. Indeed, we measured the intensity and wavefront of the target vortex superposition with a standard S–H sensor (middle left) and propagated the measured intensity to the focal plane using the transport of intensity equation^{31,32} (middle right). To quantify the result, we compute the normalized correlation coefficient [C(I_{a}, I_{b})=Σ_{i,j} I_{a}I_{b}/] of the measured intensity with the prediction: the result, C(I_{std}, I_{CCD})=0.47, confirms the inability of the standard S–H to cope with the coherence properties of the signal. This has to be compared with the result for the S–H tomography: C(I_{ρ}, I_{CCD})=0.89, which supports its advantages.
Discussion
We have demonstrated a nontrivial coherence measurement with a S–H sensor. This goes further than the standard analysis and constitutes a substantial leap ahead that might trigger potential applications in many areas. Such a breakthrough would not have been possible without reinterpreting the S–H operation as a simultaneous unsharp measurement of position and momentum. This immediately allows one to set a fundamental limit in the experimental accuracy^{33}.
Moreover, although the S–H has been the thread for our discussion, it is not difficult to extend the treatment to other wavefront sensors. For example, let us consider the recent results for temperature deviations of the cosmic microwave background^{34}. The anisotropy is mapped as spots on the sphere, representing the distribution of directions of the incoming radiation. To get access to the position distribution, the detector has to be moved and, in principle, such a scanning brings information about the position and direction simultaneously: the position of the measured signal before detection is delimited by the scanning aperture, whereas the direction the signal comes from is revealed by the detector placed at the focal plane. When the aperture moves, it scans the field repeatedly at different positions. This could be an excellent chance to investigate the coherence properties of the relict radiation. To our best knowledge, this question has not been posed yet. Quantum tomography is especially germane for this task.
Finally, let us stress that the classical estimation theory has been already applied to the raw S–H image data, offering an improved accuracy, but at a greater computational cost^{35,36}. However, the protocol used here can be implemented in a very easy, compact way, without any numerical burden.
Methods
Partiallycoherent beam preparation
Two independent vortex beams were created in the setup of Fig. 2 with two laser sources of nearly the same wavelength: a He–Ne (633 nm) and a diode laser (635 nm). The output beams were spatialy filtered by coupling them into singlemode fibres. The power ratio between the modes was controlled by changing the coupling efficiency. The resulting modes were transformed into vortex beams by different methods.
The state V_{−3}−V_{−6}› was realized using a digital hologram prepared with an amplitude spatial light modulator (OPTO SLM), with a resolution of 1,024 × 768 pixels. The hologram was then illuminated by a reference plane wave produced by placing the output of a singlemode fibre at the focal plane of a collimating lens. The diffraction spectrum involves several orders, of which only one contains useful information. To filter out the unwanted orders, a 4foptical processor, with a 0.3mm circular aperture stop placed at the rear focal plane of the second lens, was used. The resulting coherent vortex beam is then realized at the focal plane of the third lens.
The second beam V_{3}› was obtained through a planewave phase profile modulation by a special vortex phase mask (RPC Photonics). Finally, the field in equation (5) was prepared by mixing the two vortex modes in a beam splitter.
During the state preparation, special care was taken to reduce any deviation between the true and target states. This involved minimizing aberrations as well as imperfections of the spatial light modulator, resulting in distortions of the transmitted wavefront.
S–H detection
The S–H measurement involved a Flexible Optical array of 128 microlenses arranged in a hexagonal pattern. Each microlens has a focal length of 17.9 mm and a hexagonal aperture of 0.3 mm. The signal at the focal plane of the array is detected by a uEye CCD camera with a resolution of 640 × 480 pixels, each pixel being 9.9 μm × 9.9 μm in size. Because of microlens array imperfections, CCD–microlens misalignment and aberrations of the 4f processor (aberrations of the collimating optics are negligible), calibration of the detector must be carried out. The holographic part of the setup provided this calibration wave. S–H data from the calibration wave and the partially coherent beam are shown in Fig. 3. The beam axis position in the microlens array coordinates was adjusted with a Gaussian mode. The detection noise is mainly due to the background light, which is filtered out before reconstruction.
Reconstruction
The reconstruction was done in the 7D space spanned by the V_{ℓ} modes with ℓε{−9, −6, −3, 0, +3, +6, +9}. All in all, 49 real parameters had to be reconstructed. The data come from CCD areas belonging to seven microlenses around the beam axis; each one of them comprise 11 × 11 pixels, which means 847 data samples altogether. An iterative maximum likelihood algorithm^{27,28} was applied to estimate the true coherence matrix of the signal.
Dynamical range and resolution
The errors of the S–H tomography can be quantified by evaluating the covariances of the parameters of the reconstructed coherence matrix ρ. In the absence of systematic errors, the Cramér–Rao lower bound^{37,38} can be employed to that end. In practice, a simpler approach based on the singular spectrum analysis^{29} works pretty well.
Let us decompose the d × d coherence matrix ρ (d is just the dimension of the search space) and the measurement operators Π_{ij} in an orthonormal matrix basis Γ_{k} (k=1, …, d^{2}) [Tr(Γ_{k}Γ_{l})=δ_{kl}], namely
so that the Bornlike rule (2) can be recast as a system of linear equations
On using a single index α to label all possible microlens/CCD–pixel combinations α≡{i, j}, equation (8) can be concisely expressed in the matrix form
where I is the vector of measured data, r is the vector of coherencematrix parameters and P_{α k}= is the tomography matrix.
Obviously, for illconditioned measurements, the reconstruction errors will be larger and vice versa. By applying a singularvalue decomposition to the measurement matrix P=USV^{†}, equation (9) takes the diagonal form
where r′=V^{†}r and I′=U^{†}I are the normal modes of the problem and the corresponding transformed data, respectively. The singular values S_{kk} are the eigenvalues associated with the normal modes, so the relative sensitivity of the tomography to different normal modes is given by the relative sizes of the corresponding singular values. With the help of equations (9) and (10), the errors are readily propagated from the detection I to the reconstruction r.
Drawing an analogy between equation (10) and the filtering by a linear spatially invariant system, the singular spectrum S_{kk} and the sum of the singular values Σ_{k}S_{kk} are the discrete analogues of the modulation transfer function and the maximum of the point spread function, respectively. Hence we define the dynamical range (or field of view) of the S–H tomography as the set of normal modes with singular values exceeding a given threshold. The sum of the singular values then describes the overall performance of the S–H tomography setup. When some of the singular values are zero, the tomography is not informationally complete and the search space must be readjusted.
Farfield intensity
In the experiment on 3D imaging, the partially coherent vortex beam V_{4}+V_{−4}›‹V_{4}+V_{−4}+kV_{0}›‹V_{0} was generated, where k was a parameter governing the degree of spatial coherence. To this end, a coherent mixture V_{4}+V_{−4}›‹V_{4}+V_{−4} was realized by the digitalholography part of the setup, whereas the zeroorder vortex beam V_{0}› was prepared by removing the spiral phase mask. The output diameter of the beam was set to 4.9 mm.
The measurement was done in three steps. First, the S–H sensor (see Fig. 2) was replaced by a lens of 500 mm focal length and the farfield intensity was detected at its rear focal plane with a CCD camera (Olympus FView II, 1376 × 1032 pixels, 6.45 μm × 6.45 μm each). Second, the same vortex superposition was subject to the S–H tomography using the S–H sensor (Flexible Optical) and the reconstruction of the coherence matrix in the 7D subspace spaned by the vortices V_{ℓ} with ℓε{−6, −4, −2, 0, +2, +4, +6}. Once ρ is reconstructed, the farfield intensity was computed using equation (6), where the focusing is described by the Fraunhofer diffraction response function. The predicted intensity was found to be in an excellent agreement with the direct sampling by the Olympus CCD camera. Finally, the Flexible Optical S–H sensor was replaced by a HASO3 S–H detector. The intensity and wavefront of the prepared vortex beam was measured and the farfield intensity was computed by resorting to the transport of intensity performed by the HASO software. Resampling was done to match the resolution of the HASO output to the resolution of the Olympus CCD camera.
Additional information
How to cite this article: Stoklasa, B. et al. Wavefront sensing reveals optical coherence. Nat. Commun. 5:3275 doi: 10.1038/ncomms4275 (2014).
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Acknowledgements
This work was supported by the Technology Agency of the Czech Republic (Grant TE01020229), the Czech Ministry of Industry and Trade (Grant FRTI1/364), the IGA Projects of the Palacký University (Grants PRF_2012_005 and PRF_2013_019) and the Spanish MINECO (Grant FIS201126786).
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The experiment was conceived by J.R., Z.H. and L.L.S.S. and carried out by B.S. The numerical reconstruction was performed by B.S., whereas J.R. and L.M. provided theoretical analysis. Z.H. and J.R. supervised the project. The theoretical part of manuscript was written by L.L.S.S. and J.R. and the experimental part by B.S. and L.L.S.S., with input and discussions from all other authors.
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Stoklasa, B., Motka, L., Rehacek, J. et al. Wavefront sensing reveals optical coherence. Nat Commun 5, 3275 (2014). https://doi.org/10.1038/ncomms4275
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