Ferrofluidic Deformable Mirrors for Adaptive Optics

Ermanno F. Borra (left) and Denis Brousseau (right)



[This is an invited article. The authors have built the first deformable liquid mirror from a magnetic liquid or “ferrofluid”, which is set to find wide range of applications including correction of aberration in the images of telescopes and many other optical devices. -- 2Physics.com]



Authors: Denis Brousseau and Ermanno F. Borra
Affiliation: Université Laval, Département de physique, de génie physique et d’optique and Centre d’Optique, Photonique et Laser (COPL), Québec, Canada.

For roughly the last 25 years, adaptive optic (AO) systems were primarily used for astronomical applications. In the last 10 years, the range of scientific applications of AO has soared and now includes vision science, medical imaging and free space optical communications to name only a few. These new applications have stimulated research for low-cost, high-stroke deformable mirrors with a large number of actuators. Most actual deformable mirrors are expensive, costing about $1000 US per actuator. Current high-stroke deformable mirrors like the Imagine Optics 52-actuator MIRAO DM can produce large deformations (50 µm peak-to-valley tilt) but having a larger number of actuators would greatly increase its cost. Micro-Electro-Mechanical Systems (MEMS) deformable mirrors having large numbers of actuators (over 1000) and fabricated by a technique similar to surface micromachining have a great potential for low cost, but are currently limited to strokes of only a few microns.

It is well known that a liquid follows an equipotential surface to a high degree of precision. For example, the surface of a rotating pool of mercury takes a parabolic shape which can be used as a the primary mirror of a low cost telescope (http://wood.phy.ulaval.ca/what.html and http://www.astro.ubc.ca/lmt/lm/index.html). During the last few years, we have developed a new type of deformable mirror made of a ferrofluid whose surface is shaped by an array of magnetic coils. A ferrofluid is a liquid that contains a suspension of small ferromagnetic particles (Ø ~ 10 nm) within a water- or oil- based carrier liquid. In the presence of an external magnetic field, the magnetic particles react with the field and the fluid surface takes a shape that is determined by the equilibrium between the magnetic, gravitational and surface tension forces. The equation that describes the shape of the surface can be derived using equations found in [1]

where µ_r and ρ are the relative permeability and density of the ferrofluid respectively, n is a unit vector perpendicular to the liquid surface and B is the external magnetic field vector at the liquid-air interface. The external magnetic field B can be produced by an array of small current carrying coils located just under the surface of the liquid. Based on this principle, we built a 37-channel deformable mirror prototype, made of a ferrofluid whose surface is actuated by a hexagonal array of small current carrying coils.

In standard modal control of deformable mirrors, the mirror surface is shaped by the linear addition of the individual response function of the actuators. We see, from the preceding equation, that in the case of a ferrofluid deformable mirror (FDM), the liquid surface deformation is non-linear with respect to the external magnetic field, and also depends on the individual orientation of the external magnetic field components. Consequently, conventional modal control of a FDM is impossible; however we have successfully developed a custom algorithm that is able to compute the currents that must be assigned to the coils for a given mirror surface shape.

Using a commonly available ferrofluid we found that a maximum deformation of over a millimeter can be achieved before reaching instability [2]. In theory, much larger deformations (several mm) could be obtained with magnetic fields having components mostly parallel to the liquid surface and/or using ferrofluids having different physical properties.

Fig. 1. A custom ferrofluid developed in our labs is shown coated with a reflective layer of MeLLF and under the influence of a magnetic field from a permanent magnet located under the container. Picture clearly shows the very large deformation amplitudes that can be obtained.

Ferrofluids have a low reflectivity similar to motor oil and for many applications must be coated with a reflective layer. This can be done using reflective liquids based on interfacial films of silver particles known as Metal Liquid-Like Films or MeLLFs [3]. MeLLFs combine the properties of metals and liquids, can be deformed and are therefore well adapted to applications in the field of liquid optics. MeLLFs are not compatible with currently available commercial ferrofluids, which are hydrophobic, and for compatibility with MeLLFs we had to developed a custom hydrophilic ferrofluid (see Fig. 1) [4]. Our team is also considering the deposition of a chemical membrane on the ferrofluid.

Our prototype consists of 37 custom made coils (actuators) closely packed in a hexagonal array 35 mm in diameter (see Fig. 2). Each coil is made of about 200 loops of AWG28 magnet wire and has an external diameter of 5 mm. A small ferrite core is placed at the center of each coil to lower the current requirement of the device. An aluminum container (not seen) filled with a one-millimetre-thick layer of ferrofluid is placed on top.

Fig. 2. Our 37-channel prototype showing the hexagonal array of 37 coils of 5-mm diameter.

Total cost of the FDM was estimated at about $100 per actuator, including materials, electronics and shop time. Costs can certainly be reduced further with improved technology.

Using our algorithm, we have computed the required currents to produce standard Zernike polynomials (http://en.wikipedia.org/wiki/Zernike_polynomials). Those currents were then fed to the FDM and the resulting wavefronts were measured using a wavefront sensor (see Fig. 3).

Fig. 3. Experimental wavefronts representing Zernike polynomials reproduced by the FDM and measured using a Shack-Hartmann wavefront sensor. Each wavefront has a PV wavefront amplitude of about 5 μm.

Because of the vector-dependent response of our device, we suspected that trying to fit real wavefronts made from combining several Zernike polynomials would result in lower wavefront residual errors than by adding the residuals of each Zernike that made up the original wavefront. We performed experiments to test this assumption. We purposely introduced optical aberrations of 0.58 µm RMS wavefront amplitude and 2.42 µm PV wavefront amplitude in our wavefront measurement setup. PSFs before and after correction can be seen in Fig. 4. The achieved Strehl ratio of the corrected wavefront is 0.84 at a wavelength of 659.5 nm.

We also introduced much greater amplitude aberrations with PV and RMS wavefront amplitudes of 11.43 and 2.58 µm respectively. The RMS residual error of the corrected wavefront was measured to be 0.15 µm. We found that this error drops to 0.05 µm if we consider only the low order aberration terms. Correction for high spatial frequency Zernike polynomials would improve if the FDM had a greater number of actuators.

Fig. 4. Experimental result showing the PSF (log scale) of an aberrated wavefront (left) corrected by using our deformable mirror (right). Strehl ratio of corrected wavefront is 0.84 (659.5 nm).

Although we got promising results, some drawbacks remain. We need to bias the surface of the liquid to allow for a push-pull effect as the amplitude varies as the square of the current applied to a given actuator (deformations can only be positive). This reduces the available stroke of the mirror and also adds a surface residual error.

A novel way to control those liquid mirrors has recently been introduced by Iqbal and Amara, and solves most of these drawbacks [5]. The technique consists of adding a constant and uniform magnetic field whose orientation is along the direction perpendicular to the surface of the liquid. The amplitude of this constant magnetic field is about 10 times greater than what is produced by the coils (~ 2.5 gauss). The magnetic field of the actuators acts as a small perturbation of the uniform field and this linearizes the response of the liquid (as shown in Fig. 5). This also has the effect of amplifying the stroke produced by the coils, reducing the required currents, so that ferrite cores in the actuators are no longer necessary, and making negative deformations possible.

Fig. 5. Measured amplitudes of the deformations produced by a single actuator in the presence of an external uniform magnetic field, as a function of current in the coil. The red and blue curves correspond to external magnetic fields of 25 and 30 gauss respectively. Negative deformations can be produced by inverting the current flow. The actuator used in the experiment has no ferrite core.

Until recently, we thought that those liquid mirrors were limited to a time response of only a few tens of hertz, because when driven at frequencies higher than about 20 Hz, we saw a rapid loss in amplitude response of the liquid and a phase lag of over 90 degrees appeared between the driving signal and the resulting liquid deformation, quite similar to the response of a low-pass RLC filter.

We demonstrated that the amplitude loss can be overcome by overdriving the coils with a very short and high amplitude current pulse launched at the beginning of each driving signal. By using this technique, a desired surface deformation is reached faster and the remaining signal stabilize the liquid shape.

We also demonstrated that the phase lag can be countered by increasing the viscosity of the ferrofluid. The critical frequency (a 90 degrees phase lag) was improved from 20 to 450 Hz by increasing the viscosity of the ferrofluid from 6 cP to 450 cP (viscosity of water is 1 cP and SAE 50 motor viscosity is about 500 cP). Since a single square wave signal sent to the liquid corresponds to two corrections (rise and fall), this actually implies a frequency response of 900 Hz. But increasing the viscosity also increases the time required for the liquid to stabilize. However, a solution to this problem is to use overdriving pulses that give an initial velocity to the liquid as discussed in the preceding paragraph.

To conclude, we have demonstrated a liquid deformable mirror prototype that can produce standard aberration terms, and we successfully corrected a 11 µm PV amplitude aberrated wavefront, yielding a residual RMS wavefront error of 0.05 µm. A new technique linearizes the response of these new deformable mirrors and allows the use of regular control algorithms. This will simplify our goal to demonstrate closed-loop operation of these new mirrors.

We have also shown the counterintuitive result that using a liquid having a sufficiently high viscosity improves their frequency response up to 900 Hz. By using both overdriving pulses and a higher viscosity ferrofluid, utilizing these mirrors in closed-loop at a running frequency of hundreds of hertz, appears to be possible. This will enable these mirrors to be used in many more applications than we previously thought. Ongoing tests on the chemical deposition of thin chemical membranes on ferrofluids could also improve the response of FDMs. We are now building a new prototype having 91 actuators of 2-mm diameter, thus reducing the footprint and allowing a higher density of actuators.

References
[1] "Interaction of a magnetic liquid with a conductor containing current and a permanent magnet"
V. V. Kiryushin and A. V. Nazarenko, Fluid Dynamics, 23, 306–311 (1988). Abstract.
[2] R. E. Rosensweig, "Ferrohydrodynamics". (Dover, 1997).
[3] "Nanoengineered astronomical optics",
E. F. Borra, A. M. Ritcey, R. Bergamasco, P. Laird, J. Gingras, M. Dallaire, L. Da Silva and H. Yockell-Lelievre, Astron. Astrophys. 419, 777-782 (2004). Abstract.
[4] "Ethylene Glycol Based Ferrofluid for the Fabrication of Magnetically Deformable Liquid Mirrors"
J. -P. Déry, E. F. Borra, and A. M. Ritcey, Chem. Mater. 20 (2008). Abstract.
[5] "Modeling of a Magnetic-Fluid Deformable Mirror for Retinal Imaging Adaptive Optics Systems"
A. Iqbal and F. B. Amara, International Journal of Optomechatronics 1, 180-208 (2007). Abstract.

Multipolar Post Minkowskian and Post Newtonian Toolkits for Gravitational Radiation

Bala R. Iyer

[This is an invited article reviewing two decades of work by the author and his international collaborators. -- 2Physics.com ]

Author: Bala R. Iyer
Affiliation: Raman Research Institute, Bangalore, India

My work on gravitational waves (GW) began during a sabbatical I spent with Thibault Damour at DARC (CNRS-Observatoire de Paris) and Institut des Hautes Etudes Scientifiques (IHES) in France during 1989-90. I was exposed to the powerful Multipolar Post Minkowskian (MPM) formalism that Luc Blanchet and Thibault Damour had set up. Though the MPM formalism then seemed more elaborate than necessary, it is a good example of the advantage that a complete and mathematically rigorous treatment of a problem can eventually bring in the future for more demanding applications.

The wave generation formalism relates the gravitational waves observed by a detector in the far-zone of the source to the stress-energy tensor describing the source. Successful wave-generation formalisms [1] combine post-Minkowskian (PM) methods [expansions in G ], post-Newtonian (PN) methods [expansions in 1/c ], multipole (M) expansions [expansions into irreducible representations of the rotation group], and perturbations around curved backgrounds. There are two independent aspects addressing two different problems. The general method (MPM expansion) applicable to extended or fluid sources with compact support, based on the mixed PM and multipole expansion matched to some PN (slowly moving, weakly gravitating, small-retardation) source. And, the particular application to describe inspiralling compact binaries(ICB) by use of point particle models.

Starting from the general solution to the linearized Einstein's equations in the form of a multipolar expansion (valid in the external region), a PM iteration is performed and each multipolar piece is independently treated at each PM order. For the external field, the general method is not a priori limited to PN sources. However, closed form expressions for the multipole moments can be only obtained for PN sources because the exterior field may be related to the inner field only if there exists an overlapping region where both the MPM and PN expansions are valid and can be matched together. After matching, the multipole moments have a non-compact support since they depend on the gravitational field stress-energy that is distributed everywhere up to spatial infinity. To account for this correctly, the definition of the multipole moments involves a crucial finite part operation based on analytic continuation (equivalent to a Hadamard partie finie of the integrals at infinity).

The physical post-Newtonian source, for any PN order, is characterized by six symmetric and trace free (STF) time-varying moments, functionals of a formal PN expansion of the stress-energy pseudo-tensor of the material and gravitational fields. Starting from the six STF source moments one can define a different set of two canonical source moments, such that the two sets of moments are physically equivalent (i.e. lead to the same metric modulo coordinate transformations). The use of the canonical source moments simplifies the calculation of the external non-linearities and their existence shows that any radiating isolated source is characterized by two and only two sets of time-varying multipole moments.

The MPM formalism is valid all over the weak field region outside the source including the wave zone (up to future null infinity). The far zone expansion at Minkowskian future null infinity contains logarithms in the distance which are artefacts of the harmonic coordinates. One can define, order by order in the PM expansion, some radiative coordinates such that the log-terms are eliminated. One recovers the standard (Bondi-type) radiative form of the metric from which the radiative moments seen by the detector, can be extracted in the usual way.

Nonlinearities in the external field are determined by a post-Minkowskian algorithm and one obtains the radiative multipole moments as some non-linear functional of the canonical moments and thus of the actual source moments. The source moments mix with each other as the waves propagate from the source to the detector and thus the relation between radiative and source moments include many non-linear multipole interactions including hereditary (history dependent) effects like tails, memory and tails-of-tails. The radiative moments are also very convenient for the computation of the spin-weighted spherical harmonic decomposition of the gravitational waveform employed to compare analytical PN results to numerical relativity simulations for binary black holes.

The application of the above results to compact binary systems involves a new input. For compact objects the effects of finite size and quadrupole distortion induced by tidal interactions are of order 5PN. Hence, neutron stars and black holes can be modelled as point particles represented by Dirac δ-functions. The general formalism, however, applies only to smooth matter distributions and cannot be directly used for point particles since they lead to divergent integrals at the location of the particles when the energy momentum tensor of a point particle is substituted into the source moments. The calculation needs to be supplemented by a method for self-field regularisation i.e. a prescription for removing this infinite part of the integrals.

Hadamard regularisation, based on Hadamard's notion of partie finie, was employed in all earlier works and led to consistent results in different approaches up to 2.5PN order. Thus, it was surprising to discover that Hadamard regularisation at 3PN order was incomplete as signalled by the presence of four undetermined constants in the final 3PN generation results. The 3PN generation was technically more involved, and only after almost a decade of struggle and by the use of the gauge invariant dimensional regularisation, was the problem finally resolved and completed [2]. For non-spinning ICB on quasi-circular orbits, the equation of motion and the gravitational wave polarisations [3] are now known to 3PN accuracy. The radiation field determining GW phasing is known to 3.5PN order beyond the leading Einstein quadrupole formula [2]. The 3PN results for non-spinning ICB on quasi-elliptical orbits [4] and 2.5PN results for spinning binaries [5] have recently been completed. In the test particle limit results are known to order 5.5PN [1].

The PN results for ICB are the basis for the construction of all templates employed by the LIGO and VIRGO detectors [6]. Resummation methods like Pade approximants and Effective One Body models [7] going beyond the adiabatic inspiral phase to include plunge, merger and quasi-normal-mode ringing improve the convergence and extend the domain of validity of the PN approximants. In the context of the recent exciting numerical relativity simulations [8] of GW from plunge, merger and ringdown of binary black holes the best analytical PN results for inspiral (3.5PN in phase, 3PN in amplitude) are crucial for calibration and interpretation.

References:
[1] L. Blanchet, Living Rev. Relativity, 9, 4 (2006); http://www.livingreviews.org/lrr-2006-4; M. Sasaki and H. Tagoshi, Living Rev. Relativity, 6, 6 (2003) [arXiv:gr-qc/0306120]

[2] L. Blanchet, Class.Quant.Grav. 15, 1971 (1998) [arXiv:gr-qc/9801101]; ibid, 15, 113 (1998) [arXiv:gr-qc/9710038]; ibid, 15, 89 (1998) [arXiv:gr-qc/9710037]; L. Blanchet, B.R. Iyer and B. Joguet, Phys. Rev. D, 65, 064005 (2002) [gr-qc/0105098]; L. Blanchet, G. Faye, B.R. Iyer and B. Joguet, Phys. Rev. D, 65, 061501(R) (2002) [gr-qc/0105099]; L. Blanchet and B.R. Iyer, Phys. Rev. D, 71, 024004 (2005) [gr-qc/0409094]; T. Damour, P. Jaranowski and G. Schäfer, Phys. Lett. B, 513, 147 (2001) [arXiv:gr-qc/0105038]; L. Blanchet, T. Damour and G. Esposito-Farese, Phys. Rev. D 69, 124007 (2004) [arXiv:gr-qc/0311052]; L. Blanchet, T. Damour, G. Esposito-Farese and B.R. Iyer, Phys. Rev. Lett., 93, 091101 (2004) [gr-qc/0406012]; L. Blanchet, T. Damour, G. Esposito-Farese and B.R. Iyer, Phys. Rev. D, 71, 124004 (2005) [gr-qc/0503044]; T. Futamase and Y. Itoh, Living Rev. Relativity, 10, 2 (2007).

[3] K.G. Arun, L. Blanchet, B.R. Iyer and M.S.S. Qusailah, Class. Quant. Gravity, 21, 3771 (2004) [gr-qc/0404085]; L. E. Kidder, Phys. Rev. D, 77, 044016, (2008) [arXiv:0710.0614]; L. Blanchet, G. Faye, B. R. Iyer and S. Sinha, Class. Quant. Grav., 25, 165003 (2008) [gr-qc/0802.1249]; M. Favata (2009) [arXiv:0812.0069].

[4] K G Arun, L. Blanchet, B. R Iyer and M. S. S. Qusailah, Phys. Rev. D 77, 064034 (2008) [arXiv:0711.0250]; ibid, Phys. Rev. D, 77, 064035 (2008) [gr-qc/0711.0302].

[5] G. Faye, L. Blanchet and A. Buonanno, Phys. Rev. D 74, 104033 (2006) [arXiv:gr-qc/0605139]; L. Blanchet, A. Buonanno and G. Faye, Phys. Rev. D, 74, 104034 (2006) [arXiv:gr-qc/0605140]; K.G. Arun, A. Buonanno, G. Faye and E. Ochsner, [arXiv:0810.5336]; T. Damour, P. Jaranowski and G. Schafer, Phys. Rev. D 78, 024009 (2008) [arXiv:0803.0915].

[6] T. Damour, B.R. Iyer and B.S. Sathyaprakash, Phys. Rev D, 63, 044023 (2001)[gr-qc/0010009]; ibid, 66, 027502 (2002) [gr-qc/0207021].

[7] T. Damour, [arXiv:0802.4047]; T. Damour, B.R. Iyer and B.S. Sathyaprakash,Phys. Rev. D, 57, 885 (1998) [gr-qc/9708034]; A. Buonanno and T. Damour,Phys. Rev. D, 59 (1999) 084006 [arXiv:gr-qc/9811091]; T.Damour, B. R. Iyer and A. Nagar, Phys. Rev. D, 79, 064004 (2009) [gr-qc/0811.2069]; T. Damour and A. Nagar, [arXiv:0902.0136]; A. Buonanno et al, [arXiv:0902.0790].

[8] F. Pretorius [arXiv:0710.1338]; F. Pretorius, Phys. Rev. Lett. 95, 121101(2005)[arXiv:gr-qc/0507014]; M. Campanelli, C. O. Lousto, P. Marronett and Y. Zlochower, Phys. Rev. Lett. 96, 111101 (2006) [arXiv:gr-qc/0511048]; J. G. Baker et al,Phys. Rev. Lett. 96, 111102 (2006)[arXiv:gr-qc/0511103]; M. Boyle et al, Phys. Rev. D, 76, 124038 (2007) [arXiv:0710.0158].

Long-Distance Teleportation between Two Atoms

Figure 1: Teleportation Team (rear from left: Christopher Monroe, Dzmitry Matsukevich; front from left: Peter Maunz, Steven Olmschenk, David Hayes) (Photo credit: Jonathan Mizrahi)


Quantum teleportation is the faithful transfer of quantum states between systems. A team from the Joint Quantum Institute (JQI) at the University of Maryland (UMD) and the University of Michigan has succeeded in teleporting a quantum state directly from one atom to another over a distance of one meter [Figure 1]. In the Jan. 23 issue of the journal Science [1], the scientists report that, by using their protocol, atom-to-atom teleported information can be recovered with perfect accuracy about 90% of the time.


>>Link to `Trapped Ion Quantum Information Group' led by Christopher Monroe, University of Maryland
>>Link to past 2Physics article on the work of this group

Teleportation works because of a remarkable quantum phenomenon, called “entanglement.” Once two objects are put in an entangled state, their properties are inextricably entwined. Although those properties are inherently unknowable until a measurement is made, measuring either one of the objects instantly determines the characteristics of the other, no matter how far apart they are.

The JQI team set out to entangle the quantum states of two individual ytterbium ions so that information embodied in the condition of one could be teleported to the other. Each ion was isolated in a separate high-vacuum trap. [Figure 2] The researchers identified two readily discernible ground (lowest energy) states of the ions that would serve as the alternative “bit” values of an atomic quantum bit, or qubit.

Figure 2: Experimental setup. Single photons from each of two ions in separate traps interact at a beamsplitter. If both detectors record a photon simultaneously, the ions are entangled. At that point, Ion A is measured, revealing exactly what operation must be performed on Ion B in order to teleport Ion A’s information. (Image Credit: Curt Suplee, JQI)

At the start of the experimental process, each ion (designated A and B) is initialized in a given ground state. Then ion A is irradiated with a specially tailored microwave burst from one of its cage electrodes, placing the ion in some desired superposition of the two qubit states – in effect writing into memory the information to be teleported.

Immediately thereafter, both ions are excited by a picosecond laser pulse. The pulse duration is so short that each ion emits only a single photon as it sheds the energy gained from the laser pulse and falls back to one or the other of the two qubit ground states. Depending on which one it falls into, each ion emits a photon whose color is perfectly correlated with the two atomic qubit states. It is this entanglement between each atomic qubit and its photon that will eventually allow the atoms themselves to become entangled.

The emitted photons are captured by lenses, routed to separate strands of fiber-optic cable, and carried to a 50-50 beamsplitter where it is equally probable for either photon to pass straight through the splitter or to be reflected. On either side of the beamsplitter are detectors that can record the arrival of a single photon. Because of the quantum interference of the two photons [2], a simultaneous detection at both output ports of the beamsplitter occurs only if the photons are in a particular quantum state. Since state of the photons was initially correlated with the state of the atomic qubits this measurement leaves atomic qubits in an entangled state [3]. The simultaneous detection of photons at the detectors does not occur often, so the laser stimulus and photon emission process has to be repeated many thousands of times per second. But when a photon appears in each detector, it is an unambiguous signature of entanglement between the ions.

Figure 3: Quantum state of the atom is teleported by 1 meter. (Image credit: N.R. Fuller, National Science Foundation)

When an entangled condition is identified, the scientists immediately take a measurement of ion A. The act of measurement forces it out of superposition and into a definite condition: one of the two qubit states. But because ion A’s state is irreversibly tied to ion B’s, the measurement of A also forces B into a complementary state. Depending on which state ion A is found in, the researchers now know precisely what kind of microwave pulse to apply to ion B in order to recover the exact information that had originally been stored in ion A. Doing so results in the accurate teleportation of the information.

This method combines the unique advantages of both photons and atoms. Photons are ideal for transferring information fast over long distances, whereas atoms offer a valuable medium for long-lived quantum memory. The combination represents an attractive architecture for a ‘quantum repeater,’ that would allow quantum information to be communicated over much larger distances than can be done with just photons.

The work reported in Science was supported by the Intelligence Advanced Research Project Activity program under U.S. Army Research Office contract, the National Science Foundation (NSF) Physics at the Information Frontier Program, and the NSF Physics Frontier Center at JQI. This report is written by Curt Suplee.

References
[1] "Quantum Teleportation between Distant Matter Qubits," S. Olmschenk, D. N. Matsukevich, P. Maunz, D. Hayes, L.-M. Duan, and C. Monroe, Science 323, 486 (2009). Abstract.
[2] “Measurement of subpicosecond time intervals between two photons by interference,” C. K. Hong, Z. Y. Ou, L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). Abstract.
[3] “Robust Long-Distance Entanglement and a Loophole-Free Bell Test with Ions and Photons,”
C. Simon, W. T. M. Irvine, Phys. Rev. Lett. 91, 110405 (2003). Abstract.