Craig Stuart Carlson

Ryunosuke Matsumoto

Miryu Koji Fushino

Miryu Shinzato

Nobuki Kudo

Michiel Postema

Transient ink nucleation: the proof is in the pudding

Introduction

Although ultrasonic technology is most commonly used for medical imaging, 1 it is also regularly applied to manipulate, 2 deform, 3 and disrupt microscopic particles. 4 Ultrasound can disperse nanoparticles, a process which has been associated with inertial cavitation and acoustic streaming. 5 High-speed photography is commonly used to study acoustic cavitation dynamics. 4,6 Hydrophobic particles can be regarded as particles with a thin gaseous layer surrounding the solid cores. 7 They have been observed to act as cavitation nuclei that lose their gaseous layer during sonication. 8 Black tattoo ink comprises hydrophobic carbon black nanoparticles. In this study, we hypothesised that black tattoo ink exhibits similar transient inertial cavitation behaviour under sonication.

Materials and Methods

The first set of experiments was carried out in a Perspex container of 580x235x65 (mm) 3 internal dimensions, which was filled with degassed water.

A C60 5-2 MHz curvilinear probe of a SonoSite ® M-Turbo ® sonography device (FUJIFILM SonoSite, Inc., WA , USA) was clamped in the length direction of the container. The sonography device was operating in brightness mode throughout the experiments.

A 160x120x50 (mm) 3 tissue-mimicking phantom with three empty cylindrical wells, manufactured according to the recipe, 9 was positioned in the container such that its 160-mm-wide front face touched the probe surface. The cylindrical wells, with 28-mm diameter and 42-mm height, were located 20-mm from the front face (cf. Fig. 1).

Zuper Black pigment dispersion (INTENZE Products, Inc., NJ, USA) was used for black ink. 20 mL each of black ink and SABAX Pour Saline 0.9% (Adcock Ingram Critical Care (Pty) Ltd, Johannesburg, South Africa) was inserted into the centre and right wells respectively.

The ultrasound was set to a penetration depth of 6.6 cm, a mechanical index of 0.6, and a thermal index of 0.1.

The second set of experiments was carried out in the 0.2-mL observation chamber of a high-speed photography system. 10 A quantity of 0.5 μL Zuper Black pigment dispersion was pipetted into a FALCON® 15-mL High-Clarity Polypropylene Conical Tube (Corning Science México S.A. de C.V., Tamaulipas, Mexico), after which 5 mL of 049-16787 Distilled Water (FUJIFILM Wako Pure Chemical Corporation, Osaka, Japan) was added. The tube was manually shaken for 5 minutes. From the tube, 0.2 mL was pipetted into the observation chamber, which was placed under an Eclipse Ti inverted microscope (Nikon Corporation, Tokyo, Japan) with a CFI S Plan Fluor ELWD 20XC objective lens of 20x magnification and 0.45 numerical aperture. Attached to the microscope was an HPV-X2 high-speed camera (Shimadzu, Kyoto, Japan), operating at 10 million frames per second. 11 During camera recording, the materials were subjected to ultrasound pulses.

The pulses consisted of 20 cycles, each at a centre transmitting frequency of 1.0 MHz and a peak-negative pressure of 400 kPa, from a laboratory-assembled single-element transducer. 10,11 The time between subsequent pulses was at least 1 minute. The signal fed into the transducer was generated by an AFG320 arbitrary function † Correspondence to: Craig.Carlson@wits.ac.za generator (Sony-Tektronix, Tokyo, Japan) and amplified by a UOD-WB-1000 wide-band power amplifier (TOKIN Corporation, Miyagi, Japan).

A total number of 5800 images and 87 high-speed video sequences (each sequence consisted of 256 frames) were recorded during seven imaging sessions.

Results and Discussion

Brightness-mode scans of the phantom wells are shown in Fig. 2. Transient scattering was observed from individual spots inside the ink-filled well only (Fig. 2a, left). After 9 minutes, no more scattering was observed (Fig 2b ,left). This scattering can be attributed to the transient hydrophobic behaviour of carbon black, similar to the transient behaviour in an earlier study. 8 No scattering was observed in the saline-filled well. To observe this transient hydrophobicity at higher frame resolution, high-speed photography experiments were performed.

Fig. 3 shows three frames of a representative example from the high-speed video recordings of black ink. The frames show multiple ink particles before, during, and after 1.0-MHz sonication. During sonication, inertial cavities were seen to nucleate on the ink particles, after which cavity clusters were generated from interaction with other nucleations. During subsequent ultrasound pulses, the ink particles that had been involved in this nucleation cycle remained unaffected (not shown). This supports the observation from the brightness-mode scans.

Conclusions

Using brightness-mode sonography and high-speed photography, black tattoo ink was observed to transiently nucleate. Subsequent ultrasound bursts did not reactivate the ink. It is concluded that ultrasound changes the hydrophobicity of carbon black ink particles.

Fig. 1

 1 Fig. 1 Top view of the tissue-mimicking phantom with three cylindrical wells. The probe is visible on the South face. A £1 coin is included for scale. Dimensions have been stated in mm.

Fig. 2

 2 Fig. 2 Phantom wells filled with black ink (left) and saline (right), directly after filling (a) and 9 minutes later (b). The green dot corresponds to the left-hand side of the probe. Each tick corresponds to 1.1 cm.

Fig. 3

 3 Fig. 3 Zuper Black ink particles before (a), during (b), and after (c) sonication. Each frame corresponds to a 36x36 (μm) 2 area. Time stamps, in nanoseconds, are in the lower right corners.

Acknowledgement

This work was supported by JSPS KAKENHI Grant Numbers 17H00864, 20H04542.


Y Poujet

J Salvi

F I Baida

D Van Labeke

A Perentes

C Santschi

P Hoffmann

Near-field optical images of subwavelength annular aperture arrays exhibiting an extraordinary transmission

Keywords: annular aperture arrays, extraordinary transmission, guided mode, RSTOM, SNOM

In this paper, we present near-field optical images of nanostructures exhibiting an extraordinary transmission. These structures consist of annular aperture arrays engraved in a metallic film: they are quite promising structures for nanophotonics because of their high transmission directly linked to a guided mode mediated by each annular aperture. We first briefly explain our fabrication process (focused ion beam milling), then we expose the experimental setup of the near-field optical microscope working both in reflection and transmission modes. For the reflection mode, the 'coffee-bean' structure of the electromagnetic field predicted by the theory has been quite well reproduced. For the transmission mode, we present preliminary experimental results concerning the influence of the wavelength and the polarization of the incident beam on the obtained near-field images.

Introduction

It is well known now that light can substantially pass through subwavelength nano-apertures in a thick metallic film [START_REF] Ebbesen | Extraordinary optical transmission through sub-wavelength hole arrays[END_REF][START_REF] Martin-Moreno | Theory of extraordinary optical transmission through subwavelength hole arrays[END_REF], unlike the hasty predictions that can be deduced from the 'classical' theory [START_REF] Bethe | Theory of diffraction by small holes[END_REF][START_REF] Bouwkamp | Diffraction theory[END_REF]. A transmission peak in the visible range can be obtained and it reaches 10% [START_REF] Ebbesen | Extraordinary optical transmission through sub-wavelength hole arrays[END_REF]: this transmission is due to a surface plasmon resonance mediated by the nanostructures [START_REF] Popov | Theory of light transmission through subwavelength periodic hole arrays[END_REF]. Many applications made the most of this phenomenon in different domains: fluorescence [START_REF] Liu | Fluorescence enhancement from an array of subwavelength metal apertures[END_REF], nanophotonics [START_REF] Shinada | Surface plasmon resonance on microaperture vertical-cavity surface-emitting laser with metal grating[END_REF] or chemical sensors [START_REF] Brolo | Surface plasmon sensor based on the enhanced light transmission through arrays of nanoholes in gold films[END_REF]. In order to increase the light transmission in the visible range, Baida and Van Labeke proposed in 2002 a new geometry that consists of annular apertures arranged in a bi-periodic array [START_REF] Baida | Light transmission by subwavelength annular aperture arrays in metallic films[END_REF]. This structure was named AAA for annular aperture array. A similar idea has been already developed by Roberts and Mac Phedran in the case of perfect conductors [START_REF] Roberts | Bandpass grids with annular apertures[END_REF]. For specific dimensions, the AAA can theoretically lead to a transmission peak of around 90% in the visible range of the spectrum [START_REF] Baida | Three-dimensional structures for enhanced transmission through a metallic film: annular aperture arrays[END_REF]. This high value of the transmission is directly connected to the existence of a guided mode inside the coaxial apertures [START_REF] Baida | Origin of the super-enhanced light transmission through a 2-d metallic annular aperture array: a study of photonic bands[END_REF][START_REF] Baida | Subwavelength metallic coaxial waveguides in the optical range: role of the plasmonic modes[END_REF]. The aim of this paper is to study these kind of nanostructures in the near-field domain. After having explained our fabrication process and our calculation method, we present experimental results obtained with a reflection scanning tunneling optical microscope (RSTOM). As already mentioned in a previous work [START_REF] Poujet | Super-transmission of light through subwavelength annular aperture arrays in metallic films: spectral analysis and near-field optical images in the visible range[END_REF], the same 'coffee-bean' light distribution is observed with other geometries of the AAA. Finally, original experimental near-field images obtained with a scanning nearfield optical microscope (SNOM) working in transmission mode are presented for different polarizations and wavelengths.

Fabrication process

The fabrication process is the same as described elsewhere [START_REF] Elsner | Nanometer and high aspect ratio patterning by electron beam lithography using a simple DUV negative tone resist[END_REF]. The substrate consists of a 120-μm thick fused silica cover slide, on which a 5-nm adhesion layer (titanium or chromium) and a metallic film (silver or gold) are evaporated. A low evaporation rate prevents from grain formation onto the metallic layer. The surface roughness, measured by atomic force microscopy, is then about 2 nm. A NOVA 600 dual-beam system from FEI Company with installed focused ion and electron beam was used for the milling experiments. Compared to focused electron beam (FEB) lithography, focused ion beam (FIB) milling is an adequate technique for rapid fabrication of prototype optical nanostructures [START_REF] Perentes | Fabrication of arrays of subwavelength nano-apertures in an optically thick gold layer on glass slides for optical studies[END_REF]. Nanoscale structures are obtainable with perfectly tuned FIBs, and parameters such as dwell time, overlap, astigmatism and focus are determining for the final structure quality. The fine tuning of the beam focus and the stigmator was done by milling test structures and painstaking readjustments have been carried out prior to the realization of the arrays. The FIB operated at 30 keV and 10 pA (beam diameter around 12 nm) for all structures. An overlap of 50% in the x-and y-directions has been introduced to ensure a homogeneous milling. A short dwell time (200 ns) and a large number of passes have been chosen to minimize the redeposition effects. FIB drift due to charging on the underlaying silica insulating substrate was minimized by irradiating the surface using the electron beam. The adequate ion dose to reach the designated depth and to investigate the size and the shape was determined by cross-cutting of the test structures after covering them with a platinum-containing material using FEB-induced deposition. This layer protects the structure of interest during FIB milling and offers a better contrast for the subsequent scanning electronic microscope imaging.

The quality of the final result is checked by imaging the structure thanks to a scanning electron microscope (see Fig. 1). A conventional optical microscope (REICHERT-JUNG) has been used to observe three different AAAs made in the same silver layer. Figure 2 shows the three far field images obtained in transmission mode (magnifying =× 50). The whole sample was illuminated by a thermal white light source. As already shown elsewhere [START_REF] Barnes | Surface plasmon subwavelength optics[END_REF][START_REF] Kanamori | Fabrication of transmission color filters using silicon subwavelength gratings on quartz substrates[END_REF], one can notice the spectral filtering possibilities of our structure.

The far field response of such a structure depends both on the geometrical parameters of one aperture and on the period of the whole array. More quantitative experiments are in progress with these samples made in silver but we already showed and compared theoretical and experimental transmissions in the case of AAAs in gold [START_REF] Salvi | Annular aperture arrays: study in the visible range of the electromagnetic spectrum[END_REF].

Let us note here that, in the following, the presented experimental data are often compared with theoretical ones. These latest data are found through a finite difference time domain (FDTD) method that directly solves Maxwell's equations. It is a 3D home-made code where space and time are both discretized. In order to describe the fine features of the structure, a non-uniform meshing is implemented. This leads to a decrease in the total number of nodes (spatial steps: = 5 nm for the metallic film and = 25 nm for the rest of the calculation window). This mesh variation is carried out gradually to avoid parasitical reflections. In addition, the Berenger's perfectly matched layers are incorporated into this code to cancel the reflection on the external edges of the computational window. More details on the numerical simulations can be found in references [START_REF] Baida | Light transmission by subwavelength annular aperture arrays in metallic films[END_REF][START_REF] Poujet | Super-transmission of light through subwavelength annular aperture arrays in metallic films: spectral analysis and near-field optical images in the visible range[END_REF]. All the theoretical results presented in this paper are made with this code.

Near-field optics: reflection mode

The AAAs under study in this section have been made by FIB milling in a 100-nm thick gold layer deposited on a glass substrate with a thin chromium adhesion layer (5 nm). The period of the structure is 500 nm, the inner diameter of the coaxes is 200 nm and the outer diameter is 280 nm. It's theoretical spectral response is plotted in Fig. 3.

The near-field experiments are achieved with a commercial SNOM (SMENA model). The experimental setup is described in Fig. 4. The sample is illuminated at λ = 632.8 nm thanks to a dielectric tip. The light diffracted by the sample is detected by the same tip and a coupler is used to separate the incident and the backward signals. The near-field signal is then recorded by a photomultiplier (PM) and analyzed by an appropriate software.

The shear-force and the near-field optical images are shown in Fig. 5: the optical image is blurred by fringes. The origin of these fringes is not completely understood but they are very common in this kind of experiments. The interferences may be due to the interaction between the incident and the reflected light, the coupling between the tip and the sample and/or the tip shape or size. This image has been filtered in order to eliminate the fringes. This filtering simply consists in a 2D fast fourier transform (FFT), which allows us to cancel the spatial frequency corresponding to the fringes by means of a binary filtering function. A 'coffee-bean'-like distribution of the electric field then becomes apparent, as shown in Fig. 6(a). The zone marked with a white square is magnified and displayed in Fig. 6(b). We also marked with white circles the position of some annular cavities deduced from the simultaneously recorded shear-force image. This result demonstrates, if necessary, the connection between the annular cavity geometry and the two-lobe light distribution. A similar result has been already obtained with another AAA having different dimensions [START_REF] Poujet | Super-transmission of light through subwavelength annular aperture arrays in metallic films: spectral analysis and near-field optical images in the visible range[END_REF].

This two-lobe structure is typical of the TE 11 -like mode [START_REF] Baida | Origin of the super-enhanced light transmission through a 2-d metallic annular aperture array: a study of photonic bands[END_REF][START_REF] Baida | Subwavelength metallic coaxial waveguides in the optical range: role of the plasmonic modes[END_REF]. The theoretical calculation (3D-FDTD) presents the same distribution of the electromagnetic field (see Fig. 6c). This image corresponds to the square modulus of the transversal components of the electric field recorded 30 nm above the structure. The polarization of the incident field is along the two maxima of light. There is a good agreement between the experimental and the theoretical images if we consider that the dielectric tip is thin enough to act as a dipole that only extracts the tangential components of the electric field even if the normal one is larger [START_REF] Van Labeke | Probes for scanning tunneling optical microscopy: Theoretical comparison[END_REF][START_REF] Grosjean | Polarization filtering induced by imaging systems: effect on image structure[END_REF]. Note that the illuminating wavelength is out of the high transmission region of the spectrum as shown in Fig. 3 (red arrow). So, the cavity mode cannot propagate inside the annular aperture and it is completely reflected: it just settles at the entry of the cavity and its intensity exponentially decays when penetrating inside the aperture.

Near-field optics: transmission mode

The near-field optical images in transmission mode have been carried out with an AAA made in a 125-nm thick silver layer. The period is 500 nm, the inner ring and the outer ring diameters are 100, nm and 200 nm, respectively. As previously described, the AAAs have been made by following the same FIB milling process. To our knowledge, this is the first study concerning the polarization effect and the wavelength influence for such a structure. As shown in Fig. 7, a powerful white light source generated by a pulsed laser at λ = 532 nm is focused into a dispersion shifted fiber [START_REF] Mussot | Generation of a broadband single-mode supercontinuum in a conventional dispersion shifted fiber by use a nanosecond microchip laser[END_REF]. A microscope objective collimates the emerging beam, which illuminates the sample. The polarization direction is controlled with a Glan-Taylor polarizer and an interferential filter allows us to choose the incident wavelength. The images are recorded with the same SMENA commercial SNOM using a dielectric tip.

Shear-force and optical images of the same sample but for two different polarization directions are shown in Fig. 8. The theoretical spectral response of this sample is plotted in Fig. 9. The shear-force images that give the topography of the sample are identical: they correspond exactly to the same area of the sample. The optical images are recorded at λ = 622 nm. The range of intensities is the same for both optical images so that we can quantitatively compare the obtained intensities. The two polarizations used in the experiments are first along the y-axis (top image in Fig. 8) and second at 45 • from the y-axis (bottom image in Fig. 8). The 'coffeebean' light distribution is not resolved in these images. Nevertheless the two optical images are very similar, which is in good agreement with theoretical findings of reference [START_REF] Baida | Light transmission by subwavelength annular aperture arrays in metallic films[END_REF]. This is mainly due to the fact that the cylindrical symmetry of the annular apertures is insensitive to the polarization direction.

Near-field optical responses of one AAA matrix at different wavelengths (576 nm, 608 nm and 622 nm, see Fig. 9) have been also recorded in Fig. 10. The polarization is kept along the y-axis for all images. The intensity of each image is normalized to the source spectrum because the incident power can differ from one wavelength to another (Fig. 11). The grey level scale is the same for each optical image of Fig. 10. First, there is some light outside the AAA and the expected 'coffee-bean' distribution of the electric field is not resolved: as it is mentioned above, it may be due to the tip shape and/or to surface plasmons excited besides/on the matrix. Moreover, the transmitted zero order can also smear our SNOM images. Using a metallized tip could increase the resolution of our SNOM, preventing in this way the detection of the transmitted far field: the 'coffee-bean' structure of the near field could be then resolved. Second, we observe that the average value of the detected light intensity decreases when the wavelength increases. This behaviour is still unclear because recent theoretical calculations (not presented here) show a contradictory evolution for the intensity. More experimental and theoretical works are still in progress to completely understand these results.

Conclusion

In this paper, we qualitatively show the ability of AAAs to filter the light. More quantitative spectra are in progress to validate the high transmission in the visible range (80%). However, the main part of this paper is dedicated, to our knowledge, to the first experimental near-field studies of such nanostructures. For the reflection mode, the experiment is in good accordance with the theory: we experimentally highlight the existence of the TE 11 -like mode by retrieving the typical two-lobe structure of light intensity above the coaxes. For the images obtained in transmission mode, we demonstrate that the polarization has no significant influence on the nearfield response. Nonetheless, the 'coffee-bean' structure of the electromagnetic field has not been resolved. In addition, the variation of the detected intensity according to the wavelength does not fulfil the theoretical predictions. These works are still in progress in order to clarify all the underlying physical phenomena.

Fig. 1 .

 1 Fig. 1. Example of an AAA (diameters: 200/100 nm and period: 350 nm) made by FIB milling: (a) top view of two periods, (b) view with a 52 • tilt and (c) view of the whole matrix.

Fig. 2 .

 2 Fig. 2. Far field image of three different AAAs in real colours obtained by conventional microscope working in transmission mode. The AAAs are engraved on a 100-nm thick silver layer. The inner diameter (d i ), the outer diameter (d o ) and the period p are d i = 100 nm, d o = 200 nm, p = 350 nm for matrix (1) (7 × 7 μm 2 ); d i = 230 nm, d o = 330 nm, p = 500 nm for matrix (2) (10 × 10 μm 2 ) and d i = 100 nm, d o = 200 nm, p = 500 nm for matrix (3) (10 × 10 μm 2 ).

Fig. 3 .Fig. 4 .

 34 Fig. 3. Theoretical spectral response of the AAA studied in near-field optics (reflection mode): d i = 200 nm, d o = 280 nm, p = 500 nm.

Fig. 5 .

 5 Fig.5. Shear-force and optical images (14 × 14 μm 2 ) of an AAA in a 100-nm thick gold film. The inner diameter is 200 nm, the outer diameter 280 nm and the period 500 nm.

Fig. 6 .

 6 Fig. 6. (a) Filtered optical image of the near-field optical image in Fig. 5; (b) closeup of the white square in picture (a) (6 × 6 μm 2 ) and (c) theoretical response obtained by FDTD calculation (polarization in the upper right corner).

Fig. 7 .

 7 Fig. 7. Scheme of the experimental SNOM setup.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Topographic and near-field optical images (15 × 15μm 2 ) with the same wavelength λ = 622 nm at different polarizations.

Fig. 10 .

 10 Fig. 10. Topographic and near-field optical images (15 × 15μm 2 ) with the same polarization at different incident wavelengths λ; from left to right λ = 576 nm, 608 nm and 622 nm.

Fig. 11 .

 11 Fig. 11. Source spectrum. The arrows mark the wavelengths used in the experiment.

Acknowledgments

The authors thank the "Laboratoire Européen Associé Franco-Suisse" and the Network of Excellence in Micro-Optics for financial support.


Miklós Molnár

Dinh Danh Le

Jordi Perelló

Josep Solé-Pareta

Conor Mcardle

Multicast Routing from a Set of Data Centers in Elastic Optical Networks

Keywords: Data Centers, Elastic Optical Networks, Multicast Session Planning, Physical Layer Impairments, Degree-Constrained Steiner Problem, Light-Hierarchy, ILP

This paper introduces the Multi-Server Multicast (MSM) approach for Content Delivery Networks (CDNs) delivering services offered by a set of Data Centers (DCs). All DCs offer the same services. The network is an Elastic Optical Network (EON) and for a good performance, routing is performed directly at the optical layer. Optical switches have heterogeneous capacities, that is, light splitting is not available in all switches. Moreover, frequency slot conversion is not possible in any of them. We account for the degradation that optical signals suffer both in the splitting nodes, as well as across fiber links to compute their transmission reach. The optimal solution of the MSM is a set of light-hierarchies. This multicast route contains a light trail from one of the DCs to each of the destinations with respect to the optical constraints while optimizing an objective (e.g., minimizing a function). Finding such a structure is often an NP-hard problem. The light-hierarchies initiated from different DCs permit delivering the multicast session to all end-users with a better utilization of the optical resources, while also reducing multicast session latencies, as contents can be delivered from such DCs closer to end-users. We propose an Integer Linear Programming (ILP) formulation to optimally decide on which light-hierarchies should be setup. Simulation results illustrate the benefits of MSM in two reference backbone networks.

Introduction and Motivation

Cloud computing and networking are nowadays very popular and real possibilities for enterprises and end-users. In the last few years, the Data Center (DC) industry has played a key role in the development of the IT sector. Indeed, the total capacity of DCs is growing by 5 percent per year. In 2016, for example, approximatively 1710 important DC locations in North America and 1400 in Europe were counted. Providers (e.g., CenturyLink, Cogent Communications, Equinix, NTT, Tiscali, TELEHOUSE, AT&T, etc.) offer services in several continents (for more details, cf. http://www.datacentermap.com/).

Content Delivery Networks (CDNs) are distributed, server set based systems. They can be composed of dedicated replica servers located at DCs across the network, providing replication of some contents. Such a geographic distribution of servers eventually offering the same content facilitates and accelerates service delivery to end-users, as contents can be delivered to them from their closest replica server, thus minimizing network resources and experienced latencies. The latest CISCO traffic forecasts [START_REF]Cisco visual networking index: Forecast and methodology[END_REF] highlight the key role that CDNs will most probably play in the short-term future, carrying up to 71% of all global Internet traffic by 2021. Therefore, to cope with such a huge amount of traffic, it seems natural that CDNs will have to operate over ultra-high capacity network infrastructures. Service providers provide a large number of profiles, including data storage, server hosting, integrated services, voice and data communications, video and content distribution, different cloud-based solutions, etc. Each service may have a special requirement, and require a more or less rigorous QoS level or security.

To connect DCs to end-users for an effective service delivery, routing may involve unicast but also anycast and multicast communications. Multicast communication technique is important to save network resources when several endnodes are involved in the communication. Typically, big data-related broadcasting, stock information, results of environment measurements, data collected from IoT segments (traffic, meteorology, etc.), news, collaborative applications, multimedia applications, video, TV broadcasting, and games are examples of multicast-based applications. Data transfers between DCs can also be based on multicast. The required bandwidth can be very different for these applications, typically 1-10 Mbps for simple voice, 10-100 Mbps for video and TV streaming, and some Gbps for data transfers between DCs. When end-users are scattered across the network, a single route to reach them all can require a substantial amount of resources and also a high latency, a crucial figure toward successful 5G services. In this paper, we focus on the case where several equivalent servers are available in the network. This enables to organize multicast sessions based on a set of servers: Multiple Server Multicast (MSM) in contrast to the Single Server Multicast (SSM), leveraging the content replication in CDNs.

Current multicasting in CDNs is based on IP. However, IP multicasting is not scalable and the multicast implemented in the application layer introduces delays and additional traffic load. Over the last decade, Network Virtualization and Software Defined Networking (SDN) have emerged as new solutions to fa-cilitate distributed systems like CDNs. Abstractions in virtual networks permit network services to be developed and deployed independently from underlying physical infrastructure. SDN is a solution where data forwarding and control planes are separated. Generally, routing decisions are taken from a central control entity and the data follows the computed routes. SDN and virtualization are closely related to each other. Virtualization permits the coexistence of networks on a shared physical infrastructure; it can be used to perform SDNs. SDNs find viable applications in not only network virtualization but also in the optical networking domain [START_REF] Routray | Software defined networking for optical networks[END_REF]. We propose a potential SDN-based multicasting directly at optical layer.

One of the most promising solutions for implementing backbone networks is all optical networking using optical switches. Specifically, Wavelength Division Multiplexing (WDM) is the legacy solution for circuit switching in all-optical networks. Circuits are trivial elements for virtual networks and can be computed by an SDN controller. In all optical networks the used technology involves additional constraints (uniqueness of carriers in fibers, wavelength continuity, physical impairments, degree constraints in the traversed switches, etc.).

With the advent of Elastic Optical Networks (EONs), the flexibility of optical channels permits managing network resources in a more efficient manner when supporting connections at multiple bit-rates [START_REF] Christodoulopoulos | Elastic bandwidth allocation in flexible OFDM-based optical networks[END_REF]. To solve the (multicast) routing at the optical layer, the decision should entail the route computation jointly with the assignment of the frequency slots and modulation format used in the transmission, which is known as the Routing, Modulation level and Spectrum assignment (RMSA) problem. In the network, we assume heterogeneous switches and sparse splitting capability. Consequently, we focus on adequate optimal solutions corresponding to light-hierarchies (cf. Section 4.1). In the Multi-Server Multicast (MSM) approach, a set of light-hierarchies allows from multiple DCs to serve all end-users. In this way, better utilization of the resources can be achieved, while also reducing latencies, as contents can be delivered from DCs closer to end-users.

In summary, we are interested in the design of DC-based content distribution. We consider heterogeneous EONs. The network interconnects DC servers to destinations in an end-to-end fashion, taking IPTV CDNs as reference scenario. There are several DCs, which can distribute simultaneously the same content. The destinations are EON switches connecting eventual sub networks and users. The distribution uses optical multicasting for different sets of destinations. The optical multicast routes can be computed by an SDN controller or a Path Computation Element (PCE) and should satisfy the physical constraints. The objective is the optimal organization of this content delivery.

The paper presents the optimal route (which is a set of light-hierarchies and not always a set of light-trees) and proposes an Integer Linear Programming (ILP) formulation to find this set. That is, the ILP decides from which DCs routes will be started, as well as their structure to reach a specific sub-set of end-users. Moreover, the effect of the physical layer impairments on the eventual transmission reach of the optical signals is accounted in the ILP. For this, we propose the extension of a model widely accepted for unicast routing to multicasting. Once the ILP is presented, we use it to disclose some illustrative results of the benefits achievable by MSM versus SSM.

Our paper is organized as follows. In the following section the most related works are presented. Section 3 gives the definition of the problem. Section 4 briefly presents the proposed graph model and some relevant related concepts. The design of the exact ILP formulation is in Section 5 and the computation results can be found in Section 6. Conclusions and perspectives remarks close our study.

Related Work

Today, several cloud based CDNs are emerging [START_REF] Wang | An Overview of Cloud Based Content Delivery Networks: Research Dimensions and State-of-the-Art[END_REF]. Often audio and video streams, stock data, big meteorologic data bases are delivered [START_REF] Noghani | Streaming Multicast Video over Software-Defined Networks[END_REF]. With the arrival of DCs some networking paradigms need to change: 1) particular networks may be needed inside DCs to link several thousand machines in a reliable and economic manner [6] [7] 2) access to (DC based) services must be ensured with variable and often high bandwidth and must be reliable (cf. a survey in [START_REF] Wang | A Survey on Data Center Networking for Cloud Computing[END_REF]) 3) sub-sets of DCs (e.g., DCs of a company, CDNs, etc.) must be inter-connected with high capacity connections for data exchanges [START_REF] Zhang | Spectrum-efficient anycast in elastic optical interdatacenter networks[END_REF]. To ensure more efficient network management, virtualization of network functions and SDN have been deeply changing the network control and data planes. The counterpart of this evolution is that more complex problems should be solved (e.g., in routing, load balancing, security, etc). SDN enables more complex and precise network management. Often in SDN, the routing decisions are taken by a central entity (for example PCE) that configures the switches for data forwarding [START_REF] López | Elastic Optical Networks: Architectures, Technologies, and Control[END_REF]. Openflow is one of the most widely used protocol for the communication between the controller and routers. Openflow supports multicast by default [START_REF] Braun | Software-defined networking using openflow: Protocols, applications and architectural design choices[END_REF]. The structure of the network is essential for the services. In [START_REF] Wang | An Overview of Cloud Based Content Delivery Networks: Research Dimensions and State-of-the-Art[END_REF] an overview of DCNs for cloud computing is presented. The wide relation between SDN and optical networks is analyzed in a large survey [START_REF] Farhady | Software-Defined Networking: A survey[END_REF].

In an invited paper of IEEE Proceedings ( [START_REF] Develder | Optical Networks for Grid and Cloud Computing Applications[END_REF]), authors suppose that optical network based cloud computing becomes the key element of future services. Moreover as it is indicated, multicast is a good candidate for efficient communication between different DCs and users for the distribution of experimental or collected data and related results for highly expensive computing and further analysis and interpretation. Another trivial use case for multicast is IPTV where the TV is broadcased for several distributors, sites, users. The Internet-Draft "Multicast in the Data Center Overview" proposes IPTV service providers to use multicast to deliver content from the DC to end users. Redundant servers are sending multicast streams into the network and the network is forwarding the data across diverse paths [START_REF] Mcbride | Multicast in the Data Center Overview[END_REF].

As presented in [START_REF] Zhu | Dynamic service provisioning in elastic optical networks with hybrid single-/multi-path routing[END_REF], EONs provide significant benefits toward spectrumefficient service provisioning (the multi-path routing is also analyzed, which can be efficient but is out of scope of our recent study). In [START_REF] Zhang | Spectrum-efficient anycast in elastic optical interdatacenter networks[END_REF] the elastic optical inter-DC network problem is analyzed. In this study, the requests are sporadic, highly dynamic anycast type requests where any DC can satisfy any request.

Both the offline and online routing and spectrum assignment (RSA) problems are formulated as ILPs and heuristics are also proposed.

In a CDN over EON scenario, when a specific content is simultaneously requested by a set of end-users, the standard anycast solution establishes one light-path from any one of the multiple DCs for each end-user (e.g., as in [START_REF] Goscien | Distance-adaptive transmission in cloud-ready elastic optical networks[END_REF]). Instead, using multicast, the requested content can be provided from any one of the multiple DCs to all end-users together. This can be performed by setting up a light-tree [START_REF] Gong | Efficient Resource Allocation for All-Optical Multicasting Over Spectrum-Sliced Elastic Optical Networks[END_REF] or a light-hierarchy [START_REF] Zhou | Light-hierarchy: the optimal structure for multicast routing in WDM mesh networks[END_REF], being the latter the optimal solution under splitting constraints. We refer to this approach as Single Server Multicast (SSM).

Different exact formulations of the related problems exist in the literature. An ILP for manycast (unicast, multicast and anycast) routing using a set of DCs in EON has been formulated in [START_REF] Fallahpour | Energy-Efficient Manycast Routing and Spectrum Assignment in Elastic Optical Networks for Cloud Computing Environment[END_REF]. This RSA problem tries to minimize the energy usage. An additional adaptive modulation format selection can be found in [START_REF] Luo | Manycast routing, modulation level and spectrum assignment over elastic optical networks[END_REF]. Optical multicasting in EON has been analyzed in several works (e.g., [START_REF] Cai | Multicast Routing and Distance-Adaptive Spectrum Allocation in Elastic Optical Networks with Shared Protection[END_REF]). Co-joint optimization of unicast and multicast is discussed in [START_REF] Walkowiak | Joint optimization of multicast and unicast flows in elastic optical networks[END_REF]. Physical impairments should be taken into account in the route computation. For unicast communications it can be modeled by means of the well-known GNmodel [START_REF] Poggiolini | The GN-model of fiber non-linear propagation and its applications[END_REF], extending it for multicast ones. In [START_REF] Walkowiak | Optimization of Multicast Traffic in Elastic Optical Networks with Distance-Adaptive Transmission[END_REF] the optimization of multicast routing in EON (where all network nodes are equipped with multicasting) is discussed. The requests are satisfied by light-trees. The spectrum requirements are determined using the distance-adaptive transmission (DAT) principle 1 . The authors formulate two ILP models, one of them with a previous computation of candidate trees and also propose a heuristic method. A DAT based impairmentand splitting-aware cloud-ready single source multicast provisioning is presented in [START_REF] Zhu | Impairmentand Splitting-Aware Cloud-Ready Multicast Provisioning in Elastic Optical Networks[END_REF]. It is important to notice that all these works are based on light-trees and the optical switches are assumed to be multicast capable and server replications are not taken into account.

The Multi-Server Multicast Routing Problem in EON

We assume an EON represented as a directed topology graph G t = (V t , A t ), where V t is the set of nodes (i.e. Bandwidth Variable Optical Cross Connects, BV-OXCs) and A t the set of arcs corresponding to unidirectional optical fiber links. Two unidirectional fiber links in opposite directions are assumed between each connected node pair. The spectrum of each fiber is discretized as an ordered set

F = {f 1 , f 2 , • • • , f |F | } of
Frequency Slots (FS). We assume heterogeneous switches and consequently sparse splitting in the EON, namely, only a subset M C ⊆ V t have multicast capability in the network (e.g., nodes having B&S ROADM and Non-broadcast&S architectures). In contrast, the set M I = V t \ M C represents the multicast incapable nodes (e.g., R&S nodes with basic FlexGrid WSSs) [START_REF] Simmons | Optical Network Design and Planning[END_REF]. We denote as R t the set of DCs where CDN replica servers are located. Each r ∈ R t is co-located with one EON node v ∈ V t . Without loss of generality, we consider that all DCs replicate the CDN content to be delivered to end-users.

Bandwidth Variable Transponders (BV-TXPs) equipped at EON nodes can operate at a set of possible bit-rates B and employ a set of modulation formats M at any bit-rate b ∈ B. Therefore, the number of adjacent FSs required for a request using a given modulation format can be computed (cf. Section 5.1). Moreover, as we do not consider spectrum conversion in the EON, the same selected contiguous FS block will have to continuously allocated from source to destination nodes.

To decide RMSA in the EON, a controller is implemented (a PCE or an SDN controller). Although a request will generally be necessary to ask for a multicast session to the controller. In case of acceptance, the traffic will be from the DCs to the end-users. Multicast requests are served upon arrival, taking the current state of the network resources into account and trying to meet an optimization criteria. The state of the network is given by the set of available (and occupied) frequency slots in each fiber link.

To serve a multicast session request, different strategies could be employed:

• Single Server-based Multicast (SSM): The DC from which the multicast contain is delivered is chosen in an anycast fashion, taking the specific optimization criteria into account. In this case a single light-hierarchy is setup in the network.

• Multi-server-based Multicast (MSM): In this case, multiple light-hierarchies (initiated from different DCs if needed) can be setup to deliver the video content to all end-users of the request. The DCs from which the lighthierarchies are initiated can be: 1) chosen in an anycast fashion, taking a specific optimization criteria / clusterization of destinations into account 2) given by a unique joint optimization. In this paper, we propose this second MSM solution; the classification of destinations is useless applying our solution.

In both aforementioned SSM and MSM strategies, it is mandatory to deliver the requested content with the requested quality to all end-users of the request. However, we could also think about still accepting (slightly) degraded requests if they can be served given the current state of network resources, instead of directly blocking them. Such degradation could happen in terms of the number of end-users served and/or the quality of the delivered content. The optimization of the eventually limited, degraded services is a very promising subject but it is out of scope of our actual study.

Let us define our problem more precisely.

Problem 1. Multi-Server Multicast problem (MSM)

The MSM problem aims at finding a set of optical multicast routes rooted at some DC corresponding to an objective, while satisfying optical constraints, such as the availability of the FSs at fiber links, the spectrum contiguity and continuity along the communication imposed by the EON, the limited transmission reach of the optical signals (depending on the employed modulation format) and, finally, the splitting capacity constraint in the switches.

Different objectives can be followed to compute the optimal routes. Without being exhaustive, the most frequently used objectives for routing in EONs follow.

1. Minimization of the used resources (modeled by an additive cost) 2. Minimization of the highest index of FSs used permitting more possibilities for ulterior routing demands 3. Optimization of some QoS parameters (e.g., latency, etc) 4. Minimization of the energy, ... The optimization can be mono-objective or several objectives can be combined and a multi-objective optimization can serve the routing.

In this paper, we consider the objective of minimizing the highest index of the slots used and also the number of FSs used. The solution to this problem is a set H = {H i , i = 1, . . . , |H |} of directed light-hierarchies H i covering the destination set. They can use different modulation formats and FS blocks. The solution must contain at least one feasible light-trail from a DC node to each destination. Since the solution minimize the used resources, finding the optimum is NP-hard even if there is only one server [START_REF] Zhou | Light-hierarchy: the optimal structure for multicast routing in WDM mesh networks[END_REF]. The mathematical formulation is in the next section.

Proposed Models and Solutions

At first, we demonstrate that the optimal multicast route is not mandatorily a (set of) light-tree(s) but a (set of) light-hierarchy(ies). Then, we propose an adequate graph model for the computation.

Hierarchies

Without loss of generality, let us suppose there is a single source (DC) for multicasting. Usually, partial spanning trees (i.e. connected sub-graphs without cycles) are considered as the minimum cost solutions for multicast routing. However, they have some limitations. Trees can not always satisfy the constraints on node degrees representing the splitting capacity of nodes and, in some cases, there is no tree-based solution. As demonstrated in [START_REF] Molnar | Hierarchies to Solve Constrained Connected Spanning Problems[END_REF] a more flexible solution is a connected, graph related structure called hierarchy. A hierarchy is defined as follows.

Definition 1 (Hierarchy). Given an arbitrary graph G = (V, E) and a tree T = (W, F ), let x : W → V be a homomorphism which associates a node v ∈ V to each node w ∈ W . 2 The application (T, x, G) defines a hierarchy in G. Notice that a hierarchy is not a sub-graph, but it generates a sub-graph of G. This sub-graph is the image of the hierarchy and may contain cycles. Even if there is no spanning tree to cover the nodes to span and satisfying a set of degree constraints, such a hierarchy can exist. In some other cases, spanning hierarchies give better solutions than trees.

For optical multicast data forwarding, the route is rooted at the source and we consider rooted directed hierarchies (the tree T is rooted and directed). The solution will be detailed by the computational model in the next section. Here we propose the extension of light-hierarchies to EONs presented in [START_REF] Zhou | Light-hierarchy: the optimal structure for multicast routing in WDM mesh networks[END_REF] for WDM techniques.

Definition 2 (Light-hierarchy in EON). A light-hierarchy is a directed rooted hierarchy, which use the same frequency slots from the source to the destinations.

Figure 1 illustrates a light-hierarchy. The graph G is the topology graph of the network and the tree T = (W, F ) represents the routing information (the succession of nodes in the route). The splitting constraints are respected by the light-hierarchy (remember MI nodes haven't splitters).

Theorem 1. The optimal multiple server multicast route (e.g., with minimum cost, or with minimum highest index of FSs used, etc.) in EONs under the optical constraints is a set of light-hierarchies covering the destination set. Each of these directed hierarchies is rooted at one of the DC nodes. They can use different frequency slot blocks.

Proof. The optimal (e.g., minimum cost,etc.) route is a set of directed connected multicast routes. Trivially, each route of the set is directed from one of the DC nodes to a sub set of destinations. We should prove that this set of multicast routes corresponds to a set of directed hierarchies. That is, a directed graph T i can be associated with the component (route) r i of the solution. This graph T i should be connected (it should contain a directed trail from the source to each destination). If the associated connected graph T i is not a tree (the route is not a hierarchy), there is at most one useless cycle in T i and the route r i can not be optimal.

To avoid conflicts using a same arc in G by several light-hierarchies, they can use different, non-overlapping frequency slot blocks.

In the following, the complexity of the optimization is shortly discussed. Let us suppose the routing can use a given FS block and a modulation associated with. The directed topology graph of available links is G s = (V, A s ) and the source s and the destination set D are given. The modulation implicates a maximal transmission range T R for unicast communication. To simplify, suppose that T R limits the longest light-trail in the solution. Since hierarchies include trees, we are searching for light-hierarchies and the following question can be formulated.

Is there a light-hierarchy directed from s to D in G s respecting the maximal transmission range and the degree constraints imposed by the set of MI nodes? This question can be seen as the existence problem of a radius and degree constrained directed partial spanning hierarchy.

Lemma 1. It is NP-complete to decide if G s contains a partial spanning hierarchy respecting the degree and the radius constraints.

Proof. The problem is in NP, since it is easy to verify, if a hierarchy respects the constraints or not.

To prove the hardness, let us focus on the degree constraints. In a particular case, the set MC is empty and the degree bound of nodes is equal to two. That is, the solution is a walk (the tree T in the hierarchy definition is a path). The question is: is there a directed walk from the source and covering the destinations with limited total length. This problem corresponds to the directed traveling salesperson walk problem, which is NP-complete [START_REF] Lam | Traveling salesman path problems[END_REF].

The interest of the light hierarchies for multicast routing can be illustrated as follows. To simplify, let us suppose that a request needs one FS from the source a to the destinations e and g in the network indicated in Fig. 2/a). Let us suppose that two FSs (F S1 and F S2) are partially available for this routing as it is indicated in Fig. 2/b and Fig. 2/b) respectively. Let us suppose that index of F S1 < index of F S2. Using F S1 we obtain a light-hierarchy and using F S2 we have a light-tree as the figure indicate them with dotted lines. If the objective is the minimization of the maximal index of the FSs, the lighthierarchy using F S1 is better. It is not true for the costs, the light-hierarchy uses 7 arcs whereas the light-tree uses only 4 arcs.

Graph Model

To compute the optimal solution corresponding to a set of light-hierarchies, we propose some simple considerations to construct an auxiliary layered graph.

1. Since, a source can inject the light simultaneously into several fibers, DC nodes in R can be considered as MC-nodes. To take into account all combinations of modulation formats and FSs that can participate in the routing, a layered auxiliary graph can be used similarly to the proposition in [START_REF] Liu | Design integrated RSA for multicast in elastic optical networks with a layered approach[END_REF]. In our model, this layered graph is generalized to support a set of modulation formats. For each modulation format e, the capacity ns e in number of FS blocks needed for the request can be determined. Then a set of sub-graphs is generated whose cardi-

nality is |F | -ns e + 1. A sub-graph G k = (V k , A k ) is generated for each possible contiguous FS subset SB k = {b k + i, i = 0, . . . ns e -1}. For each node v ∈ V t corresponds a node v k ∈ V k . An arc a k ∈ A k is present in G k iff the subset SB k is available in the corresponding arc a ∈ A t :
f a k+i = 1, i = 0, . . . ns e -1. 3. In any valid optimal solution of our problem, there is no return to a DC, therefore incoming arcs of the data centers can be deleted. 4. An artificial source node can be added to the graph such that this source is connected to the copies of DC nodes in R by zero cost arcs having all frequencies available in the homogeneous frequency spectrum F in the network. 5. If a sub-graph G k does not contain any directed path from the source to a destination, the sub-graph is deleted. 6. Since a destination should be spanned in at least one light-hierarchy but this light-hierarchy and its associated slot block can be arbitrary, the nodes d k in the different sub-graphs corresponding to the same destination d can be connected to a virtual destination d in the auxiliary graph. In this manner, the virtual destinations should be covered. To avoid the multiple spans of a virtual destination, the incoming arcs of these virtual destinations must have the same positive costs.

The auxiliary graph is illustrated in Figure 3 and can be the target of additional practical reductions (cf. Section 5).

In this graph a directed light-hierarchy H * is computed from the artificial source to all virtual destinations. By construction, sub-hierarchies of the source pass obligatory by some DC-nodes. After deleting the artificial source and the virtual destinations, a disconnected set H * of sub-hierarchies is remained. Each sub-hierarchy is a light-hierarchy rooted at one of the DC-node. Let us suppose that H * is optimal: e.g., with the minimum highest index of FSs. Since the artificial arcs have unlimited capacities, the remained set H * is also optimal. An important property of our auxiliary graph is that each sub-graph G k corresponds to a modulation format and a slot block. Consequently, an arc of the auxiliary graph can be used in the solution at most once and its usage implicates the usage of the associated modulation format and slot block. This property simplifies the mathematical formulation in Section 5.

Transmission reach estimation for multicast

In order to account for the physical degradations experienced by optical signals along the routes, we propose to rely on the well-known Gaussian Noise (GN)-Model proposed in [START_REF] Poggiolini | The GN-model of fiber non-linear propagation and its applications[END_REF], whose reliability for nonlinear fiber propagation has been proven in most cases of interest. Nonetheless, the GN-Model does not account for the additional noise that would be introduced when amplifying optical signals after splitting them at network nodes, which degrades their eventual transmission reach. Therefore, we assume in this work that no optical signal splitting occurs at MI nodes, whereas they are split at MC ones only when strictly needed to construct the light hierarchy (or light tree).

Under these considerations, we firstly obtain the worst-case transmission reach values using the GN-Model, what we refer to as the unicast transmission reach. For this, we assume the "Link 1" scenario in [START_REF] Poggiolini | The GN-model of fiber non-linear propagation and its applications[END_REF], similarly to as in [START_REF] Rumipamba-Zambrano | Costeffective spatial super-channel allocation in Flex-Grid/MCF optical core networks[END_REF], from which we get different transmission reach values depending on the modulation format employed. A particularity of the GN-model is that, given a specific modulation format and spectral efficiency, the nonlinear interference is independent of the number of channels, that is, it is independent of the symbol rate used in the communication.

To account for the multicast nature of the communications, we rely on the fact that the transmission reach values in the GN-Model depend on the number of spans (amplification stages that introduce noise) along the route. Therefore, if an optical signal is split and amplified at an MC node, its transmission reach is reduced by a span length (L span ) as a worst-case. Hence, if T R m e (l) denotes the transmission reach value over the route from the source to leaf node l, and T R e denotes the unicast transmission reach value provided by the GN-Model, we can relate them as:

T R m e (l) = T Re -n M C (l) * Lspan (1)
where n M C (l) is the number of MC nodes crossed in the route where splitting is performed (referred to as branching nodes). To respect the QoT at the destinations, it is sufficient to verify, for each leaf node l, that the distance from the source to it does not exceed T R m e (l). Again, we would like to point out that this is a worst-case estimation of the transmission reach estimation for multicast communications. Nonetheless, given its linearity, it is very convenient for considering it in an ILP formulation as the one in the following section.

Exact Solution

Before the design of the hierarchy computations, a pre-computation step is proposed. Then we present the computations using an ILP. The notations and network parameters are summarized in Table 1. Let us notice that some parameters are the same for all arcs in a given sub-graph of the auxiliary graph. An example is the contiguous set of frequency slots available in the arcs (it is the same for all arcs in a given sub-graph). To reduce the number of parameters, the homogeneous parameters of a sub-graph are associated with the first artificial arc from the artificial source to the given occurrence of a DC node.

Pre-computation

The objective of this step is to compute the auxiliary graph, and to reduce the search space by some simplifications.

a. A sub-graph corresponds to a modulation format and a slot block capable of satisfying the multicast request. The number ns e of slots needed to satisfy the required capacity C using modulation efficiency me is

nse = C me + w GB w F S (2)
where w F S and w GB indicate the width of the slots and guard bands respectively. The slot interval F contains nbsb e = |F | -ns e + 1 slot blocks with

G t = (V t , A t )
The directed topology graph. G = (V, A)

The directed auxiliary graph. R ⊂ V

The set of DCs in the auxiliary graph. r ∈ R ⊂ V A DC node in the auxiliary graph.

M C ⊂ V

The set of MC-switches in the auxiliary graph.

M I ⊂ V

The set of MI-switches in in the auxiliary graph. s ∈ V

The artificial source. D ⊂ V

The set of nodes in G corresponding to the destinations in G t . D ⊂ V

The set of virtual destinations in the auxiliary graph G.

D d ⊂ V
The set of (duplicated) destinations connected to d .

B

the required capacity (in Gbps) of the multicast request. C the capacity (in Gbps) of an FS when using BPSK as modulation format. (u, v) ∈ A

The arc from node u to node v. l(u, v)

The length of the arc (u, v).

Out(v)

Extremities of the outgoing arcs from node v.

In(v)

Extremities of the incoming arcs to node v.

Deg -(v)
The in-degree of node v.

Deg + (v)
The out-degree of node v.

F

The set of frequency slots supported in the network.

f i A slot f i ∈ F . f i (u, v) It is 1 when the slot f i is available in arc (u, v) ∈ A m e
The modulation efficiency of the modulation format e e = 1, . . . , 4, in bits/s/Hz T R e

The maximal transmission range for unicast transmission corresponding to the modulation format e.

L span

The distance equivalent with power loss caused by a traversed splitter. ns e

The requested capacity in number of slots corresponding to the modulation format e.

SB e

The set of contiguous slot blocks with capacity ns e .

SB

The set of contiguous adequate slot blocks, SB = ∪ e=1,...,4 SB e sb k

The k th slot block, sb k ∈ SB. G k Sub-graph associated with slot block sb k . b(s, r)

The index of the highest frequency slot in the subgraph rooted at r (associated with arc (s, r)). L(u t , v t ) ⊂ A

The set of arcs in G corresponding to (u t , v t ) ∈ A t .

ns e contiguous slots in each. A slot block set SB e is then computed for each modulation. The entire available slot block set is SB = ∪ e SB e and contains nbsb = e nbsb e blocks. For each slot block

sb k ∈ SB a sub-graph G k is gen- erated. An arc (u, v) is present in G k , iff the subset sb k is available in (u t , v t ): f k+i (u, v) = 1, i = 0, . . . ns e -1.
b. The artificial (virtual) source and the set of virtual destinations are created and connected to the sub-graphs. c. The nodes for which the shortest path from the source is longer than the maximum transmission reach and the nodes without incoming arc are deleted.

d. If a sub-graph G k does not contain a path from the source to any destination, the sub-graph is deleted. (Notice these verifications may be done in polynomial time by using a simple shortest path computation.)

The solution is computed by an ILP in the auxiliary graph.

ILP formulation

Our ILP is based on flows associated to destinations. The following variables are used in the ILP.

x(u, v) Binary variable that indicates whether the arc (u, v) is used Integer variable that is the highest index of the used FSs. The ILP follows the main lines of the formulation in [START_REF] Zhou | Light-hierarchy: the optimal structure for multicast routing in WDM mesh networks[END_REF], which computes light-hierarchies (and not only light-trees) in WDM networks. Our formulation adapts the ILP to compute light-hierarchies using convenient FSs.

y d (u, v)
To avoid to produce highly congested links that could prevent the allocation of future demands, in this study we consider as first objective the minimization of the highest FS index used in the route. We also consider the minimization of the number of FSs. We should allocate routes using the fewer amount of resources (number of FSs) as possible.

Objective:

We minimize the maximal index of the FSs used first, then try to minimize of the number of FSs.

min(∆ • w + |F | i=1 s i ) (3)
∆ should be large enough to ensure the priority of the first part, so we set ∆ = |F |. Trivially w should be at most equal to the highest indexes in the different sub-graphs:

b(s, r k ) • x(s, r k ) ≤ w ∀k = 1, . . . , nbsb (4)
This objective is subject to the following constraints (5 -22). In these constraints, we use ∀d, ∀i, and ∀(u, v) to imply ∀d ∈ D , ∀i = 1, . . . , |F |, and ∀(u, v) ∈ A, respectively.

Light-Hierarchy Constraints

The number of outgoing links at the source is limited by the number of destinations.

1 ≤ v∈Out(s) x(s, v) ≤ |D| (5)
Each destination should be spanned by one incoming arc of the solution.

v∈In(d) x(v, d) = 1 ∀d (6)
Constraints ( 7), [START_REF] Wang | A Survey on Data Center Networking for Cloud Computing[END_REF] guarantee that an MC node u has at most one input arc and the number of used output arcs is at most Deg + (u) (it can be 0, when there in no input for the node).

v∈In(u) x(v, u) ≤ 1 ∀u ∈ M C (7) v∈Out(u) x(u, v) ≤ Deg + (u) • v∈In(u) x(v, u) ∀u ∈ M C (8)
For MI type nodes, which do not correspond to any destination, the number of output arcs is limited by the number of input arcs.

v∈Out(u) x(u, v) ≤ v∈In(u) x(v, u) ∀u ∈ M I \ D (9)
If the MI type node corresponds to a destination, the related virtual destination can be reached from a given occurrence of the node.

v∈Out(u) x(u, v) ≤ v∈In(u) x(v, u) + x(u, d) ∀u ∈ M I ∩ D, ∀d (10)
For the non destination nodes, the following constraint is held.

v∈Out(u)
x(u, v) ≥ v∈In(u)

x(v, u) ∀u / ∈ D (11)

Spectrum Non-overlapping Constraints

In a fiber corresponding to an arc (u t , v t ) of the topology graph, a slot can be used at most once. The following constraint guarantees that two overlapping slot blocks can not be used in a same arc.

(u,v)∈L(u t ,v t ) x(u, v) • f i (u, v) ≤ 1 ∀(ut, vt) ∈ At, ∀i (12)
A slot s i is used in the solution iff it is used in some arc (s, r), r ∈ R.

f i (s, r) • x(s, r) ≤ s i ∀i, ∀r ∈ R ( 13
)
s i ≤ r∈R (f i (s, r) • x(s, r)) ∀i (14)

Flow Conservation Constraints

v∈In(u)

y d (v, u) - v∈Out(u) y d (u, v) =      -1, u = s, 1, u = d, 0, otherwise. ∀d (15)
Eq. ( 15) ensures that if a virtual destination node d is covered in the solution, it can only be served in one light-hierarchy, and there is a single path/trail from s to d on it.

Maximum Transmission Reach Constraint

Each path and trail from the source to the destinations should satisfy the constraint on the maximum transmission reach of optical signals. Following Eq. ( 1):

u∈M C mc d (u) • Lspan + (u,v)∈A y d (u, v) • l(u, v) ≤ T Re ∀d (16)
To compute the variable mc(u) d , first we compute mc(u) by the following constraints.

mc(u) ≤ max(0, v∈Out(u)\D x(u, v) - v∈In(u) x(v, u)) ∀u ∈ M C (17) v∈Out(u)\D x(u, v) -1 ≤ Deg + (u) • mc(u) ∀u ∈ M C (18)
Eqs. ( 17) and ( 18) ensure that a node u is a branching node if it is in M C and there are at least two outgoing arcs (except the arc leading to an eventual virtual destination).

The variable mc(u) d expresses the fact that a branching node u is in the light-trail to the destination d. Its value can be computed from mc(u) as follows:

mc(u) d = mc(u) • v∈Out(u)\D y d (u, v) ∀u ∈ M C, ∀d (19) mc(u
) d = 0 ∀u ∈ M C, ∀d ∈ Out(u) ∩ D (20)
Moreover, there are trivial relations between the flow variables and the arc usages.

x(u, v) ≥ y d (u, v) ∀(u, v), ∀d (21)
x(u, v)

≤ d∈D y d (u, v) ∀(u, v) (22)
Note that the two non-linear equations ( 17) and ( 19) are transformed to linear forms in the implementation. The result is an optimal hierarchy. After the deletion of the virtual nodes, the remaining set corresponds to the set of optimal light-hierarchies.

Performance Evaluation of the Solutions

The ILP formulation presented in Section 5.2 computes the MSM solution based on light-hierarchies (LH), so we call it MSM-LH. Furthermore, it can also be extended to compute an SSM solution, even employing light-trees (LTs) in any of these two approaches, with some appropriate additional constraints. Particularly, to compute SSM solutions, each sub-graph G k should have at most one incoming arc from the virtual source, whereas to compute solutions based on LTs, there is at most one incoming arc to every node for every sub-graph. In other words, constraint [START_REF] Bilal | Green Data Center Networks: Challenges and Opportunities[END_REF] should be applied for all the nodes (not just MC nodes). With these considerations, we can easily evaluate 4 different models with the presented ILP formulation, namely, MSM-LH, MSM-LT, SSM-LH and SSM-LT, that is, MSM an SSM, either over LHs or LTs.

Moreover, the presented ILP considers one input request, which is suitable for a dynamic network scenario. However, we should point out that running an ILP program in dynamic scenarios is not realistic, given its high complexity and long execution time. We consider two main routing scenarios: incremental and single request. In an incremental scenario, a set of demands are incrementally loaded into the network, and the ILP is called to serve one demand after another. We assume infinite demand holding times, i.e. previous multicast demands are not released. In the single request scenario only one route is computed.

The load can start by an empty network state where all FSs are available for use (mainly for incremental network loading), or by an arbitrary random network state where some FSs are used for previously allocated communications.

In the random network state scenario, the availability of the FSs in the links was generated using the following algorithm. Specifically, the availability of the first two-third FSs (whose indices are from 0 to 2/3 × |F S|) are set randomly, while the last one-third FSs are all available.

Let A[k, i] the value indicating the availability of the FS of index i on arc k. Let F S Set1 be the set of the first two-third FSs.

Random Network State Generation For all arcs k:

For the FSs of index i in F S Set1: A[k, i] = a random value from {0, 1} (the probability of 1 is proportional to the index i, consequently the availability of FSs of higher index is more probable) For the FSs of index i not in F S Set1:

A[k, i] = 1 (FSs are always available)

Although the ILP formulation can work with multiple modulation formats, due to its complexity, we selected only two representative ones for the computations: PM-QPSK and PM-16QAM, whose transmission reaches are T R e = {9000, 2000}km, respectively. Moreover, the span length is set to L span = 85km. Using it, we assume the same scenario as "Link 1" in [START_REF] Poggiolini | The GN-model of fiber non-linear propagation and its applications[END_REF].

A multicast request is given by a triplet (s, D, C) where s is the source, D is the set of destinations and C is the requested capacity. In the different scenarios, we used randomly generated requests changing from one instance to another, or fixed request sets, where the same randomly generated requests are presented for different states and models. This latter permits a more accurate comparison of the models.

The simulation settings and results are detailed in the following.

Simulation Settings

A key question for the design of the simulations is the expected execution time to compute the exact solution. Due to the hardness of the problem and the exponential complexity of the ILP formulations, only small instances can be tested. Therefore we used two relatively small topologies: the 14-node NSF network and the 28-node US Longhaul topology (cf. Figs. 4a and 4b in [START_REF] Le | Minimizing blocking probability for the multicast routing and wavelength assignment problem in WDM networks: Exact solutions and heuristic algorithms[END_REF]). The size of the computation in the latter network, in which there are 90 arcs is briefly illustrated as follows. Let us suppose that there are 30 FSs in the fibers (it is not realistic but enough for the proposed computations without blocking). Supposing a small capacity demand and only one modulation format, the number of implicated layers (duplications) in the auxiliary graph is equal to 54. The number of arcs in the layered graph (without the artificial arcs) is equal to 9720. The number of decision variables is proportional to this value. Naturally, the number of constraints also strongly depends on the dimension of the problem. For this reason, the capacity of each fiber link in the simulations is limited to 20 or 30 FSs, whose bandwidth granularity is 12.5 GHz, and the guard band is set to 10 GHz. Supposing four modulation formats, the number of variables and constraints can be four times larger.

Since the diameter of the networks is small regarding the number of hops, we select no more than 3 DC nodes in MSM scenario.

To simulate EON networks with sparse splitting, we set the number of MC nodes relatively small, i.e. in the NSF topology |M C| ∈ {0, 3} and in the US Longhaul network |M C| ∈ {0, 2, 4, 6, 8}. Notice that DC nodes are all assumed to have full multicast capacity. All the remaining nodes are MI.

We generate |M R| multicast requests in a random way as follows. For each request, the desired number of destination nodes is selected discarding those nodes where a DC is co-located. The requested capacity for the multicast communication is set randomly between 10 and 100 GHz.

To evaluate the performances of the different aforementioned models with different metrics, we ensure that no request gets blocked. The detailed parameter settings can be found for each experiment in the following sub-sections.

We ran the four ILP programs using CPLEX 12.5 on a machine with Intel Core i7, 3.4GHz processor and 16GB of RAM.

Runs and Results

In the following, we present the experimentation of the exact model in the two referenced backbone networks. The optimal solutions are investigated in scenarios of incremental requests and single requests. Moreover, the effect of distance adaptive modulation selection using two modulations, and the influence of MC nodes are presented. The following notations are used in the presented tables and figures: FS max the index of the last FS used in the computed multicast route (it corresponds to the first objective of the minimization) Number of FSs the number of FSs used in the computed multicast route Total FSs the total number of FSs used in all links (it represents the total allocated resources) Spectrum gaps the sum of the number of free slots until the last allocated one in the used links; it characterizes the fragmentation Ex. time CPU time consumed by the route computation (in seconds) If a computation concerns a set of requests (for example, incremental requests or several single random requests in the same state of the network), the average values are indicated.

Incremental Requests

In the NSF network topology, we conducted 30 instances for each model. The resulted values are averaged for each metric of interest corresponding to the objective function and they are shown in Table 3 for two cases of MC nodes. The parameters used for the computation are summarized in Table 2. In the 14-node NSF network the hop distances are very limited. A second experimentation was organized in the US Longhaul network topology. In these experiments 20 random requests were generated and loaded in an incremental manner from an initial random network state (cf. the parameters in Table 4). Table 5 shows the results. The values are the averages of 10 runs and in each run 20 requests were incrementally loaded. In addition to the optimized values, the table shows the total number of FSs, the spectrum gaps at the end of the new allocations (characterizing the fragmentation), and the execution time.

Table 6 details the performance gap of the three other strategies over MSM-LH in the series obtained in the NSF network. As seen, MSM outperforms SSM, and LH is better than LT. Specifically in this series, the performance tendency of the four strategies from best to worst is: MSM-LH, MSM-LT, SSM-LH and SSM-LT. MSM-LH achieves the lowest maximal FS index used and consumes the least The values observed in the US Longhaul network confirm the findings. MSM outperforms SSM, and LH is better than LT. Notice that the MSM routing produced less not used spectrum gaps, however its execution time is more important (2-3 times more than for SSM).

The above results are reasonable. On one hand, MSM can use all possible DC nodes as sources of the multicast routes, hence it can exploit all available FSs in each slot block to reach all the destinations; whereas SSM can use only one DC as its multicast source, it may rely on multiple FS blocks to do the same. Consequently, MSM uses less FSs and lower maximal FS index than SSM. Moreover, by using SSM the FS budget gets exhausted more quickly to serve the same amount of demands compared to using MSM. On the other hand, light-hierarchies have been proved to be able to produce better solutions than light-trees as shown in previous work [START_REF] Zhou | Light-hierarchy: the optimal structure for multicast routing in WDM mesh networks[END_REF][START_REF] Le | Minimizing blocking probability for the multicast routing and wavelength assignment problem in WDM networks: Exact solutions and heuristic algorithms[END_REF] in sparse splitting optical networks. In our case, light-hierarchies can allocate some available FS blocs which can not used in a tree (because a needed branching node of the tree is an MI node). When |M C| = 0 in the first series, all strategies consume more resources. This is reasonable and aligned with the widely adopted consensus that all-optical multicasting is better with more MC nodes (cf. more elements in the sub-section 6.2.4).

The following route computations focus on the single request scenarios.

Single Requests

In this series, 250 single requests were executed for each model using the US Longhaul network topology. Different randomly generated sets of MC nodes and five previously and randomly generated initial network states were used to compute the routes for randomly generated requests (cf. Table 7). For each run, the total number of the allocated FSs, the index of the last FS, the spectrum gaps (the fragmentation) and the execution time were collected. Table 8 gives the average values and illustrates the difference between the proposed models. The average values of 250 computations illustrate the finding in the previous series of incremental requests and meet our exceptions. MSM delivery outperforms the SSM configurations and the light-hierarchies can be better than light-tree despite the fact, that the average of the total number of allocated FSs is less for light-trees than for light-hierarchies in the single server configuration. It is understandable because the first objective is the minimization of the maximal index of the allocated slots. The light-hierarchies can produce the optimum, even if the number of arcs in them may be higher than in light-trees. Moreover, the number of spectrum gaps is less using light-hierarchies and the MSM solution.

Notice that the results of another series performed in the NSF network for a set of single requests and using the possibility of the distance-adaptive modulation are presented in the next sub-section.

Distance-Adaptive Modulation

The first computations were performed using the NSF network for 30 randomly generated single requests. For the 4 models, the same requests were loaded to a random (but fixed) state of the network. Here we assumed that there is no MC node in the NSF network and two configurations of the available modulation formats were examined. At first, only the modulation PM-QPSK was available, while in the second series PM-16QAM modulation was also used. The parameter setting is as follows (Table 9). The measured values are summarized in Table 10. According these results an important gain can be seen in favor of the distanceadaptive modulation even if only two different formats are used. The maximal index and also the number of allocated FSs are significantly less with two modulations than with only one. There are less spectrum gaps using the distanceadaptive modulation but the computation of the exact solution with two modulations is more expensive.

In the following sequence we examined the effect of multiple modulations in the US Longhaul network and we found similar advantages of the modulation format adaptation with multiple formats. In the US Longhaul network topology using 3 DCs, 2 MC nodes and 5 different random network states, the routes for 50 random multicast requests were computed (10 requests for each network state). The same initial states and multicast requests were loaded at first using only the modulation PM-QPSK, and in the second case supposing two available modulations (PM-QPSK and PM-16QAM). Remember that we limited our experimentation to two modulations, since the size of the auxiliary layered graph is duplicated for each modulation format. In addition to the results presented above (namely the performance of the light-hierarchies and the multiple servers in the content delivery), the advantages of the adaptive choice of the modulation are important. The selection of the more efficient PM-16QAM modulation, if it can be used, significantly improves the quality of the routing: there is less FSs used, and the average of the maximal index is also lower. Moreover, the number of spectrum gaps responsible for the fragmentation is less using the distance-adaptive modulation. The counterpart of these advantages is the more important execution time when using the two formats.

Effect of the MC nodes

The first illustration of the effect of MC nodes on the multicast route can be found in Table 3 in the sub-section 6.2.1. A second series in the larger US Longhaul netwok gives more elements. Several runs were organized for various MC node sets using single routing requests and random network states. In this section we present one series. For the 4 models the same network state and the same 30 random multicast requests were used. The set of multicast nodes was varied as the parameter setting shows it.

The computed values are presented in Table 14. Each value is the average of 30 runs. In this configuration (network topology, DC set, network link state and 30 randomly generated but fixed requests) the effect of MC nodes is minimal.

Conclusions and Perspectives

In this paper the multi-server multicast approach for CDNs over EONs with sparse splitting constraints is investigated. We proposed an ILP formulation to compute the optimal multicast route as a set of light-hierarchies from multiple DCs to deliver a multicast content to all end-users. Unlike simple previously proposed models, the ILP, which is based on an auxiliary layered graph, takes into account the signal degradation caused not only by the EON fiber impairments but also the splitting, so as to impose precise limitations on their transmission reach. The computation also adapts to a set of different modulations. The results revealed that the MSM approach can achieve better resource utilization compared to the traditional SSM approach. In particular, the obtained results suggest that under sparse splitting constraints, light-hierarchies should be used to optimize MSM performance. Moreover, our experiments demonstrated the interest of the usage of multiple modulation formats in the multicast routing in EONs suporting CDNs. Due to the hardness of the problem, the computation of the exact solution is expensive and not scalable. Therefore, good heuristics to realize the MSM-LH approach in larger network settings will be main targets in our future work.

Figure 1 :

 1 Figure 1: Example of a light-hierarchy

Figure 2 :

 2 Figure 2: Interest of a light-hierarchy vs. a light-tree

Figure 3 :

 3 Figure 3: The directed auxiliary graph with the virtual destinations

Table 5 :

 5 Average values for 20 incremental requests in the US Longhaul network number of FSs. It is better than its counterpart MSM-LT, while outperforming SSM solutions. Accordingly to the results, MSM-LH outperforms SSM-LH by saving up to 11.3% FSs, while it is slightly better than the traditional light-tree based MSM-LT by reducing 4.2% FSs used. The combination of MSM with light-hierarchies is by far better than the traditional SSM combined with lighttrees, which can achieve up to 17.4% resource saving. We also observe a similar performance comparison in terms of the maximal FS used shown in the table.

Table 1 :

 1 Network parameters and notations

  Commodity flow variable. Equal to 1 if the flow from s to virtual destination d ∈ D passes by arc (u, v). s i Binary variable that is equal to 1 if the frequency slot f i is used in the solution mc(u) Binary variable that is equal to 1 if the MC node u is branching node in the solution mc(u) d It is equal to 1 if the MC node u is a branching node to virtual destination d ∈ D w

Table 2 :

 2 Parameter setting for incremental requests in the NSF network

	DC nodes	MC nodes	Network state |M R| per model Modulations
	3 randomly selected 0 or 3 randomly selected	empty	30	PM-QPSK

Table 3 :

 3 Results for incremental requests in the NSF network

Table 4 :

 4 Parameter setting for incremental requests in the US Longhaul network

	DC nodes	MC nodes	Network state |M R| per run Modulations
	{9, 15, 27}	0, 2, 4, 6 or 8 randomly	randomly	20	PM-QPSK
		from {7, 8, 9, 10, 13, 17, 26, 27}	generated		PM-16QAM

Table 6 :

 6 Performances of the models versus MSM-LH (comparison of the results obtained using the NSF network)

Table 7 :

 7 Parameter setting for random single requests in the US Longhaul network DC nodes MC nodes Network state |M R| per random state {9, 15, 27} 0, 2, 4, 6 or 8 randomly 5 random 50 from {7, 8, 9, 10, 13, 17, 26, 27} states

Table 8 :

 8 Random single request in the US Longhaul network

Table 9 :

 9 Parameter setting for single requests in the NSF network DC nodes MC nodes Network state |M R| per fixed state Modulations {1, 5, 12}

	∅	5 random	30	PM-QPSK
		states		PM-16QAM

Table 10 :

 10 Random single request in the NSF network

Table 11 :

 11 Parameter setting to compare the case of single modulation to the case of two modulations in the US Longhaul network DC nodes MC nodes Network state |M R| per state Modulations

	{9, 15, 27}	{8, 26}	5 random	10	PM-QPSK
			states		PM-16QAM

  Table 12 resumes the observed measurements in the Longhaul network.

Table 12 :

 12 Performances of the distance-adaptive modulation

DAT consists in selecting the most effective modulation s.t. the maximal sourcedestination distance does not exceed the modulation transmission range.

A homomorphism preserves the adjacencies of nodes; w 1 ∈ W and w 2 ∈ W can be adjacent iff the corresponding nodes v 1 ∈ V and v 2 ∈ V are also adjacent

The optimized values (the maximal index of the used FSs and the total number of FSs) diminish when increasing the MC node set. Trivially, if all the nodes are MC, the optimal solution is a set of light-trees. The number of MC nodes slightly affects the results when the objective is the minimization of the maximal index of FSs. Even if longer routes (long trees and hierarchies with less branching nodes) are found, the maximal index of the allocates FSs can be more preferable and it is not directly correlated to the length of the routes. Probably, the influence of MC nodes can be important when the goal is the cost minimization. Moreover, similarly to the other cases, the MSM model is always better than the SSM and light-hierarchies outperform light-trees.