1)
Coherent Optical Communications
2)
Ultrafast Optical Mesh Networks
3)
Centralized Optical Networks
The following notes contain a summary of the research topics I have dealt with during my career, and represent a guide to my publications record .
The publications are logically grouped by their research topic.
For each group, a table is given which contains on the left the journal publications, and on the right their corresponding conference publications.
The works are arranged by chronological order of execution (not of publication).
When the group is composed of several themes, such themes are reported and numbered in the table.
Each publication has a label, composed of the authors' initials, the acronym of the journal/conference, and the year of publication. For instance, [BGP_jlt_92] corresponds to the oldest journal article:
A. Bononi, P. Ghiggino, and G. Picchi, ``Analysis of the automatic frequency control in heterodyne optical receivers,'' J. Lightwave Technol., vol. 10, no. 6, pp. 794-803, June 1992.
Publications containing experimental work appear with the letter (E)
on the right.
For each research topic, a short summary of the innovative contents
of the publications is provided, as well as a discussion on how they
fit within the the scientific period of publication.
1) Coherent Optical Comunications |
JOURNAL | CONFERENCE |
[SBGB_ieecoh_90] (E) | |
[GSBB_octima_91] (E) | |
[BGP_jlt_92] |
[BGP_jlt_92] : ``Analysis of the automatic frequency control in heterodyne optical receivers''
stems from the specific need of Marconi S.p.A. of defining the design of the automatic frequency control (AFC) of the prototype of the coherent optical single-branch FSK system at 565 Mb/s under development at their G.E.C Wembley laboratories, UK, and described in the papers [SBGB_ieecoh_90] and [GSBB_octima_91]. During my sojourn in Wembley, I have matured both my theoretical and experimental understanding of the AFC problem, whose synthesis is reported in [BGP_jlt_92]. The paper appears several years after the climax of the interest of the scientific community in coherent communications. However, the innovative contribution of the paper lays both in its general theoretical formulation, which encompasses in a single formalism several coherent modulation formats (DPSK, ASK, FSK), and in the optimization of the AFC based on the minimization of the bit error rate of the system, in contrast with prior publications which concentrated on the minimization of the residual frequency noise of the AFC.
Citations of [BGP_jlt_92] are found in :
C. Fujihashi, ``Detection of decision error caused by the quantum noise in an optical signal system,'' IEEE J. Lightwave Technol., vol. 12, no. 6, pp. 1057-1061, June 1994.
E. Forestieri, and G. Prati, ``Analysis of delay-and-multiply optical FSK receivers with Line Coding and non-flat laser FM response,'' IEEE J. Sel. Areas Commun., vol. 13, no. 3, pp. 543-556, April 1995.
2) Ultrafast Optical Mesh Networks |
JOURNAL | CONFERENCE |
1) Routing | |
[FBP_tc_95] | [BFP_info_93] |
[BP_jhsn_96] | [BP_ofc_94] |
[BP_info_94] | |
[BPBSSB_el_94] (E) | [BPBSSB_dod_94] (E) |
[BBPSB_ecoc_94] (E) | |
[BP_tn_96] | [BP_ciss_94] |
2) Systems | |
[BFP_jlt_93] | [BFP_ofc_93] |
[FBP_efocn_93] | |
[BFP_procj_93] | [BFP_spie_93] |
[FBP_el_92] | [FBP_ofc_93] |
[BFP_el_93] | |
[FBZPPP_fio_94] |
The research was performed at a time when the trasmission potential of the optical channel/network had not yet thoroughly been explored in the presence of optical nonlinearities, which are always present whenever ultrahigh bit rate time division optical fluxes (OTDM) are transmitted.
The papers in this group are therefore leading-edge research papers, as evidenced by their presentation at such prestigious conferences in optical communications as the
Optical Fiber Conference (OFC) in the USA and the European Conference on Optical Communications (ECOC) in Europe.
The innovative contribution of such pieces of work can be singled out in the joint consideration of both the ``network'' effect of dynamic routing (i.e., variable packet route assignment according to the traffic level), and of the ``system'' effect of the accumulation of noise and distortion due to packet propagation, in a non-classical (but realistic) scenario for optical communications, such as the 100 Gb/s nonlinear soliton channel.
The two most significant publications in the group (developed more or less at the same time) are:
[FBP_tc_95] ``Analysis and comparison of hot-potato and single-buffer deflection routing in very high bit rate optical mesh networks'';
[BFP_jlt_93] ``Design and channel constraint analysis of ultra-fast multihop all-optical networks with deflection routing employing solitons''.
In [FBP_tc_95] the network tools were developed, upon which the system analysis in [BFP_jlt_93] is based.
It is worth mentioning also paper [BFP_el_93] which describes an innovative solution of the bit synchronization problem in ultrafast packet reception.
Citations of this group of papers appear in:
The work [BP_jhsn_96] ``New structures of the optical node in transparent optical multihop networks using deflection routing'' belongs instead to a later period of the Ph.D. research, in which I developed efficient routing algorithms for optical nodes with minimal hardware and control complexity, such as those imposed by the ultrafast transmission. The objective of this work is to keep a large throughput notwithstanding the simplicity of the optical hardware. One of the solutions suggested in this paper has been implemented in an experimental prototype presented in [BPBSSB_el_94], [BPBSSB_dod_94], [BBPSB_ecoc_94]. Citations of this group of papers can be found in:
[BP_tn_96] ``Analytical evaluation of improved access techniques in deflection routing networks''
provides a synthesis of the analysis techniques of meshed regular networks with nodes of reduced complexity and deflection routing in uniform traffic, that I have learned during my Ph.D. work at Princeton University, and later refined during my teaching work at SUNY Buffalo. The analysis is based on the absorbing Markov chains and on the measurement of the network parameters based on the Renewal Theory. This work is also cited (together with [BFP_el_93]) in a recent book on optical networks:
R. Ramaswami and K. N. Sivarajan, ``Optical Networks. A Practical Perspective.'' Morgan Kaufmann, San Francisco, 1998, ISBN 1-55860-445-6, ref. [BP96],[FBP95].
The work presented in the papers [BP_jhsn_96] and [BP_tn_96] was started at Princeton University, and later on completed at Buffalo University.
3) Centralized Optical Networks |
JOURNAL | CONFERENCE |
[B_jlt_98] | [B_ciss_96] |
[B_globe_96] | |
[B_lerici_97] | |
[B_el_96] |
[B_jlt_98] ``Weakly vs strongly multihop space-division optical networks''
referes to a deepening of the results of the previous group, from which one clearly learns that one of the most limiting factors in transparent optical networks is the amplified spontaneous emission noise (ASE) introduced by the optical amplifiers, which is proportional to their gain.
Therefore one of ther primary objectives in the design of transparent optical networks must be the optimal placement of the optical amplifiers and of the lumped loss elements (such as fibers, splitters, switches and so on). However, in contrast with the soliton channel of the previous group of publications, the papers in the present group consider, besides the ASE:
1) an ideal optical channel, i.e., one in which one can neglect the distortion due to fiber propagation, as is the case with optical networks of limitid geographical reach (LANs, MANs);
2) the crosstalk coming from the imperfect switching of channels in optical switches, a problem that was highly debated within the numerous international projects that were active at the time of completion of such group of works. Note that the switching crosstalk does not exist with soliton transmission, since the soliton is a constant-phase pulse. In fact, optical switching is a phase-dependent process, hence the soliton is completely switched on an optical output port of a switch, without residues (crosstalk) on the remaining ports. The novelty of the work is in the comparison between the classical meshed topologies for deflection routing on one side, and a novel topology having all the switching (routing) process centralized in its core, on the other side. The comparison (in terms of throughput/delay) is performed given the same bit error rate in the received packets.
The work stems from a deepening of the concept of ``hop'' as applied to the throughput calculation: a ``hop'' is a visit to an ``active'' node, i.e., a node able to both absorb (receive) packets from the network, and to inject (transmit) new packets into the network. Instead, a visit to an ``inactive'' switching node should not be counted as a hop in the throughput calculation.
The newly proposed centralized network comes from such understanding. The active nodes are placed at the periphery of the network, while the central switch perfoms all the routing, as opposed to the meshed networks in which all nodes are active and perform routing. When no buffers are available, the centralized topology has a small number of hops, and therefore large throughput, although the power attenuation (loss) per hop is large, and consequently so is the ASE added to the packets. In the classical meshed networks the number of hops is intrinsically large, while the per-hop attenuation is small. The paper shows under which conditions the centralized network, for a given bit error rate on received packets, has a larger throughput/smaller delay than the meshed networks.
I would also like to bring to the reader's attention the paper [B_ciss_96] which contains all the theoretical tools for network analysis, which were later used in [B_jlt_98], and where an analysis is presented of how the throughput of the centralized network improves for varying levels of complexity/completness of the central switch.
In closing this section, we noted that a network proposal identical to the cetral network in [B_jlt_98] appears in:
H. Yokoyama and H. Nakamura, ``Mechanisms and performance of reflection routing for optical packet switched networks,'' in Proc. OFC 2002, pp. 779-781, Anaheim, CA, March 2002.
thereby showing the potential interest for such a network in a metropolitan area scenario, although the authors do not cite [B_jlt_98] in their references.
4) Wavelength Routing Networks |
JOURNAL | CONFERENCE |
1) Routing | |
[BCT_jlt_99] | [BCT_icc_98] |
[CTB_el_97] | [CTB_leossum_97] |
[B_ciss_98] | |
[GKB_leos_97] | |
2) Systems | |
[CTB_procc_97] | [CTB_globe_97] |
[CTB_leos_96] | |
[CTB_icts_97] | |
[CTB_iscc_97] |
The most important publication on Routing is:
[BCT_jlt_99] ``Analysis of hot-potato optical networks with wavelength conversion''.
The theme in along the lines of the then popular research on wavelength routed optical networks (WRON). The innovative contribution of the paper is the proposal of deflection routing WRONs without buffers, in which deflection is reduced by switching (converting) to an available alternative wavelength.
The theoretical teletraffic analysis is a rigorous extension to the wavelength division multiplexing (WDM) case of the well-establised techniques of deflection routing of the previous groups of publications.
The work is cited in:
B. Mukherjee, and S. Dixit ``Advances in Photonic Packet Switching: An Overview'', IEEE Communications Magazine, Feb. 2000.
Next is a short summary of the remaining publications on Routing in this group.
The letter [CTB_el_97] quantifies the reduction of the number of hops provided by wavelength conversion for increasing number of WDM channels.
The publication [B_ciss_98] sharply differs form the preceding ones, in that it deals with slotted WDM ring networks. Such work containd interesting topological comparisond between WDM slotted rings and input-buffered slotted packet switches.
Lastly, [GKB_leos_97] is a report about WDM deflection routing slotted networks withour wavelength conversion, in which the wavelength dimension is exploited only at the injection phase in the network. Such work has been carried on jointly with the research group of Prof. Ranjan Gangopadhyay of the Indian Institute of Technology, Kharagpur, within the framework of the European-Union-funded project DAWRON, of which I was the technical coordinator.
The works on Routing provide the needed background for the system analysis of the remaining publications in this group, in which the bit error rate in such WRONs is studied. as in the previous group, the analysis addresses non-distorting channels, and non-soliton (classical) supporting pulses, and the emphasis is on the joint effect of ASE and crosstalk coming from incomplete wavelength and space switching.
5) Fiber Nonlinearities |
JOURNAL | CONFERENCE |
[BFB_oft_98] | [BFB_leos_97] |
[BVFB_ptl_98] | [BVFB_ciss_98] |
[BVFB_ecoc_98] | |
[VBBF_leos_98] | |
[BPFB_iit_98] (E) | |
[BBSB_ecoc_00] (E) | |
[VSB_jlt_02] |
The most important paper on XPM is:
[BVFB_ptl_98] ``Intensity Distortion Induced by Cross-Phase Modulation and Chromatic Dispersion in Optical-Fiber Transmissions with Dispersion Compensation''.
The original model proposed in [BVFB_ptl_98] quantifies the intensity noise generated by the joint effect of XPM and chromatic dispersion, in optical transmission systems with dispersion mapping. The analytical model is able to explain the reasons of system degradation till then only observed by experiment and simulation.
The generation of such model was made possible by the background work appearing in [BFB_oft_98], where it was made clear that the phase induced by the interplay of XPM and chromatic distortion can be accurately predicted even neglecting the intensity distortions caused by chromatic dispersion.
Curiously, two more research groups around the world independently discovered essentially the same model at about the same time (IEEE Photon. Technol. Lett., Sept. 1998, p. 1268 e p. 1271).
Altough the publication date of [BVFB_ptl_98] is December 1998, the model was developed and tuned starting from October 1997, as evidenced by the conference publications [BVFB_ciss_98], [BVFB_ecoc_98], [VBBF_leos_98], and usefully used for system design at the Alcatel Marcoussis laboratories, France, during a stage in France of my Ph.D. student G. Bellotti.
The paper is cited in:
R. I. Killey, H. J. Thiele, V. Mikhailov, and P. Bayvel, ``Prediction of transmission penalties due to cross-phase modulation in WDM systems using a simplified technique,'' IEEE Photon. Technol. Lett., vol. 12, pp. 804-806, July 2000.
S. Betti, and M. Giaconi, ``Analysis of the cross-phase modulation effect in WDM optical systems, '' IEEE Photon. Technol. Lett., vol. 13, pp. 43-45, Jan. 2001.
B. Xu, and M. Brandt-Pierce, ``Analysis of XPM-induced
intensity distortion using the VSTF method,'' in Proc.
LEOS 2001, paper TuS2, pp. 279-280, San Diego, CA, Nov. 2001.
I next summarize the remaining works on fiber nonlinearities.
The experimental work [BPFB_iit_98] , belonging to the DAWRON project, deals with experimental techniques to measure the in-band four-wave mixing power without having to suppress neither detune the channel of interest. Four-wave mixing is the dominant nonlinear effect in WDM transmission systems operating on dispesion-shifted fibers, the kind which is largely installed in the Italian long-distance telecommunications backbone.
The theoretical and experimental work [BBSB_ecoc_00] , belonging to a research collaboration with Alcatel CRC of Marcoussis, France, deals with the study of the impact of the parametric growth of the ASE at the expense of the useful signal in strongly-nonlinear terrestrial optical systems with dispersion mapping. The presented experiment has been realized in the Alcatel laboratories during the mentioned stage of G. Bellotti, while in Parma my student P. Serena developed analytical and simulation tools for parametric gain.
The innovative theoretical work
[VSB_jlt_02] ``The RP method: a new tool for the iterative solution of the nonlinear Schroedinger equation''
deals with an important theoretical understanding stemming from the thesis work of P. Serena.
In the paper, we prove that the well-known method of the regular perturbation (RP), as applied to the resolution of the differential equations governing the propagation in single-mode fibers, coincides with a Volterra series expansion of the fiber system, and the iterative nature of the calculation of the RP method allows thus to drastically accelerate the calculations of the Volterra kernels. Such study can thus have useful applications for instance in the calculation of the channel capacity of the (moderately) nonlinear channel, and in the development of compensation techniques for the nonlinear channel.
6) Doped fiber Amplifiers |
JOURNAL | CONFERENCE |
1) Theory | |
[BRT_el_97] | |
[BR_jlt_98] | [BR_icc_98] |
[BPGSCVZ_foto_99] | |
2) Burst-mode Traffic | |
[BTR_ptl_99] | [BTR_ofc_98] |
[TBR_jlt_99] | [TBR_ecoc_98] |
[BPA_ecoc_99] (E) | |
3) Control | |
[TRB_ptl_98] | [TRB_ciss_98] |
[BB_jlt_99] | [BB_icc_99] |
[KBMR_jlt_00] | |
[KBRM_procj_00] | |
[BPA_ecoc_99] (E) | |
[PBO_ptl_99] (E) | [PBO_ecoc_99] (E) |
The most important theoretical paper on EDFAs is:
[BR_jlt_98] ``Doped fiber amplifier dynamics: a system perspective''
The steady-state study and design of EDFAs are well known since the last decade at least. On the contrary, the study of power transients in saturated EDFAs is a relatively recent field of research.
Such study is necessary in the design of protection and reconfiguration mechanisms in WDM optical networks , where optical fluxes (circuits) can be suddently switched from one optical path to another.
The analytical study of such transients requires the solution of a system of 2(N+1) nonlinear coupled partial differential equations for a WDM system of N channels. It was only in 1996 that a letter of Y. Sun (ref. [1] in [BR_jlt_98]) revealed that the solution of such complex system can be simplified, under mild and practically acceptable assumptions, to the solution of a single ordinary differential equation.
The work in [BR_jlt_98] provides an interpretation and further investigation of the results of Sun in the jargon of systems theory, thereby contributing to clarify to the systems engineer the dynamic behavior of EDFAs.
The system interpretation of the dynamic equation of the saturated EDFA in terms of a state variable has generated an original circuit equivalent model of the EDFA, which can thus be implemented with commercially available circuit simulators, such as Spice. Moreover, the paper contains - to the best of my knowledge - the first theoretical investigations of problems in burst-mode transmission in WDM systems, of which Internet Protocol (IP) over WDM is a recent example.
The paper is referenced in:
M. Karasek and J. Valles, ``Analysis of channel addition/removal response in all-optical gain-controlled cascade of erbium-doped fiber amplifiers,'' IEEE J. Lightwave Technol., vol. 16, pp. 1795-1803, Oct. 1998.
M Karasek and F. Willems, ``Channel addition/removal response in cascades of strongly inverted of erbium-doped fiber amplifiers,'' IEEE J. Lightwave Technol., vol. 16, pp. 2311-2317, Dec. 1998.
H. Yoon, S. Bae, S. J. Ahn, and N. Park, ``Reference level free multichannel gain equalization and transient gain suppression of EDFA with differential ASE power monitoring,'' IEEE Photon. Technol. Lett, vol. 11, pp. 316-318, Mar. 1999.
C. Dimopoulos, D. Simeonidou and A. S. Siddiqui, ``Evolution of SNR during power transients in chains of saturated and gain equalized EDFAs,'' Electron. Lett., vol. 35, no. 20, pp. 1756-1758, Sep. 1999
S. Novak, and A Moesle, ``Analytic model for gain modulation in EDFAs,'' J. Lightwave Technol., vol. 20, pp. 975-985, June 2002
S. Novak, and A Moesle, ``Simulink model for EDFA dynamics applied to gain modulation,'' J. Lightwave Technol., vol. 20, pp. 986-992, June 2002
The model proposed in the paper has been adopted by Virtual Photonics, producer of the software package ``Photonics Transmission Design Suite'', PTDS, one among the most widespread computer simulation and design tools in the field of optical telecommunications. The model appears in the Photonics Modules Reference Manual, p. 8-58, under the heading ``Dynamic Amplifier EDFA (Improved Bononi Model)''.
The theme on burst-mode traffic in EDFAs starts with the publication of :
[BTR_ptl_99] ``Large power swings in doped-fiber amplifiers with highly variable data''
and
[TBR_jlt_99] ``Output power and SNR swings in cascades of EDFAs for circuit- and packet-switching optical networks''
where the state-variable model, originally enriched with ASE-induced saturation, is used to simulate long transients due to burst-mode packet transmission, and to collect statistics of the output power and optical signal-to-noise ratio (OSNR) fluctuations.
The value of such pieces of work lays in their tackling for the first time - to my knowledge - in a sistematic way and by simulation, the problem of power and OSNR swings in optical WDM systems with burst-mode packet transmission, such as IP over WDM.
Regarding the theme of Control of EDFAs, the stabilization technique known by the name of gain clamping has received a lot of attention in the optical communications industry because of its potential simplicity. The state-variable model has made possible an accurate analysis and an optimized design of the gain clamped EDFA.
The synthesis of such studies, carried out with my laurea thesis student L. Barbieri, appears in
[BB_jlt_99] ``Design of gain-clamped doped-fiber amplifiers for optimal dynamic performance''
This one is also a theoretical work among the first in its field (a theoretical paper on the same topic appeared on the same number of the J. of Lightwave Technol.). The innovative contribution of the paper is found in the use of techniques typically employed in automatic control theory for the analysis of dynamic systems. Further publications along this line of research are [KBMR_jlt_00] and [KBRM_procj_00] , which deal with the study of long chains of concatenated EDFAs with gain clamping, fed by burst-mode packet WDM traffic.
It is also noteworthy to mention a significant experimental activity in this EDFA field, made possible by the improved optical Lab equipment in the Telecom Department at the University of Parma.
The two experimental publications [BPA_ecoc_99], [PBO_ecoc_99] (of which [PBO_ptl_99] is the corresponding journal version), presented at ECOC 99, stem from the matured understanding of the whole experimental group of the EDFA dynamics, and witness the good synergy between theoretical and experimental research in our Department in this field.
7) Raman Amplifiers |
JOURNAL | CONFERENCE |
1) Theory | |
[BPV_el_01] | |
[BP_tops_01] | [BP_oaa_01] |
2) Burst-mode traffic | |
[BPF_oft_03] | [BP_ofc_02] |
[FBPTSC_rinem_02] (E) | |
[BF_ofc_03] |
1) [BPV_el_01] ``The impulsive pump depletion in saturated Raman amplifiers''
dealing with the analytical calculation of the gain profile of a strongly saturated Raman amplifier with counter-propagating pump, by the impulsive depletion approximation.
This paper first analytically proved - to my knowledge - that the same well-known and simple formulae for the undepleted gain apply even in the case of strong signal-induced pump saturation, provided that the pump power appearing in such formulae be suitably reduced by an amount that can be evaluated through the resolution of a simple transcendental equation.
2) [BP_oaa_01] ``Optimal placement of isolators in Raman amplified optical links''
( and its corresponding journal version [BP_tops_01] ) dealing with the analytical study of signal double Rayleigh backscattering inside the counter-pumped amplifier and the optimization of the isolator placement inside such amplifiers. In this paper it is shown that the optimal position of the isolator is roughly in the middle of the amplified final span of the distributed counter-pumped Raman amplifier.
3) [BP_ofc_02] ``Transient gain dynamics in saturated counter-pumped Raman amplifiers,''
whose novelty consists of a dynamic model of the signal-saturated Raman amplifier, in many respects similar to the one already introduced by the author for the EDFAs. Such model allows again a drastic reduction of the computational complexity of transients in Raman amplifiers, from 2N coupled partial differential equations for an N channel WDM system to a single differential equation for a state variable which physically corresponds to the relative pump depletion sensed by the signals. You can view here the slides of the OFC 2002 presentation .
4) The publication [FBPTSC_rinem_02] presents experimental and simulation results of saturation transients in co-pumped Raman amplifiers, and it shows that such transients occur on much shorter time scales than counter-pumped amplifiers. Such piece of work marks the beginning of a fruitful collaboration with the research group of Prof. S. Selleri of Parma University on experimental verifications of Raman amplifiers.
5)
[BF_ofc_03]
``Transient gain dynamics in saturated Raman amplifiers with
multiple counter-propagating pumps,''
whose novelty consists of an extension of the model in
[BP_ofc_02] to the realistic case
of multiple pumps, and the verification of the limits of the model.
You may take a look at the
slides of the OFC 2003 presentation .
6) [BPF_oft_03]
``Transient gain dynamics in saturated Raman amplifiers,''
is a long paper that synthesizes the conference presentations on transient gain dynamics. New elements not included in the OFC presentations are the application to co-propagating pumps, and a deeper study of the limits of the dynamic model in the case of multiple counter-propagating pumps.
8) Polarization Mode Dispersion (PMD) |
JOURNAL | CONFERENCE |
1) Theory | |
[VB_el_01] | |
[BV_ol_01] | |
[VB_jlt_02] | |
2) Systems | |
[VB_ofc_02] | |
[BV_oft_02] |
1) [VB_el_01] ``Extracting PMD statistics from a single emulated fibre sample''
deals with a novel method to estimate the PMD distribution from frequency measurements on a single fiber sample.
2) [BV_ol_01] ``Statistics of the Jones Matrix of fibers affected by Polarization Mode Dispersion''
(and its extended version [VB_jlt_02] ) deals with the study of the statistics of the Jones matrix of a single-mode optical fiber in the presence of PMD, starting from the stochastic differential propagation equations of the Pauli coordinates of the Jones matrix. As opposed to the vast majority of published papers on PMD, which concentrate on the principal states of polarization as the key analysis tool for systems affected by PMD, we believe that, in the presence of strong depolarization of the principal states, it is way more convenient to study another fundamental parameter of the Jones matrix, namely, the eigenmodes of the extracted matrix. The theoretical foundations behind our view appear in the long paper [BV_oft_02] A. Bononi and A. Vannucci, ``Is there life beyond the principal states of polarization?'' Optical Fiber Technology, 2002.
3) [VB_ofc_02] ``Sensitivity penalty distribution in fibers with PMD: a novel semi-analytical technique''
deals with the use of semi-analytical techniques for the estimation of the outage probability in systems affected by PMD. Even here we intensely utilized the formalism of the extracted eigenmodes.