Jones vector representation for the polarization state of characteristic cases.
\r\n\tThis book plans to focus on elaborating recent advances in the field of electrodeposition with fruitful evidence. The feasibility of electrodeposition routes to fabricate different metals and their derivatives for various uses are to be explained in detail.
",isbn:null,printIsbn:"979-953-307-X-X",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,isSalesforceBook:!1,isNomenclature:!1,hash:"207613e28b728f2b6cd3007e746cdefa",bookSignature:"Prof. Dhanasekaran Vikraman and Dr. K. Karuppasamy",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8532.jpg",keywords:"Topic 1:Thin films\r\nKeywords: Electrodeposition, Anodic deposition, cathodic deposition, Reaction kinetics\r\nTopic 2: Metals and metal oxides \r\nKeywords: II-VI elements, copper oxide, Ferrous oxide, Silver oxide\r\nTopic 3: Chalcogens \r\nKeywords: metal dichalcogens\r\nTopic 4: Alloys\r\nKeywords: Metal alloys, Solid solution, Gold - copper, Iron - Cobalt",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"August 14th 2019",dateEndSecondStepPublish:"September 4th 2019",dateEndThirdStepPublish:"November 3rd 2019",dateEndFourthStepPublish:"January 22nd 2020",dateEndFifthStepPublish:"March 22nd 2020",dateConfirmationOfParticipation:null,remainingDaysToSecondStep:"3 years",secondStepPassed:!0,areRegistrationsClosed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"199404",title:"Prof.",name:"Dhanasekaran",middleName:null,surname:"Vikraman",slug:"dhanasekaran-vikraman",fullName:"Dhanasekaran Vikraman",profilePictureURL:"https://mts.intechopen.com/storage/users/199404/images/system/199404.png",biography:"Dr. Dhanasekaran Vikraman is an assistant professor in the Division of Electronics and Electrical Engineering, Dongguk University, Seoul, Korea. He received his bachelor’s degree from Mannai Rajagopalaswamy Government Arts College, Mannargudi, affiliated with Bharathidasan University, India. He completed his master’s degree and Ph.D. at the Department of Physics, Alagappa University, India. Later, he received a visiting scientist position at KIST, Korea; a Marie-Curie Experienced Researcher fellowship at the Department of Physics, Aristotle University of Thessaloniki, Greece; and post-doc positions at Sejong University and Ajou University, Korea. He has authored more than 180 international journal articles and 3 book chapters and edited several books. He serves as a guest editor and topic editor for various journals.",institutionString:"Dongguk University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"2",institution:{name:"Dongguk University",institutionURL:null,country:{name:"Korea, South"}}}],coeditorOne:{id:"300938",title:"Dr.",name:"K.",middleName:null,surname:"Karuppasamy",slug:"k.-karuppasamy",fullName:"K. Karuppasamy",profilePictureURL:"https://mts.intechopen.com/storage/users/300938/images/system/300938.jpeg",biography:"Dr. K. Karuppasamy is currently working as an Assistant professor at Division of Electronics & Electrical Engineering, Dongguk University-Seoul, South Korea. He received his Ph.D. in the field of nanocomposite electrolytes for lithium-ion batteries in Anna University, Chennai, India in 2014. Later, he completed his Post-Doc positions at Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology Bombay, India, and Department of Chemical Engineering, Sogang University, Seoul, South Korea. His research interests mainly focus on synthesis and characterization of nanostructured materials for energy storage and conversion device applications. 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From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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These signal components arrive with a certain delay, phase, and amplitude difference relative to the LOS component. We will call these signal components multipath components (MPCs) and the phenomena multipath propagation.
Multipath propagation degrades the positioning accuracy. Moreover, in precise applications, multipath errors dominate the total error budget. Despite the different approaches developed, several aspects of multipath propagation are still not fully understood. The generally unknown number of MPCs and their path geometry, the signal characteristics, the diffraction and reflection effects as well as their changing nature together with a complex antenna and receiver design make multipath mitigation very challenging. Furthermore, the site-dependent characteristics of multipath decorrelate the errors caused by multipath propagation at different antenna locations and thus, differential techniques, like e.g. double differences (DD), cannot mitigate it.
The superposition of the MPCs and the LOS signals yields a compound signal at the receiving antenna. Depending on the relative phase between the MPCs and the LOS signal, constructive or destructive interference appears. As a result, during signal tracking the correlation output between the received signal and the local pseudorandom noise (PRN) code replica generated by the receiver is deformed. Since MPCs arrive generally at the receiving antenna with small extra paths, up to 20 m, relative to the LOS signal, the correlation output is biased and the receiver is not able to discriminate between MPC and the LOS signals. This correlation output is the fundamental input for the next iteration of the code and phase tracking loops of the receiver as well as for C/N0 estimation algorithms. As a result, the three GNSS observables code-phase, carrier-phase, and C/N0 are biased by multipath propagation. In this text, errors in code-phase and carrier-phase observations caused by multipath propagation are referred to as code multipath and carrier-phase multipath, respectively, and in general as multipath errors.
In the observation domain, multipath errors are not constant in time. They show a sinusoidal behavior which can be noticed in carrier-phase residuals from Precise Point Positioning (PPP), double differences (DD) or C/N0 time series. This behavior is due to the change of the relative phase between the direct and indirect signals as the satellite vehicle moves above the local horizon of the antenna. The magnitude of these oscillations depends on the relative amplitude of the MPC which varies as geometry changes. The C/N0 observable is the only GNSS observation type in which multipath propagation effects are directly visible without any sophisticated data pre-processing. In contrast, in the phase or code domain, residuals should be analyzed or differences should be formed in order to eliminate all other errors sources. This is one of the main reasons why signal strength measures have attracted much attention in GNSS multipath studies. Since the relative signal amplitude between the LOS and MPC signals plays a key role for the understanding of multipath propagation and also for the magnitude of the multipath error in the GNSS observables, the following contribution focuses on an extended description and proposes an analytical model for modeling GNSS signal amplitudes.
This chapter is structured as follows. A compact overview on different approaches for multipath mitigation or characterization will be presented next. The approaches are categorized into techniques in the observation domain, receiver-internal as well as antenna-related techniques and further methods. Cornerstone methods of each category will be highlighted. In the third section, the multipath phenomenon and its impact on GNSS code, carrier phase and C/N0 will be summarized. Special emphasize is given to the reflection process including signal polarization. An analytical model for GNSS signal amplitude is proposed. The equations for phase and code errors due to multipath propagation are extended so that the signal amplitude can be analytically calculated for each epoch. Finally, results from a dedicated experiment are shown in order to highlight the key features of multipath propagation.
\n\t\tIn the beginning of the 1970s multipath effects on L1 frequency were first studied by [1] and the fundamental relationships between code error due to multipath and the driving parameters were derived. In [2], it was shown that the presence of multipath can be identified by using double differenced phase observations. Since then and during the last 4 decades many researchers have been involved in the characterization and modeling of this propagation phenomenon. A large number of scientific papers have been published on this topic, where different approaches and aspects of the problem have been investigated under certain predefined assumptions. Consequently, the scientific literature on multipath propagation for GNSS positioning can appear very rich for scientists. Despite the large number of different approaches developed, a universal solution of this problem is not achieved until the time of writing. Nevertheless, different promising approaches are under consideration and development.
In almost all textbooks on geodesy or navigation, with very few exceptions (e.g. [3]), multipath propagation is presented in short texts of a couple of pages (e.g. [4, 5, 6]). Most of the time, the phenomenon is explained geometrically, while other physical or electromagnetic properties of the reflected signals are not discussed. The progress in multipath-related studies is documented in various PhD thesis, we cite here exemplarily [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].
A prominent example for the successful reduction of
One of the most popular methods to characterize and quantify code multipath, are the so-called multipath linear combinations [20]. The original code and carrier-phase observations from dual frequency receivers are combined in such a way, that the code multipath can be isolated. Due to its computational simplicity this approach is often used to assess the overall multipath contamination at continuously operating reference stations (CORS), for example, the IGS network, or to characterize the performance of new receivers or new satellite signals, like e.g., the upcoming Galileo or GPS L5. However, it should be noted that the characterization is only valid if no code smoothing was applied.
Contrary to the code observations, the
In post-processing, sidereal filtering or sidereal differencing is often applied. Taking advantage of the sidereal repetition of the GPS orbits, observation or coordinate time series of subsequent days can be subtracted in order to reduce the impact of multipath. Genrich and Bock in [22] showed that a reduction of about 80% can be obtained in this way. However, strictly speaking, each GPS satellite has its own, time-varying orbital period differing up to 10 sec with respect to the nominal sidereal period. Different approaches have been proposed to find the correct individual repeat times cf. [23, 24]. In addition, due to the nodal precession, the satellite ground track deviates over time, so that different reflectors will be illuminated, cf. for exemplary ground track variations [21, 16, 24]. Finally, changing weather conditions like rain or snow will influence the reflection properties of the antenna vicinity, so that the similarity of multipath errors is reduced. All three effects restrict the power of multipath mitigation by sidereal differencing, especially over longer time intervals.
Finally, the analysis of un-differenced carrier-phase residuals from PPP is a useful tool for accessing the impact of multipath effects [25]. For this purpose, the residuals are color-coded and depicted in a sky-plot. The variations of the residuals translate into a sequence of concentric rings in the sky-plot. But, since PPP residuals contain further remaining systematic effects, like e.g., varying tropospheric refraction, averaging strategies may be necessary [26].
Observations of the
\n\t\t\t\t
A lot of effort was given on the
Closely-spaced antennas [12, 51] are also developed for the reduction of multipath errors. They form a type of small antenna arrays. Based on a least-squares adjustment, the multipath relative amplitude α, the multipath relative phase (ΔΦ) as well as azimuth and elevation of the assumed reflector can be estimated [9]. Further developments leads to beam steering and adaptive beam forming antennas or antenna-receiver combination, like e.g., DLR’s GALANT receiver [52]. However, this interesting technology seems to be not mature enough to be installed at GNSS reference stations.
In [53] the concept of
\n\t\t\t\t
In order to get a better understanding of the physical processes involved in the multipath phenomenon, models from
In recent years, the concept of GNSS reflectometry and scatterometry has strongly evolved, cf. e.g., [63, 64, 65, 66, 67, 68, 69]. Here multipath propagation is not considered as bias but as basic information. In the context of GNSS-R, much progress has been made, especially in the mathematical and physical modeling of the wave propagation. Hence, these publications can be a valuable source for mitigating multipath in positioning.
\n\t\tAfter reaching the receiving antenna, the GNSS signals are down-converted from radio-frequency (RF) to an intermediate frequency (IF) by the RF front-end of the receiver. Afterwards, the signal is digitized, down-converted to baseband and correlated with locally-generated replicas of the PRN codes. Then, the result of the correlation is accumulated for a certain time interval. In a typical GNSS receiver architecture, three replicas for each PRN code are generated, the so-called prompt, early and late replicas. The resulting correlation outputs of the prompt replica of the PRN code in the presence of a number of k MPCs are written as in [34, 17]:
\n\t\t\t\twhere A is the amplitude of the direct signal, D is the navigation bit, R is the correlation function, τ is the code tracking error and \n\t\t\t\t\t\t
Based on the correlation outputs (IP and QP) of the prompt correlator, a GNSS receiver is able to calculate the amplitude (\n\t\t\t\t\t\t
Phase error of PLL due to one multipath component
Assuming a perfect tracking (i.e. R=1) and the relationship \n\t\t\t\t\t\t
The compound signal amplitude can be derived as a function of α, ΔΦ and \n\t\t\t\t\t\t
\n\t\t\t\t\t\n\t\t\t\t\t\t
By using trigonometric identities, the expression for the compound signal amplitude is rearranged as:
\n\t\t\t\tAn equation for pseudo-range error (ρ) is given in [13]:
\n\t\t\t\twith δ being the extra path length of the MPC expressed in meters.
From equations (3), (4) and (5) the 90° phase shift between errors in code, amplitude domains and phase domain can be noticed. When ΔΦ=0°/180° and ψ=0, then ρ, \n\t\t\t\t\t\t
The relative phase of each multipath component can be expressed as a function of the extra path length (δ) and the wavelength of the carrier signal: ΔΦ=2πδ/λ, where δ is expressed in meters. Moreover, δ is a function of station height and reflection angle, which in turns depends on the satellite elevation and the orientation of the reflector in space. In the special case of ground multipath, which we will investigate in the following parts of this chapter, the extra path delay is a function of the station height (h) and the satellite elevation (el):
\n\t\t\t\tIn conclusion, it can be stated that some of the introduced equations can be calculated based on the geometry of the scenario. The signal amplitudes for both LOS and MPC are not directly accessible. However, this information or at least the relative amplitude is a crucial part of all introduced formulas. Therefore, in the following an analytical model of the relative amplitude α is developed.
\n\t\t\tIn order to understand the amplitude relation between the LOS and the MPCs, in a first step, the polarization state of a signal is introduced and linked to the right-hand circular polarization (RHCP) of the GNSS signal. According to [70], there are many ways to represent the polarization state of a signal or of an antenna. Some are graphic in nature (e.g. Poincaré sphere) and can be easily visualized, while others can be more appropriate for specific applications, like e.g., investigation of antenna wave interaction (e.g. Stokes parameters and/or complex vector representation).
In the case of a circularly polarized (CP) planar wave, the electric field vector is propagating in a helical way (see Figure 2a), where the projection of the tip of this vector forms a circle in a fixed plane normal to the direction of propagation (z). The clockwise or counter clockwise sense of rotation of the electric field vector looking towards the direction of propagation defines whether the signal is left or right-hand circularly polarized, respectively. At each instant in time, the electric field vector is a combination of two components (see Figure 2 b, c):
\n\t\t\t\twhere E1 and E2 are the amplitudes of the instantaneous electric field [V/m] in x and y directions, respectively, ω is the angular frequency in [radians/sec] and ζ is the relative phase shift [radians] by which the y component leads the x component.
\n\t\t\t\tThe combination of the two electric field components, in horizontal and vertical direction, results in the CP electric field vector, propagating in z direction in a spiroidal way
Each of the two components represents a linearly polarized wave. When \n\t\t\t\t\t\t
Another decomposition of the electric field vector yields orthogonal circular polarization states (LHCP and RHCP). Furthermore, the electric field phasor expressed in terms of CP phasors can be written as [70]:
\n\t\t\t\twhere \n\t\t\t\t\t\t
Finally, the Jones vector is a further way for the representation of signal polarization. According to [72], the Jones vector can be used only for completely polarized signals, e.g. GNSS signals. Equation (10) can be rewritten in complex vector notation as:
\n\t\t\t\twhere the relative phase shift between the two electric field components is ζ=ζx – ζy (see Figure 3). According to [72], this can be further factorized as:
\n\t\t\t\twhich is the so-called Jones vector (\n\t\t\t\t\t\t
\n\t\t\t\t\t\n\t\t\t\t\t\t
where A2+B2=1 and the \n\t\t\t\t\t\t
\n\t\t\t\t\t\n\t\t\t\t\t\t
\n\t\t\t\t\tEq. (13) can be written in a column vector form as:
\n\t\t\t\t\n\t\t\t\t\tEq. (14) completely describes the polarization state of GNSS signals since information of the amplitude of both electric field components as well as their relative phase can be extracted. Particular cases for the Jones vector can be seen in Table 1:
\n\t\t\t\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t|
\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t | \n\t\t\t\t\t
\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t | \n\t\t\t\t\t
\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t | \n\t\t\t\t\t
\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t | \n\t\t\t\t\t
Jones vector representation for the polarization state of characteristic cases.
In multipath propagation, one or multiple reflections as well as diffraction of the transmitted signal may occur. In the following, we focus on the description of the reflection process. In the case of reflection of the incident field, the reflection coefficients will indicate how much the reflected field will be attenuated and how the polarization state of the incident field will be deformed. The reflection coefficients used in this investigation are as in [7]:
\n\t\t\t\tWith
\n\t\t\t\twhere \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t3 | \n\t\t\t\t\t\t\t2*10-5\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t20 | \n\t\t\t\t\t\t\t4 | \n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t30 | \n\t\t\t\t\t\t\t2*10-1\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
Material properties of the exemplary reflectors used in this investigation
The reflection coefficients can also be expressed as circular components of the two orthogonal polarizations (\n\t\t\t\t\t\t
For the RHCP satellite signals, Rco can be associated with the RHCP component, while Rcross represents the orthogonal LHCP component.
In Figure 3a), the circular reflection coefficients are plotted for the L1 carrier. The co-polarized components are plotted with solid lines and the cross-polarized with dashed. The material properties of the exemplary materials (see Table 2) were used for the calculation of the reflection coefficients. The reflection coefficients for a concrete reflector are plotted in black, for a sea water reflector in blue, and for a wet ground reflector in red. They indicate how the electrical field components of the incident field will be changed upon reflection, in terms of magnitude and relative phase.
Looking at reflection coefficients for the different reflection angles in Figure 3a) and for a concrete reflector, it can be seen that when the reflection angle approaches 0°, then the cross-polarized components (LHCP component) tends to 0 and the co-polarized components (RHCP component) is 1, i.e. no loss. When the reflection angle is between 0° and 30°, both components exist but still the co-polarized one is larger in magnitude and the reflected signal is right-hand elliptically polarized (RHEP), with the eccentricity of the polarization ellipse getting bigger. At around 30°, the magnitude of the reflection coefficients is equal. In geodetic literature, this angle is referred to as the Brewster angle [7, 9] for a concrete reflector, although this definition deviates from the common use of the Brewster angle in electromagnetic theory. For these material properties (concrete) and for reflection angle of 30°, the reflected signal is linear polarized. When the reflection angle is between 30° and 90°, the cross polarized component has a higher magnitude and this results in a change of the initial polarization from RH to LH. The eccentricity of the ellipse is then getting smaller as the reflection angle increases. When the reflection angle approaches 90°, the reflected signal is LHCP since the co-polarized components is zero. When the material properties change, then the changes on the reflected field are different. For example, the angle for which the magnitude of the circular reflection coefficients is the same, may vary significantly for different reflectors. Furthermore, small magnitude changes between the two different frequencies (L1 and L2) may occur. These differences may also vary between different reflectors.
\n\t\t\t\tCo- and cross-polarized circular reflection coefficients for L1 (a); and relative phase change between the electric field components due to reflection process for L1 (b). For the exemplary material properties of concrete, sea water and wet ground.
In Figure 3b), the relative phase (ζ) between the field components is illustrated for L1 for the three exemplary material properties. The relative phase between the two components is plotted for all possible reflection angles of the incident field. Without reflection (reflection angle = 0°), the relative phase (ζ) is -90°, which is the relative phase shift for RHCP signals. As the reflection angle increases, ζ is getting smaller and smaller. When the reflection angle is equal to the Brewster angle (for this particular reflector), then ζ=0°, which yields linear polarization. Finally, for reflection angles larger that the Brewster angle, ζ has an opposite sign which indicates that the reflected signal has changed polarization.
\n\t\t\tAccording to [73, 74], the signal after the receiving antenna can be written as:
\n\t\t\t\twhere SR is the received signal, \n\t\t\t\t\t\t
More information on the RHCP gain pattern of the satellite antennas can be found in [75]. The signal ellipticity should not be worst that 1.2 dB for L1 and 3.2 dB for L2, [76].
\n\t\t\t\tThe matrix \n\t\t\t\t\t\t
and for the LOS components as (23b)
where ALOS and AMPC account for the free-space loss attenuation, which for this investigation may be considered equal. \n\t\t\t\t\t\t
where \n\t\t\t\t\t\t
This expression is divided into three subsequent matrix multiplications. First, the selected base of the
Reflection geometry of ground MPC. The electric field components are expressed in spherical coordinates.
The signal amplitudes for the ground-reflected MPC and for the LOS component are calculated as the absolute values of eq. (25) and (26), where the transmitted signal \n\t\t\t\t\t\t
yielding the final formulas for the phase error, for the code error and for the compound signal amplitude (for the case of one ground MPC):
\n\t\t\t\tWhen multiple multipath components are received, the above equations can be rewritten in a similar way as:
\n\t\t\t\tIn a similar sense, eq. (1) and (2) can be also rewritten in order to model analytically the relative amplitude factor α which is present also in these equations.
\n\t\t\tIn order to illustrate the above derived formulas, various experiments were carried out. In the followings we describe one experiment in details. A dedicated experiment was conducted in a controlled environment at the PTB antenna test facility in Braunschweig, i.e. an environment where only one ground reflection would occur (Figure 5). Two antennas were set up with different heights above the reflector. The antennas were spaced about 21.3 m. The observation period lasted for about 7 hours and a cut-off angle of 0° was applied. The data rate was 1 Hz. One pair of AX1202GG LEICA antennas were used together with LEICA GRX1200+GNSS receivers.
\n\t\t\tLeft: Experimental set up of the Leica AX1202GG antenna. Right: The NOV702GG gain patterns [dB] for both orthogonal polarizations (RHCP in blue and LHCP in red) were used to model the antenna. Adopted from [
The following assumptions were used for the simulations: The receiving antenna gain patterns are taken from a NOV702GG antenna assumed to be similar to the AX1202 (see Figure 5). The normalized gain patterns were assumed symmetrical in azimuth. In this way the Jones vector of the receiving antenna for all possible azimuths and elevation angles of the incoming LOS and MPCs GNSS signals can be computed as in eq. (21). The satellite antenna was assumed here perfect RHCP and modeled as:
\n\t\t\t\tFinally, for simplicity the angles ψrec and ψtrans were assumed to be equal and constant over time. Absolute and individual antennas phase center corrections from IfE Hannover were also taken into account during data processing. Furthermore, it is assumed that the ground reflector is a perfect horizontal reflector. Using eq. (27) and (29), the simulated time series for the phase error and the compound signal amplitude (dB) were computed. Under the assumption the values are independent of the satellite azimuth; one exemplary satellite (PRN 12) is plotted covering the whole elevation range. Two different antenna heights above the reflector were used according to the measuring set up. The heights of the two antennas A1, A2 were 1.358 m and 2.053 m, respectively.
In Figure 6, simulated normalized amplitudes [dB] of the MPC, the direct and compound signals are plotted for PRN12 versus satellite elevation for both antennas of the observed baseline. The direct signal’s amplitude is plotted in red, the amplitude of the compound signal in blue, and in black the difference between the direct signal and the compound one is given, thus the impact of the multipath.
\n\t\t\t\tSimulated and normalized amplitude values versus satellite elevation for PRN 12 and reflector made of concrete. a) LOS, ground MPC and compound signal amplitudes for antenna position A2. b) LOS, ground MPC and compound signal amplitudes for the antenna position A1.
The normalized amplitude of the LOS is a smooth curve mainly determined by the gain pattern of the receiving antenna. The resulting multipath effect (black) shows the typical amplitude variations of few dB. Finally, the compound amplitude shows a superposition of both features. Furthermore, a frequency difference of the oscillations can be noticed between the antennas, depending on the height difference. For the maximum elevation angle of 90° PRN 12 has a minimum attenuation of 0 dB. For the same satellite elevation, the amplitude of the ground MPC is almost -32 dB. The reason is that the reflected MPC, at a reflection angle of about 90° and with angle of arrival near -90° with respect to the antenna horizon, is almost LHCP (see Figure 3a) and it is strongly attenuated due to the LHCP gain pattern of the antenna for such angles of arrival (see Figure 5, right side). Thus, the LOS signal is dominating. At elevation angles near 0°, the MPC is still having most of the signal energy in the RHCP component (see Figure 3) and only a small part of it in the LHCP component. The relative amplitude α of the MPC w.r.t. LOS is about 0.6. So in this case, the MPC reflection loss is minimum, the MPC RHCP component is dominating and the antenna applies a very similar gain for both LOS and ground MPC signal components.
In a next step, the impact of different reflector material properties will be investigated. We chose exemplary material properties of perfect, concrete, sea water and wet ground reflectors (see Table 2). Since the impact will be much stronger at low elevation angles, only a part of the arc of PRN 12 is plotted. Figure 7 shows the resulting curves for the compound signal amplitude for two different antenna heights.
\n\t\t\t\tNormalized amplitude in [dB] of the compound signal for different material properties. Zoom at low elevation angles. (a): antenna A2, 2.053 m above the reflector, (b): antenna A1, 1.358 m above the reflector.
The change in the material properties creates mainly a change in the amplitude of the variations. A perfect reflector creates the strongest oscillation. The concrete reflector is also creating MPC with higher amplitude than water reflector and the wet ground reflector. This is because for small reflection angles the material properties of concrete are causing a smoother change of the signal polarizations (from RH to LH) in contrast to the reflections coefficients of the other two reflectors that are creating a more abrupt change (see Figures 3).
\n\t\t\tThe C/N0 values obtained from the RINEX observation files can be compared with the simulated values. In Figure 8 for two exemplary satellites, PRN 12 and PRN 14, the corresponding time series are compared. Here, only the data for the antenna A1 at a height of 1.358 m above the reflector is considered. The reflection properties are modeled for a concrete reflector (see Table 2). Both curves show that the simulations can explain the main features present in the observations. The scale on both axes is the same. However, some disturbances occur which are due to our simplifications, e.g. neglecting a small tilting of the reflector, the approximated satellite and receiver antenna patterns.
\n\t\t\t\tObserved C/N0 values versus simulated signal amplitude for PRN 12 and 14 for antenna A1
Simulated phase error - DD values versus observed phase DD (in blue) formed from PRN 12 and 14. Simulated DD for a concrete reflector are indicated in red and for a perfect reflector in green.
Finally, the carrier phase observations are investigated. To this end, double-differences (DD) are formed between the two antennas on the short baseline. Thus, most distance-dependent systematics as well as the receiver clock errors will cancel out. Again we use PRN 12 and 14. In Figure 9, the phase DD formed by PRN 12 and 14 of the short baseline are plotted together with double-differenced simulated phase error as in eq. (27). The time series of the observed values for an observational period of about 4.5 hours are plotted in blue in Figure 9a). In Figure 9b) the time series is overlaid by the simulated DD phase errors, where the relative amplitude of each multipath component is calculated analytical. In green no reflection loss is considered and in red the material properties of a concrete reflector are used.
\n\t\t\tMultipath propagation is still limiting the accuracy of precise GNSS applications despite the four decades of intensive research in this field. Signal amplitude, both for direct and indirect GNSS signal components that arrive at the receiving antenna with small relative delays, are crucial for the resulting phase error magnitude due to multipath propagation.
After a short introduction and a discussion on the different approaches that can be found in literature, aspects of ray propagation and signal polarization modeling are introduced. Then, an extensive description of the reflection process and its impact on the polarization state of the reflected signals is presented and all involved parameters are characterized. An analytical model for GNSS signal amplitudes is presented after. The model is then integrated into the equations for the phase error and the compound signal amplitude. Simulated and real data from a dedicated experiment are used to highlight the main properties of multipath propagation on the signal amplitude and carrier-phase domains.
\n\t\tThis work has been realized within the BERTA project (50NA1012), funded by the Federal Ministry of Economics and Technology, and based on a resolution by the German Bundestag. The authors thank the BERTA partner Prof. Thomas Kürner (Institut für Nachrichtentechnik, TU Braunschweig) for the fruitful discussions. The authors would like to thank Dr. Thorsten Schrader (PTB Braunschweig) for supporting the experiments at the antenna reference open area test site at PTB.
\n\t\tFor many decades, solitons, namely robust localized waveforms that propagate undistorted in nonlinear dispersive media, have been a topic of particular interest in physics [1, 2] and applied mathematics [3, 4]. These waveforms have a variety of fascinating properties that should be mentioned: due to their particle-like nature, solitons collide elastically, thus preserving their shape after the collision process. In some cases, as e.g. in shallow water, line soliton collisions give rise to the emergence of various wave patterns, including X-, H-, or Y-shaped waves, as well as other, even more complicated, waveforms – see, e.g., Ref. [5] for a set of remarkable examples and observations. Such patterns can be described by multidimensional line soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation, a variant of the KP equation [6] with negative dispersion – as is the case of water waves with small surface tension. In particular, the KP equation can be expressed as:
where
Important to the collision-induced emergence of different types of wave patterns in shallow water is the fact that quasi (line) solitons of the KPII are stable. Furthermore, of paramount importance is the effect of soliton resonances [7, 8, 9, 10], whereby two (or more) colliding solitons resonate under certain conditions, thus creating novel stable structures. It is, thus, not surprising that the collision dynamics of many robust solitons, can give rise to a wealth of complex wave patterns [11, 12, 13, 14, 15, 16] (see also [1]).
The above discussion motivates the following question: can solitons, similar to those governed by the KPII equation and feature rich collision dynamics (as in shallow water waves), be also predicted – and, possibly, observed – in nonlinear optics? In this context, key model is the nonlinear Schrödinger (NLS) equation, which is known to support soliton solutions (see, e.g., Ref. [17] and references therein). As was shown some time ago [18], the
The scope of this work is to review the above ideas and present a variety of soliton solutions and their dynamics in the framework of a defocusing nonlocal NLS model. These solutions stem from the soliton solutions of KP models, which are derived from the nonlocal NLS via a two-step multiscale expansion method. We show that, similarly to the water wave case, the nonlocal NLS can also support complicated wave patterns, arising from “complex interactions” between two or more antidark soliton stripes that are supported in the strong nonlocality regime. In addition, we present other soliton solutions, namely dark lumps (pertinent to the KPI equation), as well as ring dark and antidark solitons (pertinent to the cylindrical KP (cKP),
First, in Section 2, we introduce the nonlocal NLS model, present its continuous-wave (cw) solution, and study its stability. Then, we perform a two-step multiscale analysis. First, at an intermediate stage, we derive a Boussinesq – or a Benney-Luke (BL) [24] – type equation, which is a long-wave, multi-dimensional and bi-directional, shallow water wave model. Then, from the Boussinesq/BL-type equation, we derive its far field, pertinent to the long time behavior, which is the KP equation, both in Cartesian and in cylindrical geometry. The reduction to the KP model allows us to construct approximate soliton solutions of the nonlocal NLS, which are presented in Section 3. In the strong nonlocality regime, we derive antidark soliton stripes, satisfying an effective KPII equation, which are shown to form patterns similar to those observed in shallow water; these include X-, Y, and H-shaped waveforms, as well as other complicated patterns arising from resonant interactions between more than two solitons. We also present dark lump solitons, as well as ring dark and antidark soliton solutions of the NLS, which are numerically found to propagate undistorted in the framework of the NLS model. Finally, in Section 4, we briefly discuss the conclusions of this work.
The model under consideration is a two-dimensional (2D) defocusing NLS, with a nonlocal nonlinearity, which is expressed in the following dimensionless form:
where subscripts denote partial derivatives,
is the Laplacian in Cartesian or polar coordinates respectively, while the parameter
As mentioned above, parameter
while for
Our analytical approach relies on the analysis of the hydrodynamic form of the model. This can be derived upon using the Madelung transformation:
where
where
The simplest, non trivial, solution of Eqs. (2) and (3) is of the form:
which corresponds to the continuous wave (cw) solution
where
The above system can straightforwardly be decoupled as follows: solving Eq. (12) for
Since the above solution is linear we can examine its stability in first order by examining the dispersion relation of plane wave solutions, i.e.,
From the above equation we can obtain the following information. First, it is observed that
where
is the squared “speed of sound”, namely the velocity of linear waves propagating on top of the cw solution. In addition, parameter
and plays the role of an effective surface tension for our original system. Indeed, as discussed in Ref. [21], this can be inferred by the fact that Eq. (17) is reminiscent of the dispersion relation for shallow water waves [4]:
Here,
Obviously, regarding the magnitude of the effective surface tension, one may identify the following two regimes:
Below it will be shown that these two regimes correspond, respectively, to a KP-II and a KP-I equation, with the former giving rise to complex interactions of line solitons (that we will study in detail).
We now proceed by analyzing the fully nonlinear regime. We start by seeking solutions of Eqs. (7)–(9) in the form of the following asymptotic expansions:
where
for the Cartesian and polar geometry, respectively (the angular coordinate
Furthermore, Eq. (14) at
which connect the amplitudes
in Cartesian and polar coordinates, respectively. In addition, Eq. (14) yields:
To this end, combining Eq. (29) with Eqs. (26) and (27), yields the following biderectional nonlinear dispersive wave equation for
where
Using a multiscale expansion method, similar to the one employed in the water wave problem [4], we now derive the KP equation, which is obtained under the additional assumptions of
where the unknown functions
for the Cartesian and polar geometry, respectively. Substituting the expansion (31) into Eq. (30), at the leading-order in
in Cartesian and polar coordinates respectively. The above equations imply that
In addition, at order
for the Cartesian geometry, and
for the polar geometry. Here, it is important to observe the following. Once Eq. (38) is integrated in
with the fields
while in the polar geometry, the relevant equations take the form of the so-called cylindrical KP (cKP), or Johnson’s equations (see Refs. [22, 23]):
It is now convenient to further simplify the equations above, namely the KP and the cKP equations, in order to express them in their standard form [4]. We consider right-going waves in the Cartesian geometry, and radially expanding ones in the polar geometry, and introduce the following transformations,
which cast the respective KP and cKP equations into the following form:
Here, the parameter
It is now clear that, in the Cartesian case, Eq. (45) includes both versions of the KP equation, KPI and KPII [33]. Indeed, for
In other words, for a fixed degree of nonlocality
Once again, it is important to highlight the fact that the existence of these regimes resembles the situation occurring in shallow water. In this context, weak surface tension corresponds to
We now employ the results of the above analysis and construct the solution of our original problem. This can be done as follows. Once a solution of Eq. (45) [or (46)] is known, an approximate solution of Eqs. (2) and (3) has the form:
It is important to note that Eq. (48) describes two different types of solitons, namely dark and antidark ones. Indeed, for
Below, we will use known solutions of the KP and cKP equations and present corresponding solutions of the nonlocal NLS. In addition, we will perform direct numerical simulations to examine the evolution and – in some cases – the interaction dynamics of the approximate soliton solutions (48).
We start with the Cartesian geometry and consider, in particular, the simplest soliton solutions of the KP equation, the so-called line solitons. Specifically, the one-line soliton solution of Eq. (45), traveling at an angle to the
where
A typical single antidark soliton solution of the nonlocal NLS (
In what follows, we choose line solitons with specific parameters, so that their angle during interaction will determine the type of pattern. In the simulations, we evolve the initial configuration up to
We start with the interaction of two solitons, and express the two-line soliton solution of KPII as follows:
with
We first focus on interacting line solitons that form Y- and X-shaped waveforms, and proceed with more complex ones later in the text. In Figure 2, we show a typical Y-shaped pattern, whereby the resonant two-soliton interaction gives rise to the emergence of one soliton, with its maximum height being four times that of the incoming solitons (this resonant Y-shaped solution was originally found by Miles [7, 8]). This feature is clearly observed in the figure. Parameter values in this case are:
A typical Y-type interaction,
Next, we proceed with the X-type interactions, which can be discriminated according to the resulting “stem”: an interaction with a
X-type interactions,
In the same category falls an X-type interaction with a long stem, with the stem height being lower than the tallest incoming line soliton, which we refer to here as an H-interaction. A typical such interaction is given in Figure 4, and parameter values:
An H-type interaction,
An immediate generalization of the above is a three-wave interaction, where now
and
so that the resulting interaction is as in Figure 5, for
A 2–3 wave interaction,
More intricate structures may still form whose mathematical description is more complex. Such novel weblike structures for the KP equation, termed
where this new
Here superscripts denote differentiation with respect to
In particular, for line solitons these are defined as:
where
A typical 2–2 wave evolution for
A typical 3–3 wave evolution, for
We start with a 2–2 interaction, by defining the matrix
and the relative real parameters
Similarly, for a 3–3 wave interaction one has:
and
It is important to note here that the matrices
So far, we have focused on structures obeying the effective KPII equation. These structures, namely the antidark stripe solitons, are quasi one-dimensional (1D) states, which are stable. The respective states that satisfy the effective KPI equation, namely dark stripe solitons, are expected to be transversely unstable – see, e.g., Ref. [36] for a stability analysis of KP line solitons.
Nevertheless, the KPI equation, that corresponds to the weakly nonlocal regime, supports stable, purely 2D, soliton states known as
where
Similarly to the case of antidark stripe solitons, once Eq. (60) is substituted into Eq. (48) gives rise to the approximate dark lump soliton of the nonlocal NLS. Using the relevant analytical expression, it is straightforward to prepare the corresponding initial condition by setting
A typical dark lump solution, at
We now turn our attention to the cKP Eq. (46), and focus on radially symmetric solutions that do not depend on
where
where
Notice that this approximate solution is a ring-shaped solitary wave, on top of cw background, which can be either dark (for
As in the Cartesian case, we have performed direct numerical simulations to examine the evolution of ring-shaped solitons in the nonlocal NLS model. First, in Figure 9, shown are 3D plots depicting the profiles of the ring dark solitons and ring antidark solitons. Here, we use
(Color online) typical ring dark (left) and anti-dark (right) solitons.
In addition, in Figure 10, we show the evolution of the ring solitons depicted in Figure 9. Once again, it is observed, that both types of ring solitons propagate undistorted up to end of the simulation (
(Color online) the evolution of the dark (top) and anti-dark (bottom) ring solitons of
In conclusion, we have studied a 2D defocusing NLS model with a spatially nonlocal nonlinearity. The considered model is relevant to optical beam propagation in a variety of physical settings, including thermal media, plasmas, and nematic liquid crystals. We have analytically treated this nonlocal NLS by means of a two-stage multiscale analysis. This led to a Boussinesq/Benney-Luke-type equation, and a KP equation (in both Cartesian and polar geometries), which are models arising in the context of shallow water waves. This analysis revealed that the nonlocal NLS is characterized by an
Apart from the type of KP (KPII or KPI), the effective surface tension also controls the type of soliton: antidark (dark) solitons on top (off) the cw pedestal can only be formed in the strong (weak) nonlocality regime. We have thus found a number of different soliton states that can be supported by the nonlocal NLS. These include antidark stripe solitons, dark lump solitons, as well as ring dark and antidark solitons (in the polar case). We also used numerical simulations to examine the existence, evolution, as well as interaction dynamics. For the latter, we focused on the case of antidark solitons, which were shown to form a plethora of patterns coming out of their collisions. These include Y-, X-, H-shaped waveforms, as well as more complicated patterns involving more than two solitons; all these, have been observed in shallow water [5], a fact that highlights the deeper connection between water waves and light! In addition, we examined, by means of direct simulations, the existence and propagation of other soliton states, namely dark lump solitons, as well as ring dark and antidark solitons. We found that all these states do exist in the framework of the nonlocal NLS, and can propagate undistorted up to times set by the simulations.
At this point, it is also relevant to comment on the possibility of observing the predicted patterns in real experiments. In that regard, first we consider the case of antidark solitons. The observation of these states took place some time ago, as it was first reported in Ref. [44]; this fact indicates that the necessary experimental setting is already available. As concerns the experimental set up needed for the observation of the Y-, X- and H-waves and other related patterns, it may be similar to that of Ref. [44]. In particular, one may employ at first a cw laser beam, which is split into two parts via a beam-splitter. One branch goes through a cavity system to form a pulse (as happens in typical pulsed lasers); this pulse branch undergoes phase-engineering, i.e., passes through a phase mask so that the characteristic phase jump of the antidark soliton is inscribed. Then, the cw and the phase-engineered pulse are incoherently coupled inside the nonlocal medium, e.g., a nematic liquid crystal. This process forms one antidark soliton, as in Ref. [44]. To observe the Y-, X- or H-patterns predicted above, two such antidark solitons have to be combined inside the crystal. The angle between the two incident beams, which should be appropriately chosen so that a specific pattern be formed, can be controlled by a rotating mirror in one of the branches of the beam-splitter.
As concerns the possibility of the experimental observation of the other soliton states that were predicted by our analysis, we note the following. First, regarding the weakly localized dark lump solitons, they can be experimentally observed as follows. In a real experiment, it is straightforward to create a dark soliton by using, e.g., a proper phase mask (see, e.g., the review [45] and references therein); this can also be done even in the case of
The obtained results relied on the formal reduction of the nonlocal NLS model to the KP (cKP) equation. It would be interesting to investigate if other types of nonlocality, arising, e.g., in the context of chromium Bose-Einstein condensates [47], or in defocusing colloidal media [48] can support relevant soliton states. Such studies are in progress and pertinent results will be presented elsewhere.
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\n\nWe will send you your price quote and after it has been accepted (by both the author and the publisher), both parties will sign a Statement of Work binding them to adhere to the agreed upon terms.
\n\nAt this step you will also be asked to accept the Copyright Agreement.
\n\n5. LANGUAGE COPYEDITING, TECHNICAL EDITING AND TYPESET PROOF
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\n\nAfter we receive your proof corrections and a final typeset of the manuscript is approved, your manuscript is sent to our in house DTP department for technical formatting and online publication preparation.
\n\nAdditionally, you will be asked to provide a profile picture (face or chest-up portrait photograph) and a short summary of the book which is required for the book cover design.
\n\n6. INVOICE PAYMENT
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\n\n7. ONLINE PUBLICATION, PRINT AND DELIVERY OF THE BOOK
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\n\nIf you feel that IntechOpen Compacts, Monographs or Edited Books are the right publishing format for your work, please fill out the publishing proposal form. For any specific queries related to the publishing process, or IntechOpen Compacts, Monographs & Edited Books in general, please contact us at book.department@intechopen.com
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