in order to satisfy the rapidly increasing demands on higher network performance , e.g. , in terms of higher quality - of - experience ( qoe ) , quality - of - service ( qos ) , and greater mobile data traffic , wireless communications using ghz radio ( often referred to as millimeter wave : mmwave communication system ) is proposed as a promising alternative to the existing lower band ( around 2 ghz ) communication . to date , there have been several academic and industrial bodies participating in the standardization of mmwave technology , e.g. , ieee 802.11ad task group , ieee 802.15.3 task group 3c , and wireless gigabit alliance ( wigig ) .there are also numerous efforts in academia that are dedicated to network design and related applications for the next generation wireless communications . however , due to the severe path loss and high oxygen absorption effect in the ghz radios , it is not yet clear if such networks can meet the desired delay and reliability requirements . furthermore , performance evaluations for the ghz wireless networks are of great importance in the system design , optimization and implementation , if the performance requirements , by the end users , are to be guaranteed . unfortunately , the traditional performance analysis methodologies , e.g. , classical queuing theory , may not be useful in analysing such networks unless some conformity assumptions on the arrival and service process are enforced .such assumptions may jeopardize the integrity of the obtained results in reference to the real system .to date , most of related study regarding the performance of ghz wireless networks departs from the perspective of physical layer ( phy ) , where a series of phy techniques , e.g. , signal modulation , channel coding and beamforming , were discussed .however , a network layer delay and backlog analysis of ghz wireless communications in terms of the channel parameters , which is crucial in evaluating the performance of qoe- or qos - driven wireless communications , does not exist .this motivates us to develop an analytical model and methodology to analyze the performance of such networks .we propose an alternative and more suitable methodology to analyze the performance of 60 ghz networks based on a network calculus approach .this approach was originally proposed by cruz for worst - case analysis of deterministic networked systems .network calculus is a system theoretic approach for the analysis of communication networks , where the network operation is described using the  algebra .a network service element , e.g. , a wired or wireless link , is modelled as a  linear , time - invariant ( lti ) system and the input / output relationship is governed by the lti system theory in the  algebra . meanwhile, the original theory for deterministic systems analysis has been extended to probabilistic settings in order to model and analyze stochastic networks .one approach to extend network calculus to stochastic settings is the mgf - based network calculus that was proposed by which extends initial results regarding the mgf arrival and service curves suggested in .more recently , a wireless network calculus based on the  algebra was proposed . in this approach , in order to simplify an otherwise intractable analysis , the network model is transferred into an alternative analysis domain ( referred to as snr domain ) by using the exponential function .probabilistic end - to - end performance bounds are provided for homogeneous multi - hop channels and numerical results for the case of multi - hop rayleigh fading channels is presented .  the mgf - based network calculus was originally developed to enable the modelling of the multiplexing gain of many independent stochastic flows .it was later used in the analysis of wireless systems by characterising the system s service process in terms of the mgf of a markov channel model of the wireless link .although network calculus has been around for some years , its applications to wireless networks are fairly recent , and the investigated scenarios are limited to rayleigh fading channels .in contrast to the existing work on analyzing the performance of wireless networks , the ghz radio channel present several challenges .first , unlike conventional wireless communications in the bands below ghz , which experience small - scale fading effects , the ghz fading channels mainly experience log - normally distributed shadowing fluctuations .second , due to the particular fading characteristic of ghz channels , the  based wireless network calculus presented in is not applicable to the analysis of this network algebra in snr domain , which is transferred from the bit domain , is not feasible . ] .therefore , we opt to use the mgf - based network calculus instead .finally , the service process characterization of the 60 ghz channel , i.e. , computing the mgf of the channel capacity , is generally intractable which makes it difficult to obtain a closed - form exact solution .we use clever manipulations and bounding of the resulted expression to obtain a tractable bound on the desired service process . in this work ,we provide analytical probabilistic backlog and delay bounds for the ghz wireless access network .this in turn corresponds to networked applications with loss- and latency - sensitivity respectively . to do that , we first derive a simplified closed - form expression for the network service curve , in terms of the fading model and parameters of the ghz channel .we then evaluate the performance of the proposed model using the mgf - based network calculus approach and derive the desired probabilistic bounds .finally , we discuss the impact of a series of factors , such as arrival rate , system gain and channel fading characteristic , on network performance .our novel application of mgf - based network calculus to the analysis of 60 ghz wireless networks reveals useful insights for the network operation and the effect of different channel parameters on network performance .it also demonstrates the ability of such networks to satisfy the stringent service requirements of modern networked applications .the remainder of the paper is organized as follows .basics for mgf - based stochastic network calculus are given in sec .[ sec : ii ] .probabilistic backlog and delay bounds for the ghz wireless network are derived in sec .[ sec : iii ] .numerical results are presented in sec . [sec : v ] .finally , conclusions are drawn in sec .[ sec : vi ] .assuming a fluid - flow , discrete - time queuing system with a buffer of infinite size , and given a time interval , , we define the non - decreasing ( in ) bivariate processes and as the cumulative arrival to , departure from and service offered by the system as illustrated in fig .[ fig : queuing model ] .we further assume that and are stationary non - negative random processes with for all .in addition , throughout the paper we only consider causal systems , i.e. , .we denote by the backlog ( the amount of buffered data ) at time .more details and the proofs for the following presented fundamental results can be found in .[ fig : queuing model ]    in network calculus , given the service process , the departure process is related to the arrival process via the convolution denoted by ( ) .another important operation in the system theory is the , deconvolution denoted by ( ) .here , and are respectively defined as  where and are arbitrary bivariate functions . for time varying systems ,we refer to a network element as _ dynamic server _ if for all it offers a time varying service that satisfies the following input - output inequality  the above holds with strict equality when the system is linear .one typical example of is a work - conserving link with a time - variant capacity , with the available service during interval .  an important and frequently used traffic envelope that considers traffic burstiness as well as the sustainable arrival rate is the affine envelope , i.e. , .this envelope was later extended to probabilistic settings as the traffic envelope , in the sense that it provides , for a given , a linear bound on the mgf of the arrival process , i.e. , \le \rho(\theta)\cdot ( t - s)+ \sigma(\theta).\ ] ] the cumulative arrival and service processes are given in terms of their instantaneous values , and respectively during the time slot , as and for all .for wireless networks , represents the instantaneous fading channel capacity during the time slot , which is given by ` nats ' per second , where is the channel bandwidth and is the instantaneous snr at the receiver during time slot .   for a given queuing system with cumulative arrival and departure and for ,the backlog and virtual delay in a first - come - first - serve ( fcfs ) scheduling system are respectively given by and where the inequalities are obtained by substituting in the above expressions and then using the definitions of and .in deterministic network calculus , only the worst - case upper bound is considered which result in pessimistic performance bounds that does not capture the multiplexing gain effect in packet networks .furthermore , deterministic analysis can not be used for wireless network performance analysis since is the only possible deterministic lower service curve due to the possibility of outage in wireless networks . the stochastic analysis on the other hand , can surpass the above limitations . in the probabilisticsetting , the backlog and delay bounds with respect to and are expressed as respectively , where and denote the probabilistic backlog and delay bounds , accordingly associated with and that are defined as _ violation probabilities_. the probabilistic bound is usually computed using the distributions of these processes , i.e. , in terms of the arrival and service processes mgfs or their mellin transforms . in general , the mgf - based bounds are obtained by applying the chernoff s bound , that is , given a random variable , it is known that \triangleq e^{-\theta x}\mathbb{m}_{x}(\theta ) ] and denote the expectation and the moment generating function ( or the laplace transform ) of , respectively , and is an arbitrary non - negative free parameter . for the bivariate stationary processes , we define its mgf as \ , .\ ] ] likewise ,  ] , where  and .first , we briefly show that  ] , i.e. , normally distributed random variable , with mean and standard deviation . its c.d.f . is given by where and the complementary error function is given by , and the process is stationary and independent in time , we immediately have by applying lemma [ lemma : inverse moment ] and substituting for the snr distribution function .generally , the bound is tighter at smaller .as the value of grows larger , the tightness of the bound becomes more sensitive to the step size parameter .thus , the selection of appropriate value for is subject to the range of interest of the parameter and it provides a trade - off between the quality of the obtained bounds and computation requirements . based on the previous descriptions of the arrival and service processes , theorem [ theorem : probabilistic bound ] below provides the desired probabilistic backlog and delay bounds for the i.i.d . case .[ theorem : probabilistic bound ] let the arrival and service processes be respectively characterized by , and as described in and in theorem [ theorem : upper bound of m_s ] .then , and are backlog and delay bounds satisfying , whenever the stability condition holds .substituting and from and theorem .[ theorem : upper bound of m_s ] , respectively , in , we obtain where and . here , equality is obtained by performing the change of variables and rearranging terms . in let .the last step is obtained by evaluating the geometric sum .the sum converges only when is satisfied .the desired backlog and delay bounds in are obtained by substituting with in and with to obtain and , respectively .in this section , we provide numerical results and simulations for selected system configurations .the objective is two - fold :  1 .  to verify the obtained performance bounds ; 2 .  to investigate and discuss system performance trends in order to draw conclusions and insights regarding the 60 ghz network performance .  in what follows ,simulations are realized via matlab .we assume constant rate arrival with rate and burst size .time is slotted with intervals of second .the rest of the system parameters are listed in table [ tab : system parameter ] . throughout this section ,the system gain is given as db , which is immediately obtained by applying and the system setting ..system parameters [ cols="^,^,^",options="header " , ]   [ tab : system parameter ]  fig .[ fig : fig1_backlog ] depicts the probabilistic backlog bound compared to simulation . the arrival rate is set to , and gbps .we use two different values for the precision parameter , and .as expected , the backlog increases as the arrival rate increases .the figure shows that the computed bounds capture the exponential decay of the backlog distribution .furthermore , the bounds become asymptotically tighter , i.e. , as .  the derived result provided by lemma [ lemma : inverse moment ] approaches its actual expectation value when . from fig .[ fig : fig1_backlog ] , we find that , the curves for and  gbps by adopting perfectly match with the actual values , i.e. , results by .nevertheless , for gbps and , the bound overestimates the simulated backlog distribution significantly which suggests that the precision of is not sufficient in this case , i.e. , more precision is required for highly utilized systems .this suggests a trade - off between computation complexity of the bounds and their tightness . on the other hand , for practical system operation , a precision of is able to provide acceptable backlog bound for moderate utilization , e.g. , gbps .v.s . targeted backlog bounds , compared to simulation with db and db , for different .,width=326 ]   v.s .targeted delay bound , compared to simulation with gbps and db , for different . , width=326 ]  in fig .[ fig : fig2_delay ] , we investigate the delay performance for different shadowing standard deviations which is given by the channel characterization in .we use in the computation of the delay bounds .the figure shows that the analytical delay bounds predict the tail distribution of the simulated delay and are are all asymptotically tight .furthermore , the delay performance of the system deteriorates as the standard deviation of the shadowing gets larger .  from the above, we can conclude that the expressions given by lemma [ lemma : inverse moment ] and theorem [ theorem : probabilistic bound ] provide asymptotically tight bounds for the probabilistic backlog and delay of  ghz wireless systems .this motivates the use of the computed analytical bounds to analyze and optimize the performance of realistic and more complex topologies of the ghz wireless communication networks .  in fig .[ fig : fig3_backlog ] , with respect to a given violation probability , we illustrate the probabilistic backlog bound against the arrival rate for four different channel shadowing distributions .the figure shows that when the arrival rate increases , the precision becomes insufficient for rigorous analysis , as it results in gross overestimation of the backlog bound compared to the finer precision , i.e. , , case .this indicates that smaller values of are required in order to maintain the precision as the network load becomes higher .we also notice that the shadowing effect , in terms of , on network performance increases rapidly as the network load increases .. arrival rate , with and  db , for different .,width=326 ]   v.s .system gain , with , and  gbps , for different .,width=326 ]  according to , the snr is characterized by the system gain .varying can be obtained by manipulating the system configurations , such as antenna gains , transmit power and tx / rx distance .[ fig : fig4_delay ] plots the probabilistic delay bound against system gain for different shadowing distributions . in general , by increasing , the delay bounds remarkably decay , while the benefits gradually diminish . for curves regarding different , an interesting finding from fig .[ fig : fig4_delay ] is that , a smaller value of surprisingly gives a higher delay bound within the region of lower system gain , i.e. , when db , while it also has a faster decay when increasing .on the other hand , when the system gain is moderate or higher , a larger will definitely result in more significant performance deterioration in terms of delay bound .  from fig .[ fig : fig3_backlog ] and [ fig : fig4_delay ] reflects the importance of considering the arrival and fading channel characteristics as well as system gain in the design and implementation of qos - driven ghz wireless networks .we investigate the probabilistic backlog and delay bounds of the  ghz wireless networks .we provide a mgf - based closed - form expression for the cumulative network service process .subsequently , probabilistic upper bounds on the backlog and delay of the  ghz network are derived using mgf - based network calculus approach .the analytical bounds are validated using extensive simulations .the obtained results demonstrated the asymptotic tightness of the analytical performance bounds .we use the obtained analytical bounds to study the impact of several network components , e.g. , arrival rate and system gain , on the network performance .the results quantify the potential of these parameters for performance improvement . in light of the above, we believe that the proposed analytical approach will have broad applications in designing , optimizing and dimensioning  ghz networks under performance constraints . the approach can be extended to address many of the issues related to the next generation wireless communications in terms of performance improvements and qos guarantees . for our future work, we are addressing service guarantees in multi - hop heterogeneous  ghz backhaul networks as well as transmit power management and optimization in such networks .e. hossain , m. rasti , h. tabassum , and a. abdelnasser , `` evolution toward 5 g multi - tier cellular wireless networks : an interference management perspective , '' _ ieee wireless communications _21 , no . 3 , pp .118127 , 2014 .s. geng , j. kivinen , x. zhao , and p. vainikainen , `` millimeter - wave propagation channel characterization for short - range wireless communications , '' _ ieee transactions on vehicular technology _ , vol .58 , no . 1, pp . 313 , 2009 .choi , m. piz , and e. grass , `` performance evaluation of gbps ofdm phy layers for 60-ghz wireless lan applications , '' in _ieee 20th international symposium on personal , indoor and mobile radio communications ( pimrc2009)_.1em plus 0.5em minus 0.4emieee , 2009 , pp .16571661 .l. rakotondrainibe , y. kokar , g. zaharia , g. grunfelder , and g. e. zein , `` performance analysis of a 60 ghz near gigabit system for wpan applications , '' in _ieee 21st international symposium on personal , indoor and mobile radio communications ( pimrc2010)_.1em plus 0.5em minus 0.4emieee , 2010 , pp .10381043 .m. fidler , `` wlc15 - 2 : a network calculus approach to probabilistic quality of service analysis of fading channels , '' in _ ieee global telecommunications conference ( globecom2006)_.1em plus 0.5em minus 0.4emieee , 2006 , pp . 16 .k. mahmood , a. rizk , and y. jiang , `` on the flow - level delay of a spatial multiplexing mimo wireless channel , '' in _ ieee international conference on communications ( icc2011)_.1em plus 0.5em minus 0.4emieee , 2011 , pp .m. williamson , g. athanasiadou , and a. nix , `` investigating the effects of antenna directivity on wireless indoor communication at 60 ghz , '' in _ the 8th ieee international symposium on personal , indoor and mobile radio communications ( pimrc97 ) _ , vol .2.1em plus 0.5em minus 0.4emieee , 1997 , pp .635639 .t. s. rappaport , g. r. maccartney , m. k. samimi , and s. sun , `` wideband millimeter - wave propagation measurements and channel models for future wireless communication system design , '' _ ieee transactions on communications _63 , no . 9 , pp . 30293056 , 2015 .m. fidler , `` an end - to - end probabilistic network calculus with moment generating functions , '' in _14th ieee international workshop on quality of service ( iwqos2006)_.1em plus 0.5em minus 0.4emieee , 2006 , pp . 261270 .g. yang , j. du , and m. xiao , `` maximum throughput path selection with random blockage for indoor 60 ghz relay networks , '' _ ieee transactions on communications _ , vol .63 , no .35113524 , oct 2015 .

Abstract: to meet the ever - increasing demands on higher throughput and better network delay performance ,  ghz networking is proposed as a promising solution for the next generation of wireless communications . to successfully deploy such networks , its important to understand their performance first . however , due to the unique fading characteristic of the  ghz channel , the characterization of the corresponding service process , offered by the channel , using the conventional methodologies may not be tractable . in this work , we provide an alternative approach to derive a closed - form expression that characterizes the cumulative service process of the  ghz channel in terms of the moment generating function ( mgf ) of its instantaneous channel capacity . we then use this expression to derive probabilistic upper bounds on the backlog and delay that are experienced by a flow traversing this network , using results from the mgf - based network calculus . the computed bounds are validated using simulation . we provide numerical results for different networking scenarios and for different traffic and channel parameters and we show that the 60 ghz wireless network is capable of satisfying stringent quality - of - service ( qos ) requirements , in terms of network delay and reliability . with this analysis approach at hand , a larger scale 60 ghz network design and optimization is possible .   ghz ; moment generating functions ; backlog ; delay ; upper bound .


recent development of cold polar - molecule sources promises to reveal many interesting , and hitherto unexplored , molecular interaction dynamics .the permanent electric dipole moment possessed by polar molecules provides a new type of interaction in the ultracold environment . this electric dipole - dipole interaction ( andcontrol over it ) should give rise to unique physics and chemistry including novel cold - collision dynamics  and quantum information processing  .  to date , cold polar - molecule samples have been produced most successfully via three different mechanisms : buffer gas cooling  ; photo- and magneto - association  ; and stark deceleration  .buffer gas cooling achieves temperatures below 1 k through thermalization of molecules with a he buffer .this technique produces relatively large densities (  ) of polar ground - state molecules .however , cooling below mk has not yet been achieved because the he buffer gas has not been removed quickly enough for evaporative cooling  .photoassociation achieves the lowest molecular temperatures of these techniques (  ) , but is limited to molecules whose atomic constituents are amenable to laser - cooling .furthermore , molecules in their ground vibrational state are not readily produced , yielding species with relatively small effective electric dipoles , although this problem can be overcome with more sophisticated laser control techniques  .stark deceleration exists as an alternative to these methods as the technique employs well - characterized supersonic beam methods  to produce large densities of ground state polar molecules (  ) , albeit at high packet velocities .one limitation of this technique for trapping of decelerated molecules is an observed drastic loss of slowed molecules at very low mean velocities in both our own group s work and the berlin group of g. meijer  .we address this problem in this article .  as the leading method for producing cold samples of chemically interesting polar molecules ,stark deceleration has generated cold samples of co  , nd  , oh  , ybf  , h  , nh  , and so  , leading to the trapping of both nd and oh  .given the importance of stark deceleration to the study of cold molecules , it is crucial that the technique be refined to achieve maximum deceleration efficiency . in this work ,we provide a detailed description of processes that limit the efficiency of current decelerators and propose methods for overcoming them .we propose possible solutions to the parametric transverse / longitudinal coupling loss originally highlighted in ref . , as well as elucidate a new loss mechanism unique to producing the slowest molecules .we restrict the described simulations and theory to stark decelerated , ground - state oh radicals , as the supporting experimental data was taken with this molecular species .however , the loss mechanisms described herein are not specific to oh , and represent a general limitation of current stark decelerators .this reduces the efficiency of stark decelerators for trapping cold polar molecules .this article is organized as follows .section ii describes the mechanisms responsible for molecular loss at low velocities .sections iii , iv , and v present methods of producing molecular packets at intermediate velocities ( m / s ) without the distributed transverse / longitudinal coupling losses of ref .however , these methods exacerbate the problem of low - velocity loss .therefore , we propose a new decelerator design in section vi that exhibits gain over conventional stark deceleration to velocities as low as 14 m / s .in much of the previous work on stark deceleration , it is assumed that all motion parallel and transverse to the decelerator axis is stable up to some maximum excursion velocity and position from the beam center  , enabling the derivation of an analytical solution to predict stable - molecule phase - space area .however , there are several important instances where the assumptions of this model become invalid . in the case of very slow molecules ( m / s ) , , where is the effective electric dipole moment and is the mass of the given molecule . ]we observe two distinct phenomena leading to reduced decelerator efficiency at the final deceleration stages : transverse overfocusing and longitudinal reflection .transverse overfocusing occurs when the decelerated molecules speed becomes so low that the decelerating electrodes focus the molecules too tightly ( transversely ) and they either make contact with the electrodes or are strongly dispersed upon exiting the decelerator .longer decelerators tend to exacerbate this effect due to the fact that molecules can travel at low speeds for many stages . nonetheless , there are several motivating factors for constructing a longer decelerator .first , longer decelerators allow less energy per stage to be removed and consequently lead to larger longitudinal phase - space acceptance .second , a longer decelerator may allow deceleration of molecules possessing an unfavorable stark shift to mass ratio .we will discuss critical issues for use of such long decelerators for slow molecule production .a second low - velocity effect , which we have denoted longitudinal reflection , " is a direct result of the spatial inhomogeneity of the electric field at the final deceleration stage . as highlighted in the context of transverse guidance in ref . , the longitudinal potential is largest for those molecules passing in the transverse dimension nearest to the decelerator rods .however , the decelerator switching sequence is generally only synchronous to a molecule on - axis traveling at the mean speed of the packet . as a result ,when the mean longitudinal kinetic energy of the slowed packet becomes comparable to the stark potential barrier at the last stage , molecules off - axis can be stopped or reflected , resulting in a spatial filtering effect .furthermore , the longitudinal velocity spread of the molecular packet at the final stage , if larger than the final mean velocity , can lead to reflection of the slowest portion of the packet .it is important that the phenomena of overfocusing and longitudinal reflection be addressed since molecule traps fed by stark decelerators require slow packets for efficient loading .  to illustrate these low - velocity effects , the number of phase - stable molecules predicted by three - dimensional ( 3d ) monte carlo simulation is shown in fig . [ cm2loss](a ) as a function of stage number for increasing phase anglewe define the deceleration phase angle , , exactly as in previous publications , where  .the length of one slowing stage is given by ( 5.5 mm for our machine ) , while the molecule position between successive stages is denoted as .we define to be exactly between two slowing stages , therefore , ( bunching ) yields no net deceleration .phase angles satisfying lead to deceleration of the molecular packet , while the maximum energy is removed for .the 3d simulation results displayed in fig . [ cm2loss](a ) include both longitudinal and transverse effects .the molecules have an initial velocity ( ) of 380 m / s , corresponding to the mean velocity of a molecular pulse created via supersonic expansion in xe .all simulations and experimental data hereafter possess this unless otherwise noted .as expected , a higher phase angle leads to a smaller number of decelerated molecules .however , there is a sharp loss of molecules in the last several deceleration stages for the highest phase of .this value of produces a packet possessing a final velocity ( ) of 14 m / s .this loss is attributed to transverse overfocusing and longitudinal reflection .these distinct effects are illustrated in fig .[ cm2loss](b ) , which displays the transverse ( trace 1 ) and longitudinal ( trace 2 ) fractional loss rate of molecules traversing the final three slowing stages at , m / s .the switching time for each stage is denoted by a vertical dashed line , which is labeled by the corresponding stage number .longitudinal reflection of molecules is clearly the dominant loss mechanism for the lowest final velocity shown in fig .[ cm2loss](a ) . nonetheless , there also exists a non - negligible rise in transverse losses at the final stage .that is , because the molecular beam is moving very slowly in the last few deceleration stages , the transverse guiding fields of the decelerator electrodes have a greater focusing effect on the molecules ( see eq . 3 of ref . ) and focus the molecules so tightly that they collide with a deceleration stage and are lost . in the case of our decelerator , this leads to loss of 20% of the decelerated molecule number between ( 50 m / s ) and ( 14 m / s ) .such a dramatic loss is not predicted by analytical theory , as the stable phase - space area decreases by only % over this range of .this number is calculated directly after the decelerator is switched - off and is thus an upper bound , since experiments employing these cold molecules require them to travel out of the decelerator where transverse spread can lead to dramatic loss of molecule number .  experimental evidence of this sudden decrease in molecule number at very low velocities is given in fig .[ tofs=1 ] , which displays time - of - flight ( tof ) spectra along with corresponding monte carlo simulation results at various phase angles .the decelerated oh molecules are in the weak - field seeking state .the first two quantum numbers of the state denote its hyperfine components , while the third number indicates the parity of the state in the absence of electric fields .figures [ tofs=1](b ) and ( c ) illustrate progressive time delays of the slowed molecular packets from the background of non - synchronous molecules .figures [ tofs=1](d ) and ( e ) display only the decelerated packets , and the vertical axes of these plots are magnified by a factor of 100 .the dual peaks correspond to molecular packets loaded into two successive stages , and the later peaks represent the intended final velocities . the total detected and simulated molecule numbers are plotted in fig .[ tofs=1](f ) as a function of final velocity , along with the theoretically expected decelerator efficiency ( dashed line )  .note that the sudden population decrease in both simulation and experimental results is not reflected in the one - dimensional theoretical model , which does not account for the transverse dynamics or field inhomogeneities that cause such behavior .this effect is detrimental to the production of dense samples of cold molecules .  to remove the overfocusing effect at low velocities , the transverse focusing of the last several decelerator stagesneeds to be reduced .different types of transverse focusing elements may be inserted into the deceleration beam line to compensate for this phenomenon .this idea is discussed in detail in sections iv - vi .we note that the proposed solutions , while successful in addressing the detrimental longitudinal / transverse coupling effects , do not mitigate the problem of longitudinal reflection at low velocities .coupling between transverse and longitudinal motion throughout the deceleration sequence invalidates the assumptions made in ref .the fact that the transverse guidance of the molecular beam comes from the same electrodes that provide the deceleration means that the longitudinal and transverse motions are necessarily coupled .while this phenomenon is well understood in the field of accelerator physics  , it was first pointed out in the context of stark deceleration in ref .the result can be seen in the left column of fig .[ pscompare ] , where the longitudinal phase space of oh packets is shown versus increasing phase angle . in fig .[ pscompare ] , the dark lines represent the separatrix , partitioning stable deceleration phase space from that of unstable motion as calculated from eq . 2 in ref .each dot represents the position in phase space of a simulated molecule . in the absence of coupling between the longitudinal and transverse motions, one would expect the entire area inside the separatrix to be occupied .therefore , the structure in these graphs is evidence of the importance of the transverse motion .  in the left column of fig .[ pscompare ] , the coupling of longitudinal and transverse motions is responsible for two effects : first , in the center of the stable area at the synchronous molecule the density of stable molecules is less than in the surrounding area .this is because molecules that oscillate very near the synchronous molecule experience little transverse guiding .this effect is not dramatic and is only discernable for an exceedingly large number of stages .furthermore , this effect is even less important for the increased phase angles typically used for deceleration , since for these switching sequences the synchronous molecule experiences more of the transverse guiding forces than it does during bunching .the second effect , which is much more evident , is the absence of molecules at intermediate distances from the synchronous molecule as shown in the left column of fig [ pscompare ] .this so - called ` halo ' is due to parametric amplification of the transverse motion and is similar to the effects seen in cold molecule storage rings  as well as charged particle accelerators  .essentially , the longitudinal oscillation frequency of a molecule in this region is matched to the transverse oscillation frequency , leading to amplification of the transverse and longitudinal motion and consequent loss .there is a compromise between decreasing longitudinal phase - space area and increasing transverse guidance for increasing . to demonstrate this effect ,we decelerate molecules to a fixed and vary the phase angle used to reach this velocity .this is done either by changing the voltage applied to the decelerator rods or by modifying the effective length of the decelerator itself for each of interest .the experimental data shown in fig . [ vlchange ] is the result of varying the voltage applied to our decelerator rods ( squares ) and the effective decelerator length ( circles ) .both lowering the decelerator voltage and using shorter lengths of the decelerator for slowing requires increasing to observe the same of 50 m / s .we are able to effectively shorten the decelerator by initially bunching the packet for a given number of stages before beginning a slowing sequence .note that we use bunching to remove any transverse / longitudinal couplings during these first stages , then switch back to slowing for the remainder of the decelerator .the parameter signifies the mode of decelerator operation and is previously defined in ref . as well as in section iv . the phase stable region of for is entirely contained within that of at , therefore no artifacts from initial velocity filtering are present in this data .we observe that , contrary to the predictions of the one - dimensional theory  , a higher phase angle can lead to greater decelerator efficiency up to some maximum .this is a direct consequence of distributed transverse / longitudinal couplings illustrated in fig .[ pscompare ] . at even larger phase angles ,the longitudinal phase - space acceptance becomes a limiting factor .the labels next to each data point correspond to either the voltage applied to the decelerator rods ( squares ) or the number of utilized slowing stages ( circles ) . figure [ vlchange ] further illustrates that the transverse / longitudinal couplings outlined by ref . reduce decelerator efficiency , and are highly dependent on phase angle .the coupling between longitudinal and transverse motion is detrimental to efficient operation of a stark decelerator , reducing the total number of decelerated molecules .this effect will be even worse for decelerating molecules with an unfavorable stark shift - to - mass ratio .fortunately , the transverse and longitudinal motions can be decoupled by introducing a transverse focusing element to the deceleration beam line that overwhelms the transverse focusing provided by the deceleration electrodes in analogy to the focusing magnets of charged - beam machines .this technique also has the advantage of providing a larger stable region in the transverse phase space which further enhances the decelerated molecule number .the remainder of this manuscript discusses methods of implementing a transverse focusing element to decouple the longitudinal and transverse motion .sections iv and v describe methods that can be implemented by modifying the timing sequences of present decelerators with moderate success at intermediate .section vi presents an improved design for a stark decelerator that solves this problem of distributed longitudinal and transverse loss and also reduces the previously described overfocusing losses at the final stage .the simplest method for introducing a transverse focusing element to the decelerator beam line is to let the molecules fly through an energized deceleration stage without removing the field . in this manner ,molecules experience the transverse focusing of the entire stage without their longitudinal motion affected .traditional longitudinal phase stability requires the switching of the fields to occur on an upward slope of the molecular potential energy , i.e. , faster molecules are slowed more while slower molecules are slowed less than the synchronous molecule . hence , it is necessary to de - sample the bunching switching rate by an odd factor ( 3,5,7 ... ) : the so - called decelerator overtones  . for convenience ,we define the quantity , where is the synchronous molecule velocity and the switching speed is given as the stage spacing divided by the switching time - interval .reference  considered only the bunching case . in this work ,we generalize to the case of actual deceleration . however , the above definition of is still valid .that is , is constant despite the fact that both and vary when . with this definitionwe see that traditional deceleration can be described by , while the method of de - sampling the switch rate by a factor 3 is described by .these two methods of deceleration can be seen in figs .[ decelerationschemes ] ( b ) and ( c ) , where their respective switching schemes are shown for . by switching at one - third the rate , the molecule packet flies through a deceleration stage that is energized and experiences enhanced transverse guiding .longitudinal phase space simulations of slowing at various phase angles are shown in the right column of fig .[ pscompare ] .no structure is present in these plots . also , the region of longitudinal phase stability for at reduced compared to .this is because the maximum stable velocity , as calculated from eqs . 2 and 6 of ref . , depends on the spacing between deceleration stages as , and thus , the separatrix velocity bound is reduced by a factor of .nonetheless , the absence of coupling to the transverse motion leads to a larger number of molecules for the case shown in the uppermost panel of fig .[ pscompare ] . in a given decelerator, slowing requires a factor of three higher phase angle than to reach the same final velocity . as illustrated in fig .[ pscompare ] , this fact severely limits the practicality of as a deceleration scheme , as it implies a dramatic reduction in velocity acceptance at the highest phase angles .figure [ tofs3 ] shows tof data ( upper curves ) taken in our 142-stage decelerator for with increasing phase angle along with the results of 3d monte carlo simulations ( lower curves ) .qualitatively , the tof signal is no different than for .  shown in fig .[ s3s1compare](a ) are experimental decelerated molecular packets for ( upper ) and ( lower ) versus increasing phase angle .for each successive packet , the phase angle increases by in the range , while the phase angle increases by . in this manner , the packetsare decelerated to roughly the same velocity .there is a slight difference at the highest phase angles shown because the total number of stages in our decelerator ( 142 ) is not an exact multiple of 3 . in fig .[ s3s1compare](b ) the de - convolved total molecule number for each of these packets is plotted versus final speed . while the method dominates over for small phase angles , its effectiveness decreases as the deceleration becomes more aggressive by 224 m / s , the molecule number is already larger than that of .this behavior is expected since the phase angle used to decelerate to 224 m / s is , while the required phase angle is only . although the longitudinal phase bucket does not exhibit structure , it is so much smaller in enclosed area than the that its total molecule number is smaller .the simulation results of fig .[ s3s1compare](c ) and the theory results of fig .[ s3s1compare](d ) support this description .figure [ s3s1compare](c ) shows that , even at , the transverse loss rate per stage is at all times greater for than . however , the calculated longitudinal phase - space acceptance of fig .[ s3s1compare](d ) mirrors the behavior observed experimentally in fig .[ s3s1compare](b ) when the initial points are scaled to the experimental ratio of 2.75 .this scaling accounts for the aforementioned ` halo ' in the slowed packet , which persists relatively unchanged over the range of phase angles used ( ) .the fact that the theory curves of fig .[ s3s1compare](d ) cross at a higher velocity than the data of fig .[ s3s1compare](b ) suggests there is increased transverse guiding of slowing at high phase angles .nevertheless , even with 142 stages of deceleration , is unfavorable for velocities below 224 m / s due to reduced stable phase - space area .  at this point, one would expect that operation at lower phase angles would permit realization of the gain produced by the method .this can be accomplished by naively using a longer decelerator . a simulation of this kind is presented in fig .[ s=3stagenumber ] , which plots the number of molecules present after deceleration at versus final velocity .is chosen because it produces the most gain over in our deceleration experiments . ]the number next to a data point represents the number of deceleration stages used .initially the decelerated molecule number is relatively flat versus final velocity .however , after about 500 stages ( 180 m / s ) , the number of decelerated molecules begins to decrease and dramatically falls off after 550 stages ( 150 m / s ) .very few molecules survive below 100 m / s .this is because for the decelerated molecules must fly through an entire stage while experiencing a guiding force in only one dimension [ see fig .[ decelerationschemes](c ) ] .once the molecules are at slower speeds they can spread out in one transverse dimension or be over - focused in the other and collide with the rods . as the mean kinetic energy of the slowed packet becomes comparable to the full potential height, the packet can be nearly stopped as it traverses the intermediate charged stage .this has two consequences : ( 1 ) longer transit time leading to more intense transverse focusing ; and ( 2 ) velocity filtering of the low - speed packet as slower molecules are longitudinally reflected from this potential .the transverse loss responsible for the extreme drop in molecule number for deceleration also occurs in traditional deceleration , but to a lesser degree .because of this decrease in molecule number at low speeds , the usefulness of slowing with is generally limited to experiments that do not require the lowest velocities , such as microwave spectroscopy and collision experiments  .we note also that our simulations predict no low - velocity gain when using slowing sequences containing combinations of deceleration at and bunching at .a natural extension of the above overtone deceleration is the use of what we have termed a modified decelerator overtone , " denoted by an additional plus sign , i.e. , .deceleration in this manner is shown in fig .[ decelerationschemes](d ) for . in this method , deceleration proceeds similarly to conventional .however , sequences yield confinement of the packet in both transverse dimensions .this is achieved by charging all slower rods for the period in which the synchronous molecule is between switching stages . in order to minimally disrupt the longitudinal dynamics of the synchronous molecule , the second set of slower rodsis charged only when the molecule is at the peak of the longitudinal potential from the first rod set .the packet then traverses two charged rod pairs before reaching the next switching stage , at which point the rod pair that was originally charged is grounded .while this does create a slight anti - bunching effect , i.e. , molecules in front of the synchronous molecule gain a small amount of energy , it provides an extra stage of transverse guidance in comparison to .experimental results of this method of slowing are shown in fig .[ s=3+loading ] for comparison to deceleration using both and . fig .[ s=3+loading](a ) displays a unique consequence of the switching sequence , where operation at = 0 leads to deceleration .figure [ s=3+loading](b ) shows that operation with provides slightly more molecules than , but the loss of molecules due to decreased longitudinal velocity acceptance remains a problem .the increase in molecule number for over is due to the extra stage of transverse guidance , which for the higher - velocity packets we measured leads to a larger transverse acceptance .  to determine whether the extra stage of transverse guidance would counter the overfocusing effects, we perform simulations of deceleration at = 20 for a varying number of deceleration stages. the results of these simulations , shown in fig .[ s=3+number ] , are similar to the results for .namely , as the decelerator length is increased and the molecules speed is reduced , there is a marked molecule number loss for velocities below 200 m / s . in our simulations, we could not observe any molecules below 100 m / s .again , transverse overfocusing and longitudinal filtering of the molecular packet by the deceleration electrodes are responsible for large losses in the decelerator , and this suggests that measures beyond modified switching schemes are required to overcome these loss mechanisms .in addition to modifying the timing sequences of existing decelerators , it is possible , and perhaps preferable , to uncouple the longitudinal motion inside the stark decelerator from the transverse motion by redesigning the decelerator electrode geometry .one simple redesign , which we call the quadrupole - guiding decelerator ( qgd ) , is shown in fig .[ alternativesdschematic](a ) . in this decelerator , a quadrupole - guiding stage ( fig .[ alternativesdschematic](b ) ) is interleaved between each deceleration stage . while it may not be necessary to have a quadrupole - guiding stage between each deceleration stage ( especially in the beginning of the decelerator ) , it simplifies the analysis and will be used here .the switching of the electric fields inside a qgd is similar to a traditional decelerator operated with .figure [ alternativesdschematic](c ) shows the potential energy experienced by a molecule decelerated at and is represented by a thick black curve . in this panel ,the quadrupole - guiding electrodes are omitted for clarity .note that the quadrupoles are always energized and their center coincides with the position , while occurs between the deceleration electrodes .deceleration with a qgd enjoys the same longitudinal phase - stability as a traditional decelerator . in a qgd , the maximum stable excursion position and velocity will differ from that of the traditional decelerator because the decelerating electrodes are most likely further apart . in other words , to prevent high - voltage breakdown , the quadrupole stages require the same inter - stage spacing as deceleration stages in a traditional decelerator .thus , the decelerating electrodes for a qgd will be twice as far apart as in our traditional decelerator , which possesses an inter - stage spacing of mm . since the dependence of the decelerating force on is less steep , the shape of the stable longitudinal phase space will change . understanding the shape of the stable longitudinal phase area is crucial for predicting the performance of the qgd , andcan be derived by examining the longitudinal forces inside the qgd .shown in fig .[ qgdlongforces](a ) is the on - axis stark shift of an oh molecule in the state inside the unit cell , defined as 3 deceleration stages of the qgd .the solid line is the stark shift due to the slowing stage centered at 11 mm , while the dashed line is the stark shift of the stages which will be energized when the fields are switched .the subtraction of these two curves , shown in fig .[ qgdlongforces](b ) as a solid line , is the amount of energy removed each time the fields are switched , .we represent as sum of sine - functions  where we have used the definition of the phase angle .a fit of the first three terms of this equation to the actual for deceleration stages spaced by 11 mm is shown as a dashed line in fig .[ qgdlongforces ] , resulting in the fit values  ,  , and  . using this fit, we derive the equation of motion of the molecules about the synchronous molecule position as    where we have used the excursion of the molecule from the synchronous molecule .the maximum stable forward excursion of a non - synchronous molecule is exactly the same as a traditional decelerator and is given as  we calculate the work done in bringing a molecule starting at this position with zero velocity to the synchronous molecule position as  )d(\delta\phi ) .\label{qgdwork}\ ] ]  integrating this equation and setting it equal to the kinetic energy yields the maximum stable excursion velocity : where is now in radians .using eqs .[ qgdn2]- [ qgdvmax ] it is possible to solve for the longitudinal separatrix , which separates stable deceleration from unstable motion inside the decelerator . these separatrices are shown ( thick black lines ) along with the results of monte carlo simulations of a qgd in the left column of fig .[ qgdpstof ] for successive phase angles .the longitudinal phase space is shown with each dot representing the position of a stable molecule .the lack of structure inside these separatrices is evidence of the lack of coupling between the transverse and longitudinal modes . the right column , which shows simulated tof curves , reveals a single stable peak arriving at later times as is increased .these simulations are for a single state of oh and do not exhibit the large background contribution of the other states of oh present in experimental tof data .  from the comparison of the simulations with the analytical results represented by the separatrices, we see that the simple theory of eqs .[ qgdn2]- [ qgdvmax ] is quite accurate in describing the longitudinal performance of the qgd .thus , by numerically integrating the area inside these separatrices , we can predict the longitudinal performance of the qgd relative to the traditional decelerator . as seen in fig .[ qgdps](a ) , the energy removed per stage of the qgd is less steep with because the decelerator stage spacing ( in this simulation ) is twice that of a traditional decelerator . for this reason, qgd deceleration with the same as in a traditional decelerator leads to a faster beam , given the same number of deceleration stages . when comparing the longitudinal acceptance of the two types of decelerators it is important to take the limit of high phase angles where both values of converge .nonetheless , the qgd shows significant gain over traditional deceleration as shown in fig .[ qgdps](b ) .this gain is primarily due to the increased physical size of the stable longitudinal phase space resulting from the larger deceleration stage spacing .it is important to note that since fig .[ qgdps ] compares only the total area inside the separatrix and deceleration does not completely fill this area due to the coupling effects this gain is actually an underestimate of the qgd longitudinal performance .furthermore , these graphs do not include transverse focusing effects , which can only be properly included through detailed simulation .the results of monte carlo simulations including these transverse effects are shown in fig .[ qgdsnloss ] .the number of decelerated molecules versus final speed is plotted for both the traditional decelerator operating at and the qgd operating at two different quadrupole rod voltages , kv and kv . while the qgd initially delivers more molecules , once the molecules are decelerated below 100 m / s , the decelerated molecule number falls off abruptly .this behavior is expected since , for these simulations , the voltage on the quadrupole - guiding stages is held constant throughout the slowing sequence . as detailed in eq .( 3 ) of ref . , the focal length of a transverse guiding element is directly proportional to the molecular kinetic energy .therefore , as the mean speed of the packet is decreased , the molecules are overfocused and collide with the decelerator electrodes .this can be prevented by lowering the voltage on the quadrupole - guiding stages during the deceleration process .figure [ loadingcurve ] displays simulation results of deceleration with this dynamically scaled voltage compared to slowing at = 30.43 . for this simulation ,the quadrupole voltages are scaled by after each deceleration stage , where is the instantaneous packet velocity directly following each stage switch .the exponent of 0.875 is found empirically to produce the most gain at m / s .for ease of simulation , the transverse forces are scaled by whenever the molecules are closer to a quadrupole - guiding stage than to a deceleration stage .while this may be a poor approximation at the lowest speeds , it will likely lead to an underestimate of the decelerated molecule number since the transverse guidance of the quadrupole - guiding stage extends beyond this regime . even if it leads to an overestimate , proper control of the quadrupole voltagesmay compensate any overfocusing introduced by the decelerating elements . as seen in fig .[ loadingcurve ] , dynamically controlling the voltage of the quadrupole - guiding stages leads to a factor of 5 increase in decelerated number for larger velocities ( m / s ) and delivers about 40% more decelerated molecules than traditional deceleration provides at the lowest final speeds ( 14 m / s ) . because the voltages applied to the quadrupole - guiding stages are relatively low, dynamic control of them should be possible using an analog waveform generator and high - voltage amplifier .it should be noted that the optimal voltage scaling may vary among decelerators since the real focal length depends sensitively on the electrode construction , and at low speeds the transverse focusing of the decelerator electrodes becomes significant .this is , presumably , because the overfocusing of the decelerator electrodes can be compensated to a certain degree by injecting molecules which are already slightly overfocused . in other words , two focusing stagescan overcome the overfocusing of a single stage .thus , it may be possible to use adaptive algorithms to optimize the quadrupole voltage or change the design of the decelerating electrodes so that they provide less transverse focusing , and maximize the number of decelerated molecules beyond what is reported here  .one important advantage of the qgd over traditional decelerators is its inherent ability to support more deceleration stages .because the molecules experience a tunable transverse focusing element after each deceleration stage , there is little loss in efficiency by extending the number of deceleration stages .in fact , as seen in fig .[ qgdsnloading ] , there is essentially no loss until the molecules are decelerated to the lowest speeds previously discussed .this low - velocity loss is due to the aforementioned transverse overfocusing and longitudinal reflection .the former loss mechanism may be overcome , while the latter presents a fundamental limit .even with this loss , the qgd outperforms both ( fig .[ s=3stagenumber ] ) and ( fig .[ s=3+number ] ) .thus , the qgd is an ideal decelerator for more efficiently producing cold molecules and , perhaps more importantly , the ideal decelerator for slowing molecules with poor stark shift to mass ratios , like h and so .in summary , we identify several loss mechanisms observed in the operation of stark decelerators and perform initial experiments and detailed monte carlo simulations to address them . while the use of decelerator overtones yields improvement over deceleration at high to intermediate speeds ( m / s ) , the loss at very low velocities remains problematic .the qgd solves the problem of coupling between the transverse and longitudinal motions inside a stark decelerator by introducing independent transverse focusing elements . by dynamicallycontrolling the focal length ( voltage ) of these guiding elements , large improvements ( factor of 5 ) in deceleration efficiency can be achieved for m / s , and gain of is predicted for the lowest velocities .furthermore , it appears that with dynamic control of the guiding stage focal length there should be no limit to the length of decelerator that can be built .this enables the deceleration of molecules with a poor stark shift to mass ratio .however , we note that none of the techniques described in this article mitigate the longitudinal low - velocity loss due to reflection , which appears to be a fundamental component of stark deceleration . building upon the strong correlation between simulation and experimental results , we are confident that the simulations presented in this work provide a solid guideline for future implementations of stark deceleration .we wish to thank h. l. bethlem and s.y.t .van de meerakker for useful discussions .we also thank m. yeo , h. lewandowski , and d. nesbitt for reading the manuscript .this work is supported by doe , nist , and nsf .b. l. lev is an nrc postdoctoral fellow .

Abstract: the transverse motion inside a stark decelerator plays a large role in the total efficiency of deceleration . we differentiate between two separate regimes of molecule loss during the slowing process . the first mechanism involves distributed loss due to coupling of transverse and longitudinal motion , while the second is a result of the rapid decrease of the molecular velocity within the final few stages . in this work , we describe these effects and present means for overcoming them . solutions based on modified switching time sequences with the existing decelerator geometry lead to a large gain of stable molecules in the intermediate velocity regime , but fail to address the loss at very low final velocities . we propose a new decelerator design , the quadrupole - guiding decelerator , which eliminates distributed loss due to transverse / longitudinal couplings throughout the slowing process and also exhibits gain over normal deceleration to the lowest velocities .   = 1


subject to the effects of wind , a flat sandy surface is unstable and evolves into a regular periodic pattern of wavelength of the order of 10 cm and height of a few centimeters ( see fig .[ death - valley ] ) .a slight sand accumulation on the surface tends to expose grains on the upwind stoss - side to the action of the wind , and shelter the grains on the downwind lee - side .dislodged grains at the stoss tend to move toward the lee of the sand accumulation , the initial perturbation is amplified , and small ripples develop , which migrate in the direction of the wind .smaller ripples travel faster than larger ripples since the migration velocity of ripples depends on the amount of grains being transported during migration so that small ripples merge with the larger ones . due to ripplemerging , the wavelength of the ripples grows in time as observed in the field and in wind tunnel experiments .  due to the action of the wind , grains fly above the bed and strike the ground at small angles and with velocities given by the wind velocity . successive impacts of grains are called _saltation_. as a result of bombardment of saltating grains , a number of ejected grains are generated from the static bed which subsequently move by surface creep .these grains ( called _ reptating _ grains ) move distances smaller than the typical saltating jump ( which is controlled by the wind velocity ) until they are captured and become part of the bed .climbing of ripples as seen in fig .[ diagram ] is described by a vector of translation which has two components : the component in the horizontal direction is the rate of ripple migration across the sediment surface , and the component in the vertical direction is the net rate of deposition , defined as the rate of displacement of the sand surface in the vertical direction ( see fig .[ climb]a ) .the angle of climbing ( in fig .[ climb]a ) defines different types of lamination structures ( see chapter 9 ) .if the angle of climb is smaller than the slope of the stoss , then the structures are called `` subcritically '' climbing ripples , and these are the sedimentary structures which are the focus of this study .  due to the climbing of ripples and segregation of grains during deposition and transport ,ripple deposits develop lamination structures which are later preserved during solidification of the rock .the basic types of the smallest stratification structures in climbing ripples and small aeolian dunes relevant to this study are summarized by hunter ( see also ) . in this paperwe will focus on two lamination structures which are commonly found in sedimentary rocks :  * inverse - graded climbing ripple lamination : one of the most common lamination structures which are formed because grains composing the ripples differ in size .large ejected grains travel in shorter trajectories than small grains , so that small grains are preferentially deposited in the trough of the ripples while large grains are deposited preferentially near the crest . due to this segregation effect ,the migration of a single ripple produces two layers of different grain size parallel to the climbing direction of the ripple as shown in fig .[ climb]a .this lamination structure is called inverse - graded according to the nomenclature of hunter since in a pocket of two consecutive layers formed by the climbing of a single ripple , the layer of large grains is above the layer of small grains ( fig . [climb]a ) .* cross - stratification : migration of ripples also produces successive layers of fine and coarse grains not in the direction of climbing but parallel to the downwind face of the ripples ( see figs .[ diagram ] and [ climb]b ) .these structures called cross - stratification or foresets are also coarser toward the troughs as opposed to the segregation in inverse - graded climbing ripples .the instability giving rise to aeolian ripple morphologies has been the subject of much study .the classic work of bagnold has been followed up by a number of studies .modern models of wind ripple deposits are usually defined in terms of the splash functions introduced by haff and coworkers ( see also for a review ) in order to model impact processes .these models have successfully reproduced the instability leading to ripple deposits when the sand grains are of only one size .when such models are defined for two species of grains differing in size , they result in patterns which resemble very closely the stratigraphic patterns found in inverse grading climbing ripple structures .it was in this context that forrest and haff proposed that grading changes in ripple lamination are related to fluctuations in wind or transport .however , anderson and bunas found using a cellular automaton model that inverse - graded ripple lamination is due to different hopping lengths of small and large grains .these models do not include the interactions between the moving grains and the static surface i.e. , grains are assumed to stop as soon as they reach the sand surface which are expected to be relevant for the dynamics in the rolling face of the ripples .recent studies by terzidis _ have reproduced the ripple instability using the theory of surface flows of grains proposed by bouchaud and collaborators .this theory takes into account the interaction between rolling and static grains .moreover , recent studies of avalanche segregation of granular mixtures show dramatic effects when such interactions are taken into account .  in this article, we use this formalism to include also the segregation effects arising when considering two type of species of different size and shape .we first formulate a discrete model for two - dimensional transverse aeolian climbing ripples which incorporates simple rules for hopping and transport . then, we incorporate the different properties of the grains , such as size and roughness , and we show that pattern formation in ripple deposits arises as a consequence of `` grain segregation '' during the flow and collisions .specifically we show that three segregation mechanisms contribute to layer formation during ripple migration ( fig .[ mechanism ] ) :  * size segregation due to different hopping lengths of small and large grains : the hopping length of the large reptating grains is smaller than the hopping length of small grains . *size segregation during transport and rolling : larger grains tend to roll down to the bottom of troughs while small grains tend to be stopped preferentially near the crest of ripples .* shape segregation during transport and rolling : rounded grains tend to roll down easier than more faceted or cubic grains , so that the more faceted grains tend to be at the crest of the ripples , and more spherical grains tend to be near the bottom .  as a result of a competition between these segregation mechanisms a richer variety of stratigraphic patterns emerges : inverse - graded ripple lamination occurs when segregation due to different jump lengths dominates , and cross - stratification ( and normal - graded ripple lamination as well ) occurs when size and shape segregation during rolling dominates .thus , a general framework is proposed to unify the mechanisms underlying the origin of the most common wind ripple deposits . in the case of inverse - graded climbing ripple laminationwe confirm a previously formulated hypothesis by bagnold and anderson and bunas that the lamination is due to hopping induced segregation due to collisions . regarding the origin of cross - stratification structures , we show that they arise under similar conditions as in recent table - top experiments of avalanche segregation of mixtures of large faceted grains and small rounded grains poured in a vertical hele - shaw cell .  in what follows wefirst define the model for the case of single species of grains .we show that the model predicts the formation of a ripple structure , and propose a simplified continuum theory to understand the onset of the stability leading to ripple structure .then , we generalize to the case of two type of species of grains differing in size and shape which give rise to the segregation mechanisms and the lamination structures seen in aeolian climbing ripples .we start by defining the model for the case of one type of grain in a two - dimensional lattice of lateral size with periodic boundary conditions in the horizontal -direction .our main assumption is to consider two different phases : a reptating or rolling phase composed by grains moving with velocity by rolling , and a static phase composed by grains in the bulk .we also consider a curtain of external saltating grains which impact at randomly chosen positions on the static surface from the left to the right at a small angle to the horizontal ( fig .[ climb]a ) .shadow effects are considered by allowing only impacts with ballistic trajectories which do not intersect any prior portion of the surface . upon impacting on the sand surface , a saltating grain dislodges grains from the static surface , and jumps a distance after which the saltating grain is incorporated to the reptating phase .the grains dislodged by the saltating grain jump a distance . upon reaching the surface ,the dislodged grains form part of the reptating phase and move with velocity .  at every time step ( )only one reptating grain ( in contact with the surface ) interacts with the static grains of the bulk according to the angle of repose  ; the remaining reptating grains move a distance to the right .the angle of repose is the maximum angle at which a reptating grain is captured on the sand bed .if the local coarse - grained angle of the surface is smaller than the angle of repose , , the interacting reptating grain will stop and will be converted into a static grains . if the angle of the surface is larger than the angle of repose , , the reptating grain is not captured and moves to the right a distance with the remaining reptating grains but ejects a static grain from the bulk into the reptating phase .the model predicts the formation of a ripple structure as seen in aeolian sand formations .the onset of the instability leading to ripples occurs with ripples of small wavelength .however , due to the fact that smaller ripples travel faster than larger ripples since smaller ripples have less amount of material to transport merging of small ripples on top of large ripples is observed .this gives rise to the change of the characteristic wavelength as a function of time , as is observed in wind tunnel experiments , as well as in field observations .we find that our model predicts an initial power law growth of the wavelength of the ripples ( see fig .[ wavelength ] ) .the wavelength of the ripples seems to saturate after this power law growth to a value determined by the saltating length .moreover , we also observe a subsequent growth of the wavelength ( not shown in the figure ) up to a new saturation value close to a multiple of the saltation length .this process continues and a series of plateaus are observed until one large ripple ( or dunes ) of the size of the system is formed .the series of plateaus was also observed in a recent lattice model of ripple merging .however , we believe that the existence of plateaus might be due to an artifact of the discreteness of the lattice , and may have no physical meaning : the rules associated with the derivatives of the sand surface break down when the local slope is bigger than a given value determined by the discretization used to define the slope of the surface . a similar artifact was observed in a discrete model of ripple morphologies formed during surface erosion via ion - sputtering in amorphous materials .moreover , we will see that when we introduce two species of grains , the wavelength saturates at a constant value and the plateaus disappear .  in fig .[ wavelength ] we compare the prediction of our model with the available experimental data from field and wind tunnel experiments .we observe a fair agreement between our model and the experiments .however , we believe that this agreement is not conclusive .in fact , the same experimental data can be also fitted with the same accuracy by a logarithmic growth of the wavelength of the ripples as shown recently by werner and gillespie , who proposed a discrete stochastic model of ripple merging and found a logarithmic growth of the wavelength ( see also ) .the onset of the instability leading to the initial ripple structure can be studied using the continuum theory proposed by bouchaud __ to study avalanche in sandpiles .recently , the set of coupled equations for surface flow of grains of has been adapted to the problem of the ripple instability . here, we propose a version of this theoretical formalism suitable to the physics of our model to study the initial ripple formation .let describe the amount of reptating grains in the rolling phase and the height of the static bed at position and time , which is related to the angle of the sand surface by for small angles .our set of convective - diffusion equations for the rolling and static phases are the following :      [ rh ]  where is the drift velocity of the reptating grains along the positive axis taken to be constant in space and time , and a diffusion constant .the interaction term takes into account the conversion of static grains into rolling grains , and vice versa .we propose the following form of consistent with our model :   + \gamma [ \theta(x , t ) - \theta_r ] r(x , t ) , \label{gamma}\ ] ]  where and are two constants with dimension of velocity and frequency respectively . is proportional to the number of collisions per unit time of the saltating grains with the sand bed , and also to the number of ejected grains per unit time , while is the number of collisions per unit time between the reptating grains and the sand bed when they creep on the downwind side of the ripple .the first term in ( [ gamma ] ) takes into account the spontaneous ( independent of ) creation of reptating grains due to collisions of the saltating grains a process which is most favorable at angles of the bed smaller than .the second term takes into account the interaction of the reptating grains with the bed of static grains : the rate of the interaction is proportional to the number of grains interacting with the sand surface . rolling grains become part of the sand surface if the angle of the surface is smaller than the repose angle ( `` capture '' ) , while static grains become rolling grains if is larger than ( `` amplification '' ) .higher order terms are neglected since we are interested in the linear stability analysis determining the origin of the ripple instability .insight into the mechanism for ripple formation is obtained by studying the stability of the uniform solution of eqs .( [ rh ] ) :   we perform a stability analysis by looking for solutions of the type : and .the linearized equations for and are      [ bar ]  where the migration velocity of the traveling wave solution is which indicates that the ripple velocity is proportional to the number of collisions per unit unit of saltating grains with the sand bed , and to the number of ejected grains per collision .  by fourier analyzing eqs .( [ bar ] ) we find a set of two homogeneous algebraic equations for and with non - trivial solutions only when the determinant of the resulting matrix is zero .we obtain a dispersion relation where the two branches correspond to the solutions of the resulting quadratic equation for .then we take the limit , which corresponds to the physical fact that the translation velocity ( [ trans ] ) is much smaller than the rolling velocity of the individual grains ( see ) , and we arrive to the following dispersion relation for ( gives rise only to stable modes ) :   = \frac { -(v_m / v)~ ( \gamma \theta_r v^2)~k^2 }  { ( \gamma\theta_r)^2 + k^2 ( 2 \gamma\theta_r d + v^2 ) + d^2 k^4 } + o (  \frac{v_m}{v})^2.\ ] ]   the final wavelength will be determined by higher order nonlinear terms arising from a more complicated interaction term than the one used in ( [ gamma ] ) . as mentioned above , using the full discrete model we find that after the appearance of initial small undulations , the wavelength of the ripples grows due to ripple merging . in the discrete model, the wavelength seems to saturate to a finite value , although this value seems to be determined by the finite size of the simulation system ( we use periodic boundary conditions in the horizontal direction ) .next we generalize the model to the case of two type of species differing in size and shape .the segregation mechanisms discussed in the introduction are incorporated in the model as follows :  * size segregation due to different hopping lengths : we define ( to replace of the mono disperse case ) as the distance a reptating grain of type travels after being collided by a saltating grain .if we call the small grains type and the large grains type , then . this effect was incorporated in the discrete stochastic model of anderson and bunas using a generalization of the splash functions proposed by haff .the interaction between the rolling grains and the surface is characterized by four different angles of repose ; for the interaction of an reptating grain and a static grain ( replacing of the mono disperse case ) .the dynamics introduced by the angle of repose are relevant at the downwind face where two extra mechanisms for segregation of grains act in the system :  * size segregation during transport and rolling : large rolling grains are found to rise to the top of the reptating phase while the small grains sink downward through the gaps left by the motion of larger grains ; an effect known as percolation or kinematic sieving . due to this effectonly the small grains interact with the surface when large grains are also present in the rolling phase .small grains are captured first near the crest of the ripples causing the larger grains to be convected further to the bottom of the ripples .we incorporate this dynamical segregation effect by considering that , when large and small grains are present in the reptating phase , only the small ones interact with the surface according to the angle of repose , while the large grains , being at the top of the reptating phase do not interact with the static grains and they are convected downward .thus , the large grains interact with the surface only when there are no small grains present in the reptating phase .a similar percolation mechanism was introduced in the discrete and continuum models of to understand the origin of stratification patterns in two - dimensional sandpiles of granular mixtures .* shape segregation during transport and rolling : rounded grains tend to roll down easier than more faceted or cubic grains , so that the more faceted grains tend to be at the crest of the ripples .this segregation effect is quantified by the angles of repose of the pure species , since the repose angle is determined by the shape and surface properties of the grains and not by their size : the rougher or the more faceted the surface of the grains the larger the angle of repose . if the large grains are more faceted than the small grains we have , while when the small grains are more faceted  .the species with the larger angle of repose have a larger probability to be captured at the sand bed than the species with smaller angle of repose .we notice that the fact that the grains have different size leads to different cross - angles of repose  .if the large grain are type and the small are type then , which in turn leads to another size segregation effect .however , by incorporating the size segregation due to the different cross - angle of repose we would get the same effect as the one incorporated in the model by the percolation effect ( the small grains are preferentially trapped at the top of the slip - face ) , so that , in what follows , we will consider only the percolation effect for simplicity i.e. , large reptating grains are allowed to interact with the sand surface only when there are no small reptating grains below the large grains at a given position .we also notice that , in general , the distance a grain travels after being kicked by a saltating grain depends on the type of colliding grain and the type of grain on the bed .thus we define as the distance a reptating grain of type travel after being collided by a saltating grain of type .if we call the large grains type and the small grains type then we have : .hoverer , in practice we will make the following approximation , and , i.e. , we do not consider which type of grain is colliding .thus , three different mechanisms size - segregation due to percolation in the reptating phase , shape - segregation in avalanches , and size segregation due to the different reptating jumps compete in the system giving rise to a rich variety of lamination patterns as we show below .we first investigate the morphologies predicted by the model in the case where the different reptating jumps play the dominant role in the segregation process .this is the case and when the grains have approximately the same shape .we start our simulations from a flat sand surface composed by a 50/50 by volume mixture of small and large grains and we observe the system to evolve into ripples traveling in the direction of the wind ( fig . [ seq ] ) . the resulting morphology seen in fig .[ seq ] resembles the most common climbing ripple structures such as those documented in field observations ( fig .[ climb]a ) .this result confirms the hypothesis of anderson and bunas that the origin of inverse - graded lamination in climbing ripples is due to the size segregation effect produced by the different hopping lengths of small and large grains . in our simulationswe observe that large grains ( dark color ) are deposited at the top of the ripples while small grains ( light color ) are deposited preferentially at the bottom of the ripples since , resulting in a lamination structure showing inverse grading ( layer of large grains on top of the layer of small grains ) .the system size used in our simulations corresponds to 256 bins , with s , grains per impact , cm , . for the case shown in fig .[ seq ] we use cm and cm , and .the efficiency of the segregation mechanisms is sensitive to the relative value of the parameters of the model .for instance we find that by decreasing the value of relative to the reptating lengths , the efficiency of segregation is greatly reduced in the case of inverse - graded lamination .the structures shown here are all subcritically climbing from left to right ( for a review of the terminology and definition of different climbing structures in ripples deposit see , and chapter 9 ) .supercritical climbing ripples are much less common and are produced in slowly translating ripples with small number of ejected grains per impact .we also notice that the shape of ripples found in nature are more asymmetric with the downwind side at a steeper angle than the upwind side , while our model predicts a more symmetric triangular shape of the ripples .more realistic asymmetric shape of the ripples can be obtained by considering the exact trajectories of the flying grains and the complex interaction of grains with the air flow as shown by previous models by anderson .next we consider the case where the difference between the reptating jumps is small , ( or when the downwind face is large compared to or ) and the grains differ appreciable in shape ( ) and also in size .then segregation in the rolling face is the relevant mechanism for segregation and we do not take into account the segregation due to different hoping lengths .we first consider the case where the small grains are the roughest and the large grains are the most rounded ( ) . in this casea segregation solution along the downwind face is possible since both , size segregation due to percolation and shape segregation act to segregate the small - faceted grains at the top of the crest and the large - rounded grains at the bottom of troughs .the result is a lamination structure ( fig .[ phase]a ) which resembles the structure of climbing ripples of fig .[ seq ] but with the opposite grading : small grains at the top of each pair of layers ( normal - grading climbing ripple lamination ) .this type of lamination is not very common in nature .  on the other hand ,when the large grains are rougher than the small grains ( ) , a competition between size and shape segregation occurs .size segregation due to percolation tends to segregate the large - cubic grains at the bottom of the downwind face , while the shape segregation effect tends to segregate the same grains at the crest of the ripples .then a segregation solution along the downwind face as in fig .[ phase]a is not possible , and the result of this instability is the appearance of layers of small and large grains parallel to the downwind face ( fig .[ phase]b ) and not perpendicular as in fig .[ phase]a .these structures correspond to the more common cross - stratification patterns found in rocks .in addition to the stratification parallel to the rolling face , the deposits are also coarser toward the bottom of the downwind face .the mechanism for cross - stratification involves two phenomena superimposed . _( i ) _ segregation in the rolling face : a pair of layers is laid down in a single rolling event , with the small grains segregating themselves underneath the larger grains through a `` kink '' mechanism as seen in .this kink in the local profile of static grains provides local stability to the rolling grains trapping them .the kink moves uphill forming layers of small and large grains . _( ii ) _ the climbing of ripples due to grain deposition and ripple migration .the conditions for cross - stratification and the dynamical segregation process found with the present model are similar to the findings of the experiments of stratification of granular mixtures in two - dimensional vertical cells performed in and also the models of .we observe the formation on the slip face of the ripples of an upward traveling wave or `` kink '' at which grains are stopped during the avalanche as was observed in . according to this ,the wavelength of the layers is proportional to the flux of grains reaching the rolling face , then it is proportional to the number of saltating grains impacting the surface .  for the case shown in fig .[ phase]a we use cm , and , and , while for fig .[ phase]b we use cm , and , and .we have also investigated the morphologies obtained when the three segregation effects ( shape segregation , percolation effect , and hopping induced size segregation ) act simultaneously .the resulting morphologies are shown in fig .[ phase2 ] along with the phase diagram summarizing the results obtained with the model in figs .[ seq ] and [ phase ] .  the case and corresponds to the inverse - graded lamination shown in fig .[ seq ] , as indicated in the phase diagram fig .[ phase2 ] . moreover , when and for any other value of the angles and , we find that the hopping induced segregation seems to dominate over rolling induced segregation .as can be seen from the upper two panels in fig .[ phase2 ] , the morphologies obtained in these cases resemble the one of fig . [ seq ] corresponding to inverse - graded climbing ripple lamination .the upper left panel shows the results when the smaller grains are the roughest and shows clearly the same lamination as in fig .[ seq ] . in this casethe hopping induced segregation dominates completely over the rolling segregation due to the angles of repose and percolation . in the upper right panelwe show the case when the large grains are the roughest . in this casewe also see the same lamination pattern as in fig .[ seq ] but we see a tenuous trace of the cross - stratification seen in fig . [ phase]b too ; the rolling induced segregation seems to have a more important role in comparison with the hopping induced segregation than in the case shown in the upper left panel .the region is discussed in figs . [ phase ] , and our model shows normal grading lamination when ( fig .[ phase]a ) and cross - stratification when ( fig .[ phase]b ) .interesting morphologies are predicted when .this might be the case of a large difference in density between the grains , i.e. , very light large grains and heavy small grains .the patterns obtained with the model are shown in the two lower panels in fig .[ phase2 ] .the late stage in the dynamics evolution of the patterns shown at the left lower panel ( ) and at the right lower panel ( ) in fig . [ phase2 ] resemble the patterns of normal grading climbing ripples ( fig .[ phase]a ) and cross - stratification ( fig .[ phase]b ) respectively .this indicates the dominance of avalanche segregation ( shape segregation and percolation effect ) over the hopping induced segregation in this region of the phase space ( this dominance is the opposite of what we find in the region ) .however , as seen in fig .[ phase2 ] , these patterns appear only in the late stages of the evolution . as seen in the two lower panels , in the early stages the lamination structure appears to be dominated by the hopping induced segregation .we observe a clear transition at intermediate time in the simulation which can be clearly observed in the change of the climbing angle of the ripples observed in the left and right lower panels in fig .[ phase2 ] .the late stage patterns appear only when the slip - face is well developed after a transient of small ripples dominated by hopping segregation .we are not aware of any wind tunnel experiment or field observation showing similar dynamical transition , so it would be interesting to explore this transition experimentally .in summary , we have shown that lamination in sedimentary rocks at around the centimeter scale is a manifestation of grain size and grain shape segregation .thin layers of coarse and fine sand are present in these rocks , and understanding how layers in sandstone are created might aid , for instance , in oil exploration since great amounts of oil are locked beneath layered rocks .our findings suggest a unifying framework towards the understanding of the origin of inverse grading , normal grading and cross - stratification patterns in wind ripple deposits .we identify the conditions under which these different laminations patterns arise in sandstone .in particular , we find that cross - stratification is only possible when the large grains are coarser than the small grains composing the layered structure in agreement with recent experiments on avalanche segregation in two - dimensional sandpiles .the fact that cross - stratification patterns are very common in nature might be due to the fact that fragmentation processes usually leads to smaller grains more rounded than larger grains .so far we have explored a two - dimensional cross - section of the ripple deposits .extensions to three dimensional systems are ongoing as new physics may emerge when taking into account the lateral motion of the grains .g . de gennes , c. r. acad .( paris ) * 321 * ii , 501 ( 1995 ) ; p .- g . de gennes , in _ the physics of complex systems _ [ proc .intl school of physics `` enrico fermi '' course cxxxiv ] , ( eds ., f. mallamace and h. e. stanley ) ( ios press , amsterdam , 1997 ) .

Abstract: many sedimentary rocks are formed by migration of sand ripples . thin layers of coarse and fine sand are present in these rocks , and understanding how layers in sandstone are created has been a longstanding question . here , we propose a mechanism for the origin of the most common layered sedimentary structures such as inverse graded climbing ripple lamination and cross - stratification patterns . the mechanism involves a competition between three segregation processes : _ ( i ) _ size - segregation and _ ( ii ) _ shape - segregation during transport and rolling , and _ ( iii ) _ size segregation due to different hopping lengths of the small and large grains . we develop a discrete model of grain dynamics which incorporates the coupling between moving grains and the static sand surface , as well as the different properties of grains , such as size and roughness , in order to test the plausibility of this physical mechanism .  2