A Derivation of collisional results

Threshold for Electron Trapping Nonlinearity in Langmuir Waves

Abstract

We assess when electron trapping nonlinearity is expected to be important in Langmuir waves. The basic criterion is that the inverse of the detrapping rate of electrons in the trapping region of velocity space must exceed the bounce period of deeply-trapped electrons, . A unitless figure of merit, the “bounce number” , encapsulates this condition and defines a trapping threshold amplitude for which . The detrapping rate is found for convective loss (transverse and longitudinal) out of a spatially finite Langmuir wave. Simulations of driven waves with a finite transverse profile, using the 2D-2V Vlasov code loki, show trapping nonlinearity increases continuously with for transverse loss, and significant for . The detrapping rate due to Coulomb collisions (both electron-electron and electron-ion) is also found, with pitch-angle scattering and parallel drag and diffusion treated in a unified manner. A simple way to combine convective and collisional detrapping is given. Application to underdense plasma conditions in inertial confinement fusion targets is presented. The results show that convective transverse loss is usually the most potent detrapping process in a single laser speckle. For typical plasma and laser conditions on the inner laser cones of the National Ignition Facility, local reflectivities are estimated to produce significant trapping effects.

nonlinear Langmuir waves; trapped electrons; laser-plasma interaction; inertial confinement fusion; stimulated Raman scattering
pacs:
52.25.Dg, 52.35.Fp, 52.35.Mw, 52.38.Bv, 52.38.-r, 52.57.-z

I Introduction

The nonlinear behavior of Langmuir waves (LWs) is a much-studied problem in basic plasma physics from the 1950s to the present. In this paper, we focus on nonlinearity due to electron trapping in the LW potential well. This intrinsically kinetic effect has motivated theoretical work such as nonlinear equilibrium or Bernstein-Greene-Kruskal (BGK) modes Bernstein, Greene, and Kruskal (1957), Landau damping reduction O’Neil (1965), nonlinear frequency shift Manheimer and Flynn (1971); Morales and O’Neil (1972); Dewar (1972), and the sideband instability Wharton, Malmberg, and O’Neil (1968); Kruer, Dawson, and Sudan (1969). Important applications of trapping occur in LWs driven by coherent (e.g., laser) light, including the laser plasma accelerator Tajima and Dawson (1979) and stimulated Raman scattering (SRS) Goldman and DuBois (1965); Drake et al. (1974); Kruer (2003). The latter allows the prospect of laser pulse compression to ultra-high amplitudes (the backward Raman amplifier) Malkin, Shvets, and Fisch (1999). In addition, SRS is an important risk to ICF Lindl et al. (2004); Atzeni and Meyer-ter-Vehn (2004), both due to loss of laser energy and the production of energetic (or “hot”) electrons that can pre-heat the fuel. Ignition experiments at the National Ignition Facility (NIF) Moses and Wuest (2005) have shown substantial Stimulated Raman backscatter (SRBS) from the inner cones of laser beams Meezan et al. (2010). The current study is prompted primarily by SRS-driven LW’s. Much recent work has focused on nonlinear kinetic aspects of SRS, including “inflation” due to Landau damping reduction Vu, DuBois, and Bezzerides (2001); Strozzi et al. (2007); Bénisti et al. (2009, 2010); Ellis et al. (2012), saturation by sideband instability Brunner and Valeo (2004), and LW self-focusing in multi-D particle-in-cell simulations Yin et al. (2009, 2007); Fahlen et al. (2011), Vlasov simulations Banks et al. (2011), and theory Rose and Yin (2008). One goal is to find reduced descriptions, such as envelope equations, that approximately incorporate kinetic effects Yampolsky and Fisch (2009); Dodin and Fisch (2012); Bénisti, Yampolsky, and Fisch (2012).

Our aim is to provide theoretical estimates for when electron trapping nonlinearity is important in LW dynamics. These allow for self-consistency checks - or invalidations - of linear calculations of LW amplitudes. This work is therefore not primarily intended to study nonlinear LW dynamics, although we do present Vlasov simulations to quantify the onset of trapping in the presence of convective transverse loss. We consider a single, quasi-monochromatic wave with electron number density fluctuation , and slowly-varying, unitless amplitude where is the background electron density. We refer to an electron as “trapped” if it is within the phase-space island centered about the phase velocity and bounded by the separatrix in the instantaneous wave amplitude, regardless of how long it has been there. The dielectric response of the plasma depends on the distribution function, and therefore manifests trapping effects only after enough time has passed for the (typically space-averaged) distribution to be distorted. We call such a distribution trapped or flattened, since trapping produces a plateau in the space-averaged distribution centered at . Deeply-trapped electrons have an angular frequency ( defines the plasma frequency in SI units), known as the bounce frequency, corresponding to a bounce period . In our language, an electron is trapped instantaneously, but a distribution becomes trapped over a time . For a process that detraps electrons at a rate , the unitless “bounce number” measures how many bounce orbits a trapped electron completes before being detrapped.

Our estimates stem from the assumption that nonlinear trapping effects are significant when is roughly unity. Trapping nonlinearity develops continuously with wave amplitude, and is not an instability with a hard threshold. Vlasov simulations presented in Sec. IV of driven LWs with a finite transverse profile demonstrate this. In addition, transit-time damping calculations Rose (2006) show the reduction in Landau damping varies continuously with and obtains a 2x reduction for . Bounce number estimates are qualitative and demonstrate basic parameter scalings. The quantitative role of trapping depends on the specific application.

We consider two detrapping processes: convective loss and Coulomb collisions. For a LW of finite spatial extent, electrons enter and leave the wave from the surrounding plasma (assumed here to be in thermal equilibrium, i.e. Maxwellian). Trapping will only be effective if these electrons complete a bounce orbit before transiting the wave. We find the detrapping rate for both longitudinal end loss, which can be important in finite-domain 1D kinetic simulations, and for transverse side loss in 2D and 3D. To quantify the effect of trapping in a LW with finite transverse extent, we perform 2D-2V simulations with the parallel Vlasov code lokiBanks et al. (2011); Banks and Hittinger (2010) of a LW driven by an external field with a smooth transverse profile. Our results are in qualitative agreement with Sec. IV of Ref. Banks et al., 2011. That work considered a free LW excited by a driver of finite duration, while we consider a driver that remains on.

We present a unified calculation of collisional detrapping due to electron-ion and electron-electron collisions, including both pitch-angle scattering and parallel slowing down and diffusion. This relies on the fact that (see the Appendix) the distribution in the trapping region can be Fourier decomposed into modes for , and the diffusion rate of mode is proportional to . After a short time, only electrons in the fundamental mode remain trapped. The collisional detrapping rate scales as , since the trapping width in velocity increases with wave amplitude. We discuss two ways to compare the relative importance of detrapping by side loss and collisions, which is complicated by their different scaling with .

Our calculations are applied to ICF plasma conditions, particularly LW’s driven by stimulated Raman backscatter (SRBS) on the NIF. Transverse side loss out of laser speckles in a phase-plate-smoothed beam is generally a more effective detrapping process than collisions. The threshold for trapping to overcome side loss decreases with density and increases with temperature, while the collisional threshold decreases with density and slightly increases with temperature. For conditions typical of backscatter on NIF ignition experiments, namely =2 keV and with the critical density for laser light of wavelength 351 nm, a reflectivity of W cm produces linear Langmuir waves above the side loss threshold. Such values are likely to occur in intense speckles. We also show that smoothing by spectral dispersion (SSD) Skupsky et al. (1989) is ineffective at detrapping in NIF-relevant conditions.

The paper is organized as follows. Section II provides some general considerations on our detrapping analysis. We present in Sec. III convective loss calculations for both longitudinal (end) and transverse (side) loss. Section IV contains Vlasov simulations with the loki code which study the competition of trapping and side loss. Detrapping by Coulomb collisions is treated in Sec. V. Our results are applied to SRBS in underdense ICF conditions in Sec. VI. We conclude in Sec. VII. The Appendix presents details of our collisional derivation and discusses the validity of our Fokker-Planck model.

Ii General Considerations

This section presents our overall framework for estimating the trapping threshold, and lays out some definitions. Consider the trapped electrons in a LW field, attempting to undergo bounce orbits. There is a time-dependent condition for trapping to distort the distribution significantly, even in the absence of any detrapping process. For instance, if a LW is suddenly excited in a Maxwellian plasma, electrons execute bounce orbits according to what we call the dynamic bounce number

(1)

The time dependence of allows for a slowly-varying wave amplitude . Vlasov simulations presented in Sec. IV show that trapping starts to significantly affect the dielectric response when . That is, it takes a finite time for the distribution to reflect trapping. The early works of Morales and O’Neil O’Neil (1965); Morales and O’Neil (1972) indicate such behavior, where the damping rate and frequency shift evolve over several bounce periods until approaching steady values as the system reaches a Bernstein-Greene-Kruskal (BGK) state Bernstein, Greene, and Kruskal (1957).

To estimate the threshold for trapping to overcome a detrapping process, we assume the wave has been present long enough that . The distribution has had enough time to become flattened, to the extent the detrapping process allows. For flattening to occur, an appreciable fraction of trapped electrons must remain so for about a bounce period before being detrapped. We are interested in the number of electrons in the trapping region, and how long they stay there.

We define the “trapping region” to extend from where is the full width of the phase-space trapping island and with . Throughout this paper, we use

(2)

to denote the scaled velocity for various subscripts . Let denote the fraction of electrons in the trapping region at the initial time , that continuously remain so to some later time (note ). At we take the electron distribution to be Maxwellian. The fact that only some electrons in the trapping region lie within the separatrix (depending on their initial phase ) is not relevant, since all the detrapping processes considered here are insensitive to the electron’s phase in the wave. That is, the rate at which electrons leave the trapping region is independent of .

The detrapping rate is defined by assuming exponential decay for the trapped fraction: . We allow for several independent detrapping processes to occur simultaneously, in that the overall detrapping rate is the sum of the rates for each th process considered separately. Since a detrapping process generally does not strictly follow exponential decay, we choose a critical fraction , which obtains for a critical time , and let . is independent of for exponential decay. We set in what follows. Given the approximate nature of our calculation, further refinement of has little value.

In the literature, detrapping processes are sometimes approximated by a 1D kinetic equation with a Bhatnagar-Gross-Krook relaxation (or simply a Krook) operator Bhatnagar, Gross, and Krook (1954):

(3)

The linear electron susceptibility for this kinetic equation is

(4)

where and is the plasma dispersion function Fried and Conte (1961). The Krook operator relaxes the electron distribution function to an equilibrium , and locally conserves number density . The above operator does not conserve momentum or energy, although it can easily be generalized to do so. In a 1D-1V system, a Krook operator can mimic detrapping by transverse convective loss (a higher space-dimension effect) or Coulomb collisions (a higher velocity-dimension effect), such as in Ref. Rose and Russell, 2001. Any perturbation from decays exponentially at the rate , so for such an operator. This is especially useful for a detrapping process which has independent of wave amplitude; this is the case for convective loss but not for collisions (as shown below). SRS simulations with a 1D Vlasov code and Krook operator, and its suppression of kinetic inflation, are presented in Ref. Strozzi et al., 2011a. In this paper, we do not use a Krook operator to model detrapping, although we do use one in our 2D Vlasov simulations to make them effectively finite in the transverse direction (a purely numerical purpose), and to include collisional LW damping in our application to ICF conditions in Sec. VI.

We take the bounce period of all trapped electrons to be , the result for deeply-trapped electrons. The actual period slowly increases to infinity for electrons near the separatrix. We then define the bounce number for process as

(5)

We have expressed as a ratio of the LW amplitude to a “threshold” amplitude , to some power . Recall that trapping effects like the Landau damping reduction develop continuously with , so the threshold for trapping nonlinearity is not a hard one. Besides the dependence of , also depends on in a process-dependent way. For independent of wave amplitude, which we show below is the case for convective loss, the power . This is not the case for detrapping by Coulomb collisions, which is shown in Sec. V to have . The overall detrapping rate , gives an overall bounce number via . We also define an overall threshold amplitude such that ; it is not generally true that .

Iii Convective Loss: Theory

In a LW of finite spatial extent, electrons remain in the trapping region only until they transit the wave. This detrapping manifests itself by longitudinal loss out of the ends of the wavepacket (the direction for our field representation ), as well as transverse loss out the sides. End loss is found by considering a wavepacket of length and infinite transverse extent. We work in the rest frame of the wavepacket, which may differ from the lab frame depending on application. For instance, a free LW propagates at group velocity for , while a LW driven by a driver fixed in the lab frame (such as the ponderomotive drive in SRS) will essentially be at rest. For we can treat all trapped electrons as moving forward at . Thus for end loss . To find , we take , which gives and with . The bounce number for end loss is , with exponent and threshold amplitude . In practical units where is in cm, is in keV, and is in m.

For transverse side loss, consider a cylindrical wavepacket of transverse diameter and infinite longitudinal length. In total spatial dimensions, the cylinder has an dimensional cross-section. Electrons with a Maxwellian distribution are transiting the cylinder, with unnormalized distribution where is the transverse speed and is the number of electrons per . The average for , indicating that detrapping is faster in 3D than in 2D.

Figure 1: (Color online.) Trapped electron fraction due to transverse side loss for a 2D plane (blue) and 3D cylinder (red) region. The dashed curves are the appropriate early and late time limits. See Eqs. (6) through (12).

We find the number of initially trapped electrons , that remain so after time , by summing the fraction of electrons with a given that remain trapped, times . All electrons with with have escaped, so this sets the limits of integration. In 2D, the trapped fraction is for , and the total trapped fraction is

(6)
(7)

In 3D we obtain

(8)

The factor in square brackets is the trapped fraction. The limiting forms are

(9)
(10)
(11)
(12)

In both limits the decrease is more rapid in 3D than in 2D. Figure 1 displays the various formulas for .

The resulting detrapping rate, based on , is

(13)

with in (2D, 3D). As expected, the 3D detrapping rate is faster. The 3D detrapping rate exceeds the 2D one by a larger factor than the average transverse speed because the faster electrons leave first, and the relative surplus of electrons in 3D over 2D (proportional to ) increases with transverse speed. A wavepacket with asymmetric (e.g. elliptical) cross-section should have a rate between the 2D and 3D result with taken as the shortest transverse length. In a laser beam smoothed with phase plates, elliptical speckles can be produced by certain polarization-smoothing schemes or a non-spherical lens; Langmuir waves driven by SRS in such speckles would also acquire an elliptical cross-section.

Comparing the end loss and side loss rates gives

(14)

is in the wavepacket frame. For the LW to not experience strong Landau damping, we have . depends on the physical situation (laser speckles are discussed in Sec. VI). The bounce number for side loss is analogous to end loss: , with exponent and threshold amplitude . In practical units and for the 3D , .

Iv Vlasov simulations of convective side loss

In this section, we quantify the competition between convective side loss and electron trapping in a driven Langmuir wave. We use the parallel, 2D-2V Eulerian Vlasov code lokiBanks and Hittinger (2010). This code employs a finite-volume method which discretely conserves particle number. The discretization uses a fourth-order accurate approximation for well-resolved features, and smoothly transitions to a third-order upwind method as the size of solution features approaches the grid scale. This construction enables accurate long-time integration by minimizing numerical dissipation, while retaining robustness for nonlinearly generated high frequencies. As a result, the method is not strictly monotone- or positivity-preserving, nor does it eliminate the so-called recurrence problem. This occurs at a recurrence time of when further linear evolution of a sinusoidal perturbation cannot be represented on a given grid.

Our simulations are 1D or 2D, with the longitudinal coordinate as above, and the transverse coordinate. Only electrons are mobile, there is a fixed, uniform neutralizing background charge, and there is no magnetic field. The total electric field is , where the internal electric field and . The external driver field is with

(15)

There is no component to the driver field, which would be needed if the driver were derived from a scalar potential. The temporal envelope ramps up from zero to unity over a time and then stays constant. The transverse profile is

(17)

.

The numerical aspects of our runs are as follows. The domain extends for one driver wavelength, with periodic boundaries for fields and particles. zones in was used for all runs in this paper, except for two cases in Fig. 2(a). 2D runs had periodic boundaries for fields and particles at . A Krook operator with for and rising rapidly in the boundary region was used to relax the distribution to the initial Maxwellian near the transverse boundaries. The runs were thus effectively finite in . We used 11 to 45 zones in , with more used for larger and to check convergence. The and grids both extended to . zones in were used throughout. is set by two requirements: the trapping region must be adequately resolved, and recurrence phenomena must not be significant. We found was sufficient to give converged results. loki’s advection scheme is designed to mitigate aliasing problems, and we only saw modest effects related to it when comparing runs with different . The convergence of our numerical results is shown in Fig. 2(a). The black curve is typical: it uses and has a typical , which we kept similar by varying with wave amplitude and .

We first present 1D runs with and , which are detailed in Table 1. From linear theory with , where

(18)

is the linear electron susceptibility from Eq. (4) with , evaluated at the driver and . We chose to give nearly the maximum for a given . For , a linearly resonant exists where ; the maximum then occurs close to this point. No linear resonance exists for , which is called the loss of resonance Rose and Russell (2001). Some still maximizes in this regime. The non-resonant case differs from the resonant one, in that reducing and Landau damping, e.g. by flattening the distribution at the phase velocity by electron trapping or some other means, does not lead to a large enhancement in the Langmuir wave response to an external drive. The term in Eq. (18) keeps finite even if . For the parameters of the run 1D.7a, we find for the full, complex , while setting slightly increases it to .

Similar logic applies to kinetic inflation of stimulated Raman scattering. Electron trapping and the resultant Landau damping reduction can greatly increase the scattering at a resonant wavelength. However, scattering at a non-resonant wavelength is not subject to inflation, and can even decrease, due to reducing . Non-resonant SRS can occur in a situation seeded away from resonance Ellis et al. (2012), or if the plasma conditions are such that no resonance exists for any scattered wavelength, namely high and low .

Figure 2 presents the results of our 1D runs. Panel (a) shows the time evolution of the amplitude of for , normalized to the linear value from Eq. (18). Early in time () the linear response is achieved, which validates the linear dispersion and properties of loki when using the chosen grid resolution. As time progresses the response increases due to the damping reduction, and then oscillates due to the interplay of the frequency shift and the fixed driver. Similar behavior was seen in Ref. Yampolsky and Fisch, 2009. We plot the results vs. the dynamic bounce number from Eq. (1), using the time-dependent , in the center and right panels. is thus a trapping-based re-scaling of time. The other runs from Table 1 are included as well. The driver strength was chosen in runs 1D.35b, 1D.5a, and 1D.7a to give similar bounce periods. In all cases, the linear response is achieved after a transient period related to driver turn-on, until . After this point the response increases, until the frequency shift develops at . As increases, the enhancement above linear response decreases. This is likely due to the rapid increase of the frequency shift with , as shown by most theoretical calculations, e.g. Ref. Morales and O’Neil, 1972. For , there is a slight enhancement to 1.3x the linear response, followed by a dip to about 0.7x and subsequent oscillation about unity. This lack of significant trapping nonlinearity agrees with the above discussion of the non-resonant regime.

Figure 2: (Color online.) (a) Amplitude of mode of , scaled to linear response, vs. time for 1D loki case 1D.35b and four different resolutions: (black), (green), (blue), and (red). (b) vs. dynamic bounce number from Eq. (1) using from 1D loki runs in Table 1 (see table for curve meanings). (c) panel (b) for expanded domain. The black curve is the only run in (a) that appears in (b) and (c).
Run plot curve
1D.35a 0.35 1.22 11.9 1.25 871 red dash
1D.35b 5 436 solid black
1D.35c 2 218 blue dash
1D.5a 0.5 1.44 3.22 1.3 434 black dash
1D.5b 5.2 217 green dash
1D.7a 0.7 1.79 1.80 1.7 430 black dot
Table 1: 1D loki runs with no transverse driver profile . . is found using .
Run plot curve
2D100 100 0.113 red
2D200 200 0.226 dark blue
2D400 400 0.452 green
2D800 800 0.905 magenta
2D1200 1200 1.357 blue
Table 2: 2D loki runs with transverse driver profile . All runs have , , and , the same as run 1D.35c.

From Eq. (13), the 2D side loss rate is , where we have taken , the full-width at half-max of . The side loss bounce number is then

(19)

Recall that electrons feel the total electric field (drive plus interal), and is an equivalent density fluctuation. Gauss’s law gives , where is the amplitude of the Fourier mode of the on-axis field , and denotes a normalized field. Using the linear response from Eq. (18), we obtain the linear estimate

(20)

The 2D loki runs are listed in Table 2. All runs used , , and , the same as run 1D.35c. For these values, our linear estimate becomes .

The field magnitude is plotted vs. the dynamic bounce number found using for the 2D runs in Fig. 3. The black curve is the analogous 1D run 1D.35c. For there is a continuous increase in the response with profile width . This allows us to quantify trapping nonlinearity vs. , which we do in Fig. 4. The abscissa in that figure is the side loss bounce number, , computed with linear response as in Eq. (20). The ordinate is the field enhancement due to trapping, scaled to the same quantity for the 1D run. This is shown at times corresponding to several values of ranging from 0.75 to 2. These times are early enough that the amplitudes have been mostly increasing, with little oscillation due to the frequency shift. The curves agree well, and demonstrate the continuous development of trapping effects with wide profiles. Slightly more than half the 1D trapping effect obtains for , which vindicates our approximate threshold for trapping.

Figure 3: (Color online.) Amplitude of mode of for 2D loki runs with transverse driver profiles with various . Run parameters and curve meanings are given in Table 2. Black curve is 1D run 1D.35c.
Figure 4: (Color online.) Departure from linear response for 2D loki runs, scaled to the same quantity from the 1D run 1D.35c. The colored curves are taken at times when the dynamic bounce number has reached the value indicated by the colored text. is found from linear response, using Eq. (20).

The plasma response to a driver with transverse profile differs from the 1D case. This can be seen in the ordinate of Fig. 4 falling below zero for the smallest . There have been several linear calculations of transit-time damping in LWs of finite extent, mostly by integration along particle orbits Short and Simon (1998); Skjæraasen, Robinson, and Melatos (1999). Ref. Short and Simon, 1998 showed that, for a potential with a step-function profile in space, the transit-time damping exceeds that for an infinite plane-wave for , while for it can be less. We adopt the alternative approach of writing the response as a superposition of responses to the Fourier modes comprising the drive. This is particularly convenient for our , which (when periodically repeated) is composed of only two Fourier modes. For simplicity we present the result for periodically repeated, instead of the actual loki profile with compact support over . The compact case would lead to a continuous Fourier transform rather than discrete series, and introduce a line width around the dominant modes. This does not change the qualitative result. Unlike Ref. Short and Simon, 1998, our compact profile is not a step function but smooth, with and continuous at all points (although is not).

The drive , made periodic in , is

(21)

A standard kinetic calculation, accounting for the fact that has no component and thus does not come from a potential, gives the field at :

(22)
(23)

Note that the linear for our . is a real phase. is the collisionless susceptibility for from Eq. 4, which depends only on and . and with . For , we recover the 1D result Eq. (18). Physically, the higher- modes induced by the transverse profile are more Landau damped (as well as being slightly off resonance for the fixed ), which reduces the response. For the parameters of Table 2, we find for where is the value for . We obtain a slight decrease in the linear response for our sharpest profile (), and an insignificant change for wider ones. This is borne out by Fig. 3. The red curve for shows no signs of trapping, and reaches a steady level slightly more than 0.8 times the 1D linear value. The blue curve () shows a slight trapping enhancement, and reaches a steady level slightly above 1.2x linear after about 2 bounce periods.

V Coulomb Collisions

Collisions remove electrons from the trapping region via pitch-angle scattering (from electron-ion and electron-electron collisions) as well as parallel drag and diffusion (from only electron-electron collisions since ). We adopt a Fokker-Planck collision operator, and discuss its validity in the Appendix:

(24)

where is the pitch angle between and the direction, and . is a thermal electron-electron collision rate:

(25)

and ( in cm, in eV) is the electron-electron Coulomb logarithm appropriate for eV (Ref. Huba, 2007, p. 34). The effective charge state is

(26)

where is the total ion density, with ; , and is the electron-ion Coulomb logarithm Huba (2007).

In section VI we apply our results to Langmuir waves generated by Raman scattering in underdense ICF plasmas, which are typically low-Z. For instance, NIF ignition hohlraum designs currently use an He gas fill (with H/He mixtures contemplated), and plastic ablators (57% H, 42% C atomic fractions). This gives when fully-ionized and . Be and diamond ablators are also being considered. For illustration, we take as the lowest reasonable value (fully-ionized H), and use (fully-ionized Be) to represent an ablator plasma.

It is useful to define a unitless time (different from the side loss used above), which demonstrates some of the basic collisional scaling:

(27)
(28)

Our collisional calculation of the trapped fraction is detailed in the Appendix. The key observation is that the distribution in the trapping region can be decomposed into Fourier modes for , and the diffusion rate of mode is proportional to . After a short time, only electrons in the mode remain trapped, so it suffices to consider just the number in the mode. At , this is 81% of the total (the other 19% rapidly diffuses out). The upshot is that , the fraction of initially trapped particles remaining in the fundamental mode after time , is

(29)

is given in Eq. (53).

Eq. (29) is an implicit, integral equation for as a function of , , and . We find the “exact” solution by performing the integral numerically, and interpolating for a desired . We derive an approximate solution, valid for , for in the Appendix. The result is

(30)

and are both positive and depend only on , so decreases with increasing . Figure 5 plots for several and , using the exact results (solid curves) and the approximate form for of Eq. (60) (dashed curves). Few electrons remain trapped at . The approximate forms are quite good, even though is not that large.

Figure 6 displays the relative error between for computed two ways. The exact is found numerically, and is from Eq. (30), with Eq. (61) for and Eq. (66) for . The agreement is excellent, within 1% for most of parameter space.

Figure 5: (Color online.) Trapped fraction due to collisions vs. unitless time defined in Eq. (27). Solid curves are exact results from Eq. (29). = (2,1), (4,1), and (4,4) for black, red, and blue, respectively. Dashed red and blue curves are approximate results for from Eq. (60), which depend only on and not . Green dashed curve is , the approximate form neglecting the term proportional to in the denominator. and not unity due to electrons not initially in the fundamental mode.
Figure 6: (Color online.) Relative error between the exact and approximate results from Eqs. (29,30), respectively, for collisional ; see text for details. The green curves are .

The collisional detrapping rate is

(31)

Note that since : the larger the wave amplitude, the wider the trapping region extends in velocity, and collisions take longer to remove the electron velocity from this region. Recall that depends slightly on the choice of due to the non-exponential decay of with ; as with convective loss we choose .

The collisional bounce number is

(32)

The amplitude exponent for collisions is , unlike the convective loss value of 1/2. This stems from the fact that for collisions is amplitude-dependent while for convective loss it is not. We now construct the overall bounce number for convective side loss and collisions, as outlined above. Assuming that separate detrapping processes are independent, and their detrapping rates add, yields

(33)

We define an overall threshold amplitude such that . Eq. (33) gives a cubic equation for :

(34)

There are two ways to compare the relative importance of side loss and collisions. One is: for which process must the wave amplitude be larger for trapping to be significant ()? The other is: for a given , which process will detrap more effectively? The two views are not equivalent, due to the different dependence of the side loss and collisional detrapping rate on . The first amounts to comparing the thresholds and , which can be computed just from plasma and wave properties without knowing . The ratio of detrapping rates can be written in terms of a critical amplitude :

(35)

Vi Parameter study for ICF underdense plasmas

We now apply our analysis to ICF conditions where stimulated Raman scattering (SRS) can occur, namely the underdense coronal plasma. SRS is a parametric three-wave process where a pump light wave such as a laser (we which label mode 0) decays to a scattered light wave (mode 1) and a Langmuir wave (mode 2). We restrict ourselves to exact backscatter (SRBS; anti-parallel to ), as this generates the largest (smallest ) and thus makes trapping effects more important (small transverse components to have little effect on the phase velocity). Both measurements and simulations with the paraxial-envelope propagation code pf3d Berger et al. (1998) have shown backscatter to be the dominant direction for SRS. With , the phase-matching conditions are and with . We employ the (cold) light-wave dispersion relation for modes 0 and 1, and use the vacuum wavelength . Frequency matching thus requires , with the critical density for mode , and . For specific examples we choose = 351 nm, appropriate for frequency-tripled UV light currently in use on NIF. Specific plasma conditions thought to be typical for SRBS on NIF ignition targets, during early to mid peak laser power, are and keV ( nm) Strozzi et al. (2011b). The scattered wavelength continuously increases during a NIF experiment, consistent with the hohlraum filling to higher density.

An important case for this paper is LW’s driven by SRBS in the speckles of a phase-plate-smoothed laser beam Kato et al. (1984). For a laser wavelength and square RPP with optics F-number , the intense speckles have and (see Ref. Garnier and Videau, 2001). A speckled beam is not the only situation where SRS can occur; for instance, there has been recent interest in re-amplification of backscatter by crossing laser beams Kirkwood et al. (2011) and backward Raman amplifiers Yampolsky and Fisch (2011). However, for a single laser beam, experiments at Omega and pf3d simulations show speckle physics, and its modification by beam smoothing, must be accounted for to accurately model SRS Froula et al. (2009, 2010). Experiments have also verified the increase in backscatter with increased gain per speckle length, by changing the laser aperture and thus the effective Froula (). We therefore focus on speckles. On NIF, four laser beams, each smoothed by a phase plate and with an overall square aperture, are grouped into a “quad” which yields an effective square aperture of . We thus use for illustration. As the beams of a quad propagate through a target, they can separate from one another, refract, and undergo other effects that change the shape of their effective aperture and speckle pattern. We do not pursue this further here, but it should be born in mind when applying our analysis. Also the ratio is so small that (3D) is small for essentially all speckles of interest. Thus side loss is a more potent detrapping mechanism than end loss, in speckles.

Figure 7: (Color online.) Wavelength in nm for SRBS light (increments of 50 nm), for a pump wavelength = 351 nm. Black solid: phase-matched with a natural Langmuir wave, satisfying the dispersion relation Eq. (36) with . Red dash: for the maximum local SRBS spatial gain rate.
Figure 8: (Color online.) Langmuir wave phase-matched for SRBS with different choices, for a pump wavelength = 351 nm. Black solid: phase-matched with a natural Langmuir wave, as in black solid curves of Fig. 7. Red dash: =550 nm (’s fixed, ’s vary). Blue dot: =550 nm black-solid contour from Fig. 7.

To quantify detrapping rates, we consider the threshold amplitudes and . Unlike , depends on and of the Langmuir wave. For a given set of plasma conditions, the choice of is not unique but depends on the application. For SRS developing locally, one can choose the LW corresponding to the largest growth rate for those conditions. Another approach is to consider a single scattered-light frequency as it propagates through a target. We consider only variations induced by spatial profiles and not variations due to temporal plasma evolution Dewandre, Albritton, and Williams (1981) (which is mostly relevant to stimulated Brillouin scattering). In this case, the matching conditions given the local plasma properties dictate how varies.

Figure 7 presents the local for SRBS computed in two ways. The black curves are found by phase-matching with a “natural” LW, by which we mean where complex satisfies

(36)

with real . To find , we set and recover the usual collisionless . We use below as a simple way to include collisional LW damping when Landau damping is negligible. The red curves in Fig. 7 are the which maximizes the local spatial SRBS gain rate in the strong damping limit Strozzi et al. (2008):