Intense Laser-Plasma InteractionsDouglass SchumacherDepartment of PhysicsThe Ohio State University

FSC Summer SchoolJuly 15-19, 2013Columbus, Ohio

Based on Linn Van Woerkom’s(OSU) presentation at the 2011 FSC Summer School.

Many figures were provided by Andy Krygier (OSU).

Outline

1. Goals, Scope and Motivation2. High Intensity Lasers3. Single Electron In An EM Field4. Laser Plasma Interaction (LPI)5. Transport 6. Topic: B-field Generation7. Topic: TNSA8. Summary

2

Big Picture• Intense short laser pulse strong electromagnetic fields• Initial coupling to relativistic electrons• Energetic electrons

– Carry energy into/through material– Lose energy to ions– Lose energy to radiation

• Heated material then expands/explodes at long times

3PIC Simulation; Gremillet, et. al, POP 9, 941 (2002)

Goals• Provide introduction to ultra-intense laser matter

interactions• Describe related issues in laser-driven electron transport

necessary to understand experiment• Describe a variety of phenomena to give a flavor for the

subject area.

4

Two texts that are great for getting started and for reference:• Short Pulse Laser Interactions with Matter: An Introduction

Paul Gibbon, ISBN-10 1860941354• High-Energy-Density Physics: Fundamentals, Inertial Fusion, and

Experimental Astrophysics (Shock Wave and High Pressure Phenomena)Paul Drake, ISBN-10 9783540293149

Motivation and Scope• Basic physics of High Energy Density Science

– Many fundamental questions that have been raised are unanswered– Laboratory Astrophysics– Materials in extreme conditions

• Laser Driven Applications– Fusion energy – Fast Ignition– Ion beams

• Medical therapies• Materials studies

– Electron beams• Possible accelerator schemes

– Photon beams• Bright x-ray and gamma sources for radiography

5

For this talk I’ll define “intense” as relativistic so I ≥ 1018 W/cm2.

I’ll also focus on short pulse lasers so á ns.

Summary• Ultraintense, short pulse lasers drive electrons relativistically with

far reaching consequences for most processes.• The ponderomotive force/energy scale is a key figure of merit.• For such short pulses, hydrodynamic evolution can be limited

during the time the pulse is on.• Most diagnostics depend on the hot electron transport, directly or

indirectly, and that must be accounted for when interpreting the diagnostics.

• Current balance of hot and cold electrons and temperature dependent transport coefficients must be understood.

• Hot electrons drive the creation of quasi-static magnetic and electric fields that are intense with important effects (TNSA).

• Intense laser matter interactions and the subsequent transport problem offers a “rich” environment with many key questions still unanswered.

6

Intensity• Irradiance (“intensity”):

so W/m2 in SI, but usually

W/cm2 in practice.

– I determines E and B which determines the EM force.

– (I λ2) provides a good figure of merit for many quantities in HEDP, like the electron energy distribution.

– Power: . Popular figure of merit for laser systems – determines their “class”.

– For meaningful experiments, the intensity needs to be well characterized. More on this in a moment.

Two lasers at the same intensity can be doing extraordinarilydifferent experiments:

Scarlet: 15 J, 50 fs, few μm spot size ~½ PW, >1021 W/cm2

NIF: 1.8 MJ (192 beams), ns, dist over ~mm ~ ½ PW, <1016 W/cm27

Typical Intense Lasers

• Get high intensity from high energy– Lamp-pumped glass-type wavelength = 1 m– Energies from 10 - 1000 Joules– Durations ~500 fs to 10 ps– Titan @ LLNL, Trident @ LANL, MTW & EP @ LLE, Vulcan @ RAL– Repetition rate several shots per day

• Get high intensity from short pulses– Laser-pumped Ti:Sapphire wavelength = 0.8 m– Energies from 0.1 – 100 Joules– Durations ~30 – 200 fs– Scarlet @ OSU, Callisto @ LLNL, Astra Gemini @ RAL, Hercules @ UM– Repetition rate several per day up to 1 Hz

• Of course, you can do both– Mixed glass, 150 fs, 200 J Texas Petawatt @ UT

8

Chirped Pulse Amplification

http://en.wikipedia.org/wiki/File:Chirped_pulse_amplification.png 9

Spatial Mode

x

I(x)

• Total pulse energy = 120 J• Assume 30% of energy between dashed lines @ xo = 10 m• Fluence = Energy/area = 0.3(120J)/(xo

2) = 1.1x107 J/cm2

• Assume pulse duration FWHM = 700 fs• Peak intensity = 1.6x1019 Wcm-2

10

Specify peak or average or comparable:• Pick an amount of energy• Area containing this energy• Time duration containing this energy

Can be FWHM, 1/e, 40% of energy, etc.Be careful to be clear especially when comparing intensities

Real Example Of Spatial Mode

Focusing Laser

microscope andcamera View actual focus at low power

and map pixels by brightness into a circular gaussian beam

From Daniel Hey thesis 11

Temporal Mode

12

• Plots like this can be determined experimentally, but frequently measuring over many shots.

• The energy in the wings can be measured on each shot using a water cell.• All neutral matter ionizes for I = 1012-1013 W/cm2. At 1021 W/cm2, you

need 1010 contrast to minimize front surface target damage.• Ablators are often placed on the front surface to protect it.

Short pulse red “pancake”

Pre-pulse pink

Pre-plasma from the pre-pulse

13

Interferogram from Daniel Hey thesis.

Irradiation by 0.8 J, 120 ps Ti:Sapphire laser withI = ~1012 W/cm2, derived from measured interferograms.Grava et al, PRE 78, 016403 (2008)

Laser Diagnostics

14

Multiple diagnostics are needed to characterize all facets of a laser pulse, preferably operating simultaneously with each other and the experimental shot.• “On-shot” diagnostics are harder since they must be single-shot measurements

(by definition), whereas many measurements are greatly facilitated by scanning or integrating over many shots.

• Measuring the full power pulse can be very difficult, and a proxy is often used.

• Energy: meter or measuring system (harder than it sounds at high energies)

• 2D spatial profile: Focal plane imaging camera to measure 2D spatial profile

Hard to measure at full power and “equivalent” focal spot monitors are frequently used. Spot profile is often not known well.

Wavefront sensorMore for alignment and improvement of laser.

Laser Diagnostics

15

• Temporal profile: Autocorrelators

Time gated measurement using laser pulse to measure itself (“auto”). Cannot determine the temporal profile, but can constrain it –

often sufficient. Can measure fast pre-pulse

FROG (and SPIDER and colleagues)Spectrally resolved, time-gated measurement that can completely determine the temporal profile. Not frequently used.

Water cell monitorUses water to block the main pulse so a photodiode can measure the slow pre-pulse.

Knowing the intensity to within a factor of 2 is generally pretty good!The pre-pulse is frequently not known or, at least, reported.

Ponderomotive Force (NR case)

-

16

Consider an electron oscillating in an EM field with an intensity gradient on a scale larger than λ.

, ,

To start, neglect the magnetic field. To first order, we have:

Ponderomotive Force (NR case)

17

Now include the effects of a gentle variation in the electric field amplitude and the magnetic field by:

∙

Using the identity:

Substituting in for r(1) and v(1) and taking the time average:

2 ∙

12 ∙

We get:

4 o

Ponderomotive Force (NR case)

18

This is sometimes written as:

4

I

• Up is the so-called Ponderomotive potential. It is basically a kinetic energy re-labled as a potential energy.

• The ionization potential of an atom will shift by Up, making it harder to ionize (although the reason is more subtle than I’m indicating).

• Up ~ Iλ2 and likewise the force. CO2 lasers exert a large ponderomotive force!Note NIF operates in the UV: λ = 1/3 λlaser.

• Up is of order 1 eV at 1013 W/cm2 for 1 μm light.

Ponderomotive Force (general case)

19

(For a very good treatment, see Jeong-Hoon Yang’s thesis, available via download from the University of Rochester. Figures and movie from Andy Krygier.)

Derivation of the relativistic ponderomotive force is harder, so we’ll satisfy ourselves with a plausibility argument (see Gibbon Ch. 3 and references, see Quesnel, PRE 58, 3719 (1998) for a better treatment:

1

1 1 2

I in W/cm2, λ in μm

Note we get Iλ2 squared dependence for Up in the non-relativistic limit, as before.

We’ll still want to refer to Up:

1 1.37 10 1

1 1.37 10

Some representative values…Iλ2 (W/cm2 μm2) ao g Up (MeV)

1018 0.85 1.2 0.2

1019 2.7 2.2 1.0

1020 8.5 6.1 3.9

1021 27 19 13

20

• ao = 1 is usually taken as the border between non-relativistic and relativistic laser intensities (1.4 x 1018 W/cm2 for 1 μm light). At this value, the electron becomes sufficiently relativistic during a single optical cycle that a NR treatment is insufficient.

• Although the motion is oscillatory, you can reasonably use g for the relativistic mass and similar. This has far ranging effects.

• Up provides an energy scale, and this “ponderomotive” scale is frequently referred to. (See: Wilks, et al, PRL 69, 1383 (1992)).

“JxB Force”

21

Origin of the “JxB force” Direct Laser Acceleration

“Figure of 8” motion?

Single Electron In A Field

22

(Andy Krygier has posted this on youtube: http://www.youtube.com/watch?v=d0AywfEs6WA)

An electron starting at rest at a node in a planewave with ao = 3 (~1019 W/cm2.

LPI - Outline

1. Plasma de-phasing2. Self-focusing3. Intensity dependent critical

density4. Hot electron excitation5. Ion motion, fluid behavior.6. Case study7. Angular distribution of hot

electrons

23

1. Goals, Scope and Motivation2. High Intensity Lasers3. Single Electron In An EM Field4. Laser Plasma Interaction (LPI)5. Transport 6. Topic: B-field Generation7. Topic: TNSA8. Summary

Plasma De-phasing

241

≡

Self-Focusing

25

1

Lensing

Plasma focusing

Relativistic focusing

Optical thickness varies laterally

Intensity Dependent Critical Density

When laser frequency exceeds the plasma frequency (w < wp)• Index becomes complex• k-vector becomes complex and there is no propagation

past the “critical surface” (evanescent wave)• light is reflected (or absorbed)

For the non-relativistic case, we had:

For light w/ wavelength = 1 micron nc = 1021 cm-3

For the relativistic case: rc grc

2 221 2 3

2 2 1.1 10 ( )4

o L Lc

m mn x m cme e

26

1

Pre-plasma profile• The laser pulse (or at least its peak) never encounters a

sharp interface – there is always some plasma density profile leading up to target density (usually solid)

• Light will propagate up to the critical surface• Primary coupling to hot electrons occurs here (more on this

shortly)• Below critical – “underdense”• Above critical – “over dense”

x

r

rc

27

non-relativistic “classical”

critical surface

Solid Density

The Shaped Critical Surface

28

L = 3 m L = 1 mL = 0.3 m

By [gauss] AT Peak

PIC simulations for 110 fs, 1019 W/cm2, Gaussian spatial profile pulse

incident on singly charged ion. L = pre-plasma scale length: r = roexL

(Schumacher et al, POP 18, 013102 (2011).

Note beam structure.

Fast Ignition

29

Cone‐guided Fast Ignition:Fast Ignition relaxes requirements for fusion burn by separating the fuel assembly from the ignition/burn –the idea is analogous to a gasoline engine where the spark plug initiates the explosion. (Tabak, et al. POP 1,1626 (1994))

“Fast” because a ps laser is used for ignition after the ns drive lasers have compressed the target.

Getting the ignition laser to the fuel through the corona is hard. Using a cone isn’t sufficient, but might help. There are other strategies (Li, et al, PRL 100, 125002 (2008)).

A not uncommon electron spectrum measured far from the target

30Link et al, POP 18, 053107 (2011)

Electrons from a discharging phase and not directly correlated to LPI source.

Electrons strong downshifted in energy, but still correlated to LPI source.

Collisionless Heating• Vacuum (or Brunel: not-so-resonant, resonant) Heating

– Ignore B-field– E-field accelerates electrons near surface– Requires some p-component of light (E poking surface)

– Electrons slammed into surface w/ v ~ vossinEL

31

• Normal skin effect – For skin depth distance ls = c/p and electrons wiggle in

laser field & energy loss via collisions– Electron mean free path (lmfp) < skin depth

• Anomalous Skin Effect– Electrons get hotter higher speed, longer lmfp > ls– Electrons carry energy further into plasma

J x B Heating

• Requires high intensities• Accelerates electrons along k-vector• Accelerates electrons at twice laser frequency

ELk

JxB electrons

Brunel electrons

32

Scaling Laws vs. Experiment(Robert Mitchell, OSU)

Other scalings have been proposed or observed. See Beg, et al, POP 4, 447 (1997).

(references for previous slide)

1. X.X. LIN, Y.T. LI, B.C. LIU, et. al., Laser and Particle Beams 30, 39–43, 20122. P. M. Nilson, J. R. Davies, W. Theobald, et. al., PRL 108, 085002, 20123. B. Westover, C. D. Chen, P. K. Patel, et. al., PoP 18, 063101, 20114. Aghapi G. Mordovanakis, Paul-Edouard Masson-Laborde, James Easter, et. al., Applied

Physics Letters 96, 071109, 20105. T. Tanimoto, H. Habara, K. A. Tanaka, et. al., Journal of Physics: Conference Series 244,

022067, 2010 ; and T. Tanimoto, H. Habara, K. A. Tanaka, et. al., Phys. Plasmas 16, 062703, 2009;

6. A. L. Lei, K. A. Tanaka, R. Kodama, et. al., PoP 16, 056307, 20097. Hui Chen, S. C. Wilks, W. L. Kruer, et. al., PoP 16, 020705, 2009 (Kluge Ref 18)8. F. Beg, A. Bell, A. Dangor, et. al., PoP 4, 447, 1997 (Kluge Ref 19)9. P. Gibbon, “Short Pulse Laser Interactions with Matter”, 200710. Patrizio Antici, Julien Fuchs, Thomas Grismayer, et. al., IEEE Transactions on Plasma

Science, Vol. 36, No. 4, 200811. H. Chen, R. Shepherd, H. K. Chung, et. al., PRE 76, 056402 200712. Jian Zheng, K. A. Tanaka, T. Sato, et. al., PRL 92, 16, 200413. K. B. Wharton, S. P. Hatchett, S. C. Wilks, et al., PRL 81, 4, 199814. J. Yu, Z. Jiang, J. C. Kieffer, PoP 6, 1318, 1999 (Kluge Ref 17)

Long vs Short Pulses

• Long duration laser pulses ~ns (10-9 sec)Fluid approximations tend to be good, and hydrodynamic treatments dominate.

• Short duration laser pulses ~fs-ps (10-15-10-12 s)Short pulses tend to be intense and kinetic treatments are usually needed with many electrons having v @ c (perhaps travelling through a colder sea of “background” electrons) and PIC/Monte Carlo methods are often used; ions may not move much during the laser (esp. for fs case).– Typically large Iλ2

– vhot-electron à vthermal

– Large mean free path for hot electrons

35

Length ScaleOne figure of merit: sound speed

Use pulse duration & speed of sound to get distance ions move while laser is present

• Example: Aluminum - A=26, Z*~7-9, Te~100eV = 0.1keV• D (nm) ~ 0.053 t(fs)

Long pulse: t ~ 1ns = 106 fs D ~ 5.3x104 nm = 53 microns– Much longer than wavelength (~1 micron)

Short pulse: t ~1ps = 103 fs D ~ 53 nm– Sub-wavelength for any laser (except x-ray)

Ultrashort pulse: 30 fs D ~ 1.6 nm

1

* 212( ) 0.3 ( ) ( )s e

ZD nm c t T keV t fsA

36

1 1

* 2 * 217 23.1 10 ( ) /B es e

i

Z k T Zc x T keV cm sm A

Laser Interactions w/ solid density

• D 0 can use Fresnel equations from optics– Absorption given by 1-R

• D º solve Helmholtz eqns– Assume Drude type dielectric function– Assume harmonic time dependence (use linear approx)

• D > and p-polarized light (E-field pokes into surface)– Resonance absorption

• D < & short pulses & high intensity– collisional heating (inverse Bremsstrahlung) inefficient– “collisionless” heating begins

37

Case study: Kemp and Divol,PRL 109, 195005 (2012)

38

PIC simulationLight: I = 1.37 x 1020 W/cm2, λ = 1 μm, spatial profile: 40 μm flat top, 13 kJ over 10 ps (Fast Ignition), 200 fs rise time.Target: Deuterium ions and electrons (already ionized) with pre-plasma profile.

Case study: Kemp and Divol,PRL 109, 195005 (2012)

39

The pre-plasma profile reshapes dramatically over ~3 ps. The classical critical surface moves significantly.

This changes hot electron generation correspondingly, so there is no one “Thot”.

Collisionless Electrostatic Ion Shocks

Laser

ne

ni

ExEx0

λD

Sentoku et al., Phys. Plasmas (2003)Haberberger et al, Nature Phys 8, 95 (2012)

• Started by an intense laser sweeping up electrons in front of the pulse, where the ponderomotivepotential balances the electrostatic potential

• This sets up a electric field in the laser propagation direction, which accelerates the ions

• This ion source might have significant advantages over TNSA depending on the appliation.

nc

Buried layer K imaging

CCD

LaserK fluor

Bragg crystal

K (10 m res.)

41

Credit: K.U. Akli

Akli, et al, PRE 86, 026404 (2012)

See Meyerhoferslides 16+ on Kα.

Mike Storm Review Of Various Experimental MeasuresOf The Electron Divergence

½ (degrees)

Dia

gnos

tic

Transport Issues• The hot electrons can easily travel into and through the target. Escaping

electrons will charge the target creating large electric fields.• If the target is thin, most of the hot electrons will oscillate about the target

many times (refluxing).• The ion and (interior) target electrons provide an initially cold background.• The hot electrons (and the radiation they excite) provide crucial

diagnostics and may drive many exciting applications.

laser150 J0.7 psI2 ~ 8x1019

+

--

- --

-

- ++++

´

e-

ions+

e-

e-

solid target

B > 10 MG

sc ~ MV

43

Charge Separation• Starting with a neutral solid target with pre-plasma profile• The laser (perhaps with additional ionization) excites hot

(relativistic) electrons• High speed electrons begin to stream into the target from

the LPI region but…• …if nothing else happens, that would leave net positive

charge behind with associated quasi-static electric fields. Instead, the fields drive current flows that “attempt” to maintain local charge neutrality.

• This current (at least initially) requires a contribution from the cold, background electrons.

44

Total Current

ALFVEN LIMT

Current I increases, the B-field intensifies, until electrons bent back upon themselves by v x B forces. In vacuum 17 kA

Confined current made up of fast moving charges

RETURN CURRENT

1 ps laser pulse focused to spot ~30 µm, absorbed intensity of 1018 W/cm2 energy per pulse ~7J, (1014 fast e- @200keV); bunch ~60 μm in length (RMS 200 keV fast e- range in Al); magnetic field on surface of cylinder ~3200 MG magnetic field energy of 5 kJ!A.Bell, et al., Plasma Phys Control Fusion 39 653 (1997)

Energetics require a return

current

Self consistent B field of current I

45

Return Current• There will be a “return” current yielding a net current ~ 0.

• However, the hot electron density is very small compared to the cold

• The hot electrons are not collisional, whereas the cold electrons are. Now, from Ohm’s Law,

0 net hot return hot hot hot return return returnj j j j n ev j n ev

46

≫ ≪

Jhotlaser

Jreturn

E (ohmic field)

Ohmic Inhibition

From King et. al; Phys. Plasmas 16, 020701 (2009)

Measure x-ray fluorescence as a probe of the hot electron current. • Hot electrons ionize inner K-

shells• Holes fill & emit Ka radiation• ~8 keV for Copper

47

Assuming the laser successfully injects the same amount of charge into the wire, the current density will be higher in the thinner wire.

When is Ohmic Inhibition Important?

• Must have large current density for large Ohmic stopping

• How to minimize Ohmic stopping?– Decreasing current density

• Beam divergence laser focused into slab• Area grows & current density drops

– Start w/ lower current density• If get high intensity w/ short pulse, low energy• Remember more laser energy coupled into more electrons

– Fast Ignition scale lasers necessarily will have high current density, but…

return return hot hotE j j j E j

48

Remember Material Resistivity

10 g/cc

1 g/cc

100 g/cc

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

0.1 1 10 100 1000Temperature eV

Res

istiv

ity O

hm m

AuAlCD 1 g/ccD2 1 g/ccCD 10 g/ccD2 10 g/ccCD 100 g/ccD2 100 g/cc

Spitzer resistivity for high T collision dominated: ~T-3/2

Full scale conditions will have hot dense material for background plasma and reduced ohmic field

return return hot hotE j j j E j

49

Hybrid PIC model ( Paris)L Gremillet G Bonnaud,

F Amiranoff POP 9,941,(2002)

The Picture

Laser Ionization creates fast forward electron stream

Large number of slow electrons are drawn in to neutralize the fast electrons

If original hot electron current exceeds Alfven limit, filaments into many small components, each separated by return currents

50

Resistive Effect• Maxwell Eqns (transport problem)

• E driven by return current

BEt

0 0 0netEB Jt

return return hot hotE j j j E j

Ignore displacement current

( ) ( )return hot hot hotBE j j j jt

Spatial variation in resistivity

Spatial variation in hot electron current

51

This generated B-field will then subsequently guide the hot electrons, dynamically reshaping the electron spatial profile – this effect is being examined as a means to improve transport for FI.

Hybrid PIC model ( Paris)L Gremillet G Bonnaud,

F Amiranoff POP 9,941,(2002)

Strong variation in jhot is a certainty

Spatial variation in jhot can lead to B-generation via(1) spatially varying heating leading to “h (first term)(2) “ x jhot (second term)

52

Induced B Fields Near Critical Surface

Field pinches expandingPlasma electrons

en I

ponderomotive

en

II

nc

53

BEt

eT n thermoelectric

en

T T

nc

n /

/

en I

x

y

( )yE x

( )zB x

( )yE x

( )zB x

inducedjinducedj

Target surface Assume uniform

eT n thermoelectric

ponderomotive

Incident laser

hotv coldv hotj coldj

inducedjhot cold

cold hot

yz

o induced

j j

E j j

BEt

EBt x

B j

Magnetic Fields

54

55

Magnetic Field Evolution (By): 1019 W/cm2, 55 fs

50fs                                   100fs                                150fs

320fs                                  620fs                                920fs

Ion Acceleration Via TNSA

~10 mE-field

TNSATarget Normal Sheath Acceleration

• Laser accelerates electrons some escape, most reflux (~99.9%)

• Very high fields produced at target surfaces and edges (~ MV/μm)

• Atoms near and on the surface are ionized and accelerated by these sheath fields

• Although electron motion is compelx, heavy ions see an averaged motion and field pointing away from the target and normal to it.

• Pre-plasma is actually helpful here! Short duration, ~ps Small source, ~200 microns High Brightness

57See: Maachi, RMP 85, 751 (2013); other major ion accel. mechanisms also discussed.See: Wilks, et al, POP 8, 542 (2001).

Great Proton (ion) Beam Source

From P. Patel

58

Since the TNSA ion pulse is intense, short pulsed and synchronized to the pump laser, it’s useful as a diagnostic.

The protons can even be focused

Measure visible light from heating as a function of time.

Patel, et al, PRL 91 125004; see also: Bartal, et al, Nature Phys 8, 139 (2012)

59

The target doesn’t stand still

Debye Sheath where

ion(local) ≤ Debye (local)

Ion frontNe, hot

Ne, cold

Nion

Ion charge sheet

Nion ~ exp z ion

Ne,hot + Ne, cold = Nion

Electric Field (constant) ~ Thot/e lion

REFLUXING REGION: Vhot is max at ion charge sheetAnd is zero at ion front

60

Electrons co-moving with ions (on average) reduce the quality of the ion beam, especially if you want a focus.

Large Effort To Study TNSA

61

Zeil, et al, NJP 12, 045015 (2010).

Results using Trident, Vulcan, NovaPW, and other laser system.

Thinner targets tend to do better (until you prematurely destroy it). Target preparation is crucial, especially if you want anything other than protons.See: Morrison, et al, POP 19, 030707 (2012).

Summary• Ultraintense, short pulse lasers drive electrons relativistically with

far reaching consequences for most processes.• The ponderomotive force/energy scale is a key figure of merit.• For such short pulses, hydrodynamic evolution can be limited

during the time the pulse is on.• Most diagnostics depend on the hot electron transport, directly or

indirectly, and that must be accounted for when interpreting the diagnostics.

• Current balance of hot and cold electrons and temperature dependent transport coefficients must be understood.

• Hot electrons drive the creation of quasi-static magnetic and electric fields that are intense with important effects (TNSA).

• Intense laser matter interactions and the subsequent transport problem offers a “rich” environment with many key questions still unanswered.

62