Computer modeling of a subfemtosecond photoelectron gun with timedependent electric field for TRED experiments Mikhail Monastyrskiya, Sergey Andreeva, Dmitry Greenfieldb, Gennadiy Bryukhnevicha , Victor Tarasova, and Mikhail Scheleva a)
A.M. Prokhorov General Physics Institute, Russian Academy of Science 38 Vavilov St., Moscow 119991, Russia b)
Research and Production Center ‘Orion’ 9 Kosinskaya St., Moscow 111538, Russia ABSTRACT In the paper, theoretical and numerical studies on temporal focusing of photoelectron bunch in time-dependent fields are continued. Presented are the results of computer modeling on electron-optical system with combined time-dependent electric and static magnetic fields to ensure both spatial focusing and temporal compressing of photoelectron bunch down to sub-femtosecond level. The peculiarity of space charge effect contribution to the bunch broadening in the case of time-dependent electric field is discussed. Keywords: temporal resolution, temporal compressing of photoelectron bunch, spatial and temporal aberrations, timedependent electric field, space charge effects 1. INTRODUCTION There are two principally different ways of how the bunch of electrons emitted from a photocathode can be used. The first way is measuring temporal structure of incident optical radiation with the use of streak tubes. In this case the photoelectron bunch is considered as a carrier of information on temporal structure of incident optical pulse from the photocathode to image receiver, and the term “temporal resolution” is referred to the events that happen upon the photocathode. It automatically implies that static fields only can be used to measure temporal resolution in streak tube because any time-dependent field can unavoidably destroy initial temporal structure of the bunch. It is well known that the first-order temporal chromatic aberration (the so-called Zavoisky-Fanchenko aberration1), which is proportional to the spread of initial velocities of photoelectrons and inversely proportional to the field intensity nearby the photocathode, is the main factor that restricts temporal resolution in streak tubes. During more than fifty years, since the first Courtney-Pratt’s time-analyzing image converter tube2 was invented, the increase of field intensity nearby the photocathode has served as the main approach to diminish the Zavoisky-Fanchenko aberration and thus to improve temporal resolution in streak tube. We find ourselves in quite different situation if we consider a photoelectron bunch not as a carrier of information on the structure of incident optical radiation but as a probing tool for interaction with matter, as it is the case in timeresolved electron (TRED) experiments. In this case we are not interested in preserving the “maternal” temporal structure inside the bunch - our main goal is to get electron bunch temporally compressed at the target. Thus we can speak here about temporal resolution with respect to the target. It is important that such statement of the problem opens to us a possibility of temporal compressing of photoelectron bunches with the use of time-depending fields. As shown in3,4 , the most remarkable thing here is that, in full similarity to the case of spatial focusing, the first-order temporal chromatic aberration can be made strictly equal to zero at the point of temporal focusing in a properly chosen time-dependent field.
a
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26th International Congress on High-Speed Photography and Photonics, edited by D. L. Paisley, S. Kleinfelder, D. R. Snyder, B. J. Thompson, Proc. of SPIE Vol. 5580 (SPIE, Bellingham, WA, 2005) · 0277-786X/05/$15 · doi: 10.1117/12.567644
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At the point of temporal focusing, the bunch duration is determined by second-order and higher-orders temporal chromatic aberrations. In this paper the studies3,4 on temporal compressing of photoelectron bunches with non-stationary electromagnetic fields are continued. A system of two linear differential equations describing evolution of the first-order temporal aberration and the first variation of the particle full energy inside the bunch is derived. The analysis of this system has clearly shown the first-order temporal focusing to be conditioned by the redistribution of the full energy between the particles inside the bunch when traveling through the time-dependent field region. The contribution of the second-order temporal aberrations to the bunch duration and some aspects of quantum limitations for the bunch compressing are analyzed. Presented are the results of computer modeling of a photoelectron gun with combined time-dependent electric and static magnetic fields. It is shown that, owing to partial correction of the second-order temporal aberrations, the system proposed ensures both spatial focusing of the bunch and its temporal compressing down to the subfemtosecond level. The problem of space charge effects contribution to the temporal compression of the bunch is discussed. 2. THE MASTER EQUATION FOR FIRST – ORDER TEMPORAL FOCUSING IN TIME-DEPENDENT ELECTRIC FIELDS Considering a bunch of emitted photoelectrons which is traveling in a an axially symmetric electron-optical system along the main optical axis in a time-dependent electric field with the axial potential Φ ( z , t ) towards a target, we can describe the evolution in time of the axial coordinate z (t ;τ , ε z , ε r ) and arrival time T (t ;τ , ε z , ε r ) of an electron started at initial time moment τ with axial ( ε z ) and radial ( ε r ) components of initial energy in terms of aberration expansions
z (t ; τ , ε z , ε r ) = z 0 (t ) + zτ (t ) τ + z z (t ) ε z + K (1)
T (t ;τ , ε z , ε r ) = t + Tτ (t )τ + Tz (t ) ε z + K Most important fact established in2,3 is that the first order aberration coefficients Tτ (temporal magnification) and Tz (first-order chromatic aberration) are being proportional in any arbitrary non-stationary electromagnetic field:
Tz = −
1 E0
2m Tτ , e
(2)
and, consequently, can be made zero simultaneously. If we denote E and W full and kinetic energy of a particle in volt units, correspondingly, so that E = W − Φ , we can derive a system of two differential equations
Eτ & Tτ = − 2 W 2 E& = 2e ∂ Φ T W τ τ m ∂t ∂z
, Tτ (0) = 1 (3)
,
Eτ (0) = 0
with respect to the first variations Tτ and Eτ , which holds true on the principal trajectory corresponding to zero values
of the small initial parameters τ , ε z , ε r . The value Eτ may be referred to as first-order term of the full-energy dispersion, which arises from the bunch’ traveling in the non-stationary field. The system (3) can be called the master equation of the first-order temporal focusing. In the static case, when the axial potential distribution Φ does not depend
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on time, we get from (3) the trivial solution Eτ = 0 and Tτ = 1 for all t , which gives the well-known ZavoiskyFanchenko formula
(Tz )static = −
1 E0
2m e
(4)
and confirms the fact that the first-order chromatic aberration is unavoidable in static case and can be decreased only at the expense of increasing the field intensity E0 nearby the photocathode. On the contrary, the time-depending electric fields offer us a possibility of making Tτ (and, as a sequence, Tz ) strictly zero at the point of first-order temporal focusing. 3. TEMPORAL FOCUSING AS THE PHASE VOLUME TRANSFORMATION To clarify the peculiarities of temporal focusing of photoelectron bunch in time-depending fields, let us consider the evolution of the shape of the ( E , T ) phase volume that the bunch occupies during its motion in a simple electronoptical system comprising a photocathode (a), a thin temporal lens (c), and a target (d) as depicted in Fig.1. Here E is full energy, and T is arrival time of the particles at a plane perpendicular to the z-axis. Due to the fact that the equations of charged particle motion in time dependent field remain Hamiltonian, the total value of this volume is invariable while the bunch is traveling from the photocathode to the target. This immediately implies that an additional spread in energies of the particles is required to compress the bunch temporally. Fig. 2 shows the bunch phase volume evolution on a sequence of stages from the photocathode to the target.
Fig.1. Principal scheme of the electron-optical system containing a thin temporal lens. The letters indicate various stages of the bunch motion (see the phase diagrams in Fig.5).
Obviously, now we cannot restrict ourselves by consideration of the linear terms only in the temporal aberration expansion (1), and therefore let us extend the expansion by the second-order terms with respect to the parameters τ and εz :
[
]
T (t ;τ , ε z ) = t + F (t ) τ − χ ε z + Tzz ε z + Tτzτ ε z + Tττ τ 2 + K ,
(5)
Here χ denotes the Zavoisky-Fanchenko constant (1 / E0 ) 2m / e , and F = Tτ . Disengaging ourselves from the effects of temporal broadening of the photoelectron bunch inside the photocathode (according to known estimations that this time constitutes about ∼ 10-15 s), we can accept with sufficient accuracy that the duration of the new-born photoelectron bunch is approximately equal to the incident laser pulse duration δτ . Assuming the time of emission of photoelectrons into vacuum and their initial energies mutually uncorrelated, we can represent the initial phase volume of the bunch at the photocathode as a rectangle shown in Fig.2a. In a rather short time interval, when average energy of the bunch would only several times exceed its initial energy spread, the fastest particles would overtake the slow ones, and the bunch duration would be determined by the Zavoisky-Fanchenko effect with the coefficient (4).
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Within the second order approximation, the time of particle arrival at the plane z = z1 positioned in the static field in between the photocathode and the temporal lens appears as T1 = T (0) ( z1 ) + τ − χ ε z + λ ( z1 )ε z ,
(6)
where T (0) is arrival time on the principal trajectory and λ is the second-order chromatic aberration coefficient. Provided that the Zavoisky-Fanchenko term is dominant in the bunch temporal spread, the bunch’ duration at can be estimated as δT1 = χ δε z . The corresponding phase distribution of particles is shown in Fig.2b. Unlike the first-order coefficient χ , the second-order one λ is changing with time. Indeed, at the points where the kinetic energy is rather large ( W >> δε z ), the velocity spread of the particle velocity (calculated in a given plane) due to the axial energy spread ε z is
∆v = 2e(W + ε z ) / m − 2eW / m ≈ e / 2mW ε z
(7)
has the order of ε z , and this spread results in changing of the second-order term dλ 1 δv 1 . =− =− dt z& δε z 2W
(8)
Let us denote the value of λ at the entrance of the temporal lens as λ 2 = λ ( z 2 ) and write down the formula for the corresponding deviation of the arrival time from that on the principal trajectory: ∆T2 = τ − χ ε z + λ 2 ε z
(9) Fig.2 The evolution of photoelectron bunch in phase space:
a) at the photocathode, b) before entering the temporal lens, c) after exiting the temporal lens, d) at the point of temporal focusing
As a matter of fact, the action of the time-dependent field in a thin temporal lens is that different particles of the bunch gain or loose different amount of energy being mainly determined by the particle’s location inside the bunch, e.g. by the value (8). After a particle has passed through the temporal lens area, its full energy
[
]
becomes E (τ , ε z ) = ε z + E * T2 (τ , ε z ) , where the full energy increment E * (⋅) depends on the lens geometry and voltages. To analyze the temporal aberration coefficients of the first and second order, we have to expand the increment
E * within the second-order accuracy into the Taylor series
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E* E * (∆T2 ) ≈ E0* + Eτ*∆T2 + ττ ∆T2 2 + ... , 2
(10)
the coefficients of which can be calculated with the tau-variation technique. The corresponding phase diagram is presented in Fig.2c. To the right from the temporal lens, the bunch is again traveling in static field. Quite similar to the photocathode-temporal lens region, a term proportional to the full energy variation appears in the expression for arrival time at the target, with the only difference that this variation is not the target ( 0)
T3 (τ , ε z ) = T3 ( 0)
where T3
εz
but ε z + E * [∆T2 (τ , ε z )] − E0* . Thus, we get at
E* + τ − χ ε z + λ × ε z + Eτ* (τ − χ ε z + λ 2ε z ) + ττ (τ − χ ε z + λ 2ε z ) 2 + K 2
is arrival time at the target on the principal trajectory. The coefficient λ
(11)
still obeys the equation (9),
however behind the temporal lens the corresponding term is the first-order one with respect to ε z and may compensate for the Zavoisky-Fanchenko term in a point of temporal focus z 3 , where the value of λ gets
λ3 = −1 / Eτ* . The corresponding phase diagram of the bunch at the target is shown in Fig.2d. Comparing then (11) with the general expansion (5), we eventually come to the relations for the second-order temporal aberration coefficients at the point of temporal focusing:
Tzz = −
1+
χ2 2 Eτ*
Eττ*
,
Tτz =
χEττ* Eτ*
,
Tττ = −
* 1 Eττ . 2 Eτ*
(12)
At the point of temporal focusing, the three second-order aberration coefficients are completely determined by the first and second variations of the function E * or, in other words, by the geometry and working modes of the temporal lens along with the time derivative of the field intensity at the photocathode. It is worth to emphasize that those coefficients does not depend upon the static field regions positioned either to the left or to the right from the temporal lens. Provided that χ δε z >> δτ , the term Tzz ε z gives the most significant contribution of those three terms to the bunch’ duration, while the next is the term Tτzτ ε z . Suppose, we have constructed a lens, in which the term Tzz vanishes at the point of temporal focusing. It means that
( )
* * * the condition χEττ + χτ Eτ = −2 / χ holds true, and the coefficient Tτz = 2 / χEτ being necessarily different from zero determines the bunch’ duration:
δT ≈
2
χ Eτ*
δτ δε z .
(13)
Thus, it is impossible to simultaneously eliminate all the second-order temporal aberrations. The temporal spread of the particles at the lens’ entry is mainly determined by the first-order temporal aberration and is approximately equal to
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δT2 ≈ χ δε z . It means that the energy spread inside the bunch appears as δE ≈ Eτ∗ χ δε z , and the relation (13) leads to the estimation
δTδE ≈ 2δτδε z
(14)
which binds together the minimum bunch’ duration δT and its energy spread δE at the target with the incident laser pulse duration δτ and the initial energy spread δε z of photoelectrons at the photocathode. The relation 13 expressing the law of phase volume conservation is extremely important for temporal focusing theory because it determines theoretical limit of photoelectron bunch compressing. It is to be noticed that the coefficient 2 in (14) reflects the fact that the phase volume of the rectangular region δT × δE (Fig.2d) is approximately two times larger than the real phase volume of the bunch. More accurate consideration which does not rely upon the thin temporal lens approximation leads to the relation
δ 2T δE =2, δτδ ε z δ ε z
(15)
which holds true at the point of temporal focusing, provided that the coefficients F and Tzz vanish. In this connection it is worth to touch upon the quantum limitations for the bunch duration. It is well known that the electromagnetic pulse generated by a femtosecond laser is non-monochromatic because of the duration δτ of the pulse is extremely short. The spectrum width of such a pulse is δω ≈ 2π / δτ , and, correspondingly, the quantum-mechanic uncertainty of the photon energy appears as
eδε quant = hδω ≈
2πh
δτ
.
(16)
It is clear that the energy uncertainty of emitted photoelectrons cannot be less this value, which leads to the inequality
δTδE ≥ 4πh / e .
(17)
The value eδε quant is about some tenths of electron-volt. In particular, at δτ = 20 fs the quantum uncertainty of energy is e δε quantum ≈ 0.2 eV. In reality, due to the fact that photons knock out electrons from different energy levels, and electrons scatter their energy on the crystalline pattern oscillations, both the initial energy spread of emitted particles and the bunch’ duration at the target commonly exceed the quantum minimum.
4. ELECTRON-OPTICAL SYSTEM WITH THE FIRST-ORDER TEMPORAL FOCUSING AND PARTIALLY CORRECTED SECOND-ORDER TEMPORAL ABERRATIONS This section is devoted to numerical calculation of an axially-symmetric electron-optical system capable of forming electron bunches of approximately one femtosecond duration. Temporal aberrations are calculated with the use of the tau-variation technique which allows efficient constructing of aberration expansions with respect to any set of previously chosen small parameters. In the case under consideration such a set include the initial time moment τ of photoelectron emission from the photocathode into vacuum, the values
ε z and ε r , which with the accuracy to the
2e / m relatively coincide with the axial and radial components of photoelectron’ initial velocity, and the initial radial coordinate Rc of a photoelectron start from the photocathode. With regard to the axial symmetry of the system in
factor
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question, the third-order temporal aberration expansion associated with the set of small parameters { τ ,
εz , εr }
appears as 1 T ( z;τ , ε z ) = T0 ( z ) + F ( z ) τ − E0
2m ε z + Tzz ε z + Trr ε r + Tτzτ ε z + Tττ τ 2 + Tcc Rc2 + e
+ Tzzz ε z3 / 2 + Tzrr ε r ε z + Tτzzτε z + Tτrrτε r + Tττzτ 2 ε z + Tτττ τ 3 + Tcr Rc ε r +
(18)
+ Tτccτ Rc2 + Tτcrτ Rc ε r + K
Here the function T0 ( z ) represents the traveling time along the principal trajectory. The temporal aberration coefficients determining the bunch’ duration in (18) are represented as functions of z-coordinate along the principal trajectory. As we have seen above, the predominant contribution to the bunch’ duration in static fields (constituting, as has been shown, hundreds femtoseconds) is given by the first-order temporal chromatic aberration. The modern pulse lasers are capable of generating much shorter pulses (about few femtoseconds), and the contribution of the first-order term F (z )τ appears to be rather small in this case. The same holds true for the higher-order terms: the most significant contribution is given by those of them which do not depend upon τ , or contain the lowest powers of this small parameter. In order to ultimately shorten the electron bunch’ duration at the target, it is desirable to make zero the most influential terms F , T zz , Trr in the expansion (18). The largest of the rest second-order terms is the term Tτzτ ε z , which, as it has been shown above, cannot be eliminated coincidentally with the term Tzz ε z . Nevertheless, it by no means signify that we are within our right to completely neglect the third-order terms contribution. In particular, the third-order terms Tzzz ε z3 / 2 and Tzrr ε r ε z do not contain the parameter τ , while the rest second-order terms do contain this parameter, and their contribution to the bunch’ duration may be even less as compare with the contribution of the third-order terms. The example presented below illustrates a possibility of simultaneous eliminating all the three temporal aberration coefficients F , Tzz , and Trr in the crossover plane, where the bunch’ cross-section is minimal. The requirement of elimination of the secondorder temporal chromatic aberration Tzz ε z has brought us to the temporal lens comprising two ‘time-dependent’ diaphragms, with one voltage increasing and another one decreasing in time. The rate of the voltage change in time should be not higher than 10 kV per nanosecond which is attainable for modern high voltage pulse technique. Spatial focusing of the bunch in radial direction might be, in principle, ensured by a proper choice of the diaphragms’ aperture, or with a supplementary electrostatic lens introduced into the system. Nevertheless, as our calculations have shown, the radial chromatic aberration coefficient Trr sharply increases in the electrostatic focusing mode. From this point of view magnetic focusing seems to be more promising. The matter is that the magnetic lens contributes much less to the coefficient Trr and does not affect the coefficients F and Tzz . It is interesting to notice that in the case that the longitudinal magnetic field
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Fig.3. Principle scheme of the electron-optical system with two “time-dependent” diaphragm and magnetic focusing (1 – photocathode; 2,3 – “time-dependent” diaphragms; 4 - exit diaphragm; 5,6 – magnetic focusing system; 7 - target). The arrows indicate how the voltage changes in time: down – decrease, up – increase.
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spatially coincides with the time-dependent electric one, the coefficient Trr , having sharply increased in the magnetic field region, acquires then a negative derivative and decreases in value. With an appropriate choice of geometry parameters, it proves to be possible to make this coefficient zero at the point of temporal focusing. Fig.3 displays the principal electron-optical scheme of the device. The main electrical and electron-optical characteristics are as follows: Near-cathode electric field at the initial time moment τ = 0
700 V/mm
Voltage change rate ns)
7.5 KV/ns (within the time interval from 0 to 1.2
Maximum magnetic field
250 Gauss
Target voltage (with respect to photocathode)
8000 V
Electron energy at the target
9600 eV
Crossover position
170 mm
Image position
350 mm
Energy dispersion at the target
30 eV (0.3% of the electron energy at the target)
In Fig.4 the first-order aberration coefficients and the traveling time T0 ( z ) along the principal trajectory are shown as functions of z-coordinate. The first-order aberration coefficient F and the limiting paraxial trajectory W (z ) determining both the crossover plane and electron-optical magnification vanish at the same point. Thus, in the system under consideration the point of temporal focusing and the crossover point do coincide. Fig.5 illustrates the fact that the second-order aberration coefficients Tzz and Trr vanish at the same point.
Fig.4. The first-order aberration coefficients: 1 - traveling time along the principal trajectory (ns) 2 - limiting paraxial trajectory W (dimensionless) 3 - limiting paraxial trajectory V (relative units) 4 - the coefficient F (dimensionless)
Fig.5. The second-order temporal aberration coefficients:
Tzz 2 – transversal chromatic aberration coefficient Trr 1 – longitudinal chromatic aberration coefficient
Fig.6 displays the particle arrival time distribution at the target, provided that all the particles have started from the photocathode center. The time along the horizontal axis is counting off the arrival time on the principal trajectory. For
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comparatively large laser pulse duration ( δτ = 100 fs (A) and δτ = 50 fs (B)) the main contribution to the bunch duration at the target is given by the second order aberration Tτzτ ε z term, with Tτz = −0.03 V-1/2. In this case the bunch duration proves to be proportional to the laser pulse duration, with the proportionality coefficient about 1/100. Due to the presence of the third-order aberrations being independent of δτ , it seems rather difficult to produce a bunch with the duration of much less than one femtosecond. In the case in question those coefficients are Tzzz = −3 fs V-3/2 and Tzrr = 9 fs V-3/2. At δτ = 15 fs the corresponding aberrations give a prevalent contribution to the bunch duration.
Fig.6. The distribution of particles in arrival time at the target for maximal initial energy of photoelectrons ε 0 = 0.6 eV and various laser pulse durations δτ : A - 100fs, B - 50fs, and C - 15fs.
Among the temporal aberrations caused by finiteness of the emission region, the main contribution to the bunch duration make the aberrations Tcc Rc2 and Tcr Rc ε r , with the coefficients Tcc = -470 fs mm-2 and Tcr = 230 fs mm-1 V-1/2. The work area which does not essentially contribute to the bunch duration is about 100 microns.
5. ESTIMATION OF SPACE CHARGE EFFECTS CONTRIBUTION TO TEMPORAL COMPRESSING OF ELECTRON BUNCH IN TIME-DEPENDING FIELDS In conclusion, let us make some estimation regarding the contribution of space charge effects to the bunch duration. First, it is important to note that static focusing electric field that compresses the bunch in radial direction is able to significantly compensate for Coulomb broadening of the bunch in this direction. Roughly speaking, the counteraction of these two factors simply results in a shift of the image plane position compared to that for the bunch whose space charge is assumed to be negligibly small. On the contrary, as we have seen above, in static case there are no forces compressing the bunch in axial direction, and as a sequence, there are no factors that could prevent the axial space charge broadening of the bunch. In time-depending field the situation is quite different: due to the fact that the first variation Eτ of full energy of the particles with respect to the start time moment τ is no longer zero, in a properly chosen time-dependent field the bunch tail is moving faster that its leading edge, so that the bunch is getting compressed at the point of the first-order temporal focusing. As it can be easily seen from the second of equations (3), such a compressing effect is proportional to 2
the mixed derivative ∂ Φ / ∂z∂t which characterizes the rate of the field intensity change in time at a fixed point of z-axis. By analogy with charged particle optics
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Fig.7. Approximate axial and radial sizes of a photoelectron bunch
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of static fields, in which the second spatial derivative d 2 Φ / dz 2 determines the optical force (and, correspondingly, the focal length) of static lens, we can say that in charged particle optics of time-dependent fields, the mixed derivative
∂ 2 Φ / ∂z∂t determines the optical force (and, correspondingly, the focal length) of temporal lens. Thus, it can be expected that the temporal compressing of the bunch would prevent its space charge broadening in axial direction, and the estimation for dynamic range of bunch duration in the case of time-dependent field would be looking more optimistic than in the static one. Simple calculation confirms this fact. The first variation of the extra energy spread due to Coulomb repulsion inside the bunch with respect to the initial time moment τ as applied to the electron-optical system presented above, with approximate axial and radial sizes of the bunch given in Fig. 7, can be estimated as
2W 2e 2 N Eτ (t ) ≈ − × m ∆T0 c
t
dτ
∫ R(τ )
2
≈ N × 108 eV / s
(19)
0
where ∆T0 - initial photoelectron bunch duration, R (τ ) - effective radius of the bunch, N - number of particles within the bunch. The first efficient in (19) is the bunch velocity, while the second is the first variation of momentum in initial time moment τ . The fact is that this linear part of the energy spread is small compared to that contributed by the temporal lens, and can be compensated with tuning the lens. Seemingly, more essential is contribution of the secondorder terms connected with non-homogeneity of the space charge inside the bunch. Assuming the bunch shape to be Gaussian, we get the extra temporal dispersion due to the space charge effects δT c ≈ 4 × 10 −3 fs × N , which, in turn, gives δT c ≈ 1 fs for N=250, and δT c ≈ 10 fs for N=2500.
6. CONCLUSIONS Summarizing the main results of this work, we may conclude the following: 1. The up-to-date streak tubes and photoelectron guns with static focusing are capable of providing the minimal bunch duration of about 200 fs which is insufficient for the needs of some applications of the TRED technique. Nevertheless the use of time-dependent electric field may provide the effect of temporal focusing which leads to essential increase of temporal resolution. 2. A system of two linear differential equations describing the evolution of the first-order temporal aberration and the first variation of the full energy of a particle inside the bunch traveling in an axially-symmetric time-dependent electric field is derived. The analysis of this system makes clear the mechanism of ideal first-order focusing in timedependent electric fields. It has been shown that the temporal focusing is conditioned by the redistribution of full energy between the particles when traveling through the time-dependent field region. 3. Special design of the “temporal” lenses and the use of additional magnetic focusing make it possible to compensate for some second-order temporal aberrations and gain the bunch duration of less then one femtosecond nearby the photoelectron gun crossover. 4. It has been shown that space charge repulsion in the dynamic photoelectron gun is not so crucial as in the static one. A bunch may contain thousands of particles within the temporal resolution of about 10 fs. Better resolution would require accumulation of multiple shots to form a visible diffraction pattern. 7. ACKNOLEDGEMENTS This work has been supported by the ISTC Grant # 2643 as well as by the RFBR Grants #02-02-17548 and #02-0239017.
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Е.К. Zavoisky, and S.D. Fanchenko. Physical Basis of Electron-Optical Photography. The Reports of USSR Academy of Sciences, 1956, 108 (2), pp. 218-221 (in Russian).
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S. V. Andreev, D.E. Greenfield, M.А. Monastyrskiy, V.А. Tarasov, and M.Ya. Schelev. Formation of subfemtosecond photoelectron bunches in time-dependent fields. SPIE Proceedings, vol.5398, 2004, p.p. 1-15.
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