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method based on coherent transition radiation to measure the bunch length of femtosecond electron bunches, and then the improved experimental setup and results of the bunch length measurement are given. Finally, we analyze the e?ects of humidity in air on bunch length mea- surements and explain the plan for future investigations.
2 Theoretical background
2.1 Coherent transition radiation
Radiation from a relativistic electron bunch such as synchrotron radiation, transition radiation, etc. intrinsically has a broad spectrum. If the wavelength of the radiation is shorter than the electron bunch length, the phases of the radiation emitted by the electrons di?er from each another, so the radiation is incoherent. On the other hand, if the wavelength is longer than the bunch length, the radiation is coherent and the intensity of the radiation is proportional to the square of the electron numbers per bunch. The spectral intensity emitted by a bunch of N particles is given by
Itot(?)?NI1(?)?N(N?1)I1(?)|f(?)| (2-1)
where I1(?) is the intensity radiated by a single electron and f(λ) is the bunch form factor [3, 4] , which is the Fourier transform of the normalized electron density distribution S(z). For a relativistic bunch whose transverse dimension is small compared to the length, the form factor becomes
f(?)??S(Z)exp[i2?(nz)?]dz (2-2)
where n is the unit vector pointing from the center of the bunch to the observation point and z is the position vector of the electron relative to the bunch center. Obviously, a measurement of the radiation spectrum will give the form factor and the electron density distribution through the Fourier transform.
2.2 Electron bunch length measurement
Measurement of electron bunch length is often done by examination of the autocorrelation of the CTR signal with a Michelson interferometer [5] . The interferometer is composed of a beam splitter, a fixed flat mirror and a movable flat mirror. The light en- tering the Michelson interferometer is split into two parts by the beam splitter. The two parts travel in two di?erent directions and are reflected back by the mirrors. After reflection the two radiation pulses are combined again and transmitted into a Golay detector to measure the light intensity.
The interferogram is obtained by measuring the detector signal as a function of the path di?erence in the two arms. The measured energy of the recombined radiation pulses are the
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radiation pulses from the fixed mirror, Efix?TRE(t), and the radiation from the movable mirror delayed in the time ?/c, Emove?RTE(t??/c). Here R?R(?) and T?T(?) are the reflection and transmission coe?cients of the beam splitter. The intensity measured at the detector can be expressed as
I(?)??|TRE(t)?RTE(t?)|dt???c2?|RT|2E(t)E*(t?)dt?2|RT|?2|RT|2c???(2-3)
??????|E(t)dt|2where ? is the optical path di?erence, c is the speed of light. Alternatively, a similar expression can be obtained in the frequency domain by adding an extra phase di?erence e?i??/c
to the radiation from the movable arm at angular frequency ??2?f. Thus, the total energy measured at the detector is expressed as
2I(?)??|TRE(?)?RTE(?)e?????i??/c|d????2
2RE?|RT||E(?)|e????2(2-4)
22?i??/cd??2?|RT||E(?)|d???and Eqs. (3) and (4) are related by the Fourier transform
E(?)?12??????E(t)ei?tdt(2-5)
The baseline is defined as the intensity at δ → ±∞, where the two pulses are totally separated, hence, we have
?2|RT|2??|E(t)|2dt????I??????2?|RT|2|E(t)|2d????时域频域(2-6)
By definition, the interferogram can be written as
????2*2|RT|ReE(t)E(t?)dt????cS(?)?I(?)?I????2Re??|RT|2|E(?)|2e?i??/cd?????时域频域(2-7)
Solving for |E(?)| in Eq.(7) yields
21|E(?)|?4?c|RT|22?????S(?)ei??/cd?(2-8)
where |E(?)|?|E(??)| . Using Eq. (1) and the relation Itotal(?)?|E(2?c/?)|2 the bunch form factor can be obtained from
??11 (2-9) f(?;n)?[?S(?)ei2???/cd??1]22N?14?c|RT|2NIe(?)9
???
hence, the interferogram contains the frequency spectrum of coherent transition radiation and can be used to derive the bunch length.
For a bunch with Gaussian longitudinal distribution,
the interferogram becomes
21??2/2?zf(z)?e2??z(2-10)
S(?)??????f(z)f(z??)dz?*12??ze2??2/4?z(2-11)
and the FWHM of this Gaussian interferogram is 4In2?z . Therefore, the equivalent bunch length for a Gaussian bunch distribution is FWHM.
?/In2?0.7527 times the interferogram
3 Description of the equipment
The present experiment was performed at the Femtosecond Accelerator in the THz Research Center of SINAP, which mainly consists of a thermionic RF gun, an ? magnet, and a SLAC (Stanford Linear Accelerator Center) type accelerating tube. The ? magnet is used to compress the bunches produced by the thermionic RF gun. Then the electron beam is transported through the gun-to-linac beam line and finally accelerated up to 20—30 MeV by a SLAC type tube. Coherent THz radiation with high brightness will be emitted when super-short bunches pass through the aluminum foil [6].
It is well known that the resolution of a Michelson interferometer is mainly determined by the maximum optical path di?erence of the two parts of coherent light. However, this is only true if the planes of the mirrors remain in good alignment throughout the entire scan and if the light that passes through the interferometer is su?ciently collimated. However, the moving flat mirror tends to tilt or wobble as it is retarded and, as such, will not always be perpendicular to the incident beam. This causes the light originating from the reflection of the movable mirror to deviate o? the optical axis of the detector, as shown in Fig. 1.
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Fig. 1. Schematic diagram of the Michelson optics showing how tilting the moving mirror
causes the recombinant beams to diverge from the optical axis.
In order to calculate the maximum allowable mirror tilt, we introduce first the modulation e?ciency in the case of the circular shape of the light spot on the mirrors, which could be written as
?(m)?2[J1(a)/a] (3-1)
where ?(m) is the modulation e?ciency, J1(a) is the first order Bessel function, with a given by
a?4???? (3-2)
? is the wave number of interest (cm ?1 ), ? is the tilt angle (radians) and ? is
the radius of the light spot (cm).
As a general rule, a satisfactory modulation e?ciency must satisfy [7]
?(m)?0.9 (3-3)
2[J1(a)/a]?0.9 (3-4)
According to Cohen[8] , Eq. (12) can be approximated by
2J1(4????)?1?A?2?24????(3-5)
where A?2?2?2 . Assuming ?= 100cm?1，r = 2.5cm requires that the allowable
?4mirror tilt be kept at a value of ??2.85?10rad, or ??58.7 arc seconds. According
to the above analysis, ? must therefore be less than 58.7 arc seconds throughout the entire scan.
Fig. 2. The retroreflection property of the hollow retroreflector.
To overcome the e?ect of tilt in our former Michelson interferometer, our solution is to replace the flat mirror by a hollow retroreflector. A hollow retroreflector is a device made up of three mutually orthogonal reflective mirrors. For our experiment the hollow retroreflector was made by the Edmund corporation (NT46-189); because gold has a good reflectivity in
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