#### 3.1. Coherent Synchrotron Emission

Figure 2 shows the coherent enhancement above the bursting threshold for a 46 pC bunch (3 × 10

^{8} electrons) and a synchrotron frequency of

f_{s} = 1.75 kHz measured with the vacuum FT spectrometer. Assuming a Gaussian bunch form, the experimental data fit neither the number of electrons nor the nominal

rms bunch length of 1 mm. A pure Gaussian electron distribution with the above parameters would produce a spectral power. Below 60 GHz the finite dimensions of the dipole chamber of a free height of 35 mm at BESSY II almost shield the radiation. Additional diffraction losses from the beamline limit a realistic detection of the CSR even below 0.1 THz.

Figure 2 cartoons this spectral region as well. If one uses the Equations (1) and (2) with the nominal

rms bunch length to describe the CSR process, bunch shapes different from Gaussian have to be assumed (e.g., saw tooth shape). Such bunch shapes would contain higher frequency components explaining the broad experimentally observed amplification factor. However, the measured coherent enhancement fits perfectly a Gaussian over four orders of magnitudes indicating that the CSR is emitted from a longitudinal micro bunch structure with an

rms micro bunch length of 0.17 mm. Moreover, only less than 0.1‰ of the electrons from the nominal bunch charge are involved in the detected coherent process.

Below a charge threshold of about 30 pC, the bunch distortion is stable in time and so the CSR power does. Above this threshold, the bunch distortion becomes longitudinally unstable and modulates the THz emission in the time-domain.

Figure 3 shows the power spectrum of the temporal behavior of the CSR for two different bunch charges, below and above the threshold. The synchrotron frequency

f_{s} is clearly seen in the power spectra for the coherent and incoherent part of the synchrotron radiation. For the coherent radiation below the bursting threshold, a longitudinal instability develops in a very regular repetition of a fundamental frequency

f_{f} at about 3.75 kHz. This frequency is about twice the synchrotron frequency

f_{s}. As it is also observed at other storage ring facilities [

24,

30], the fundamental frequency of instability shifts to higher frequencies and gains in strength with increasing bunch charge and even mix with different other modes. A higher bunch charge produces saw-tooth type instability and the duration of a saw-tooth determines the side-band structure. A further increase of bunch charge leads to a turbulent longitudinal bunch distortion causing CSR bursts emitting randomly shown in

Figure 3 for the CSR power spectrum for a bunch charge of 46 pC. A more detailed picture of the temporal structure of CSR bursts at BESSY II is published elsewhere [

31,

32].

The existence of two thresholds in the electron bunch spatio-temporal dynamics while increasing the total charge of the bunch has been pointed out in [

21]. The first threshold found indicates the presence of micro-structures drifting in the bunch profile and is signaled by the appearance of a resonance in the power spectral density at high frequencies (above 30 kHz) which is outside the range of our measurement. The second threshold indicates that the micro-structures are strong enough to persist after about half a revolution period of the electron bunch in the longitudinal phase space. It is linked with the appearance of another resonance in the power spectral density at about twice the

f_{s}, in perfect agreement with our observation. An analytical link between the structure of the longitudinal phase space of the electrons and the frequencies seen in the power spectrum of the CSR emission during instabilities is derived in [

33].

#### 3.2. Cross-Correlation of Subsequent Bunches

Next, we like to draw a simple picture of the mechanism of the longitudinal instability by assuming a longitudinal oscillation with frequency

f_{f} of the emitting micro bunch. Despite this scheme is extremely simple, it reproduces well the mechanism described in [

21], where the longitudinal phase space of the electron bunch, after the first threshold, presents some sub-structures that evolve in time with a rotating movement due to the synchrotron rotation. Those sub-structures translate into a micro-bunching in the associated bunch profile. Each micro-bunch oscillates harmonically in a region whose length depends on how long the sub-structure persists in the bunch.

In contrast to established temporal approaches we follow a statistical approach to describe the bunch dynamics. Our simple model assumes that the micro bunch has an oscillating phase against the nominal bunch position and tries to describe the amplitude of this phase oscillation in respect to the bunch charge.

The oscillating phase shift

τ (t) of the individual micro bunch in time units can be expressed by

with

τ_{o} as the maximum oscillation phase which is a function of the bunch charge

q. With the assumption that the instability is a single bunch effect each micro bunch behaves independently and experiences all possible phase shifts over the integration time of the detector. Therefore, a measured cross-correlation of CSR pulses from pairs of successive micro-bunches can mathematically be treated as the auto-correlation of the mean from all CSR pulses. The mean electrical field of the pulse is given by the convolution integral of the single CSR pulse shape with the probability

P(τ, dτ) = p(τ) dτ that the individual pulse shift can be found in the interval

[τ, τ + dτ]:

In analogy to the harmonic oscillator model (e.g., [

34]) we assume that the probability density function

p(τ) is proportional to the time

dt in which the CSR pulse phase changes by

dτ and can be written in its normalized form to:

In

Figure 4 the probability density function multiplied by the maximum of the oscillating phase is presented. It is obvious, that the turning points at

±τ_{o} yield the highest probability for the phase shift. The inset of this figure shows a cartoon of the longitudinal projection of the resulting mean pulse trace under investigation. As stated above the theoretical cross-correlation of the successive CSR pulses can be obtained by the correlation of the mean CSR pulse from Equation (5):

The experimental cross-correlation was measured utilizing the Martin–Puplett interferometer as shown in

Figure 1 with a nominal optical pass difference between the two interferometer arms equal to the distance between subsequent bunches in the bunch trail. For experimental and mathematical simplicity, we restrict our consideration not to the entire CSR spectrum of the micro bunch but to a limited bandwidth achieved by passing the radiation through a band-pass filter.

Figure 5 shows the power spectrum of the electrical field of the new created CSR pulse when passing the filter and which is according to the Wiener–Khinchin theorem [

35] the squared Fourier transform of the measured field auto-correlation. A Gaussian fit to the spectrum yields an

rms spectral width of 8.5 GHz (0.29 cm

^{−1}) and a center frequency of 0.22 THz (7.4 cm

^{−1}). These values are then used to construct an analytical transform-limited Gaussian pulse

E(t) by using the time-bandwidth product [

36].

Figure 5 also shows this pulse with a

rms temporal width for the Gaussian envelope of 9.4 ps.

The amplitude of the cross-correlation as the function of the time delay between the two spectrometer arms is exemplarily shown in

Figure 5. For the calculation

τ_{o} was set to zero and the result compares well with the measured cross-correlation for a pulse emitted from a bunch below the bursting threshold. Note that the experimentally obtained cross-correlation function is superimposed by a constant

dc voltage signal. This

dc component corresponds to the mean of the power of the two pulses (e.g., [

37]) and therefore is used for power normalization of the experimental cross-correlation function.

From both the theoretical and measured cross-correlation functions, shown in

Figure 6, a contrast calculates as the difference between the maximum and the minimum of the modulation amplitudes. This contrast, shown in

Figure 7, is independent of the bunch charge in the stable mode below 28.6 pC (vertical dashed line in

Figure 7) where no oscillating instabilities appear. Above that charge, instabilities start to appear in the nominal bunch and the contrast drops rapidly to zero and develops with further increasing charge into an expiring oscillation. The oscillation period in the contrast function relates to the center frequency and the shape of the band path filter applied. In order to fit the theoretical contrast given as a function of the phase shift amplitude to the ring current (and accordingly to bunch charge units), distinctive data points were chosen from the theoretical and measured contrast to scale the abscissa. The inset of

Figure 7 shows the polynomial fit function to this scaling procedure.

The theoretical contrast as a function of ring current describes remarkably well the measured data as shown in

Figure 7. This holds, even the experimental data were obtained at several temporal experiment shifts over one year of observation at the storage ring and proofs the stability and reproducibility of the BESSY II ring and the

low α set up offered to THz users. Note, that for a given center frequency of the band-pass filter the relation between the contrast of the cross-correlation function and the phase shift of the pulse is completely determined and the only assumption made is that the micro pulse is oscillating in longitudinal direction. With the scaling relation shown in

Figure 7 one has a ruler in hands to measure the longitudinal phase shift amplitude of the micro-bunch directly from the ring current stored. It is obvious that the model discussed above is not only valid for one oscillating micro bunch but also applies to higher instability modes where more than one micro bunch is involved. The projection of higher azimuthal modes, for example quadrupole or sextupole modes, are very similar to two or three harmonically oscillating micro-bunches also described by Equation (6).

In [

28] an empirical relation is given to describe the measured

rms length of the nominal bunch at BESSY II for

low α operation and different synchrotron frequencies as a function of the bunch charge. Here, the nominal lengths were measured by means of a streak camera for bunches longer than 1.5 ps. As an example, this empirical relation results an

rms length of about 3.9 ps for a bunch of 28.6 pC and a synchrotron frequency of 1.75 kHz just when the oscillating instability becomes observable by our measurements. As introduced before,

2τ_{ο} describes a longitudinal region in the nominal bunch profile of the oscillating instability whose length depends on how long the sub-structure persists in the bunch.

Figure 8 shows this characteristic length for given bunch charges in units of the

rms length of the nominal bunch taken from [

28]. The characteristic micro bunch length increases stronger than the nominal bunch length. It reaches about the 2.5-fold of the nominal

rms length for the limit of the highest bunch charge analyzed by the cross-correlation method proposed. Further higher bunch charges define the limit of our model when the bunch instabilities become chaotic and a cross-correlation signal is not anymore detectable.