Experimental demonstration of optical stochastic cooling – Nature

Measurement with synchrotron-radiation monitors

IOTA is equipped with synchrotron-radiation monitors at each main dipole. These stations, which use Blackfly-PGE-23S6M-C complementary metal–oxide–semiconductor (CMOS) cameras and have approximately unity magnification, are used to record direct transverse images of the beam distribution. For the OSC experiments, the M2R station was upgraded for higher resolution using two hardware improvements: the rejection of vertically polarized light (Thorlabs PBSW-405) and the use of a narrowband filter (Thorlabs FBH405-10). Although the emitted vertically polarized light is weak, it increases the diffraction-limited spot size by about 20%. The wavelength of the narrowband filter was made as short as possible while maintaining a high quantum efficiency in the camera’s sensor. The filter reduced the diffraction contribution of long-wavelength radiation and the contribution of lens chromaticity from short-wavelength radiation. Calculations indicate that these measures reduced the diffraction contribution by almost a factor of two. To focus the beam image, the longitudinal position of each camera was adjusted to minimize the measured beam size; preference was given to the vertical size as depth-of-field effects are more substantial in the horizontal size owing to the horizontal curvature of the beam trajectory. ‘Hot’ pixels in the M2R projections were selected using a peak-detection algorithm and replaced by the average of their nearest neighbours. The diffraction corrections to the measured beam sizes were determined experimentally for the M2R and M1L measurement stations by fitting a single correction to the cooling, heating and no-OSC configurations for various beam currents. Examples of the measured and corrected beam sizes are shown for sample data in Extended Data Fig. 1. The experimentally determined corrections are close to the theoretical estimates of 15 μm and 31 μm for M2R and M1L, respectively.

Alignment of the OSC system

The OSC insertion is equipped with a laser-based alignment system for supporting the alignment of diagnostic systems and manipulation of the CO for OSC tuning. The system comprises a helium–neon laser (632.8 nm) with its axis aligned through two surveyed pinholes at either end of the OSC insertion. The transverse positioning errors of the pinholes are approximately ±50 μm. A set of air-side matching optics is used to focus the alignment laser to the centre of the PU. The laser then relays through the in-vacuum optics and all downstream diagnostic lines to produce submillimetre images on the undulator radiation (UR) cameras described below. Before the OSC experiments, the CO was corrected to a few-hundred micrometres using quad-centring, and the lattice functions were corrected to several percent using standard techniques^40,41.

In-vacuum light optics

The in-vacuum lens and delay plates are fabricated from CORNING-HPFS-7980, which was chosen primarily for its low-group-velocity dispersion in the wavelength range of interest. The lens is anti-reflection coated for the fundamental band of the UR (950–1,400 nm) and has a clear aperture of about 13 mm, which corresponds to an acceptance angle for the PU radiation of about 3.5 mrad. The lens position can be adjusted in six degrees of freedom (<±10 nm in positions and <±15 μrad in angles) using an in-vacuum, piezo-electric manipulator (Smaract Smarpod 70.42) operating in a closed-loop mode. The delay plates have a central thickness of 250 μm and the typical variation over the 25-mm plates was measured via Haidinger interferometry to be about 100 nm. The nominal orientation of the delay plates is near the Brewster angle to reduce reflection losses of the PU light. The delay system uses two closed-loop, rotary piezo stages (Smaract SR-2013) to provide independent rotation of the two delay plates. The delay can be tuned over a full range of approximately 0.1 mm with a precision of about 10 nm. Although the absolute angles (that is, relative to the OSC alignment axis) of the plates are not known, a delay model can be fit to the periodicity of the OSC cooling force for a continuous-rate angular scan.

Undulator-radiation diagnostics

Imaging of the radiation from the PU and KU provides important diagnostic capabilities. The undulators produce an on-axis radiation wavelength given by λ_r = l_u(1 + K²/2)/2nγ², where l_u is the undulator period, n is the harmonic of the radiation, γ is the Lorentz factor, K = q_e Bl_u/2πm_ec is the undulator parameter, q_e is the electron charge, B is the peak on-axis magnetic field, m_e is the electron rest mass and c is the speed of light. A lightbox located on the M4L dipole contains all diagnostic systems for the KU and PU radiation. Two Blackfly-PGE-23S6M-C CMOS cameras are used in combination with a filter wheel to image the fundamental, second or third harmonic from the KU and PU. These UR cameras are located at two separate image planes corresponding to different locations inside the KU. As the in-vacuum lens is in a 2f-relay configuration, the PU light is mapped into the KU with an approximately negative-identity transformation. The imaging system then produces a single, relatively sharp image of the beam, for both the PU and KU, from the corresponding source plane. In conjunction with the alignment system, these images can be used to estimate the trajectory errors of the closed orbit in both undulators. To validate the concept, the propagation of realistic UR through the entire imaging system was performed in the Synchrotron Radiation Workshop⁴². In practice, the simultaneous imaging of the KU and PU radiation from the same source plane in the KU enables straightforward, rough alignment of the KU closed orbit with the PU radiation, which along with longitudinal alignment is the principal requirement for establishing the OSC interaction. The cameras have sufficient infrared quantum efficiency to directly image the interference of the fundamental radiation, which indicates successful longitudinal alignment.

Longitudinal-beam measurements

A Hamamatsu model C5680 dual-sweep streak camera with a synchroscan (M5675) vertical deflection unit was used to measure the beam’s longitudinal distribution during the OSC experiments. A Blackfly-PGE-23S6M-C CMOS camera was used as the detector element, and the system was installed above the M3R dipole. A 50/50 non-polarizing beam splitter was used to direct half of the SR from the existing M3R SR beam-position monitor to the entrance slit of the streak camera. An external clock generator was phase locked to IOTA’s fourth-harmonic radio frequency (30 MHz) and used to drive the streak camera’s sweep at the eleventh harmonic (82.5 MHz) of the beam’s circulation frequency (7.5 MHz). To calibrate the streak-camera image (picoseconds per pixel), IOTA’s radio-frequency voltage was first calibrated using a wall current monitor to measure shifts in the synchronous phase of the beam for different voltage settings. Measurements of the synchrotron frequency (by resonant excitation of the beam) were then made as a function of voltage setting, which yielded a small correction factor for the momentum compaction (+15%). This value is highly sensitive to focusing errors in the low-emittance lattice designed for the OSC studies. Finally, the streak camera’s calibration factor was determined by fitting the measured longitudinal-beam position as a function of voltage. A slight nonlinearity was observed at the edge of the system’s field of view. It was corrected by making the beam’s longitudinal distribution symmetric relative to its centre for OSC in the heating mode, where the amplitudes were largest and the bunch lengths barely fit within the field of view. Extended Data Fig. 2 presents an example of the longitudinal distribution before and after correction.

Specialized power systems

The chicane dipoles are powered in pairs using special current regulators (BiRa Systems PCRC) with a ripple-plus-noise at the 1 × 10⁻⁵ level (r.m.s.) and a long-term stability of a few parts per million. This ensures that the nominal phase of the OSC force is stable as the beam particles sample it over many turns. It is noted that regulation at the mid 10⁻⁵ level corresponds to effective momentum errors comparable to the beam’s natural momentum spread. The regulation of the main power supply that feeds IOTA’s dipoles was also improved to comparable stability. This was required for IOTA because, as an electron synchrotron with fixed radio frequency, the beam energy is directly related to the magnetic field in the ring’s dipoles; therefore, variations of the bending field result in variations of the particle delay.

Coupling of the phase-space planes

Energy exchange between the particles and their PU radiation fields in the KU is a longitudinal effect; however, as described in the next section, the presence of dispersion in the undulators can be used to couple the cooling force into the transverse phase planes. In the system reported here, this coupling (longitudinal to horizontal) is smoothly adjustable by excitation of a single quadrupole in the centre of the OSC bypass. Operation of the storage ring on a transverse coupling resonance, in our case a difference resonance with betatron tunes (that is, number of betatron oscillations per revolution) of Q_x = 5.42 and Q_y = 2.42, splits the beam emittance and cooling and heating between the horizontal and vertical planes. This combination of bypass and lattice coupling enables full three-dimensional cooling of the beam using OSC. To couple the lattice, the ring optics were corrected to minimize the split between the fractional part of the betatron tunes (ΔQ < 0.005), and then strong transverse coupling was introduced by excitation of a single skew quadrupole in a region with zero dispersion.

Analysis of OSC rates and ranges

For the derivation of the OSC cooling rates, we refer the reader to ref. ⁹. Here we only summarize the major results needed for analysis of the experimental data. For relatively small momentum deviation, the longitudinal kick experienced by a particle can be approximated as

where κ is the maximum kick value, k₀ = 2π/λ_r is the radiation wavenumber, s is the particle’s longitudinal displacement on the way from the PU to the KU relative to the reference particle, which obtains zero kick, and u(s) is an envelope function, with u(0) = 1 and u(s) = 0 for |s| > N_uλ_r, that accounts for the bandwidth of the integrated system. The effects of the envelope function are observed in Fig. 3c. In the linear approximation, one can write

where M_5n are the elements of 6 × 6 transfer matrix from pick-up to kicker, and x, θ_x and Δp/p are the particle coordinate, angle and relative momentum deviation in the PU centre. To find the longitudinal cooling rate for small-amplitude motion, we leave only the linear term in ks in equation (1) and set u(s) = 1. The longitudinal cooling rate is straightforwardly obtained as

with D and D′ = dD/ds being the ring dispersion and its longitudinal derivative at the PU. Here we also include that for pure longitudinal motion x = D(Δp/p) and θ_x = D′(Δp/p). Then, using symplectic perturbation theory and the rate-sum theorem⁴³, one obtains that the sum of cooling rates (in amplitude) is equal to the longitudinal cooling rate in the absence of x–s coupling:

where λ₁ and λ₂ are the cooling rates of the two betatron modes, λ_s is the cooling rate of longitudinal motion and f₀ is the revolution frequency in the storage ring.

In the general case of arbitrary x–y coupling, the cooling rates for the transverse modes have lengthy expressions; however, for the case of operation at the coupling resonance the cooling rates of two transverse modes are equal and have a compact representation. In this case, combining equations (3) and (4), one obtains

The harmonic dependence of the cooling force on momentum deviation, presented in equation (1), results in a reduction of the cooling rates with increasing amplitude. Averaging over betatron (transverse) and synchrotron (longitudinal) oscillations yields the dependence of cooling rates on the particle amplitudes:

where J_n is the nth-order Bessel’s function of the first kind, a₁, a₂ and a_s are the dimensionless amplitudes of the particle’s longitudinal displacement in the kicker related to the oscillations in the corresponding plane. Expressed in units of the phase of the electromagnetic field they are given by

where (Δp/p)_max is the amplitude of the synchrotron motion, ε₁ and ε₂ are the generalized Courant–Snyder invariants (single-particle emittances), and β_1x, β_2x, α_1x, α_2x and u are the four-dimensional Twiss parameters defined in Section 2.2.5 of ref. ⁴³. The cooling rates in equation (6) oscillate with the particle amplitudes, and as a result, particles may be trapped at large amplitudes by the OSC force. The requirement to have simultaneous damping for all degrees of freedom determines the cooling acceptances so that a_i ≤ μ₀₁ ≈ 2.405, i = 1, 2, s; where μ₀₁ is the first root of Bessel function J₀(x). If oscillations happen in only one degree of freedom, then the cooling range is larger: a_i ≤ μ₁₁ ≈ 3.83, where μ₁₁ is the first non-zero root of Bessel function J₁(x).

Although the small-amplitude OSC rates greatly exceed the SR cooling rates, accounting for SR is important to understand the observed beam behaviour. In this case the total cooling rate for the nth degree of freedom is:

where R_nτ is the ratio of the small-amplitude OSC rate to the SR cooling rate for the nth degree of freedom, m and k are the other degrees of freedom, and the label τ is used to indicate a ratio of cooling rates. In our measurements with the antidamping OSC phase, the dimensionless amplitudes of betatron motion are much smaller than one. For longitudinal OSC in the antidamping mode, one then obtains the dependence of the longitudinal cooling rate on the dimensionless amplitude of the synchrotron motion as:

Consequently, the equilibrium amplitude is determined by the following equation: a_s = 2R_sτJ₁(a_s). Extended Data Fig. 3 presents the dependence of the longitudinal cooling rates for OSC in the damping and antidamping modes for the measured parameters of OSC. For both modes, there is only one equilibrium point: a_s = 0 for the damping mode, and a_s = 3.273 for the antidamping mode.

For very small beam current, where IBS is negligible, the r.m.s. emittance growth rate of small-amplitude motion is determined by the following equation:

Here B_n is the diffusion driven by fluctuations from SR emission and scattering from residual gas molecules, and thus does not depend on the beam parameters. In equilibrium, equation (10) determines the cooling rate, ({lambda }_{n}={B}_{n}/2{varepsilon }_{n},) and a straightforward way to compute the cooling rate from the ratio of r.m.s. beam sizes with (σ_n) and without (σ_n0) OSC:

where λ_n0 is the damping rate in the absence of OSC. Although all reported OSC measurements were done with a small beam current (about 50–150 nA), for a large fraction of the measurements IBS was not negligible; therefore, we use a simplified IBS model to calculate corrections to the cooling rates that are largely independent of exact beam parameters⁴⁴. We add the IBS term to the right-hand side of equation (10):

where ε_⊥ = ε₁ = ε₂ is the r.m.s. transverse emittance, ε_s is the r.m.s. longitudinal emittance and the constant A_n is determined from the measurements.

To find A_n, we use the fact that the r.m.s. beam sizes are different at the beginning and at the end of OSC sweep, which continues for about 1,000 s. For the measurement presented in Fig. 3, the measured beam lifetime of 17 min yields the ratio of beam currents at the beginning and at the end of the measurements to be R_N = 2.5. With some algebraic manipulations, one can express the ratio of the measured beam sizes at the sweep end, σ_n2, to the beam sizes that would be measured in the absence of IBS, σ_n0, through the ratios of other measured parameters as:

Here (σ_v1/σ_v2) is the ratio of the initial and final vertical beam size, (σ_s1/σ_s2) is the same measure for the bunch lengths, the emittances of both transverse modes are taken to be equal and we have used ε_⊥1/ε_⊥2 = (σ_v1/σ_v2)². For these experiments, the approximate ratios of the beam sizes (before and after sweep) are: (σ_v1/σ_v2) = 1.09 and (σ_s1/σ_s2) = 1.1; that yields (σ_v0/σ_v2)² = 0.745.

In the next step, we find the ratio of cooling rates with and without OSC. Similar manipulations with equation (12) yield an improved version of equation (11):

Here σ_vSR/σ_vOSC is the ratio of the measured beam sizes without and with OSC, respectively, σ_sSR/σ_sOSC is the same for the bunch lengths. For the measurements presented here, σ_vSR/σ_vOSC = 1.51 and σ_sSR/σ_sOSC = 2.5, which yields λ_vOSC/λ_vSR = 2.94. Similar calculations for the longitudinal cooling yield λ_sOSC/λ_sSR = 8.06. The typical day-to-day variability of the rates was around the 10% level owing to variations in the overall tuning and alignment of the OSC system and CO; however, the performance was stable during any given operations session with only infrequent, minor tuning required.

Gas scattering and ring acceptance

Fits to the vertical distributions in Fig. 4 reveal two features of note: (1) the equilibrium size in the OSC-off case is approximately two-times larger than anticipated⁹ and (2) the distributions have non-Gaussian tails for both cases, with and without OSC. Both observations are consistent with scattering from residual gas molecules. The average vacuum pressure in the storage ring, which is estimated from the beam size increase, is about 3.7 × 10⁻⁸ torr of atomic hydrogen equivalent and coincides with the vacuum estimate of the previous IOTA run to within about 15% accuracy⁴⁵. The vertical acceptance of the storage ring is smaller than the horizontal acceptance and is estimated from the beam lifetime to be about 3 μm, which is about 50% of the design value and exceeds the cooling range (Table 1) by a factor of about 40.

Cooling-rate and coupling discrepancies

One likely source of the apparent cooling-rate discrepancy is a reduced physical aperture for the PU light owing to misalignments of the vacuum chambers and the beam. Preliminary simulations based on three-dimensional scans of the integrated apparatus suggest that on the order of 30–40% of the difference might be accounted for in this way. Other potential sources may include nonlinearities in the bypass mapping, the possibility of distorted CO trajectories in the undulators owing to saturation in the steel poles, and reduced energy exchange owing to the finite radiation spot size in the KU, which is exacerbated by the beam’s increased size owing to residual gas scattering.

Regarding longitudinal-to-transverse coupling, the measured ratio (1:0.34) would correspond to excitation of the coupling quadrupole at approximately half its nominal strength. This suggests the presence of additional coupling terms in the bypass and/or deviations of the lattice optics functions from the model; however, in two-dimensional OSC experiments, which are not reported here, the coupling-quad excitation was doubled and a coupling ratio closer to unity was achieved.