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A photoinjector is device used to produce an electron beam tailored for specific use in a particle accelerator application. They work by precisely controlling the time structure, emittance, and brightness of each electron beam bunch. A general scheme for a photoinjector includes (1) a cathode to produce electrons; (2) a gun to perform initial acceleration; (3) beam optics for additional bunch manipulation; and (4) timing controls for synchronization. There are a number of different options for each one of these sub systems but in general we can use the same theories to explain them collectively.

Introduction
For historical context the photoinjector was originally introduced as a more specific instance of this now more general category of accelerating devices. It was originally introduced as a high brightness, low emittance, high peak current source for free electron lasers in work at Los Alamos National Laboratory. The original design came about as a result of an experiment which showed that a photocathode could deliver current densities in excess of $$200 A cm^{-2}$$. This initial photoinjector then used thuis high charge photocathode which was accelerated by a normal conducting radiofrequency gun cavity and then measured downstream with diagnostics to measure energy spread and emittance. Since then the photoinjector has grown to include cathodes utilizing not only photoemission but also thermionic and field emission and electron guns that utilize both oscillating field (usually in radiofrequencies) and constant DC field accelerating gradients.

Photoemission
The most ubiquitous source of electrons for photoinjectors are cathodes based on photoemission. This is to say materials which emit electron via absorption of photons (of specified wavelength most often from lasers) a la the photoelectric effect. Existing information on photoemission is extensive so it is sufficient here to mention they both metallic and semiconductor cathodes are used in photoinjectors with semiconductor cathodes often preferred due to their high quantum efficieny (QE). QE is defined with respect to the laser beam's energy as follows

$$QE=\frac{n_e}{n_{\gamma}}=\frac{h\nu[\text{eV}]}{E_{laser}[\text{J}]}q[\text{C}]$$

Where $$n_e$$ is the number of electrons produced per number of photons absorbed, $$n_{\gamma}$$. The QE for copper cathodes at an accelerating gradient of $$100MV m^{-1}$$ is about $$0.014\%$$. This can be increased to up to $20\%$ for semiconductor cathodes such as alkali antimonides.

Thermionic Emission and Field Emission
Less common but still considered are cathodes which rely on thermionic emission or field emission. Thermionic emission relies on the heating of cathode bulk electrons such that they thermal energy is enough to escape over the surface potential barrier. Field emission are those cathode which due emit electron by tunneling through a potential barrier significantly reduced due to the presence of a strong electric field such as that in a very intense laser pulse.

Oscillating Field
In many ways the workhorse of the photoinjector community is the normal conducting oscillating field or radiofrequency (RF) photogun, so-called for historical reasons due the utilizing frequencies most often in the radiofrequency range (1-10 GHz). Typically the main difference between linear accelerating cavities and the photoinjector guns is that photoinjector guns often have non-integer numbers of cells. This is necessary as the emission site of the cathode is often most optimally found at a maximum field value in the gun cavity. Depending of the modes present in the cavity, especially for a standing wave mode, the first gun cavity is often close to a half cell such that the antinode is at or near the cathode plane. This changing slightly for gun cavities that intentionally permit traveling wave modes but to begin this is a decent approximation. The guns themselves are powered by high power signal amplifying devices such as klystrons and so depending on the direction where the RF power is fed often contain different geometries of coupling. If the photoinjector device being used for a higher energy accelerator application, care must be exercised in order to ensure that the beam produced by the photoinjector is matched to the frequency of the linear accelerating section into which it is being injected. This is especially challenging for circular machines, even more so than linear machines.

Cavities used in oscillating field guns are often either in the pillbox or reentrant style geometry. For acceleration, the useful modes are the transverse magnetic (TM) modes. So for the simplest pillbox geometry e fields can be solved as

$$E_z=E_0J_m(k_{mn}r)\cos(m\theta)\cos(pk_zz)\exp(i(\omega t+\phi_0))$$

$$E_r=-p\frac{k_z}{k_{mn}}E_0J'_m(k_{mn}r)\cos(m\theta)\sin(p k_z z)\exp\left(i(\omega t +\phi_0)\right)$$

$$E_{\theta} = -mp\frac{k_z}{k_{mn}}E_0J'_m(k_{mn}r)\cos(m\theta)\sin(pk_z-z)\exp(i(\omega t-\phi_0))$$

$$B_z=0$$

$$B_r = \frac{-i\omega m}{k_{mn}^2c^2r}E_0J_m(k_{mn}r)\sin(m\theta)\cos(pk_z z)\exp(i(\omega t+\phi_0))$$

$$B_{\theta} = \frac{-i\omega}{k_{mn}c^2}E_0J_m'(k_{mn}r)\cos(m\theta)\cos(p k_zz)\exp(i(\omega t+\phi_0))$$

Perhaps the most notable protypical oscillating field geometry is the BNL 1.5 cell pillbox design. The operating frequency is in the American S-band range (2.856 GHz). Pillbox cells are basically simple cylinders with coupling ports from to adjacent cells and in the case of the photoinjectors, some RF power feed along the circumference. The gun uses the so-called $$\pi$$ mode where adjacent cells are $$\pi$$ radians separated in phase such that every other cell is in phase. The design also uses the $$TM_{011}$$ mode which has non zero field components as the following

$$E_z=E_0J_0(k_{01}r)\cos(k_z z)\exp(i(\omega t +\phi_0))$$

$$E_r=\frac{-k_z}{k_{01}}E_0J'_0(k_{01}r)\sin(k_z z)\exp(i(\omega t +\phi_0))$$

$$B_{\theta}=\frac{-ik_z}{k_{01}c}E_0J'_0(k_{01}r)\cos(k_z z)\exp(i(\omega t +\phi_0))$$

The design has often been modified to real additional uses including for example, this was modified into the BNL/SLAC/UCLA 1.6 cell gun used for the the LCLS xray free electron laser at SLAC National Laboratory. One of the main results was using numerical studies to show that for more complicated beam dynamics reasons, operating 1.6 cells actually improves performance over the 1.5 cell simple version.

Constant (DC) Field
In certain cases, a high dc gradient gun which feeds directly into an RF accelerating cavity is often preferable. Such is the case for photoinjectors which need to produce high average power beam with modest charge requirements. Such can be the case for energy recovery linac based light sources and free electron lasers. Typically these high voltages are in the hundreds of kV range.

Electron Optics
For guns with sufficiently high accelerating gradients, as if often the case in photoinjectors, there is strong defocusing at the exit which must be controlled using electromagnetic components to refocus and potentially further manipulate the beam. These are often electromagnetics such as coiled solenoids. In the case of an oscillating field gun we can consider the radial force

$$F_r=e(E_r-\beta c B_{\theta})$$

Using the approximation that $$\beta\approx 1$$ as is the case of an electron accelerating to near $$c$$ and using the pillbox equations above we can find that

$$F_r=-eE_0\frac{k_zr}{2}\sin(\omega t+\phi_0-k_z z)$$

This radial force can be used to determine an RF focal length for a gun cavity which looks like the following

$$f_{RF}=\frac{-2\beta\gamma mc^2}{eE_0\sin(\phi_e)}$$

The negative sign here denotes that this is a defocusing focal length. Taking some example parameters such as $$100 MV m^{-1}$$ and exiting the gun with $$6 MeV$$ means a defocusing length of $$12 cm$$. We can thus see why focusing elements such as solenoids are necessary. This is not all however since a realistic solenoid is on the same order of magnitude in width as an representational defocusing length, the solenoid will not focus perfectly and introduce additional optical aberrations that should be dealt with. These effects fall into the category of geometric optics, such as the well known 3rd order spherical aberration.

However, there are also errors from energy spread, called chromatic aberration. Field asymmetries in realistic gun cavities have a tendency to introduce quadrupole field kicks on the beam. Even beams in otherwise perfect setups if they are high enough charge will experience space charge repulsion expanding the beam and lowering the emittance. These effects also must be compensated for with the addition of tuned optics.

Synchronization
One additional very important system for a photoinjector is the control system responsible for the synchronization. This is especially true for photoemission based high gradient RF guns where the laser pulse and RF oscillation but be in the precise correct phase with respect to one another.