Beam emittance

When a particle moves through a circular accelerator or storage ring, the position ${\displaystyle x}$ and angle ${\displaystyle x'}$ of the particle in the x direction will trace an ellipse in ${\displaystyle x/x'}$ phase space. (All of this section applies equivalently to ${\displaystyle y}$ and ${\displaystyle y'}$) This ellipse can be described by the following equation:[1]^: 81

${\displaystyle {\frac {\varepsilon }{\pi }}=\gamma x^{2}+2\alpha xx'+\beta x'^{2}}$

where x and x′ are the position and angle of the particle, and ${\displaystyle \beta ,\alpha ,\gamma }$ are the Courant–Snyder (Twiss) parameters, calculated from the shape of the ellipse.

The emittance is given by ${\displaystyle \varepsilon }$, and has units of length × angle. However, many sources will move the factor of ${\displaystyle \pi }$ into the units of emittance rather than including the specific value, giving units of "length × angle × ${\displaystyle \pi }$."[2]^{: 335–336}

This formula is the single particle emittance, which describes the area enclosed by the trajectory of a single particle in phase space. However, emittance is more useful as a description of the collective properties of the particles in a beam, rather than of a single particle. Since beam particles are not necessarily distributed uniformly in phase space, definitions of emittance for an entire beam will be based on the area of the ellipse required to enclose a specific fraction of the beam particles.

If the beam is distributed in phase space with a Gaussian distribution, the emittance of the beam may be specified in terms of the root mean square value of ${\displaystyle x}$ and the fraction of the beam to be included in the emittance.

The equation for the emittance of a Gaussian beam is:[1]^: 83

${\displaystyle \varepsilon =-{\frac {2\pi \sigma ^{2}}{\beta }}ln(1-F)}$

where ${\displaystyle \sigma }$ is the root mean square width of the beam, ${\displaystyle \beta }$ is the Courant-Snyder ${\displaystyle \beta }$, and ${\displaystyle F}$ is the fraction of the beam to be enclosed in the ellipse, given as a number between 0 and 1. Here the factor of ${\displaystyle \pi }$ is shown on the right of the equation, and would often be included in the units of emittance, rather than being multiplied in to the computed value.[2]^{: 335–336}

The value chosen for ${\displaystyle F}$ will depend on the application and the author, and a number of different choices exist in the literature. Some common choices and their equivalent definition of emittance are: [1]^: 83

${\displaystyle \varepsilon }$	${\displaystyle F}$
${\displaystyle {\frac {\sigma ^{2}}{\beta }}}$	0.15
${\displaystyle {\frac {\pi \sigma ^{2}}{\beta }}}$	0.39
${\displaystyle {\frac {4\pi \sigma ^{2}}{\beta }}}$	0.87
${\displaystyle {\frac {6\pi \sigma ^{2}}{\beta }}}$	0.95

While the x and y axes are generally equivalent mathematically, in horizontal rings where the x coordinate represents the plane of the ring, consideration of dispersion can be added to the equation of the emittance. Because the magnetic force of a bending magnet is dependent on the energy of the particle being bent, particles of different energies will be bent along different trajectories through the magnet, even if their initial position and angle are the same. The effect of this dispersion on the beam emittance is given by:

${\displaystyle \varepsilon _{x}={\frac {\sigma _{x}^{2}}{\beta _{x}(s)}}-{\frac {D(s)^{2}}{\beta _{x}}}({\frac {\sigma _{p}}{p_{o}}})^{2}}$

where ${\displaystyle D(s)}$ is the dispersion at location s, ${\displaystyle p_{o}}$ is the ideal particle momentum, and ${\displaystyle \sigma _{p}}$ is the root mean square of the momentum difference of the particles in the beam from the ideal momentum. (This definition assumes F=0.15)[1]^: 91

The geometrical definition of longitudinal emittance is more complex than that of transverse emittance. While the ${\displaystyle x}$ and ${\displaystyle y}$ coordinates represent deviation from a reference trajectory which remains static, the ${\displaystyle z}$ coordinate represents deviation from a reference particle, which is itself moving with a specified energy. This deviation can be expressed in terms of distance along the reference trajectory, time of flight along the reference trajectory (how "early" or "late" the particle is compared to the reference), or phase (for a specified reference frequency).

In turn, the ${\displaystyle z'}$ coordinate is generally not expressed as an angle. Since ${\displaystyle z'}$ represents the change in z over time, it corresponds to the forward motion of the particle. This can be given in absolute terms, as a velocity, momentum, or energy, or in relative terms, as a fraction of the position, momentum, or energy of the reference particle. [1]^: 32

However, the fundamental concept of emittance is the same—the position of the particles are plotted along one axis of a phase space plot, the rate of change in that position over time is plotted on the other axis, and the emittance is a measure of the area occupied on that plot.

One possible definition of longitudinal emittance is given by:

${\displaystyle \varepsilon _{\phi }=\int _{S}{\frac {\Delta E}{\omega _{rf}}}d\phi }$

where the integral is taken along a path ${\displaystyle S}$ which tightly encloses the beam particles in ${\displaystyle E/\phi }$ phase space. Here ${\displaystyle \omega _{rf}}$ is the reference frequency and the longitudinal coordinate ${\displaystyle \phi }$ is the phase of the particles relative to a reference particle. Longitudinal equations such as this one often must be solved numerically, rather than analytically.[3]^: 218

The geometric definition of emittance assumes that the distribution of particles in phase space can be reasonably well characterized by an ellipse. In addition, the definitions using the root mean square of the particle distribution assume a Gaussian particle distribution.

In cases where these assumptions do not hold, it is still possible to define a beam emittance using the moments of the distribution. Here, the RMS emittance (${\displaystyle \varepsilon _{\text{RMS}}}$) is defined to be,[5]

${\displaystyle \varepsilon _{\text{RMS}}={\sqrt {\langle x^{2}\rangle \langle x^{\prime 2}\rangle -\langle x\cdot x^{\prime }\rangle ^{2}}}}$

where ${\displaystyle \langle x^{2}\rangle }$ is the variance of the particle's position, ${\displaystyle \langle x^{\prime 2}\rangle }$ is the variance of the angle a particle makes with the direction of travel in the accelerator (${\textstyle x^{\prime }={\frac {\mathrm {d} x}{\mathrm {d} z}}}$ with ${\displaystyle z}$ along the direction of travel), and ${\displaystyle \langle x\cdot x^{\prime }\rangle }$ represents an angle-position correlation of particles in the beam. This definition is equivalent to the geometric emittance in the case of an elliptical particle distribution in phase space.

The emittance may also be expressed as the determinant of the variance-covariance matrix of the beam's phase space coordinates where it becomes clear that quantity describes an effective area occupied by the beam in terms of its second order statistics.

${\displaystyle \varepsilon _{\text{RMS}}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle \\\langle x\cdot x^{\prime }\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle \end{vmatrix}}}}$

Depending on context, some definitions of RMS emittance will add a scaling factor to correspond to a fraction of the total distribution, to facilitate comparison with geometric emittances using the same fraction.

It is sometimes useful to talk about phase space area for either four dimensional transverse phase space (IE ${\displaystyle x}$, ${\displaystyle x^{\prime }}$, ${\displaystyle y}$, ${\displaystyle y^{\prime }}$) or the full six dimensional phase space of particles (IE ${\displaystyle x}$, ${\displaystyle x^{\prime }}$, ${\displaystyle y}$, ${\displaystyle y^{\prime }}$, ${\displaystyle \Delta z}$, ${\displaystyle \Delta z^{\prime }}$). The RMS emittance generalizes to full three dimensional space as shown:

${\displaystyle \varepsilon _{{\text{RMS}},6D}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle &\langle x\cdot y\rangle &\langle x\cdot y^{\prime }\rangle &\langle x\cdot z\rangle &\langle x\cdot z^{\prime }\rangle \\\langle x^{\prime }\cdot x\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle &\langle x^{\prime }\cdot y\rangle &\langle x^{\prime }\cdot y^{\prime }\rangle &\langle x^{\prime }\cdot z\rangle &\langle x^{\prime }\cdot z^{\prime }\rangle \\\langle y\cdot x\rangle &\langle y\cdot x^{\prime }\rangle &\langle y\cdot y\rangle &\langle y\cdot y^{\prime }\rangle &\langle y\cdot z\rangle &\langle y\cdot z^{\prime }\rangle \\\langle y^{\prime }\cdot x\rangle &\langle y^{\prime }\cdot x^{\prime }\rangle &\langle y^{\prime }\cdot y\rangle &\langle y^{\prime }\cdot y^{\prime }\rangle &\langle y^{\prime }\cdot z\rangle &\langle y^{\prime }\cdot z^{\prime }\rangle \\\langle z\cdot x\rangle &\langle z\cdot x^{\prime }\rangle &\langle z\cdot y\rangle &\langle z\cdot y^{\prime }\rangle &\langle z\cdot z\rangle &\langle z\cdot z^{\prime }\rangle \\\langle z^{\prime }\cdot x\rangle &\langle z^{\prime }\cdot x^{\prime }\rangle &\langle z^{\prime }\cdot y\rangle &\langle z^{\prime }\cdot y^{\prime }\rangle &\langle z^{\prime }\cdot z\rangle &\langle z^{\prime }\cdot z^{\prime }\rangle \\\end{vmatrix}}}}$

In the absences of correlations between different axes in the particle accelerator, most of these matrix elements become zero and we are left with a product of the emittance along each axis.

${\displaystyle \varepsilon _{{\text{RMS}},6D}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle &0&0&0&0\\\langle x^{\prime }\cdot x\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle &0&0&0&0\\0&0&\langle y\cdot y\rangle &\langle y\cdot y^{\prime }\rangle &0&0\\0&0&\langle y^{\prime }\cdot y\rangle &\langle y^{\prime }\cdot y^{\prime }\rangle &0&0\\0&0&0&0&\langle z\cdot z\rangle &\langle z\cdot z^{\prime }\rangle \\0&0&0&0&\langle z^{\prime }\cdot z\rangle &\langle z^{\prime }\cdot z^{\prime }\rangle \\\end{vmatrix}}}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle \\\langle x^{\prime }\cdot x\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle \\\end{vmatrix}}}{\sqrt {\begin{vmatrix}\langle y\cdot y\rangle &\langle y\cdot y^{\prime }\rangle \\\langle y^{\prime }\cdot y\rangle &\langle y^{\prime }\cdot y^{\prime }\rangle \\\end{vmatrix}}}{\sqrt {\begin{vmatrix}\langle z\cdot z\rangle &\langle z\cdot z^{\prime }\rangle \\\langle z^{\prime }\cdot z\rangle &\langle z^{\prime }\cdot z^{\prime }\rangle \\\end{vmatrix}}}=\varepsilon _{x}\varepsilon _{y}\varepsilon _{z}}$

Although the previous definitions of emittance remain constant for linear beam transport, they do change when the particles undergo acceleration (an effect called adiabatic damping). In some applications, such as for linear accelerators, photoinjectors, and the accelerating sections of larger systems, it becomes important to compare beam quality across different energies. Normalized emittance, which is invariant under acceleration, is used for this purpose.

Normalized emittance in one dimension is given by:

${\displaystyle \varepsilon _{n}={\sqrt {\langle x^{2}\rangle \langle \left(\gamma \beta _{x}\right)^{2}\rangle -\langle x\cdot \gamma \beta _{x}\rangle ^{2}}}={\sqrt {\begin{vmatrix}\langle x\cdot x\rangle &\langle x\cdot \gamma \beta _{x}\rangle \\\langle x\cdot \gamma \beta _{x}\rangle &\langle \gamma \beta _{x}\cdot \gamma \beta _{x}\rangle \end{vmatrix}}}}$

The angle ${\textstyle x^{\prime }={\frac {\mathrm {d} x}{\mathrm {d} z}}}$ in the prior definition has been replaced with the normalized transverse momentum ${\textstyle {\frac {p_{x}}{mc}}=\gamma \beta _{x}}$, where ${\displaystyle \gamma }$ is the Lorentz factor and ${\textstyle \beta _{x}=v_{x}/c}$ is the normalized transverse velocity.

Normalized emittance is related to the previous definitions of emittance through ${\displaystyle \gamma }$ and the normalized velocity in the direction of the beam's travel (${\textstyle \beta _{z}=v_{z}/c}$):[6]

${\displaystyle \varepsilon _{n}=\gamma \beta _{z}\varepsilon }$

The normalized emittance does not change as a function of energy and so can be used to indicate beam degradation if the particles are accelerated. For speeds close to the speed of light, where ${\displaystyle \beta =v/c}$ is close to one, the emittance is approximately inversely proportional to the energy. In this case, the physical width of the beam will vary inversely with the square root of the energy.

Higher dimensional versions of the normalized emittance can be defined in analogy to the RMS version by replacing all angles with their corresponding momenta.

One of the most fundamental methods of measuring beam emittance is the quadrupole scan method. The emittance of the beam for a particular plane of interest (i.e., horizontal or vertical) can be obtained by varying the field strength of a quadrupole (or quadrupoles) upstream of a monitor (i.e., a wire or a screen).[4]

A schematic of the accelerator optics used in the quadrupole scan technique.

The properties of a beam can be described as the following beam matrix.

${\displaystyle \Sigma ={\begin{bmatrix}\langle x\cdot x\rangle &\langle x\cdot x^{\prime }\rangle \\\langle x\cdot x^{\prime }\rangle &\langle x^{\prime }\cdot x^{\prime }\rangle \end{bmatrix}}={\begin{bmatrix}\sigma _{11}&\sigma _{12}\\\sigma _{21}&\sigma _{22}\end{bmatrix}}}$

where ${\textstyle x^{\prime }={dx \over dz}}$ is the derivative of x with respect to the longitudinal coordinate. The forces experienced by the beam as it travels down the beam line and passes through the quadrupole(s) are described using the transfer matrix (referenced to transfer maps page) ${\displaystyle R}$ of the beam line, including the quadrupole(s) and other beam line components such as drifts:

${\displaystyle R=S_{1}QS_{2}={\begin{pmatrix}r_{11}&r_{12}\\r_{21}&r_{22}\end{pmatrix}}}$

Here ${\displaystyle S_{1}}$ is the transfer matrix between the original beam position and the quadrupole(s), ${\displaystyle Q}$ is the transfer matrix of the quadrupole(s), and ${\displaystyle S_{2}}$ is the transfer matrix between the quadrupole(s) and the monitor screen. During the quadrupole scan process, ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ stay constant, and ${\displaystyle Q}$ changes with the field strength of the quadrupole(s).

The final beam when it reaches the monitor screen at distance s from its original position can be described as another beam matrix ${\displaystyle \Sigma _{s}}$:

${\displaystyle \Sigma _{s}={\begin{bmatrix}\langle x_{s}\cdot x_{s}\rangle &\langle x_{s}\cdot x_{s}^{\prime }\rangle \\\langle x_{s}\cdot x_{s}^{\prime }\rangle &\langle x_{s}^{\prime }\cdot x_{s}^{\prime }\rangle \end{bmatrix}}={\begin{bmatrix}\sigma _{s,11}&\sigma _{s,12}\\\sigma _{s,21}&\sigma _{s,22}\end{bmatrix}}}$

The final beam matrix ${\displaystyle \Sigma _{s}}$ can be calculated from the original beam matrix ${\displaystyle \Sigma }$ by doing matrix multiplications with the beam line transfer matrix ${\displaystyle R}$:

${\displaystyle \Sigma _{s}=R\Sigma R^{T}}$

Where ${\displaystyle R^{T}}$ is the transpose of ${\displaystyle R}$.

Now, focusing on the (1,1) element of the final beam matrix throughout the matrix multiplications, we get the equation:

${\displaystyle \sigma _{s,11}=r_{11}^{2}\sigma _{11}+2r_{11}r_{12}\sigma _{12}+r_{12}^{2}\sigma _{22}}$

Here the middle term has a factor of 2 because ${\displaystyle \sigma _{12}=\sigma _{21}}$.

Now divide both sides of the above equation by ${\displaystyle r_{12}^{2}}$, the equation becomes:

${\displaystyle {\frac {\sigma _{s,11}}{r_{12}^{2}}}={\frac {r_{11}^{2}}{r_{12}^{2}}}\sigma _{11}+2{\frac {r_{11}}{r_{12}}}\sigma _{12}+\sigma _{22}}$

Which is a quadratic equation of the variable ${\textstyle {\frac {r_{11}}{r_{12}}}}$. Since the RMS emittance RMS is defined to be the following.

${\displaystyle \varepsilon _{\text{RMS}}={\sqrt {\langle x^{2}\rangle \langle x^{\prime 2}\rangle -\langle x\cdot x^{\prime }\rangle ^{2}}}}$

The RMS emittance of the original beam can be calculated using its beam matrix elements:

${\displaystyle \varepsilon _{\text{RMS}}={\sqrt {\sigma _{11}\sigma _{22}-\sigma _{12}^{2}}}}$

To obtain the emittance measurement, the following procedure is employed:

1. For each value (or value combination) of the quadrupole(s), the beam line transfer transfer matrix ${\displaystyle R}$ is calculated to determine values of ${\displaystyle r_{11}}$ and ${\displaystyle r_{12}}$.
2. The beam propagates through the varied beam line, and is observed at the monitor screen, where the beam size ${\textstyle \sigma _{s,11}}$ is measured.
3. Repeat step 1 and 2 to obtain a series of values for ${\textstyle {\frac {\sigma _{s,11}}{r_{12}^{2}}}}$ and ${\displaystyle {\frac {r_{11}}{r_{12}}}}$, fit the results with a parabola ${\textstyle {\frac {\sigma _{s,11}}{r_{12}^{2}}}=A\left({\frac {r_{11}}{r_{12}}}\right)^{2}+B\left({\frac {r_{11}}{r_{12}}}\right)+C}$.
4. Equate parabola fit parameters with original beam matrix elements: ${\textstyle A=\sigma _{11}}$, ${\displaystyle B=2\sigma _{12}}$, ${\displaystyle C=\sigma _{22}}$.
5. Calculate RMS emittance of the original beam: ${\textstyle \varepsilon _{\text{RMS}}={\sqrt {\sigma _{11}\sigma _{22}-\sigma _{12}^{2}}}}$

If the length of the quadrupole is short compared to its focal length ${\displaystyle f=1/K}$, where ${\displaystyle K}$ is the field strength of the quadrupole, its transfer matrix ${\displaystyle Q}$ can be approximated by the thin lens approximation:

${\displaystyle Q={\begin{pmatrix}1&0\\K&1\end{pmatrix}}}$

Then the RMS emittance can be calculated by fitting a parabola to values of measured beam size ${\displaystyle \sigma _{x}^{2}}$ versus quadrupole strength ${\displaystyle K}$.

By adding additional quadrupoles, this technique can be extended to a full 4-D reconstruction.[7]

A schematic of mask based reconstruction. A charged particle beam is blocked by a grid and the profile is analyzed at a screen to the right.

Another fundamental method for measuring emittance is by using a predefined mask to imprint a pattern on the beam and sample the remaining beam at a screen downstream.  Two such masks are pepper pots[8] and TEM grids[9].  A schematic of the TEM grid measurement is shown below.

By using the knowledge of the spacing of the features in the mask one can extract information about the beam size at the mask plane.  By measuring the spacing between the same features on the sampled beam downstream, one can extract information about the angles in the beam.  The quantities of merit can be extracted as described in Marx et al.[10]

The choice of mask is generally dependent on the charge of the beam; low-charge beams are better suited to the TEM grid mask over the pepper pot, as more of the beam is transmitted.