# Presentation: 212985

## Charged Particle Optics. Matrix

**Charged Particle Optics. Matrix**
**Representation of the Accelerator Elements**

**David Robin**

## What are the Optics?
Magnet Definitions
Magnet Functions
Particle motion in accelerator
Coordinate system
Beam guidance
Dipoles
Beam focusing
Quadrupoles
Hill’s equations and Transport Matrices 
Matrix formalism
Drift
Thin lens
Quadrupoles
Dipoles
Sector magnets
Rectangular magnets
Doublet
FODO

**Magnet Definitions**
**Magnet Functions**
**Particle motion in accelerator**
**Coordinate system**
**Beam guidance**
**Dipoles**
**Beam focusing**
**Quadrupoles**
**Hill’s equations and Transport Matrices **
**Matrix formalism**
**Drift**
**Thin lens**
**Quadrupoles**
**Dipoles**
**Sector magnets**
**Rectangular magnets**
**Doublet**
**FODO**

## The Optics are the distribution of elements (typically magnetic or electrostatic) that guide and focus the beam - sometimes referred to as the lattice.

- Focusing Elements

- Bend Element

## Choice of the design depends upon the goal of the accelerator
Small spot size
High brightness
Small divergence
Obey certain physical constraints (building or tunnel)
...

**Small spot size**
**High brightness**
**Small divergence**
**Obey certain physical constraints (building or tunnel)**
**...**

## The motion of each charged particle is determined by the electric and magnetic forces that it encounters as it orbits the ring:

**  ****Lorentz Force**

**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## Given an existing lattice, determine the properties of the beam.

For a desired set of beam properties, determine the design of the lattice.

The first problem is in principle straight-forward to solve.

The second problem is not straight-forward – a bit of an art.

**For a desired set of beam properties, determine the design of the lattice.**
**The first problem is in principle straight-forward to solve.**
**The second problem is not straight-forward – *****a bit of an art.***

- Quadrupoles

- Sextupoles

- Dipoles

## 2n-pole:
          dipole      quadrupole       sextupole        octupole ...

n:          1                   2                     3                      4    ...               

Normal: gap appears at the horizontal plane
Skew: rotate around beam axis by /2n angle
Symmetry: rotating around beam axis by /n angle, the field is reversed (polarity flipped)

-           dipole      quadrupole       sextupole        octupole ...
**n**:          1                   2                     3                      4    ...
- Normal: gap appears at the horizontal plane
- Skew: rotate around beam axis by ******/2n** angle
- Symmetry: rotating around beam axis by ******/n** angle, the field is reversed (polarity flipped)

## There are several magnet types that are used in storage rings:

-  **Dipoles**** ******** used for guiding**
**	**** ****B****x**** = 0 **
**	**** ****B****y**** = B****o**
**Quadrupoles**** ******** used for focussing**
**	**** ****B****x**** = Ky **
**	**** ****B****y**** = Kx**
**Sextupoles**** ******** used for chromatic correction**
**	**** ****B****x**** = 2Sxy**
**	**** ****B****y**** = S(x****2**** – y****2****)**

**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## Slide 10

## In the Lorentz Force Equation as written below, the dependent variable is time, t

**  ****Lorentz Force**

**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## Consider a storage ring for particles with energy E with N dipoles of length l
The bending angle is 

The bending radius is

- The bending angle is
- The bending radius is

-    The integrated dipole strength will be
-    By fixing the dipole field, the dipole length is imposed and vice versa
-    The highest the field, shorter or smaller number of dipoles can be used
-    Ring circumference (cost) is influenced by the field choice

## Magnetic element that deflects the beam by an angle proportional to the distance from its centre (equivalent to ray optics) provides focusing.
For a focal length f the deflection angle is 
A magnetic element with length l and with a gradient g has a field	                         so that the deflection angle is

- For a focal length f the deflection angle is
- A magnetic element with length **l** and with a gradient **g **has a field	                         so that the deflection angle is

-    The normalised focusing strength
-    In more practical units, for **Z=1**
**  **
**   **The focal length becomes          and the deflection angle is

## Quadrupoles are focusing in one plane and defocusing in the other
The field is
The resulting force
Need to alternate focusing and defocusing in order to control the beam, i.e. alternating gradient focusing
From optics we know that a combination of two lenses with focal lengths f1 and f2 separated by a distance d

If f1 = -f2, there is a net focusing effect, i.e.

- Quadrupoles are focusing in one plane and defocusing in the other
- The field is
- The resulting force
- Need to alternate focusing and defocusing in order to control the beam, i.e. **alternating gradient focusing**
- From optics we know that a combination of two lenses with focal lengths** f1 **and** f2 **separated by a distance** d**
- If f1 = -f2, there is a net focusing effect, i.e.

## Coordinate system used to describe the motion is usually locally Cartesian or cylindrical

**Typically the coordinate system chosen is the one that allows the easiest field representation**

**Change dependent variable from time, *****t*****, to longitudinal position, *****s***

**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## Integrate through the elements

**Integrate through the elements**

**Use the following coordinates***
****Note sometimes one uses canonical momentum rather than x’ and y’***

**Notes:**
Hello my name is David Robin and I will be describing the Superbend project --- a project that the ALS successfully completed last October.  

Superbends are a cost effective way of producing bright hard X-ray beams using high-field superconducting bending magnets. With  Superbends, the ALS has been transformed --- substantially enhancing the facility’s ability to serve the hard X-ray community. 

In this talk I will give a brief introduction to the Superbend project --- present the history --- and finally discuss the impact of Superbends on the ALS

## The equations of motion within an element is

- The fields have to be defined

## The equations become

- Inhomogeneous equations with s-dependent coefficients
- Note that the term **1/****ρ****2**** **corresponds to the dipole **weak focusing**
- The term **Δ****P/(P****ρ)** is present for off-momentum particles** **

## Solutions are combination of the ones from the 	            homogeneous and inhomogeneous equations

- Consider particles with the design momentum. 	                        The equations of motion become
- 	with
-  **Hill’s equations of linear transverse particle motion**
- Linear equations with s-dependent coefficients (harmonic oscillator with time dependent frequency)
- In a ring or in transport line with symmetries, coefficients  are periodic
- Not feasible to get analytical solutions for all accelerator

## Consider K(s) = k0 = constant

- Consider K(s) = k0 = constant
- Equations of harmonic oscillator with solution
- with
- 				for k0 > 0
- 				for k0 < 0

-  Note that the solution can be written in **matrix** form

## General transfer matrix from s0 to s

- Note that
- 	which is always true for conservative systems
- Note also that
- The accelerator can be build by a series of matrix multiplications

- from **s****0** to **s****1**

- from **s****0** to **s****2**

- from **s****0** to **s****3**

- from **s****0** to **s****n**

-  System with normal symmetry

**S**

## 	to get a total 4x4 matrix

- Combine the matrices for each plane

## Consider a drift (no magnetic elements) of length L=s-s0

- Consider a drift (no magnetic elements) of length L=s-s0
- Position changes if there is a slope. Slope remains unchanged

## Consider a lens with focal length ±f

- Consider a lens with focal length **±f**
- Slope diminishes (focusing) or increases (defocusing). Position remains unchanged

## Consider a quadrupole magnet of length L. The field is

- Consider a quadrupole magnet of length **L**. The field is
- with normalized quadrupole gradient (in **m****-2**)
- The transport through a quadrupole is

## Consider a dipole of length L. By setting  in the focusing quadrupole matrix

- 	the transfer matrix for a sector dipole becomes** **
- 	with a bending radius
- In the non-deflecting plane
- This is a **hard-edge** model. In fact, there is some **edge focusing** in the vertical plane

## Consider a rectangular dipole of length L. At each edge, the deflecting angle is

- It acts as a thin defocusing lens with focal length
- The transfer matrix is				           with
- For **θ<<1**, **δ=θ/2**.
- In deflecting plane (like **drift**)  in non-deflecting plane (like **sector**)

## Consider a quadrupole doublet, i.e. two quadrupoles with focal lengths f1 and f2 separated by a distance L. 

- Consider a quadrupole doublet, i.e. two quadrupoles with focal lengths **f****1** and **f****2** separated by a distance **L**.
- In thin lens approximation the transport matrix is

- 	with the **total focal length**
- Setting **f****1** = - **f****2**** = f**
- Alternating gradient focusing seems overall focusing
- This is only valid in thin lens approximation!!!

## Consider a defocusing quadrupole “sandwiched” by two focusing quadrupoles with focal lengths f.

- The symmetric transfer matrix from center to center of focusing quads

- 	with the transfer matrices
- The total transfer matrix is

**L**

**L**

## From Gauss law of magnetostatics, we construct a vector potential

Assuming a 2D field in x and y, the vector potential has only one component As
The Ampere’s law in vacuum (inside the beam pipe) 

Using the previous equations one finds the conditions which are Riemann conditions of an analytic function.

- Assuming a 2D field in **x** and **y**, the vector potential has only one component **A****s**
- The Ampere’s law in vacuum (inside the beam pipe)
- Using the previous equations one finds the conditions which are Riemann conditions of an analytic function.

-   There exist a complex potential of **z = x+iy** with a power series expansion convergent in a circle with radius |z| = rc (distance from iron yoke)

## From the complex potential we can derive the fields

Setting				    we have

Define normalized units 					     

on a reference radius, 10-4 of the main field to get

Note: n’=n-1 the American convention

- Setting				    we have
- Define normalized units
- on a reference radius, 10-4 of the main field to get
**Note**: **n’=n-1** the American convention

## Wish to thank Y. Papaphilippou and N.Catalan-Lasheras for sharing the tranparencies that they used in the USPAS, Cornell University, Ithaca, NY 20th June – 1st July 2005

**Wish to thank Y. Papaphilippou and N.Catalan-Lasheras for sharing the tranparencies that they used in the USPAS, Cornell University, Ithaca, NY 20th June – 1st July 2005**

##  Derive the thin matrix representation for a focusing quadrupole starting from the  “thick” element matrix. Hint: calculate the limit for the matrix when the quadrupole length approaches zero while the integrated magnetic field is kept constant. 

** ****Derive the thin matrix representation for a focusing quadrupole starting from the  “thick” element matrix. Hint: calculate the limit for the matrix when the quadrupole length approaches zero while the integrated magnetic field is kept constant. **

** ****Suppose that a particle traverses, first, a thin focusing lens with a focal length F; second, a drift of length L; third, a thin defocusing lens with focal length F; and, fourth, another drift of length L. Calculate the matrix for this cell.**
** ****Consider a system made up of two thin lenses each of focal length *****F*****, one focusing and one defocusing, separated by a distance *****L*****. Show that the system is focusing if *****|F|>L*****.**