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Elements

(of Math and stuff useful for Beam Diagnostics)

Randy Thurman-Keup

Fermilab

Euclid

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Outline

Basic Math

Vectors
Matrices
Complex Numbers

Fourier Transforms

Theory
Fast Fourier Transforms (FFT)
Common Transforms

Electromagnetics

Electrostatics
Magnetostatics

Miscellaneous

R. Thurman-Keup --- USPAS Hampton, VA

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Vectors

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(x,y) Cartesian coordinates

r is the length of the vector

y

x

r

θ

θ gives the direction of the vector

( r, θ ) Polar coordinates

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Matrices

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Vector

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Vector Transformations

Given a vector to point A, a linear transformation to point B necessitates equating each new coordinate of B to a linear combination of the old coordinates of A

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y

x

A

B

Rotation Matrix

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Vector Operations

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Unit Vector

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Vector Operations

Cross Product: Result is a perpendicular vector

Can also be written as

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Determinant

Right Hand Rule

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Matrices and the Accelerator

Every particle has a position and a momentum

So… every particle can be described by a pair of vectors
One can combine the two vectors into one big vector and use matrices to mathematically move the particles around the accelerator

Uncoupled beams (no correlation between x,y,z) have non-zero elements only near the diagonal (block diagonal)

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Complex Numbers

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y

x

r

θ

Re

Im

z=x+jy

Amplitude

Phase

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Outline

Basic Math

Vectors
Matrices
Complex Numbers

Fourier Transforms

Theory
Fast Fourier Transforms (FFT)
Common Transforms

Electromagnetics

Electrostatics
Magnetostatics

Miscellaneous

R. Thurman-Keup --- USPAS Hampton, VA

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Fourier Transforms

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Orthogonal Functions

(sin and/or cos)

Weights

Sum

Arbitrary Function

Orthogonal Functions

(sin and/or cos)

Weights

Sum

Arbitrary Function

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Periodic Functions

Since the function is periodic, it has implicit boundaries where the function repeats

Naively expect the sine and cosine functions from which the function is built to also repeat
Only certain wavelengths satisfy these boundary conditions

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Fourier Transforms

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Discrete Fourier Transforms

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Dirac Delta Function

‘Sampling’ Function

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Discrete Fourier Transforms

Switch order of integration and summation

Integrate

Not all terms are necessary (beyond M are redundant)

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Discrete Fourier Transforms

Number of ‘unique’ frequencies = number of time values

Not quite true…
Representing M real time values�by M complex frequency values

More information than needed
Only half the complex information is unique

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Nyquist

Nyquist Sampling Theorem

The sampling period must be less than half the period of the highest frequency in the signal to accurately determine the frequencies in the signal

- or conversely -

The sampling frequency must be greater than twice the highest frequency in the signal

If sampling is too infrequent, confusion reigns

Data points can be satisfied by many higher frequencies

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Nyquist

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Slightly less than�Nyquist frequency

Slightly more than�Nyquist frequency

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Aliasing in Discrete Transforms

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Frequency

Must analog filter�higher frequencies

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Fast Fourier Transform (FFT)

Just an algorithm for doing discrete Fourier transforms quickly

O(n log n) for any n, not just powers of 2

Frequencies above Nyquist are ‘duplicates’ of those below Nyquist

Actually, they represent the negative frequencies

R. Thurman-Keup --- USPAS Hampton, VA

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Time Domain

Frequency Domain

Same

Nyquist Frequency

DC

Magnitude

Magnitude is symmetric about DC

Phase is antisymmetric about DC

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Fast Fourier Transform (FFT)

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Fourier Transform Theorems

Parseval’s (Plancherel’s) Theorem

Convolution Theorem

Convolution in one domain is equivalent to multiplication in the other

Propagation of electromagnetic waves is convolution (Huygens, Kirchoff)

Can be much faster to use FFTs and multiply

R. Thurman-Keup --- USPAS Hampton, VA

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Normalization constant from slide 11

Convolution

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Common Transform Pairs

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Delta Function

Constant Spectrum

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Common Transform Pairs

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Common Transform Pairs

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Convolution Theorem at work!! Triangle is convolution of square pulses.

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Transform Pair Examples

Infinite train of gaussian bunches

What does the spectrum look like?
First: Construct a train of gaussians using the common transform pairs

Convolution of gaussian with train of delta functions

Train of delta functions is a replicator under convolution
Result is a train of gaussians

Second: Apply convolution theorem in the frequency domain

Product of gaussian transform and delta function train transform

Result is

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Example 1

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Outline

Basic Math

Vectors
Matrices
Complex Numbers

Fourier Transforms

Theory
Fast Fourier Transforms (FFT)
Common Transforms

Electromagnetics

Electrostatics
Magnetostatics

Miscellaneous

R. Thurman-Keup --- USPAS Hampton, VA

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Vector Calculus

Gradient

Divergence

Laplacian

Curl

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Identities

Divergence Theorem

Stoke’s Theorem

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Equations of Electrostatics

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Total charge�within surface

Closed surface

Divergence

Theorem

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Equations of Electrostatics

Coulomb’s Law

Using vector calculus identity

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Scalar Potential

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Equations of Electrostatics

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Laplace’s Equation if ρ = 0

e.g. Successive �over-relaxation (SOR)

Me (1988)

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Magnetostatics

Biot-Savart Law

Magnetic field of line current

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Magnetostatics

Ampere’s Law

Magnetic field of line current

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B is constant by symmetry at R = const.

Transformer�Input

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Electromagnetics

Faraday’s Law

Vector form via Stoke’s Theorem

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Voltage!

Varying Magnetic Flux!

Transformer�Output

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Maxwell’s Equations

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The Wave Equations

Inhomogeneous E and B fields

Inhomogeneous vector and scalar potentials (A and φ)

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Solutions to the homogeneous equations (J = ρ = 0)

are plane waves in cartesian coordinates

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Reciprocity and Equivalence

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Actual�Sources

Surface�Sources

BPM Response

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Outline

Basic Math

Vectors
Matrices
Complex Numbers

Fourier Transforms

Theory
Fast Fourier Transforms (FFT)
Common Transforms

Electromagnetics

Electrostatics
Magnetostatics

Miscellaneous

R. Thurman-Keup --- USPAS Hampton, VA

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Relativity

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Natural Units

Maybe…

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Example

Relativistic electric field compression

Many detectors rely on fields of the particle
The further away the detector is, the longer the signal is

Detector at beampipe wall with 3 cm radius
What’s the duration of the electric field pulse from a single electron with 5 MeV of kinetic energy?
If we have a bunch of electrons of length 10 ps, what’s the total field pulse duration?

R. Thurman-Keup --- USPAS Hampton, VA

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Circuit Review

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Voltage around a closed loop is ZERO

Current must be conserved at junctions

Time Domain

Frequency Domain

Impedance

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Extras

R. Thurman-Keup --- USPAS Hampton, VA

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Approximations

When first trying to understand a complex problem, always approximate!!!!
The Taylor Series is your friend

R. Thurman-Keup --- USPAS Hampton, VA

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