Package xal.tools.beam.em

Class AxialFieldSpectrum

java.lang.Object

xal.tools.beam.em.AxialFieldSpectrum

public class AxialFieldSpectrum
extends Object

Class that representing the spatial spectral properties of a time-harmonic, axial electric field. The most important properties from a beam physics standpoint are the transit time factors. These are the components of the Fourier transform of the axial field, and their resulting Hilbert transforms. The Hilbert transform of a transit time factor turns out to be the transit time factor of axial field times the signum function. The spectral pre- and post-envelopes are formed from a transit time factor and its Hilbert transform. These pre- and post-envelopes are the primary entities for computing the pre- and post-gap energy gain and phase jump, respectively.

Partial Field Model

This class attempts to maintain backward compatibility between the RF acceleration model produced by Los Alamos and CERN. There the T_q and S transit time factors are zero. Although usually labeled S, the "sine" transit time factor is actually its quadrature conjugate S_q. That model also requires an "offset" Δz which is the distance between the coordinate origin and the field center (the point of symmetry). The assumptions are that the field is symmetric about the axial location Δz, if not this information is lost.

In the partial field model the provided spectra, T, S_q, and their derivatives, are assumed to be functions of normalized particle velocity β ≜ v/c, where v is particle velocity and c is the speed of light. Thus, the functions T(β), dT(β)/dk, S_q(β), and dS_q(β)/dk are provided to the partial field model constructor.

Note the important difference that in the full-field model the arguments of the spectra quantities are the particle wave number k. That is, the constructor for the full-field model takes spectral functions T(k), dT(k)/dk, S_q(k), and dS_q(k)/dk, along with all the other spectral quantities.

Full Field Model

The current full-field model includes all four transit time factors and their derivatives. Thus, all the field information is kept, no offsets are necessary, and there is information enough to compute the post gap energy gain and phase jump. (In the above model the post gap quantities are assumed to be equal to the pre-gap quantities.) When function objects are provided for all four transit time factors this class assumes that the new model is being used. When only T and S are provided the class assumes that the old model is being used. Further assumptions are

· The sine transit time factor S is actually the conjugate S_q (see below)
· The offset Δz must also be provided

The class then makes the appropriate conversions from the quantities (T,T',S,S',Δz) to (T,T',T_q,T'_q,S,S',S_q,S'_q). Of course the later set is incomplete.

Axial Electric Fields

Let the longitudinal electric field along the beam axis z be denoted E_z(z). Let the total voltage gain long the field be denoted V₀, that is,

V₀ ≜ ∫E_z(z) dz .

Note then that the total available energy gain ΔW₀ for a particle falling through the field E_z(z) is qV₀. The value V₀ is used to create the normalized electric field e_z(z) given by

e_z(z) ≜ (1/V₀)E_z(z) .

All spectral quantities in this class are with respect to this normalized field.

Next, let sgn(z) denote the signum function, that is,

     sgn(z) ≜ -1    z  < 0
              +1    z  > 0
 

Finally, define the quadrature field E_q(z) as

E_q(z) ≜ sgn(z) E_z(z) .

Of course the quantities here will be with respect to the normalized quadrature field eq(z) defined as

     ez(z) ≜ (1/V0)Eq(z) ,
           = sgn(z)ez(z) .
 

Transit Time Factors

There are four (4) transit time factors. These transit time factors are described as follows:

· T(k) - The Fourier cosine transform of axial field E_z(z)
T(k) ≜ (1/V₀)∫E_z(z) cos kz dz

· S(k) - The Fourier sine transform of axial field E_z(z)
S(k) ≜ (1/V₀)∫E_z(z) sin kz dz

· T_q(k) - The Fourier cosine transform of the axial field sgn(z)E_z(z)
T_q(k) ≜ (1/V₀)∫sgn(z)E_z(z) cos kz dz

· S_q(k) - The Fourier sine transform of axial field sgn(z)E_z(z)
S_q(k) ≜ (1/V₀)∫ sgn(z)E_z(z) sin kz dz

where k is the synchronous particle wave number. Sometimes the arguments to the transit time factors is the synchronous particle velocity β. This includes that for the derivative functions as well (see below). One needs to check the method documentation for the argument type.

The derivatives of the transit time factors are also available. These are the derivatives with respect to wave number k and will be denoted T'(k), S'(k), T'_q(k), and S'_q(k).

Hilbert Transform

The transit time factors are related to each other via the Hilbert transform ℋ. Specifically,

T_q(k) = -ℋ[S(k)] ,
S_q(k) = +ℋ[T(k)] .

This is a transitive relation which follows from the anti-selfadjointness of the Hilbert transform, thus,

T(k) = -ℋ[S_q(k)] ,
S(k) = +ℋ[T_q(k)] .

The Hilbert transform also relates the field spectra, as shown below.

Field Spectra

Denote by ℰ_z(k) and ℰ_q(k) the Fourier transforms of the axial field e_z(z) and its conjugate e_q(z), respectively. That is, the field spectra are

ℰ_z(k) ≜ ℱ[e_z](k) ,
ℰ_q(k) ≜ ℱ[e_q](k) ,

where ℱ[·] is the Fourier transform operator. The Fourier transforms of the fields have the decomposition

ℰ_z(k) = T(k) - iS(k) ,
ℰ_q(k) = T_q(k) - iS_q(k) ,

where i is the imaginary unit. The spectra are then related by the Hilbert transforms

ℋ[ℰ_z(k)] = iℰ_q(k) ,
ℋ[ℰ_q(k)] = iℰ_z(k) ,

Thus, we see ℰ_z(k) and ℰ_q(k) are conjugates of each other.

Pre- and Post-Envelope Spectra

The pre- and post-envelope spectra can be formed from the field spectra. First, denote by ℰ^-(k) and ℰ⁺(k) the pre- and post-envelope spectra, respectively. They are defined

ℰ^-(k) ≜ (1/2)[ ℰ_z(k) + iℋ[ℰ_z(k] ] = (1/2)[ ℰ_z(k) - ℰ_q(k) ],
ℰ⁺(k) ≜ (1/2)[ ℰ_z(k) + iℋ[ℰ_z(k] ] = (1/2)[ ℰ_z(k) + ℰ_q(k) ] ,

Let φ be the synchronous particle phase at the gap center. Then the quantities e^-iφℰ^-(k) and e^-iφℰ⁺(k) contain the pre- and post-gap energy gain ΔW^-, ΔW⁺ and phase jump Δφ^-, Δφ⁺, respectively. For example, the real part of e^-iφℰ^-(k) tracks the pre-gap energy gain while the imaginary part tracks the phase jump. We have

ΔW^-(φ,k) = qV₀ Re ℰ^-(k)e^-iφ ,
Δφ^-(φ,k) = d/dk Im K_iℰ^-(k)e^-iφ ,

where q is the unit charge and K_i is the quantity

K_i ≜ k₀(qV₀/mc²)(1/β_i³γ_i³) .

The subscript i indicates initial, pre-gap values. The post-gap quantities have analogous expressions

ΔW⁺(k,φ) = (q/2) Re e^-iφℰ^-(k) ,
Δφ⁺(k,φ) = (K_f/2) Im d/dk e^-iφℰ^-(k) ,

The subscript f indicates final, post-gap quantities. Methods to compute these values are provided.

Since:

Sep 23, 2015

Version:

Sep 23, 2015

Author:

Christopher K. Allen

Constructor Summary

Constructors

Constructor	Description
AxialFieldSpectrum(double dblFreq, double dblOffset, IRealFunction fncTz0, IRealFunction fncDTz0, IRealFunction fncSq0, IRealFunction fncDSq0)	Constructor for creating a partial field model object using a central field offset.
AxialFieldSpectrum(IRealFunction fncTz, IRealFunction fncDTz, IRealFunction fncSz, IRealFunction fncDSz, IRealFunction fncTq, IRealFunction fncDTq, IRealFunction fncSq, IRealFunction fncDSq)	Constructor for creating the full field model using the sine and cosine transform and their conjugates.

Method Summary

Modifier and Type	Method	Description
Complex	cnjSpectrum(double k)	Computes and returns the complex spectral of the conjugate spatial field at the given wave number k.
Complex	dkCnjSpectrum(double k)	Computes and returns the derivative, with respect to the wave number k, of the quadrature field spectrum at the given wave number k.
Complex	dkFldSpectrum(double k)	Computes and returns the derivative, with respect to the wave number k, of the field spectrum at the given wave number k.
Complex	dkPostEnvSpectrum(double k)	Compute and return the derivative of the spectral post-envelope ℰ+(k), that is dℰ+(k)/dk.
Complex	dkPreEnvSpectrum(double k)	Compute and return the derivative of the spectral pre-envelope ℰ-(k), that is dℰ-(k)/dk.
double	dkSq(double k)	Returns the derivative dS(k)/dk of the conjugate sine transit-time factor w.r.t. the particle wave number k.
double	dkSz(double k)	Returns the derivative of the sine transit-time factor dSz/dk with respect to the particle wave number k .
double	dkTq(double k)	Returns the derivative dT(k)/dk of the conjugate cosine transit-time factor w.r.t. the particle wave number k.
double	dkTz(double k)	Returns the derivative of the cosine transit time factor Tz w.r.t. the wave number k.
Complex	fldSpectrum(double k)	Computes and returns the complex spectra of this spatial field at the given wave number k.
Double	getFieldOffset()	Returns the offset between the axis origin and the center of the field, however that is defined.
double	getFrequency()	Returns the time-harmonic frequency of this electric field.
boolean	isPartialFieldModel()	Returns true when this object has been initialized to use the partial field model.
Complex	postEnvSpectrum(double k)	Compute and return the spectral post-envelope ℰ+(k).
Complex	preEnvSpectrum(double k)	Compute and return the spectral pre-envelope ℰ-(k).
double	sq(double k)	Returns the conjugate sine transit-time factor Sq, proportional to the Fourier sine transform of the field sgn(z)Ez(z).
double	sz(double k)	Returns the sine transit-time factor Sz, proportional to the Fourier sine transform of the axial field, for the given wave number.
double	tq(double k)	Returns the conjugate cosine transit-time factor Tq, proportional to the Fourier cosine transform of the field sgn(z)Ez(z).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

AxialFieldSpectrum

public AxialFieldSpectrum(double dblFreq,
 double dblOffset,
 IRealFunction fncTz0,
 IRealFunction fncDTz0,
 IRealFunction fncSq0,
 IRealFunction fncDSq0)

Constructor for creating a partial field model object using a central field offset.

In this case the given field spectra are assumed to be functions of normalized particle velocity β. Thus, the RF frequency is needed to convert from wave number k to normalized velocity β in order to evaluate the functions.

Parameters:

dblFreq - frequency of the time-harmonic electric field (Hz)

dblOffset - offset between axial origin and central field position (meters)

fncTz0 - central cosine transform (transit time factor)

fncDTz0 - derivative of the central cosine transform w.r.t. k

fncSq0 - conjugate central sine transform (transit time factor)

fncDSq0 - derivative of the conjugate central sine transform w.r.t. k

Since:

Sep 23, 2015 by Christopher K. Allen

AxialFieldSpectrum

public AxialFieldSpectrum(IRealFunction fncTz,
 IRealFunction fncDTz,
 IRealFunction fncSz,
 IRealFunction fncDSz,
 IRealFunction fncTq,
 IRealFunction fncDTq,
 IRealFunction fncSq,
 IRealFunction fncDSq)

Constructor for creating the full field model using the sine and cosine transform and their conjugates. The pre- and post-envelope spectra are build from these objects and used to compute acceleration parameters.

Note that frequency f is not needed here since the given arguments are assumed to be functions of k. RF frequency is only needed to convert from wave number k to normalized velocity β in the case of spectral functions that are functions of β (e.g., T(β), S(β), etc.).

Parameters:

fncTz - field cosine transform (transit time factor)

fncDTz - derivative of the cosine transform w.r.t. k

fncSq - field conjugate sine transform (transit time factor)

fncDSq - derivative of the sine transform w.r.t. k

fncTq - field conjugate cosine transform (transit time factor)

fncDTq - derivative of the conjugate cosine transform w.r.t. k

fncSz - field sine transform (transit time factor)

fncDSz - derivative of the field sine transform w.r.t. k

Since:

Sep 28, 2015 by Christopher K. Allen

Method Details

isPartialFieldModel

public boolean isPartialFieldModel()

Returns true when this object has been initialized to use the partial field model.

Returns:

Since:

Sep 28, 2015 by Christopher K. Allen

getFrequency

public double getFrequency()

Returns the time-harmonic frequency of this electric field.

Returns:

the RF frequency of the electric field (Hz)

Since:

Sep 28, 2015 by Christopher K. Allen

getFieldOffset

public Double getFieldOffset()

Returns the offset between the axis origin and the center of the field, however that is defined. That is, it could be defined as the point of maximum field, or the center of mass, etc.

If the full field model is being used then this parameter is not defined and a null value will be returned.

Returns:

offset between field center and axis origin in partial field model (meters), or null if the full field model is being used

Since:

Sep 28, 2015 by Christopher K. Allen

fldSpectrum

public Complex fldSpectrum(double k)

Computes and returns the complex spectra of this spatial field at the given wave number k. The value returned has the formula

ℰ_z(k) = T_z(k) - i S_z(k)

where ℰ_z is the returned spectral value, T_z is the cosine transit-time factor, and S_z is the sine transit-time factor.

NOTES:

· We assume that the electric field is normalized by its total potential drop V₀ ≜ ∫E_z(z)dz so that the resulting total potential of the field is 1.

Parameters:

k - particle wave number (in radians/meter)

Returns:

the Fourier spectrum ℰ(k) of the field at the given wave number k

Since:

Sep 30, 2015, Christopher K. Allen

dkFldSpectrum

public Complex dkFldSpectrum(double k)

Computes and returns the derivative, with respect to the wave number k, of the field spectrum at the given wave number k. The value returned has the formula

ℰ_z'(k) = T'_z(k) - i S_z'(k)

where ℰ_z' is the returned spectral derivative, T_z' is the cosine transit-time factor derivative, and S_z' is the sine transit-time factor derivative of the field.

NOTES:

· We assume that the electric field is normalized by its total potential drop V₀ ≜ ∫E_z(z)dz so that the resulting total potential of the field is 1.

Parameters:

k - particle wave number (in radians/meter)

Returns:

Fourier spectrum derivative ℰ'(k), w.r.t. k, of the field at the given wave number k (meters/radian)

Since:

Sep 30, 2015, Christopher K. Allen

cnjSpectrum

public Complex cnjSpectrum(double k)

Computes and returns the complex spectral of the conjugate spatial field at the given wave number k. The value returned has the formula

ℰ_q(k) = T_q(k) - i S_q(k)

where ℰ_q is the returned spectral value, T_q is the cosine transit-time factor of the quadrature field, and S_q is the sine transit-time factor of the quadrature field.

NOTES:

· We assume that the electric field is normalized by its total potential drop V₀ ≜ ∫E_z(z)dz so that the resulting total potential of the field is 1.

Parameters:

k - particle wave number (in radians/meter)

Returns:

the Fourier spectrum ℰ_q(k) of the quadrature field at the given wave number k (unitless)

Since:

Sep 30, 2015, Christopher K. Allen

dkCnjSpectrum

public Complex dkCnjSpectrum(double k)

Computes and returns the derivative, with respect to the wave number k, of the quadrature field spectrum at the given wave number k. The value returned has the formula

ℰ_q'(k) = T_q'(k) - i S_q'(k)

where ℰ_q' is the returned spectral derivative, T_q' is the cosine transit-time factor derivative of the quadrature field, and S_q is the sine transit-time factor derivative of the quadrature field.

NOTES:

· We assume that the electric field is normalized by its total potential drop V₀ ≜ ∫E_z(z)dz so that the resulting total potential of the field is 1.

Parameters:

k - particle wave number (in radians/meter)

Returns:

Fourier spectrum derivative ℰ_q'(k) w.r.t. k of the quadrature field at the given wave number k (meters/radian)

Since:

Sep 30, 2015, Christopher K. Allen

preEnvSpectrum

public Complex preEnvSpectrum(double k)

Compute and return the spectral pre-envelope ℰ^-(k). The spectral pre- and post-envelopes turn out to be equal to odd and even combinations, respectively, of the field spectra. Specifically, the pre- and post-envelopes ℰ^-(k) and ℰ⁺(k) are given by

ℰ^-(k) = (1/2)[ ℰ_z(k) - ℰ_q(k) ] ,
ℰ⁺(k) = (1/2)[ ℰ_z(k) + ℰ_q(k) ] ,

where ℰ_z(k) is the Fourier spectrum of the axial field and ℰ_q(k) is the spectrum of the quadrature field (the conjugate Fourier transform).

NOTES:

· We assume that the electric field is normalized by its total potential drop V₀ ≜ ∫E_z(z)dz so that the resulting total potential of the field is 1.

Parameters:

k - the particle wave number (radians/meter)

Returns:

the pre-envelope ℰ^-(k) at the given wave number (unitless)

Since:

Sep 30, 2015, Christopher K. Allen

dkPreEnvSpectrum

public Complex dkPreEnvSpectrum(double k)

Compute and return the derivative of the spectral pre-envelope ℰ^-(k), that is dℰ^-(k)/dk. For a description of the pre-envelope see preEnvSpectrum(double) or the class documentation.

NOTES:

· We assume that the electric field is normalized by its total potential drop V₀ ≜ ∫E_z(z)dz so that the resulting total potential of the field is 1.

Parameters:

k - the particle wave number (radians/meter)

Returns:

derivative of the pre-envelope spectrum dℰ^-(k)/dk at the given wave number (in meters/radian)

Since:

Oct 1, 2015, Christopher K. Allen

See Also:

• preEnvSpectrum(double)

postEnvSpectrum

public Complex postEnvSpectrum(double k)

Compute and return the spectral post-envelope ℰ⁺(k). The spectral pre- and post-envelopes turn out to be equal to odd and even combinations, respectively, of the field spectra. Specifically, the pre- and post-envelopes ℰ^-(k) and ℰ⁺(k) are given by

ℰ^-(k) = (1/2)[ ℰ_z(k) - ℰ_q(k) ] ,
ℰ⁺(k) = (1/2)[ ℰ_z(k) + ℰ_q(k) ] ,

where ℰ_z(k) is the Fourier spectrum of the axial field and ℰ_q(k) is the spectrum of the quadrature field (the conjugate Fourier transform).

NOTES:

· We assume that the electric field is normalized by its total potential drop V₀ ≜ ∫E_z(z)dz so that the resulting total potential of the field is 1.

Parameters:

k - the particle wave number (radians/meter)

Returns:

the post-envelope ℰ⁺(k) at the given wave number (unitless)

Since:

Sep 30, 2015, Christopher K. Allen

dkPostEnvSpectrum

public Complex dkPostEnvSpectrum(double k)

Compute and return the derivative of the spectral post-envelope ℰ⁺(k), that is dℰ⁺(k)/dk. For a description of the post-envelope see postEnvSpectrum(double) or the class documentation.

NOTES:

· We assume that the electric field is normalized by its total potential drop V₀ ≜ ∫E_z(z)dz so that the resulting total potential of the field is 1.

Parameters:

k - the particle wave number (radians/meter)

Returns:

derivative of the post-envelope spectrum dℰ^-(k)/dk at the given wave number (in meters/radian)

Since:

Oct 1, 2015, Christopher K. Allen

See Also:

• postEnvSpectrum(double)

tz

public double tz(double k)

Returns the cosine transit time factor T_z, proportional to the Fourier cosine transform, for the given wave number.

Parameters:

k - particle wave number with respect to the RF frequency (radians/meter)

Returns:

the value of T(k) (unitless)

Since:

Sep 28, 2015 by Christopher K. Allen

dkTz

public double dkTz(double k)

Returns the derivative of the cosine transit time factor T_z w.r.t. the wave number k.

Parameters:

k - particle wave number with respect to the RF frequency (radians/meter)

Returns:

the value of dT_z(k)/dk (meters/rad)

Since:

Sep 28, 2015 by Christopher K. Allen

sz

public double sz(double k)

Returns the sine transit-time factor S_z, proportional to the Fourier sine transform of the axial field, for the given wave number.

Parameters:

k - particle wave number with respect to the RF frequency (radians/meter)

Returns:

the value of S_z(k) (unitless)

Since:

Sep 28, 2015 by Christopher K. Allen

dkSz

public double dkSz(double k)

Returns the derivative of the sine transit-time factor dS_z/dk with respect to the particle wave number k .

Parameters:

k - particle wave number with respect to the RF frequency (radians/meter)

Returns:

the value of dS_z(k)/dk (meters/radian)

Since:

Sep 28, 2015 by Christopher K. Allen

tq

public double tq(double k)

Returns the conjugate cosine transit-time factor T_q, proportional to the Fourier cosine transform of the field sgn(z)E_z(z).

Parameters:

k - the particle wave number w.r.t. the RF frequency (radians/meter)

Returns:

the conjugate cosine transit-time factor T_q(k) (unitless)

Since:

Sep 29, 2015 by Christopher K. Allen

dkTq

public double dkTq(double k)

Returns the derivative dT(k)/dk of the conjugate cosine transit-time factor w.r.t. the particle wave number k.

Parameters:

k - the particle wave number w.r.t. the RF frequency (radians/meter)

Returns:

the value of dT_q(k)/dk (meters/rad)

Since:

Sep 29, 2015, Christopher K. Allen

sq

public double sq(double k)

Returns the conjugate sine transit-time factor S_q, proportional to the Fourier sine transform of the field sgn(z)E_z(z).

Parameters:

k - the particle wave number w.r.t. the RF frequency (radians/meter)

Returns:

the conjugate cosine transit-time factor S_q(k) (unitless)

Since:

Sep 29, 2015 by Christopher K. Allen

dkSq

public double dkSq(double k)

Returns the derivative dS(k)/dk of the conjugate sine transit-time factor w.r.t. the particle wave number k.

Parameters:

k - the particle wave number w.r.t. the RF frequency (radians/meter)

Returns:

the value of dS_q(k)/dk (meters/rad)

Since:

Sep 29, 2015, Christopher K. Allen