waves and beams division - · waves and beams division motivation! we wish to be able to...

PPPS01 06/22/01 1

Waves and Beams Division

PHASE RETRIEVAL OF GYROTRON MICROWAVE BEAMS FROM INTENSITY

MEASUREMENTS BASED ON IRRADIANCE MOMENTS

J. P. Anderson, M. A. Shapiro, R. J. Temkin, and D. R. DenisonMassachusetts Institute of Technology

Plasma Science and Fusion CenterCambridge, MA 02139

PPPS01 06/22/01 2

Waves and Beams DivisionOutline

! Motivation! Gyrotron and Reflectors! Cold Test Setup and Sample Data! Iteration Method! Irradiance Moment Theory

! Propagating the moments! Stating the initial phase as a polynomial! Forming linear equations

! Examples: Gaussian Beam! Case 1 -- Ideal Beam! Case 2 -- Beam with Offset Error

! Results Using Gyrotron Data! Summary

PPPS01 06/22/01 3

Waves and Beams DivisionMotivation

! We wish to be able to accurately and reliably design the phase-correcting reflectors of a gyrotron.

! The reflectors shape the output microwave beam by correcting foraberrations in the phasefront.

! To synthesize these phase-correcting surfaces requires knowledge of both amplitude and phase.

! Amplitude may be determined directly from intensity measurements in the lab. The phase, at high frequencies, cannot be directly measured.

! Need a formulation which retrieves the phase from intensity measurements.

PPPS01 06/22/01 4

Waves and Beams DivisionPhase Retrieval

! We wish to develop an accurate and reliable method to retrieve the phase of a gyrotron gaussian-like beam from a series of intensity measurements.

! Given two or more amplitude distributions along the direction of propagation (y) the phase must be retrieved.

! Once the phase is known, we can predict the wave’s behavior at any point.

! This process is useful in designing the phase-correcting reflectors of the gyrotron.

! Phase retrieval may be accomplished by using either an iteration method or a formulation based on the irradiance moments of the beam.

),(0 zxΦ),(0 zxA ),(1 zxA ),(2 zxA, , ,K

),(),(),,( zxiy

yezxAzyx Φ=ψ

),(0 zxA ),(0 zxΦie=),0,( zxψ

Paraxial beam (general):

Beam at y = 0:

PPPS01 06/22/01 5


Communication and Power Industries (CPI, Inc.)

CollectorCoil

DiamondWindow Superconducting

Magnet

Dimpled-WallLauncher

Cavity

ElectronGun

Mirrors (4)

Collector

RF Energy

Electron Beam

Vac-ion Pumps

110 GHz, 1 MW Gyrotron

PPPS01 06/22/01 6


! Internal reflectors are used to shape the emitted microwave beam such that it is Gaussian with a small waist near the window.

! Perturbations in surfaces correct for phase aberrations calculated from phase-retrieval methods.

! These internal reflectors have been designed using the iteration method.

Internal Reflectors

Mirror 3 (M3)

Mirror 4 (M3)

PPPS01 06/22/01 7

Waves and Beams DivisionExternal Reflectors

! Must focus the outgoing beam for injection into corrugated waveguide for guided transmission.

! May use one or two simple focusing mirrors to match beam waist to radius of waveguide.

! Microwave power may be coupled more efficiently by using phase-corrrecting reflectors instead of focusing mirrors to shape the beam.

! With irradiance moment method, it may be possible to design a system with only one external phase-correcting mirror.

PPPS01 06/22/01 8

Waves and Beams DivisionCold Test Setup

cold testmeasurement

plane locations

M4z

measurementplanes

window

M3

M4

M2

M1

z

y

launcher

gyrotroncold test

setup

(TE22,6 mode)

Gaussianbeam

PPPS01 06/22/01 9


propagation (y) axis

measurementplanes

y

z

x

! To be consistent with gyrotron geometry, the propagation axis is the y axis.

! Intensity is measured using a spatial scanner or infrared camera.

! Data set is a series of discrete points sampled over a square grid.

! Measurement errors are common due to detector misalignment.

Plane 0at y = y0

Plane 1at y = y1

Plane 2at y = y2

Intensity Data Planes

PPPS01 06/22/01 10


32 34 36 38 40 42 44

32 34 36 38 40 42 44

6

4

0

4

6

32 z (cm) 44-6

6

x (c

m)

0

32 z (cm) 44-6

6

x (c

m)

0

32 z (cm) 44-6

6x

(cm

)0

32 z (cm) 44-6

6

x (c

m)

0

Intensityat

y = -10 cm

Intensity at

y = -5 cm

Intensity at

y = +5 cm

Intensity at

y = 0 cm

-3

-3

-3

-3

-21 -21

-21 -21

“Toto” Intensity Data

PPPS01 06/22/01 11


32 34 36 38 40 42 4432 z (cm) 44-6

6

x (c

m)

0

32 z (cm) 44-6

6

x (c

m)

0

32 z (cm) 44-6

6x

(cm

)0

32 z (cm) 44-6

6

x (c

m)

0

Intensityat

y = 10 cm

Intensity at

y = 20 cm

Intensity at

y = 60 cm

Intensity at

y = 40 cm

-3

-3

-3

-3

-21 -21

-21 -21

“Toto” Intensity Data

PPPS01 06/22/01 12

Waves and Beams DivisionIteration Method

! Iteration method attempts an initial guess for the phase .

! The guess is refined during each iteration by updating the propagated amplitude with the measured amplitude.

),(2 zxΦ),(2 zxA ie),(1 zxΦ),(1 zxA ie

zx

y

Plane 1 Plane 2Algorithm (one iteration):

=),()0(1 zxψ.1 ),(1 zxA ),()0(

1 zxΦie

.2 ),()0(2 zxA=),()0(

2 zxψ ),()0(2 zxΦie }{ )0(

112 ψP=

),()0(1 zxΦ′=′ ),()0(

1 zxψ.4 ),()0(1 zxA′ ie }{ )0(

221 ψ ′= P

=′ ),()0(2 zxψ ),(2 zxA ),()0(

2 zxΦie.3

=),()1(1 zxψ

≡.5 ),(1 zxA ),()0(

1 zxΦ′

),(1 zxA

ieie ),()1(

1 zxΦ

PPPS01 06/22/01 13


! The moments are expectation values based on the normalized weighted intensity integrated over the measurement plane at a particular axial location, y:

! The Fresnel integral describes a paraxial beam at a distance y in terms of the initial wavefunction:

! By applying the Fresnel integral to the definition of moments, the y dependence of each moment is shown by the following formula:

∫∫≡= dxdzzxAzxzxM yqp

yqp

pq ),(2

∫∫ ×

+−

−=

+

yzxkizxdxdz

kyiyM

qp

pq 2)(exp),0,()(

22ψ

+∂∂

∂∂ ∗

yzxkizx

zx q

q

p

p

2)(exp),0,(

22ψ

Irradiance Moment Theory

( ) ( ) ( ) ( )[ ]∫∫

−′+−′−′′′′=y

zzxxkizxzdxdyizyx

2exp,0,,,

22ψ

λψ

PPPS01 06/22/01 14


! From Fresnel integral applied to the definition of moments, each moment is a polynomial in y of order p+q:

1st OrderMoments

2nd OrderMoments

3rd OrderMoments

)1(10C y)0(

10C +=)(10 yM)1(

01C y)0(01C +=)(01 yM

)1(20C y)0(

20C +=)(20 yM + 2y)2(20C

)1(11C y)0(

11C +=)(11 yM + 2y)2(11C

)1(02C y)0(

02C +=)(02 yM + 2y)2(02C

)1(30C y)0(

30C +=)(30 yM + 2y)2(30C + )3(

30C 3y)1(

21C y)0(21C +=)(21 yM + 2y)2(

21C + )3(21C 3y

)1(12C y)0(

12C +=)(12 yM + 2y)2(12C + )3(

12C 3y)1(

03C y)0(03C +=)(03 yM + 2y)2(

03C + )3(03C 3y


PPPS01 06/22/01 15


M10, M01

Moments

mom

. val

ue (c

m)

M20, M11,M02

Moments

mom

. val

ue (c

m2 )

M30, M21,M12, M03

Moments

mom

. val

ue (c

m3 )

M40, M31,M22, M13,

M04

Moments

mom

. val

ue (c

m4 )

y (cm) y (cm)

y (cm) y (cm)


PPPS01 06/22/01 16


! From the Fresnel diffraction theory, each moment is a polynomial in y of order p+q:

! The coefficients of each moment are determined from the series of intensity measurements by:! Normalizing each moment.! Calculating the moment at each measurement plane (y location).! Fitting the moment data to a polynomial of the appropriate order. For example, the

Mpq(y) moment is fitted to a polynomial in y of order p+q.

! If it is not possible to exactly fit a polynomial to the moment data then measurement error has occurred.

mqp

m

mpq

qp

m

mmpqpqpq yCyCMyM ∑∑

+

=

+

=

=+=0

)(

1

)()0()(


PPPS01 06/22/01 17


! For any moment, it can be shown from the Fresnel equation that the coefficient of first rank (slope) may be expressed as:

! For the first and second order moments, we have:

0A2)1(10C ∫∫ ∂

∂−=x

dxdzk1 0Φ

0A∫∫ ∂∂−=

zdxdz

k1)1(

01C 0Φ 2

∫∫

∂∂−=

xxdxdz

k2)1(

20C 0Φ0A2

∫∫

∂∂+

∂∂−=

xz

zxdxdz

k1)1(

11C 0Φ 0Φ0A 0A2 2

∫∫

∂∂−=

zzdxdz

k2

0A2)1(02C 0Φ

( ) ( )∫∫

∂∂

∂∂+

∂∂

∂∂−= qpqp zx

zzzx

xxdxdz

k1)1(

pqC 0A0Φ 20Φ

Slopes of1st OrderMoments

Slopes of2nd OrderMoments


PPPS01 06/22/01 18


! For a spatially directed microwave beam, the initial phasefront can be expanded as a 2D polynomial in the xz-plane with coefficients φij:

! Using this expansion, the moment terms of first rank, C(1)pq, become linear

functions of the phase coefficients. It is possible to solve for the phase coefficients since the moment terms have already been determined from the intensity data.

! The set of linear equations which determines the phase coefficients, φij, is closed and solvable by truncating the phase expansion series to an appropriate order, N.

! The number of moments which are calculated in the algorithm are specified by N.

! The minimum number of planes required to fit the moments is N+1.

( ) ( ) Njiqp =+=+ maxmax

=),(0 zxΦ 10φ 01φ 20φ 11φ 02φ 30φx + z + 2x + xz + 2z + 3x +K


PPPS01 06/22/01 19


! Substituting our 2D polynomial expansion of into our equations for the moment slopes, we have:

Slopes of1st OrderMoments

Slopes of2nd OrderMoments

( ) ( )[ ]∫∫ ∫∫ ∫∫ +++−= Kzdxdzxdxdzdxdzk

2110φ 0A 0A 0A20φ 11φ2 2 2)1(

10C


21)1(01C 01φ 2

0A 11φ 20A 0A2

02φ

( ) ( ) ( )[ ]∫∫ ∫∫ ∫∫ +++−= Kxzdxdzxdxdzxdxdzk

22)1(20C 10φ 0A2

20φ 0A211φ 0A22

( )[ ( ) ( )∫∫∫∫∫∫ +++−= xzdxdzxdxdzzdxdzk

21)1(11C 10φ 0A2

01φ 0A220φ 0A2

( ) ( ) ]K+++ ∫∫∫∫ xzdxdzzxdxdz 2220A2

0A211φ 02φ

( ) ( ) ( )[ ]∫∫∫∫∫∫ +++−= K222 zdxdzxzdxdzzdxdzk 01φ 0A2

11φ 0A202φ 0A2)1(

02C

),(0 zxΦ


PPPS01 06/22/01 20


! We recognize that the integrations are simply moments at the initial plane. We simplify our equations by making the substitution .

! For example, the slope equation:

! Similarly, for the slope equation:

( ) =0pqM )0(pqC



2110φ 0A 0A 0A20φ 11φ2 2 2)1(

10C

( ) ( ) ( )[ ]∫∫ ∫∫ ∫∫ +++−= Kxzdxdzxdxdzxdxdzk

22)1(20C 10φ 0A2

20φ 0A211φ 0A22

10φ )0(10C20φ 11φ)1(

10C )0(01C( )1[

k1−= + 2 + + ]K

Becomes:

)1(10C

)1(20C

Becomes:)1(

20C 10φ )0(10C )0(

20C20φ 11φ )0(11C+ 2[

k2−= + + ]K

PPPS01 06/22/01 21


! Truncating the phasefront expansion to second order (N = (i+j)max = 2) will yield a set of 5 equations with 5 unknowns: , , , , and .

! The moment coefficients and are determined from the polynomial fits generated from the intensity data.

[ ]++−= 21k

)1(10C 10φ 20φ 11φ)0(

10C )0(01C

[ ]21 ++−=k

)1(01C 01φ 11φ 02φ)0(

10C )0(01C

[ ]++−= 22k

)1(20C 10φ )0(

10C 20φ )0(20C 11φ )0(

11C

[ ]22 ++−=k

)1(02C 01φ )0(

01C 11φ )0(11C 02φ )0(

02C

[ +++−= 21k

)1(11C 10φ )0(

01C 01φ )0(10C 20φ )0(

11C

( ) ]2++11φ )0(20C )0(

02C )0(11C02φ

10φ 01φ 20φ 11φ 02φ

)1(pqC )0(

pqC

2 Equations from 1st

Order Moments

3 Equations from 2nd

Order Moments


PPPS01 06/22/01 22


Choose appropriate order, N, for phase expansion

Irradiance Moment AlgorithmFormulate linear equations by

substituting phase expansion into moment slope equations

Numerically calculate moments up to Nth order from data at each plane

Solve linear equations for phase expansion coefficients Analytical phasefront solution

Determine values of slopes, , and intercepts, , from fitting moments

to polynomials

)1(pqC

)0(pqCIs there measurement

error?

Discard measurement data planes containing error

YES

NO

PPPS01 06/22/01 23

Waves and Beams DivisionPhase Retrieval Methods

! Approximates an initial guess for the phasefront.

! Numerically propagates beam. Through iteration, phase solution converges.

! Requires computationally-intensive iterative calculations.

! Produces a phase solution in numerical form. Ambiguous process of phase unwrapping is required.

! Does not compensate for measurement errors. For large errors, method does not converge to a solution.

Iteration Method Irradiance Moment Method! Calculates the irradiance moments of

the intensity data.

! Forms a set of linear equations from Fresnel integral applied to moments.

! Previously limited to phase-retrieval problems in the optical regime.

! Produces an analytical form for the phasefront solution. Does not require phase unwrapping for shaping mirrors.

! May be used to locate measurement errors and compensate for them.

PPPS01 06/22/01 24


! A series of 5 intensity measurement planes is generated for an ideal Gaussian beam with λ = 0.27 cm and wo = 2.0 cm.

! Planes are at y = 20, 30, 40, 50, and 60 cm from the beam waist.

! We assume a second order phasefront (N = 2) in xz-plane for the irradiance moment solution.

! The retrieved wavefunction (at y = 20 cm) may be propagated to an observation plane at y = -30 cm.


zx

y

yo = 20cm y1 = 30cm y2 = 40cm y3 = 50cm y4 = 60cmyobs = -30cm

),(0 zxΦ),(0 zxA ie ),(3 zxΦ),(3 zxA ie ),(4 zxΦ),(4 zxA ie),( zxobsΦ),( zxAobsie

Example 1: Ideal Gaussian

PPPS01 06/22/01 25


! A series of 5 intensity measurement planes is generated for an ideal Gaussian beam with λ = 0.27 cm and wo = 2.0 cm.

! Coefficient values are determined from polynomial fits of the intensity data.

M10

(cm

)

y (cm) 600-0.5

1

0

0)1(10 =C

0)0(10 =C

M10 Moments

y (cm) 600

M20

(cm

2 )

0

3

1

M20 Moments

019.0)1(20 =C

2.1)0(20 =C

00047.0)2(20 =C


PPPS01 06/22/01 26


! Recall our coefficients from the polynomial fits:

! Our set of linear equations describing the moment slopes, for N = 2:

! The irradiance moment solution:,


[ ]++−= 21k

)1(10C 10φ 20φ 11φ)0(

10C )0(01C

[ ]21 ++−=k

)1(01C 01φ 11φ 02φ)0(

10C )0(01C

[ ]++−= 22k

)1(20C 10φ )0(

10C 20φ )0(20C 11φ )0(

11C

[ ]22 ++−=k

)1(02C 01φ )0(

01C 11φ )0(11C 02φ )0(

02C

[ +++−= 21k

)1(11C 10φ )0(

01C 01φ )0(10C 20φ )0(

11C

( ) ]2++11φ )0(20C )0(

02C )0(11C02φ

2 Equations from 1st

Order Moments

3 Equations from 2nd

Order Moments

0)1(10

)1(10 == CC

0)0(01

)0(10 == CC

019.0)1(02

)1(20 == CC

2.1)0(02

)0(20 == CC

00047.0)2(02

)2(20 == CC

0)1(11 =C

0)0(11 =C

0)2(11 =C

0110110 === φφφ 20220 091.0 −== cmφφ ( )zx,0Φ ( )2091.0 −cm ( )2091.0 −cm+2x 2z=

PPPS01 06/22/01 27


! A series of 5 intensity measurement planes is generated for a Gaussian beam with λ = 0.27 cm and wo = 2.0 cm.

! Planes are again at y = 20, 30, 40, 50, and 60 cm from the beam waist. An offset error is introduced at y = 40 cm of +0.7cm (> 2.5 λ) in x.

! The retrieved wavefunction (at y = 20 cm) may be propagated to an observation plane at y = -30 cm.

! The results are compared with the iteration method solution.


zx

y

yo = 20cm y1 = 30cm y2 = 40cm y3 = 50cm y4 = 60cmyobs = -30cm

),(0 zxΦ),(0 zxA ie ),(3 zxΦ),(3 zxA ie ),(4 zxΦ),(4 zxA ie),( zxobsΦ),( zxAobsie

offseterror

Example 2: Offset Gaussian

PPPS01 06/22/01 28


! Irradiance moments yield the following phase solution:

! Recall the previous solution with no offset:

M10

(cm

)

y (cm) 600-0.5

1

0

0)1(10 =C

14.0)0(10 =C

M10 Moments

y (cm) 600

M20

(cm

2 )

0

3

1

M20 Moments

019.0)1(20 =C

2.1)0(20 =C

00047.0)2(20 =C

OffsetNo offset

( )zx,0Φ ( )1026.0 −− cm ( )2093.0 −cm ( )2091.0 −cm= +x +2x 2z

( )zx,0Φ ( )2091.0 −cm ( )2091.0 −cm+2x 2z=


PPPS01 06/22/01 29


! Intensity is reconstructed at observation plane, y = -30 cm.

! Irradiance Moment Tech. intensity is only slightly shifted and distorted.

! Distortion and shifting is more noticeable for the Iteration Method intensity.

! May recover exact solution by using irradiance moments to eliminate the data planes containing error from the data set.

Inte

nsity

(dB

)

x (cm) 2-2-3

0

Intensity Comparisonat y = -30 cm

Moment Tech.Iteration Meth.

Gaussian


PPPS01 06/22/01 30


! A series of 8 intensity data planes are measured from cold test of CPI’s “Toto” 1 MW, 110 GHz gyrotron.

! Retrieval planes are at y = -10, -5, 0, 5, 10, 20, and 40 cm from the window. The last plane, y = 60 cm, is the observation plane.

! We assume a fourth order phasefront (N = 4) in xz-plane for the irradiance moment solution.

! The retrieved wavefunction (at y = -10 cm) may be propagated to the observation plane at y = 60 cm and compared with measured data.

),(7 zxΦ),(7 zxA ie

zx

y

-5cm yobs = 60cmyplane = -10cm

),(0 zxΦ),(0 zxA ie ),( zxobsΦ),( zxAobsie

0cm 5cm 10cm 20cm 40cm

Results from Gyrotron Data

PPPS01 06/22/01 31


32 34 36 38 40 42 44

-6

-4

-2

0

2

4

6

Measured intensity dataat y = 60 cm

Momentmethodintensity

Iterationmethodintensity

-6

6

x (c

m)

32 z (cm) 44

0-3

-21

32 z (cm) 44-6

6

x (c

m)

0

32 z (cm) 44-6

6

x (c

m)

0

-3

-21

-3

-21


PPPS01 06/22/01 32


! A normalized amplitude error, E, is calculated between the measured and computed amplitudes at the observation plane:

! Errors are compared with iteration method:

! The iteration method has slightly less error for this case.

! Is it possible to compensate for measurement error to reduce E?

∫∫∫∫∫∫

−=22

2

1dxdzdxdz

dxdzE

),()( zxA my

),()( zxA my

),()( zxA cy

),()( zxA cy

Moment Tech.Iteration Meth.

Measured

Intensity Comparisonat y = 60 cm

Inte

nsity

(dB

)

x (cm) 7-7-30

0

Phase Retrieval Scheme

Error, E, at y = 60 cm

7-plane Moment 0.019 7-plane Iteration 0.015


PPPS01 06/22/01 33


! Deviation from straight line and quadratic indicate some measurement error has occurred.

! As a result of this error, the phase solution is affected. The beam is predicted to be narrower at y = 60 cm than it actually is.

! There are too few planes to properly correct for measurement error in this case.

M10

(cm

)

y (cm) 40-10-0.2

0.15

0 0045.0)1(10 =C

15.0)0(10 −=C

M10 Moments

y (cm) 40-10

M20

(cm

2 )

0

3

1

M20 Moments

0047.0)1(20 −=C

15.1)0(20 =C

00062.0)2(20 =C


PPPS01 06/22/01 34


! The Irradiance Moment Theory is derived from the Fresnel equation. The method first estimates the moment coefficients by fitting each moment (which are calculated from the intensity measurements) to a polynomial.

! The method assumes a 2D polynomial phasefront. A set of linear equations are formed from the equations describing the slopes of the moments.

! The Irradiance Moment Method is confirmed using an ideal Gaussian beam.

! The effects of measurement error were examined using a Gaussian beam with an offset. It is possible to correct for measurement error by examining the moments.

! The method is tested using cold test gyrotron intensity data. The iteration method is used as a benchmark.

! Results show that the irradiance moment method has a comparable error to the iteration method.

Summary

waves and beams division - · waves and beams division motivation! we wish to be able to...

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