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Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188

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A new two-step tuning procedure for a photocathode gun

Shankar Lal�, K.K. Pant, S. Krishnagopal

Beam Physics and FEL Laboratory, Raja Ramanna Center for Advanced Technology, Indore 452013, India

Received 4 February 2008; received in revised form 13 March 2008; accepted 2 April 2008

Available online 11 April 2008

Abstract

An important aspect of the development of multi-cell RF accelerating structures is tuning the resonant frequency f of the operating

mode, field balance eb, and waveguide to cavity coupling coefficient b to the desired values. Earlier theoretical analyses have not been

able to predict all three parameters simultaneously for a coupled-cavity system. We have developed a generalized circuit analysis to

predict f, eb, and b of a coupled structure, based on the RF properties of the individual, uncoupled, cells. This has been used to develop a

simplified two-step tuning procedure to tune a BNL/SLAC/UCLA type 1.6 cell S-band photocathode gun by varying RF properties of

individual half and full cells, which are easily measurable. This procedure has been validated by tuning two true-to-scale prototypes made

of aluminum and ETP copper to the desired values of the RF parameters.

r 2008 Elsevier B.V. All rights reserved.

PACS: 29.25.Bx; 41.75.Fr

Keywords: Photocathode gun; LCR circuit; Tuning; Field balance; RF coupling coefficient

1. Introduction

The development of multi-cell RF accelerating structuresfor different applications involves tuning their RF proper-ties, particularly the resonant frequency f, the waveguide-to-cavity coupling coefficient b, and the field balance eb(defined as the ratio of the on-axis field in the cell intowhich RF power is coupled to that in other cells), to pre-determined values. For a fixed RF source power and shuntimpedance of the structure, the accelerating gradientduring operation depends on the power coupled into thestructure, which is determined by b [1,2]. Standing wavestructures are typically operated in the p mode to obtainhigh shunt impedance, and with equal amplitudes of theon-axis electric field in each cell (eb=1).

High-power microwaves are transported to the accel-erating structure through waveguides and are coupled intothe cavity through a small coupling slot located on its outerwall. Under the approximation that the aperture is small

e front matter r 2008 Elsevier B.V. All rights reserved.

ma.2008.04.005

ing author. Tel.: +91731 2488065; fax: +91 731 2488000.

esses: shankar@cat.ernet.in,

at@rediffmail.com (S. Lal).

compared to the wavelength, the coupling can be modeledas a combination of radiating electric and magnetic dipoleswith moments proportional to the electric and magneticfields of the incident microwaves [3]. Many three-dimen-sional simulation codes have the capability to predict thedimensions of the cells and the coupling slot required toachieve the desired RF properties, including b [4–6].However, since all these codes consider ideal structures,and do not take into account machining errors andimperfections, there is significant deviation between simu-lated and experimentally measured values of the RFproperties. Tuning is usually achieved by employing aniterative cut-and-measure technique, where machining cutsare taken on the dimensions of the cell and the couplingslot. After each machining cut, the RF properties aremeasured using a vector network analyzer (VNA).Depending on the convergence of the particular RFparameter toward the desired value, the next cut is takeneither on the cell dimension, or on the coupling slot, or onboth, and this procedure is continued till the desired valuesare achieved for all RF parameters. The strong inter-dependence of f, b, and eb makes it difficult to tune thestructures by this method, which is time consuming and

ARTICLE IN PRESSS. Lal et al. / Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188 181

tedious. An understanding of this inter-dependence cansimplify the tuning procedure by enabling one to predictthe values to which the RF parameters of independent cellsshould be tuned, in order to achieve the desired RFparameters for the coupled structure.

Schriber [7] has analyzed a coupled-cavity structureoperating in the p mode, including the effect of errors incell frequencies and inter-cell coupling constants, todetermine f and eb for the coupled structure. Palmer [8]has studied the dependence of coupled mode frequencies ofa 1.6 cell photocathode gun on the frequency of theindependent full cell, and the dependence of field balanceon the separation between the coupled modes of the gun,using equivalent LC circuits to represent the gun. Theseanalyses help in tuning the structures to the desiredoperational mode frequency with a desired field balance.However, both analyses do not address the dependence ofb on these parameters, making it difficult to simultaneouslytune the structures for the desired b. Gao [9] has given ascaling law that predicts the dimensions of the RF couplingslot needed to obtain the desired b for a single cavitycoupled to a waveguide. However, this analysis fails whenemployed for a coupled-cavity system, where power is fedinto one cell through a waveguide, while fields are set up inall cells on account of the inter-cell coupling.

We have developed an analysis of a coupled-cavitysystem using an equivalent LCR circuit that includes thecontribution of the waveguide-to-cavity coupling coeffi-cient b. This analysis has been successfully used to establisha two-step tuning procedure for a 1.6 cell photocathodegun, where power is fed by a waveguide into the full cellthrough a coupling slot on its outer wall, while fields are setup in the full and half cells. The analysis is able to predictthe dependence of the coupled mode frequencies, eb, and bon the independent-cell frequencies and on the waveguideto independent-cell coupling coefficient. Employing thisprocedure, two prototypes of the gun have been success-fully tuned to the desired p-mode frequency with eb and bboth close to unity.

In the next section we perform an equivalent circuitanalysis of a coupled-cavity system and obtain the generalset of equations for a system of N coupled cells. For a 1.6cell photocathode gun, this reduces to a simplified set ofequations, which are solved to obtain expressions for thecoupled mode frequencies, eb, and b. In Section 3 weexplain the two-step tuning procedure that was developedbased on this analysis, and present results of our

Fig. 1. LCR equivalent circu

experiments to tune two prototypes. We conclude inSection 4 with a discussion of these results.

2. Equivalent circuit model for coupled cells

A single RF cavity can be represented by an LCRresonant circuit [10], where L and C are the equivalentinductance and capacitance of the cavity, respectively, andR is its shunt impedance. The amplitude of the current inthe equivalent circuit gives the amplitude of the on-axiselectric field in the cavity [7]. The values of L, C, and R canbe derived from experimentally measurable parameters f,Q, and R/Q, using the expressions f ¼ o=2p ¼ 1=2p

ffiffiffiffiffiffiffiLCp

and Q ¼ oL/R. The coupling of a cavity to a generator(power source), or to another cavity, is represented by atransformer with the coupling strength determined by itsturns ratio or mutual inductance.Using this analogy, the LCR equivalent circuit repre-

sentation of a system of N coupled cells, including thecoupling of the RF source to the first cell through awaveguide, is shown in Fig. 1, where the first circuit on theleft, labeled ‘G’, refers to the generator (in this case thewaveguide and the RF source), and circuits 1–N refer tothe N coupled cells. Quantities with subscript ‘g’ refer tothe generator circuit while those with subscripts n ( ¼ 1�N)refer to the nth cavity, and Mij refers to the mutualinductance between the ith and jth circuits. Note thatSchriber’s analysis takes into account only circuitsn ¼ 1–N, while Gao’s analysis takes into account onlycircuits G and n ¼ 1.Applying Kirchoff’s law to the nth cell, we get

In�1ZMn�1;n þ InZn þ Inþ1Z

Mn;nþ1 ¼ 0 (1)

Here, In is the current in the nth circuit,

Zn ¼ Rn þ j oLn �1

oCn

� �

¼ Rn 1þ jQn

oon

ðo2 � o2nÞ

� �

ZMi;j ¼ joMij ¼ jokij

ffiffiffiffiffiffiffiffiffiLiLj

p¼ jokij

ffiffiffiffiffiffiffiffiffiffiQiRi

oi

s ffiffiffiffiffiffiffiffiffiffiQjRj

oj

s; i ¼ n; j ¼ i � 1

Qn and Rn are the quality factor and shunt impedance,respectively, on ¼ 2pfn where fn is the resonant frequency

it of N coupled cavities.

ARTICLE IN PRESSS. Lal et al. / Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188182

of the independent nth cell, and kij ¼Mij=ffiffiffiffiffiffiffiffiffiLiLj

pis the

inter-cell coupling coefficient between the ith and jth cells.Note that o is the frequency of the coupled mode and isdifferent from on.

Similarly, by applying Kirchoff’s Law to the other cells,we get a set of N equations:

IgZg þ I1ZMg;1 ¼ Vg

IgZMg;1 þ I1Z1 þ I2ZM

g;2 ¼ 0

..

.

In�1ZMn�1;n þ InZn þ Inþ1Z

Mn;nþ1 ¼ 0

..

.

IN�2ZMN�2;N�1 þ IN�1ZN�1 þ INZM

N�1;N ¼ 0

IN�1ZMN�1;N þ INZN ¼ 0 (2)

These equations can be used to study an acceleratingstructure with any number of cells. As a specific case, weconsider the 1.6 cell, BNL/SLAC/UCLA type S-bandphotocathode gun [8,11–14], a cross-section of which isshown in Fig. 2. For this gun, N ¼ 2, and we obtain a set ofthree equations which can be solved analytically to studythe dependence of operating mode frequency, eb and b onthe independent-cell properties:

IhZh þ I fZMf ;h ¼ 0 (3a)

IgZMg;f þ I fZf þ IhZM

f ;h ¼ 0 (3b)

IgZg þ I fZMg;f ¼ Vg (3c)

where subscripts f and h refer to the full and half cells,respectively. Solving Eqs. (3a) and (3b), assuming each cellis lossless with very high Q [8], and neglecting thecontribution of the generator circuit to determine thecoupled mode frequencies, we get

ðo2 � o2f Þðo

2 � o2hÞ ¼ o4k2

fh (4)

Fig. 2. Cross-section of the 1.6 cell photocathode gun.

The positive roots of Eq. (4) give the coupled modefrequencies o0 and op as

o0 ¼ðo2

f þ o2hÞ

2ð1� k2fhÞ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðo2

f þ o2hÞ

2� 4o2

fo2hð1� k2

fhÞ

q2ð1� k2

fhÞ

24

351=2

(5a)

op ¼ðo2

f þ o2hÞ

2ð1� k2fhÞþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðo2

f þ o2hÞ

2� 4o2

fo2hð1� k2

fhÞ

q2ð1� k2

fhÞ

24

351=2

(5b)

The field balance, eb, is given by

eb ¼jI f j

jIhj¼ �

Zh

ZMf ;h

����������

¼Rh½1þ ðQ

2h=ðo

2o2hÞÞðo

2 � o2hÞ

2�1=2

okfh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQhRh=oh

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQfRf=of

p (6)

The dependence of the mode frequencies and fieldbalance on the independent-cell parameters is evident fromEqs. (5a, b) and (6). These equations can be used to predictthe frequencies to which the independent cells should betuned, in order to simultaneously obtain the desiredcoupled-mode frequency fp, as well as a field balance ebclose to unity.To study the dependence of b on eb, we solve the set of

Eqs. (3a–c) simultaneously to get

IgZg þ IgZeff ¼ Vg (7)

where Zeff=�(Zg, fM )2/[Zf�(Zf, h

M )2/Zh].Eq. (7) represents Kirchoff’s Law for the circuit shown in

Fig. 3, which is the equivalent circuit of a photocathodegun coupled to a generator with the impedance of the guntransferred toward the generator side. Here Zg is thecharacteristic impedance of the waveguide and Zeff is theeffective impedance of the half and full cells coupledtogether, as seen by the generator.The waveguide-to-gun coupling coefficient b is defined as

b ¼Pext

Pc¼

ReðZeff Þ

Zg(8)

where Pc is the power dissipated in the gun and Pext is thepower dissipated in the external load (waveguide) [9,15].

Fig. 3. Equivalent circuit model of a photocathode gun for calculation of b.

ARTICLE IN PRESSS. Lal et al. / Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188 183

Using Eq. (7), we get

b ¼ Re ðZMg;f Þ

2

,Zf �

ðZMf ;hÞ

2

Zh

!Zg

" #(9)

The second term in the denominator of Eq. (9) shows thedependence of b on eb. Using Eq. (6) and the expression forZg, fM in Eq. (9), we get

b ¼o2M2

gf

Re½Zf þ Z̄h=e2b�Zg

(10)

where Z̄h is the complex conjugate of Zh. Eq. (10) relates bto the experimentally measurable quantities o, Zf, eb, Zh,and Zg, for the desired mode of operation. The value ofMgf, which is not directly measurable, can be determinedusing Eq. (10) for the independent full cell by detuning thehalf cell, resulting in zero coupling between the half andfull cells. In this case, Zf ¼ Rf at o ¼ of, and Zf, h

M¼ 0.

Eq. (10) now reduces to bf ¼ of2Mgf

2 /RfZg, which is thewaveguide to independent full-cell coupling coefficient (bf).Using this in Eq. (10) we get

b ¼ðo=of Þ

2bfRf

Re½Zf þ Z̄h=e2b�(11)

Substituting the expressions for op, 0 from Eqs. (5a, b) intoEq. (11) gives the dependence of b on eb for the desiredmode.

Eqs. (5b), (6), and (11) give the dependence of thecoupled response of the gun, i.e. the dependence offrequency, eb, and b for the p mode, on parameters ofthe independent full and half cells, which are easilymeasurable. While the resonant frequency of the indepen-dent half and full cells can be fine tuned by taking smallmachining cuts on the cell IDs, this causes a very smallchange in their Q and R, as is evident from standardanalytical expressions for the dependence of Q and R onthe radius of pill-box cavities. The coupled-mode proper-ties can therefore essentially be fine tuned by tuningresonant frequencies of the independent cells.

It must be noted here that the Q and R of the half cell arenot the same as that of the full cell because of theirdifferent lengths and port openings. Ignoring the portopenings and considering ideal pill-box type structures, Q

and R are given by [16]

Q ¼2:405m0c

2rs½1þ ðr=LÞ�(12)

R ¼Z2

0

prsJ21ð2:405Þ

L

r

T2

½1þ ðr=LÞ�(13)

Here T is the transit time factor, r ¼ 2.405c/2pf the radiusand L the length of the pill box. For a 1.6 cell gun, Lf ¼ l/2and Lh ¼ 0.6(l/2), and Eqs. (12) and (13) give Qh/Qf�0.8and Rh/Rf�0.9. Therefore, to a first approximation, we canassume that both the half and full cells have the same valueof Q and R. Then, Eqs. (5a, b) and (6) show that fp=2856MHz with eb unity when ff=fh=f00=2854.54MHz.

In this case, Eqs. (5b), (6), and (11) reduce to

f p ¼f 00ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� kfh

p ; eb ¼ 1; b ¼bf2

(14)

Eq. (14) gives a simple way of obtaining the right fp, eb, andb for the coupled structure, and is therefore very usefulwhile tuning the gun. Of course, Eq. (14) assumes that Q

and R are equal for the two cells—an approximation that isgood only to within 10–20%. A more accurate method is toactually measure the values of Q and R for the independenthalf and full cells and to use these values in Eq. (5b), (6),and (11) to predict the required independent-cell frequen-cies and bf to obtain fp=2856MHz with eb and b bothequal to 1. In the discussion of the experimental resultsbelow, we compare those results with both of the analyticalmodels.

3. Tuning and RF measurements on prototypes

3.1. Tuning procedure

The analysis of the previous section shows that it ispossible to relate the parameters of a complete gun to theparameters of the independent cells. This could be veryuseful in tuning the photocathode gun. Tuning the guninvolves fixing three parameters: (i) the p-mode frequencyfp to 2856MHz; (ii) the field balance eb to unity; and(iii) the coupling factor b also to unity (though occasionallya higher value may be preferred in order to reduce the filltime). It is difficult and tedious to tune the gun once thecells have been brazed together. The earlier analysis showsthat once the value of the coupling kfh is determinedexperimentally, Eqs. (5a, b) and (6) can be used to relatethe desired RF parameters of the complete gun, to those ofthe individual cells.The first task is therefore to determine the values of the

independent-cell parameters required to tune the gun to thedesired values of fp, b, and eb. This can be done as follows:

(1)

Measure the resonant frequency f and quality factor Q

of the independent half and full cells, and thewaveguide to independent full-cell coupling coeffi-cient bf.

(2)

Assemble the half and full cells together and measurethe p-mode frequency and field balance. Using thesevalues in Eq. (6), the value of kfh can be determined.Since no machining cut is taken on the inter-cellcoupling iris during the tuning process, kfh is un-changed during the tuning process.

(3)

Assuming a constant fh, Eqs. (5b) and (6) are used toscan the variation of resonant frequency and eb,respectively for the p mode with ff. The value of eb isnoted at the desired value of fp. A simple program hasbeen written to do this.

(4)

If eb at the desired fp differs from the target value, thevalue of fh is changed and step 3 is repeated until thedesired value of eb is obtained.

ARTICLE IN PRESSS. Lal et al. / Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188184

(5)

Knowing the required values of ff and fh for the desiredfp and eb, Eq. (11) is used to determine the value of b/bf,using which the required values of bf can be determinedfor any target value of b.

Fig. 4. Experimental setup for measurement of RF parameters of the

independent full cell. A plunger is inserted through the laser port to detune

the half cell.

Table 1

Quality factor and shunt impedance of independent full and half cells for

AGUN and ETPGUN

Parameters Qh Qf Rh (MO) Rf (MO)

AGUN 4618 4186 0.76 1.06

ETPGUN 9500 7544 1.79 2.17

With all required independent-cell parameters predicted,a two-step procedure can now be followed for actual tuningof the gun:

Step 1: Tune the half-cell frequency to the target value.Step 2: Tune the full-cell frequency to the target value

while simultaneously obtaining the required bf.When theindependent cells are now coupled, the analysis of theearlier section predicts that the coupled gun will have thedesired RF properties: fp ¼ 2856MHz, eb ¼ 1, and bf ¼ 1.

We now turn to actual measurements to implement thistuning procedure. In these measurements the RF para-meters of the gun prototypes were measured using a VNA(Agilent, 8753ES) in S11 mode. The value of b wasdetermined using the Smith Chart technique and theprofile of the on-axis accelerating field in the two cells,which gives the field balance, was determined by the beadpull technique [1,15,17] with a 5mm diameter, 4mm longcylindrical Teflon bead.

3.2. Measurement on the independent cells

To confirm the predictions of our analysis, we built twotrue-to-scale prototypes of a photocathode gun, one madeof aluminum (AGUN) and the other of ETP copper(ETPGUN). Both prototypes were fabricated with innerdiameters (ID) of the half and full cells slightly undersized,as compared to dimensions predicted by simulations[18,19], for flexibility in tuning individual cell frequencies.The length of the waveguide to cavity coupling slot wasalso initially smaller than the size required for criticalcoupling. These prototypes were fabricated with relaxedgeometric and dimensional tolerances to reduce machiningcosts.

Since the gun has to be tuned prior to brazing togetherthe half and full cells, a fixture was designed to hold thegun assembly, including the helicoflex seal that is proposedto be used for vacuum sealing at that cathode plate–halfcell joint of the gun. Fig. 4 shows a picture of theexperimental setup used to measure independent-cellparameters. As is well known, the two cells of aphotocathode RF gun exhibit independent behavior whentheir independent resonant frequencies are wide apart suchthat their ‘Q’ curves do not overlap. This has beenexploited to measure the independent-cell parameters of a1.6 cell photocathode gun. A plunger was inserted into thehalf cell through one of the laser ports to move itsindependent-cell frequency away from that of the full cell,and the independent full-cell RF parameters Qf and Rf

were measured in this condition. Similarly, RF parametersQh and Rh of the independent half cell were measured byinserting a plunger through one of the tuner ports todetune the full cell.

The measured values of Q and R of the independent cellsfor both prototypes are shown in Table 1. For thesemeasured values Qh/Qf ¼ 1.1 for AGUN and 1.25 forETPGUN, and Rh/Rf ¼ 0.71 for AGUN and 0.82 forETPGUN. Thus, for both guns, the measured Qs and Rs ofthe two cells vary by 10–30%, which compares well withthe analytical values of 10–20% calculated earlier usingEqs. (12) and (13). The machined full cell has five portswhile the machined half cell has two ports. The largernumber of ports in the full cell causes a greater reduction inits Q value as compared to that of the half cell. Hence, themeasured Qh/Qf is greater than unity while the valueobtained from analytical calculations is less than unity.

3.3. Simulated tuning of AGUN

For AGUN, the analysis of the previous section predictsthat for a p-mode frequency of 2856MHz with eb and bequal to unity, the required values of the independent-cellparameters are: ff=2854.8MHz, fh=2854.5MHz, andbf=1.69.Before taking machining cuts on the independent cells to

change their frequencies, the tuning procedure wassimulated by employing plungers to modify the indepen-dent-cell frequencies. As is well known, the full and halfcells are uncoupled when their resonant frequencies are

ARTICLE IN PRESS

Table 2

A comparison of experimental results for AGUN, with predictions of our LCR circuit analysis

Parameters Predicted by LCR circuit analysis with

Qh ¼ Qf and Rh ¼ Rf

Measured Predicted by LCR circuit analysis,

using measured Qh, Qf & Rh, Rf

Measured

fp (MHz) 2856 2855.79 2856 2856.35

eb 1.00 0.8 1.00 1.02

b/bf 0.5 0.28 0.59 0.60

ff (MHz) 2854.54 2854.54 2854.8 2854.8

fh (MHz) 2854.54 2854.54 2854.5 2854.5

fp�f0 (MHz) 2.94 2.7 2.96 3.1

Fig. 5. Variation of f0 and fp with independent full-cell frequency ff for a

constant half-cell frequency fh for AGUN.

Fig. 6. Variation of eb with ff�fh for AGUN.

S. Lal et al. / Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188 185

wide apart. Since power is fed only into the full cell, thefield in the full cell is high while the field in the half cell isalmost zero, resulting in a very high value of eb. The valueof b at this point is just bf. As the independent-cellfrequencies come closer, the two cells start showingcoupled behavior and the measured values of eb and b atthis stage are those of the coupled mode.

To simulate step 1 of the two-step tuning procedure, aplunger (plunger #1) was inserted into the full cell throughone of the tuning ports to shift its frequency far from thatof the independent half cell. Using another plunger(plunger #2) inserted into the half cell through a laserport, the half-cell frequency was adjusted to the predictedvalue of 2854.5MHz.

For simulating step 2, the half cell was now detunedusing another plunger (plunger #3) inserted through theother laser port without disturbing plunger #2. The full-cellfrequency was now tuned to the required value of2854.8MHz by inserting another plunger (plunger #4)through another tuning port and adjusting the position ofplungers #1 and #4 in the full cell.

Since no machining cut was taken on the RF couplingslot, bf could not simultaneously be modified from its smallinitial value of 0.04 to the targeted valued of 1.69.However, from Eq. (11) we can see that b/bf is independentof the RF slot. Therefore, it is adequate to compare themeasured values of b/bf with the prediction in order tovalidate our theory.

At this stage, when plunger #3 was removed from thehalf cell, the desired frequency for the pmode was obtainedwith eb ¼ 1.02 and b/bf ¼ 0.60, which agrees well withpredictions. The small deviations may be due to theperturbations in Q and R caused by the plungers. Acomparison of results obtained experimentally with thosepredicted by our circuit analysis is given in Table 2.Columns 2 and 4 show the analytically calculated values offf, fh, and b/bf required to obtain fp ¼ 2856MHz, eb ¼ 1,and b ¼ 1 for AGUN for the two cases: (1) assuming equalvalues of Q and R for the two cells, and (2) using measuredvalues of Q and R of the half and full cells, respectively.Columns 3 and 5 show the measured values on AGUN,where the plungers were adjusted to tune ff and fh to thevalues predicted by the LCR analysis, and the cells werethen coupled together by removing plunger #3 and thevalues of fp, eb, and b/bf measured. While there is good

agreement between the measured and predicted values of fpand eb for both cases, the agreement in b/bf is good only inCase 2. Eq. (11) shows that b/bf varies as 1=ðZf þ Zh=e2bÞ,and since eb ¼ 0.8 in Case 1, it leads to a rather largedeviation in the value of b/bf.This simulation was further extended to study the

evolution of frequency, eb and b for the coupled modesas a function of change in the independent-cell frequencies.The variation in the ‘0’ and ‘p’ mode frequencies with theresonant frequency of the independent full cell, for a fixedresonant frequency of the independent half cell, agrees wellwith results predicted by our circuit analysis using the

ARTICLE IN PRESSS. Lal et al. / Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188186

measured values of Q and R for the independent full andhalf cells, as shown in Fig. 5. The variation of eb with theseparation between independent full and half-cell frequen-cies is shown in Fig. 6, where a value of eb ¼ 1 is obtainedfor ff�fh ¼ 300 kHz, which is also in very good agreementwith predictions from the circuit analysis.

Similar results have been reported earlier by Palmer [8].However, since he has not considered the coupling of thefull cell to the waveguide, he does not study the dependenceof b on eb for the p mode, which is an important aspect inthe tuning of a photocathode gun. We have predicted thisdependence using our analysis and obtained good agree-ment with results obtained during simulated tuning ofAGUN, as shown in Fig. 7.

3.4. Two-step tuning

With the confidence gained from simulated tuning, weproceeded to the actual tuning of AGUN. Employing thetwo-step procedure, AGUN was tuned by taking machin-

Fig. 7. Variation of bp/bf with eb for AGUN.

Table 3

A comparison of experimental measurements with predictions of our LCR cir

Parameter AGUN

Predicted by LCR

analysis

Measured Verified by

analysis

fp (MHz) 2856 2856.28 2856.48

eb 1.00 1.04 1.06

bp 1.00 1.06 1.02

bf 1.69 1.67 1.67

ff (MHz) 2854.8 2855.22 2855.22

fh (MHz) 2854.5 2854.75 2854.75

fp�f0 (MHz) 2.96 2.92 2.98

Coupling-slot

length (mm)

– 25.2 –

R (MO) – 1.13 –

Q0 – 4465 –

ing cuts on the full and half cells to obtain the requiredvalues for independent-cell parameters: ff ¼ 2855.22MHz,fh ¼ 2854.76MHz, and bf ¼ 1.67. When the two cells werecoupled together, we obtained fp ¼ 2856.28MHz withb ¼ 1.06 and eb ¼ 1.04.The two-step tuning procedure was subsequently em-

ployed successfully on ETPGUN to study the repeatabilityof the tuning procedure. Since the Q and R for theindependent cells of ETPGUN are different from those ofAGUN (c.f. Table 1), while the inter-cell couplingcoefficient kfh is the same, our analysis predicts that for ap-mode frequency of 2856MHz with a field balance and bequal to unity, the required values of independent-cellparameters are: ff ¼ 2854.6MHz, fh ¼ 2854.5MHz, andbf ¼ 1.72.A comparison of these experimental results with the

predictions of our circuit analysis is given in Table 3. Forboth prototypes, AGUN and ETPGUN, the agreementbetween the circuit analysis and the measured values of thevarious RF parameters is excellent. The ‘Predicted by LCRanalysis’ columns show the values of ff, fh, and bf that theindividual cells need to be tuned to, in order to obtainfp ¼ 2856MHz, eb ¼ 1, and bp ¼ 1. The ‘Measured’columns are the experimental results: the cells were firsttuned (by taking machining cuts on the IDs of the full celland half cell, and filing the coupling slot) to the predictedvalues of ff, fh, and bf, and then coupled, and the resultingfp, eb, and bp were noted—these differ from the targetedvalues by only a few percent. From Table 3 it can be seenthat the actual values of ff, fh, and bf we achieved differslightly from the values predicted by our LCR analysis.This could contribute to the observed deviation of thecoupled-cell parameters from the targeted values. Toconfirm this we used the measured values of theindependent-cell parameters in our LCR circuit modelto predict the coupled-cell parameters (fp, bp, and eb),shown in the columns marked ‘Verified by LCR analysis’.It can be seen that these are now closer to the measuredvalues.

cuit analysis

ETPGUN

LCR Predicted by LCR

analysis

Measured Verified by LCR

analysis

2856 2856.29 2856.34

1.00 1.06 1.09

1.00 1.02 1.00

1.72 1.69 1.69

2854.6 2854.97 2854.97

2854.5 2854.76 2854.76

2.94 2.93 2.95

– 23.5 –

– 2.30 –

– 7777 –

ARTICLE IN PRESS

Fig. 8. Frequency spectrum of the tuned ETPGUN.

Fig. 9. On-axis accelerating field profile of the p mode in ETPGUN (a)

before tuning and (b) after tuning. Here Z is the position of the bead from

the outer wall of the cathode plate and EZ is the accelerating field, in

arbitrary units.

Fig. 10. Smith Chart for the tuned ETPGUN.

Fig. 11. Three-dimensional plot of ff, ff�fh, and fp.

S. Lal et al. / Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188 187

Fig. 8 shows the spectrum of modes supported byETPGUN with the p mode at 2856.294MHz and a modeseparation of 2.93MHz. Fig. 9 shows the on-axis accel-erating field profile in ETPGUN before and after tuning.Before tuning, the field in the full cell is much higher thanthat in the half cell with eb�3. After tuning, the fields in thetwo cells are almost equal, and eb�1. Fig. 10 shows theSmith Chart for the tuned ETPGUN structure with bpslightly greater than unity.

It must be mentioned here that since the aim of theexercise was to establish a two-step tuning procedure for aphotocathode RF gun, the targeted frequency for the pmode was chosen as 2856MHz even though the two cellswere not actually brazed together. Brazing of the cells hasbeen reported to cause a change in the RF parameters ofthe p mode. However, as discussed below in Section 3.5,the procedure discussed in this paper can be used to targetany desired frequency and other RF parameters for the pmode of a photocathode RF gun, to compensate for effectsof brazing.

As is clear from Table 3, a small difference is observed inthe size of the coupling slot required for attaining b�1 inthe aluminum and ETP copper prototypes. This could bedue to the difference in the conductivities of aluminum andcopper, resulting in different shunt impedances for the twoprototypes. There is not much difference, however, in thedimensions of the half and full cells of the aluminum andcopper prototypes for similar independent-cell frequencies.

3.5. A graphical method of tuning

In principle, for any gun for which Qf, h and Rf, h areknown, Fig. 6 can be used to predict the required value offf�fh for any desired value of eb, and Fig. 7 can be used topredict the value of b/bf for the chosen value of eb. Withthis value of b/bf, the required value of bf for the desiredvalue of b can be predicted. In addition to these twoparameters, tuning of a gun involves tuning fp to thedesired value. While the value of eb depends only on thedifference (ff�fh), fp depends on the individual values of ff

ARTICLE IN PRESSS. Lal et al. / Nuclear Instruments and Methods in Physics Research A 592 (2008) 180–188188

and fh, which cannot be extracted from Figs. 6 and 7. Thethree-dimensional plot shown in Fig. 11, which has beengenerated by numerically solving Eqs. (5a) and (5b), givesthe required value of ff, and therefore the value of fh, toobtain the desired value of fp for any given value of ff�fh.Thus, Figs. 6, 7, and 11, can together be used to tune thephotocathode gun to any desired value of fp with anydesired value of eb and b. For example, if it is desired totune the gun for fp=2856MHz with eb=1.2 and b=1.5,Fig. 6 shows that the required ff�fh ¼ 0.86MHz, whileFig. 7 gives the required value of bf ¼ 2.25. From Fig. 11,ff ¼ 2854.9MHz, giving fh ¼ 2854.04MHz.

4. Discussion and conclusions

Results predicted by our LCR analysis agree very wellwith those obtained from simulated tuning of AGUN andfrom actual experiments with prototypes. As discussed inSection 3, the starting point for the two-step tuningprocedure is the machining of independent cells withslightly undersized IDs. Since the machining cuts on theIDs during the tuning of prototypes were not taken on aprecision lathe, the independent-cell frequencies deviatedfrom the targeted values, resulting in small deviations of fpand eb from the ideal values of 2856MHz and unity,respectively. With precise machining, the deviations areexpected to be very small and well within the bandwidth oftunability that can be achieved by varying the temperatureof the cooling water.

It is sometimes preferred to slightly overcouple thestructure, say with b ¼ 1.5, in order to reduce the fill timeof the gun. Similarly, it is sometimes desired to operate thegun with field balance different from unity [20]. The two-step tuning procedure we have developed can still be usedin such cases.

Our analysis, and the tuning procedure, can also beextended to multi-cell structures, in a manner that isformally straightforward, using Eq. (2). However, theconsequent complexity of the equations may requirenumerical solutions to get the required independent-cellfrequencies and b of the independent cavity to which RFpower is coupled.

Finally, though we have examined an S-band photo-cathode gun, it is clear that the analysis is general and canbe applied to guns at any frequency.

In summary, the variation of fp, eb, and b of a 1.6 cellphotocathode gun with independent-cell frequencies hasbeen studied using a generalized LCR equivalent circuitanalysis. Results predicted by this analysis agree very wellwith those obtained from a simulated tuning experimentemploying plungers. Based on this, a two-step tuning

procedure has been established and successfully employedto tune two gun prototypes. This two-step proceduremakes it relatively easier to tune a photocathode gun ascompared to the iterative cut-and-measure techniquenormally adopted.

Acknowledgments

We are grateful to S. Chouksey for help in design andfabrication of the prototypes and to V. Kodiarasan fortechnical assistance in tuning the gun. We also thank VinitKumar and Akhil Patel for useful discussions.

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