corrugated waveguide mode content analysis using...
TRANSCRIPT
July, 2014
Plasma Science and Fusion Center Massachusetts Institute of Technology
Cambridge MA 02139 USA This work was supported in part by NIH and NIBIB under Grant EB001965 and Grant EB004866, in part the US ITER Project managed by Battelle / Oak Ridge National Laboratory, and in part the U.S. Department of Energy, Office of Fusion Energy Sciences. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.
PSFC/JA-14-47 Corrugated Waveguide Mode Content Analysis Using
Irradiance Moments
Jawla, S. K., Shapiro, M. A., Idei* , H., Temkin, R. J. *Advanced Fusion Research Center, Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8560, Japan
1
Corrugated Waveguide Mode Content AnalysisUsing Irradiance Moments
Sudheer K. Jawla, Michael A. Shapiro, Hiroshi Idei, and, Richard J. Temkin, Fellow, IEEE
Abstract—We present a novel, relatively simple methodfor determining the mode content of the linearly polarized(LP) modes of a corrugated waveguide using the momentsof the intensity pattern of the field radiated from the end ofthe waveguide. This irradiance moment method is based oncalculating the low order irradiance moments, using mea-sured intensity profiles only, of the radiated field from thewaveguide aperture. Unlike the phase retrieval method, thismethod does not use or determine the phase distributionat the waveguide aperture. The new method was bench-marked numerically by comparison with sample mode mix-tures. The results predict less than ±0.7% error bar in theretrieval of the mode content. The method was also testedusing high resolution experimental data from beams radi-ated from 63.5 mm and 19 mm diameter corrugated waveg-uides at 170 GHz and 250 GHz, respectively. The resultsshowed very good agreement of the mode content retrievedusing the irradiance moment method vs. the phase retrievaltechnique. The irradiance moment method is most suitablefor cases where the modal power is primarily in the funda-mental HE11 mode, with less than 8% of the power in highorder modes.
Index Terms—Corrugated Waveguide, Quasi-OpticalBeam, Waveguide Modes, Irradiance Moments and PhaseRetrieval.
I. Introduction
LOW loss oversized corrugated waveguides are used forhigh power millimeter-wave transmission in electron
cyclotron heating systems of fusion devices [1], [2], [3], [4].The 24 MW ECH system of the International Thermonu-clear Experimental Reactor (ITER) utilizes 63.5 mm diam-eter corrugated waveguides to transmit continuous wave(CW) microwave power over more than 100 meters of dis-tance at a frequency of 170 GHz [5], [6]. The JAEA groupuses a 170 GHz, 1 MW gyrotron to test the ITER relevanttransmission lines [7].
The critical issue in transmitting high power CW mi-crowaves, generated by the gyrotron, is the coupling ofthe Gaussian-like quasi-optical beam into the corrugatedwaveguide. Ideally, one expects the excitation of a purelinearly polarized HE11 (or LP01) mode in the waveguideby the Gaussian-like beam because of its very low ohmiclosses as the mode propagates along the waveguide [8], [9],[10], [11]. However, other higher order modes (HOMs) arealso generated in the corrugated waveguide because thegyrotron output beam is not an ideal Gaussian beam, and,also due to the experimental misalignment in the couplingassembly. Experimental [12], [13], [14] and theoretical
S. Jawla, M. Shapiro and R. Temkin are with the PlasmaScience and Fusion Center, Massachusetts Institute of Tech-nology, Cambridge, MA 02139 USA (e-mail: [email protected];[email protected]; [email protected];).H. Idei is with the Advanced Fusion Research Center, Research In-stitute for Applied Mechanics, Kyushu University, Kasuga 816-8580,Japan (e-mail:[email protected]).
studies [15], [16] have been conducted to optimize the cou-pling of the gyrotron beam into the corrugated waveguideusing a mirror optics unit. This requires an accurate de-termination of mode contents in the corrugated waveguide.At MW power levels, accurate infrared measurements anddata analysis of the field radiated at several locations infree space from the corrugated waveguide aperture wereperformed [12] and the mode contents were retrieved fromthe intensity patterns at the waveguide aperture. This re-quires full complex field information which is retrieved us-ing the phase retrieval technique [17], [18], [19], [20]. Theaccuracy of these techniques for high average power gy-rotrons is limited by the signal to noise ratio and spatialresolution of the infrared measurement technique. In coldtest, the mode contents are evaluated to higher accuracyusing higher resolution data from a scanner [18], [19], [21],[22], [23], where intensity profiles are measured down to -50dB or less. Another approach is to measure the mode con-tents using a monitor installed with the miter bend assem-bly on the transmission line [24], [25]. These experimentsindicate that other than the fundamental HE11 mode, thedominant modes excited in the corrugated waveguide dueto quasi-optical Gaussian beam coupling are primarily thefour next lowest order modes of the waveguide, LP11, LP02,LP03 and LP21 [26]. The LP11 is excited due to a tilt/offsetin the Gaussian beam, and, the LP02 (also called the HE12)is excited due to the phase curvature mismatch of the beamand HE11 mode, and, also due to mode conversion in themiter bends installed in the transmission line system [10],[12]. The HE11 and LP11 mode superposition results in thewaveguide field energy center bouncing in the waveguide[26] and the HE11 and LP02 mode superposition results inthe field intensity distribution oscillating along the prop-agation direction with the period depending on the beatwavelength of the two modes. Recently, an analytical ap-proach has been proposed to correct these modes usingsimple mirror correctors [27].
Mode content retrieval requires full complex field infor-mation which utilizes the phase retrieval technique. In thispaper, we present a simple but effective method to deter-mine the mode contents in a corrugated waveguide usingonly the intensity information, skipping the process of ob-taining phase using the phase retrieval technique. Underthe assumption that the gyrotron Gaussian beam excitesonly the low order modes of the guide, the low order irra-diance moments of the radiated intensity can be used todetermine the mode content. We show how the mode con-tents can be determined from the irradiance moments asa function of propagation distance z from the waveguideaperture. The irradiance moments propagation has been
2
previously studied in [28], [29], [30], [31], [32]. The tech-nique of phase retrieval at the waveguide radiating aper-ture using irradiance moments was proposed in the past[33].
The method proposed in this paper can also be usedfor the mode content determination in overmoded opticalwaveguides [34]. In this paper, we will assume that themode entering the waveguide is linearly polarized. We usethe formulation of the waveguide modes as a set of lin-early polarized (LPmn) waves following ref. [26]. In theLPmn mode notation, the lowest mode of the waveguide,the LP01 mode is the same as the HE11 mode and theLP0n modes are the same as the HE1n modes. In the text,we will use the LP0n mode and the HE1n mode notationinterchangeably.
This paper is organized as follows, Section II describesthe rigorous analytical equations of the superposition of thedominant mode HE11 and the higher order modes based ontheir symmetry. In Section III we reduce these equations tothe modes of interest generated in a corrugated waveguidefrom coupling of a gyrotron quasi-optical beam. Section IVnumerically benchmark the method by using combinationsof the known modal powers of the dominant modes, and,in section V and VI we used the experimentally measureddata from two different corrugated waveguide setup andcompared the results obtained using the proposed methodin this paper to the phase retrieval method [23].
II. Superposition of HE11 and higher orderLPmn Modes
Higher order modes are generated in addition to thefundamental mode HE11 when an overmoded corrugatedwaveguide is powered by a quasi-optical output beam froma gyrotron. In order to determine the power coupled intothese HOMs, we investigate the irradiance moments of thefield radiating in free space after the waveguide aperture asa function of propagation distance. We analytically studythe propagation of these moments based on the superposi-tion of the fundamental mode HE11 and the HOMs. As weknow from previous work [26], [27], one can predict the be-havior of the field intensity propagating in the waveguidebased on the HOM symmetry. The LP11 mode can be pre-dicted very accurately by analyzing the tilt and offset in thebeam, and, LP0n (n≥ 2) can be predicted by analyzing thefield intensity distribution oscillating in the waveguide as afunction of distance. LP1n and LP0n modes contribute in-dependently of each other to the intensity propagation andthus can be studied separately by analytically investigatingtheir superposition with the fundamental mode HE11.
A. Superposition of HE11 and LP1n (n≥ 1) Modes
We assume that the field is linearly polarized, and theelectric field has Ex-component. The electric field at thewaveguide aperture is represented as a superposition ofHE11 and LP1n modes,
E =√P01 exp(iθ01)u01 +
∑n
√P1n exp(iθ1n)u1n (1)
where the HE11 mode field distribution is given by,
u01 =1√πa
J0(ν01a r)
J1 (ν01)(2)
and the LP1n mode field distribution is given by,
u1n =
√2√πa
J1(ν1na r)
J0 (ν1n)cosϕ (3)
where (r,ϕ) are the polar coordinates, ν01 = 2.405 is thefirst zero of the Bessel function J0 , ν1n is the nth zero ofJ1, for example ν11 = 3.832, P01 and P1n are the modalpowers, and θ01 and θ1n are the modal phases.
We assume that the LP1n modes are even modes, andthe superposition results in a shift of the beam energy cen-ter in the x-direction. The superposition with the LP1n-odd modes causes a shift in the y-direction and can beanalyzed similarly. The first order moment 〈x〉, which isthe x-coordinate of the energy center, can be representedas a function of the propagation distance z, of the radiatedmicrowave beam, from the waveguide aperture [33],
〈x(z)〉=
∫∫x |E|2 dxdy
− z
2ik
∫∫ (E∗
∂E
∂x−E∂E
∗
∂x
)dxdy
(4)
where, k is the free space wavenumber and the electric fielddistribution E at z = 0 is integrated over the waveguideaperture of radius a. Using (2),(3) and (4) we get,
〈x(z)〉=2∑n
b1n√P01P1n cosψ1n
+ 2z∑n
d1n√P01P1n sinψ1n
(5)
where ψ1n is the phase difference between the HE11 andLP1n modes given by ψ1n = θ1n− θ01, and
b1n =√
2a
∫ 1
0
ξ2J1 (ν1nξ)J0 (ν01ξ)
J0 (ν1n)J1 (ν01)dξ
= −2√
2ν01ν1na
(ν21n− ν201)2 (6)
d1n =
√2
ka
ν01ν1nν21n− ν201
(7)
B. Superposition of HE11 and LP0n (n≥ 2) Modes
The electric field at the waveguide aperture is rep-resented as a superposition of the HE11 mode and theazimuthally-symmetric LP0n modes,
E =√P01 exp(iθ01)u01 +
∞∑n=2
√P0n exp(iθ0n)u0n (8)
where the LP0n mode field distribution is given by,
u0n =1√πa
J0(ν0na r)
J1 (ν0n)(9)
3
where, for example, ν02 = 5.52. The variance⟨r2⟩
=⟨x2 + y2
⟩can be represented by the following equation [33],
⟨r2(z)
⟩=2π
∫ a
0
r3 |E|2 dr
− 2πz
ik
∫ a
0
r2(E∗
∂E
∂r−E∂E
∗
∂r
)dr
+ 2πz2
k2
∫ a
0
r
∣∣∣∣∂E∂r∣∣∣∣2 dr
(10)
Using (9) and (10) we derive the equation for effectivebeam waist w2 = 2
⟨r2⟩
as
w2(z) =2F11P01 + 2
∞∑n=2
FnnP0n
+ 4
∞∑n=2
B1n
√P01P0n cosψ0n
+ 4z
k
∞∑n=2
D1n
√P01P0n sinψ0n
+ 2z2
k2a2
(ν201P01 +
∞∑n=2
ν20nP0n
)(11)
where the phase difference ψ0n = θ0n− θ01, and
Fnn = 2a2∫ 1
0
ξ3J20 (ν0nξ)
J21 (ν0n)
dξ =ν20n − 2
3ν20na2 (12)
B1n = 2a2∫ 1
0
ξ3J0 (ν01ξ)J0 (ν0nξ)
J1 (ν01)J1 (ν0n)dξ
=8ν01ν0n
(ν20n− ν201)2 a
2 (13)
and,
D1n =
2
∫ 1
0
ξ2ν0nJ0 (ν01ξ)J1 (ν0nξ)− ν01J0 (ν0nξ)J1 (ν01ξ)
J1 (ν01)J1 (ν0n)dξ
=− 4ν01ν0nν20n− ν201
(14)
C. Superposition of HE11 and LP2n (n≥ 1) modes
The electric field at the waveguide aperture is repre-sented as the following superposition
E =√P01 exp(iθ01)u01 +
∑n
√P2n exp(iθ2n)u2n (15)
where the LP2n mode field distribution is
u2n =
√2√πa
J2(ν2na r)
J1 (ν2n)cos 2ϕ (16)
where for example, ν21 = 5.1356. The second order mo-ment
⟨x2− y2
⟩can be represented by the following equa-
tion [33]
(17)
⟨x2(z)− y2(z)
⟩=∫∫
(x2 − y2) |E|2 dxdy
− z
ik
∫∫ [x
(E∗
∂E
∂x− E∂E
∗
∂x
)− y
(E∗
∂E
∂y− E∂E
∗
∂y
)]dxdy
+z2
k2
∫∫ [∣∣∣∣∂E∂x∣∣∣∣2 − ∣∣∣∣∂E∂y
∣∣∣∣2]dxdy
Using (16) and (17) we derive the equation⟨x2(z)− y2(z)
⟩=∑n
B2n
√P01P2n cosψ2n
+ 2z
k
∑n
D2n
√P01P2n sinψ2n
+z2
k2a2
∑n
4√
2ν01ν2nν22n− ν201
√P01P2n cosψ2n
(18)
where the phase difference ψ2n = θ2n− θ01, and
B2n = 2√
2a2∫ 1
0
ξ3J0 (ν01ξ)J2 (ν2nξ)
J1 (ν01)J1 (ν2n)dξ
=− 8√
2ν01ν2n
(ν22n− ν201)3
(4 + ν22n− ν201
)a2
(19)
D2n =
−√
2
∫ 1
0
ξ2ν2nJ0 (ν01ξ)J1 (ν2nξ) + ν01J1 (ν01ξ)J2 (ν2nξ)
J1 (ν01)J1 (ν2n)dξ
=2√
2ν01ν2n
(ν22n− ν201)2
(4 + ν22n− ν201
)(20)
Note that the LP2n modes are of different orientation(even and odd). Here we characterize the HE11 andLP2n even mode mixture by the moment
⟨x2(z)− y2(z)
⟩=⟨
r2cos(2φ)⟩. The mixture of the HE11 and LP2n odd
modes can be characterized by the moment 2〈x(z)y(z)〉=⟨r2sin(2φ)
⟩.
III. Reduced Equations for Dominant Modes
The equations derived in the previous section can beused to determine the mode contents if the powers of higherorder modes are small (total HOM power of about 8% orless) compared to the fundamental mode HE11. Based onthe previous works [11], [12], [13], [14], [23] and [27], it isclear that the dominant modes generated in a corrugatedwaveguide from coupling of a gyrotron quasi-optical beamare low index HOMs; HE11, LP11, LP02, LP03, and, LP21.Therefore, we can simplify the equations derived in theprevious section for these dominant modes only.
4
A. Calculating the LP11 even and odd mode content
From the z-dependence of the first order moment, cal-culated from the propagated intensity profiles at severallocations from the waveguide aperture, we can write
〈x〉 = 〈x〉z=0 + αxz (21)
and following (5), the LP11-even mode power, P e11, can bewritten as
P e11 =
(〈x〉z=0
2b11
)2
+
(αx
2d11
)2
(22)
where b11 = 0.329a and d11 = 1.464/(ka). Substituting αxand 〈x〉z=0 in (22) one can calculate P e11. Similarly, thepower of LP11 odd mode can be calculated from 〈y〉z=0
and αy. As a check of this approach, we note that equa-tion (22) is in agreement with the tilt/offset conservationtheorem [26] for a superposition of HE11 and LP11 modesin a corrugated waveguide.
B. Calculating the LP02 and LP03 mode content
From the propagated intensity profiles one can calculatethe z-dependence of the effective beam waists using thesecond order moments
⟨x2(z)
⟩and
⟨y2(z)
⟩as
w2 = w20 + Z1z +
β21
k2a2z2 (23)
and, from (11) the LP02 mode power, P02, can then be de-termined by neglecting the smaller valued terms as follows
P02 =
(w2
0 − 2F11
4B12
)2
+
(kZ1
4D12
)2
(24)
where F11 = 0.218a2,B12 = 0.174a2 and D12 =−2.151 aredefined by (12), (13), (14). Z1 is the coefficient of the linearterm in (23). Once P02 is obtained, the quadratic term in(11) and (23) can be used to determine the power, P03, inthe higher order mode LP03 from the following equationusing the coefficient β2
1 of the quadratic term in (23)
P03 =β21 − 2ν201 − 2ν202P02
2ν203(25)
C. Calculating the LP21 mode content
From the z-dependence of the second order moments⟨x2⟩
and⟨y2⟩, we can also write
⟨x2(z)− y2(z)
⟩=⟨x2 − y2
⟩z=0
+ Z2z +β22
k2a2z2 (26)
where, Z2 is the coefficient of the linear term. The powerin the LP21 mode, P21, can be determined using (18) asfollows
P21 =
(⟨x2 − y2
⟩z=0
B21
)2
+
(kZ2
2D21
)2
(27)
where B21 = −0.3936a2 and D21 = 2.026 are defined by(19), (20). The quadratic term in (26) is very small if thehigh order mode content is small and can be neglected.
0 1 2 3 4 50
1
2
3
4
5
6
7
LP
11, 0
2, 0
3, 2
1 ret
riev
ed (
%)
LP11
(%)0 1 2 3 4 5
91
92
93
94
95
96
97
98
HE
11 r
etri
eved
(%
)
0 1 2 3 4 591
92
93
94
95
96
97
98939495969798HE
11 (%)
939495969798
HE11
LP02
LP03
LP11
LP21
Fig. 1. Retrieved mode contents are plotted using the known inputmode contents. In this case, the known input values of the HE11
and LP11 mode contents are varied between 93 to 98% and 5 to 0%respectively, while the other mode contents are fixed at small inputvalues of LP02 = 0.5%, LP03 = 0.0% and LP21 = 1.5%. The retrievedmode contents are plotted as solid lines while the input values areplotted as dashed lines, HE11 (red), LP11 (black), LP02 (blue), LP03
(magenta) and LP21 (green).
94 95 96 97 98 990
1
2
3
4
5
6L
P02
, 03 r
etri
eved
(%
)
HE11
(%)94 95 96 97 98 99
94
95
96
97
98
99
100
94 95 96 97 98 9994
95
96
97
98
99
100
HE
11 r
etri
eved
(%
)
012345LP
02 (%)
012345
LP03
HE11
LP02
Fig. 2. Retrieved mode contents are plotted using the known inputmode contents. In this case, the known input values of the HE11
and LP02 mode contents are varied between 94 to 99% and 5 to 0%respectively, while LP03 is kept at 1.0%. The retrieved mode contentsare plotted as solid lines while the input values are plotted as dashedlines, HE11 (red), LP02 (black) and LP03 (blue).
IV. Benchmarking using Known Mode Contents
To numerically validate the method, we generated an ar-tificial field at the waveguide aperture using several differ-ent combinations of the known modal powers of the abovementioned modes and propagate the complex field in freespace to several locations using a full diffraction integralapproach [35]. We then evaluate the irradiance moments ofthe radiated field as a function of the propagation distanceusing the following expression [33]
Mpq ≡ 〈xpyq〉z ≡∫∫
(x)p(y)qIz(x, y)dxdy∫∫Iz(x, y)dxdy
(28)
Here, p and q represent the order of the moment (p+ q)and Iz(x,y) is the intensity profile at a given z position.
5
The equations described in the previous section are thensolved for mode content calculation using these irradiancemoments. From the formulation of equations in section-II it is clear that the first and second order moments areenough to evaluate the power in the fundamental modeand the dominant modes. The calculations done with thereduced set of equations are based on the assumption thatthe power in the HE11 mode is large compared to the powerin the other modes. The results of the retrieved mode con-tents are plotted in Figures (1), (2) and (3) showing threedifferent cases. In first case shown in Figure 1, the knownvalues of HE11 and LP11 mode powers are varied between93 to 98% and 5 to 0% respectively, and, the modes LP02
and LP21 are kept fixed at 0.5% and 1.5% respectively.For the higher HE11 input mode content (∼≥ 95%), theretrieval error bar is less than ∼ 0.5%. A small amount ofLP03 is also retrieved for lower HE11 mode input. It shouldbe noted here that the modes with different symmetry donot affect their retrieval, e.g. the retrieval of LP11 modecontent is not affected by the presence of LP0n modes andvice versa. They are however affected by the presence ofthe next HOMs of same symmetry. It is also important tomention the fact that while the LP11 mode is generatedmainly due to the experimental errors of misalignment bytilt and offset during the coupling to the corrugated waveg-uide, the excitation of LP02 modes is inherent to the beamitself. If a perfect Gaussian beam couples into a corru-gated waveguide, it will excite higher order LP0n modesbecause it does not match perfectly to the HE11 mode.In addition, the gyrotron output beam is never a perfectGaussian beam. Therefore, the LP02 modes will always beexcited in the waveguide. In Figure (2), we have shown thecase for symmetric modes only where HE11 and LP02 arevaried between 94 to 99% and 5 to 0% respectively whileLP03 is kept fixed at 1.0%. In this case, the error bar inthe retrieval of HE11 is slightly higher, ∼ 0.7%, comparedto the previous case. Another case, shown in Figure (3),also predicts an error bar of ∼ 0.7%.
Detailed investigation based on the presented equationsand numerical analysis shows that in the proposed methodof irradiance moments the effect of having different phasesbetween the modes is very small.
V. Experimental Setup and Results
In order to experimentally test the new method, theradiation pattern from the 63.5 mm diameter corrugatedwaveguide, used for the ITER transmission line setup, wasmeasured at the frequency of 170 GHz using a 3-axis scan-ner. The high purity linearly polarized HE11 mode in thecorrugated waveguide was generated using a combinationof waveguide transitions and a tapered mode generator de-signed and built by General Atomics. An Agilent VectorNetwork Analyzer (VNA) was used to measure the inten-sity over a 28 cm × 28 cm cross section of the measurementplane. The receiving waveguide and the VNA extensionhead were placed on an automated 3-axis scanner to mea-sure the field intensity patterns on a plane parallel to thewaveguide aperture at many locations along the waveguide
0 1 2 3 4 50
1
2
3
4
5
6
LP
11, 0
2, 0
3 ret
riev
ed (
%)
LP11
(%)0 1 2 3 4 5
91
92
93
94
95
96
97
HE
11 r
etri
eved
(%
)
0 1 2 3 4 591
92
93
94
95
96
97012345LP
02 (%)
012345
LP02
HE11
LP03
LP11
Fig. 3. Retrieved mode contents for benchmarking are plotted usingthe known input mode contents. In this case, the known input val-ues of the LP11 and LP02 mode contents are both varied between 0to 5%, while the HE11 is fixed 95.0% and LP03 and LP21 are zero.The retrieved mode contents are plotted as solid lines while the in-put values are plotted as dashed lines, HE11 (red), LP11 (black),LP02 (blue) and LP03 (green) are plotted vs. the known input modecontents.
Fig. 4. Experimental setup for measuring the Intensity patternsafter the waveguide aperture.
axis. The experimental setup is shown in Figure (4). Asexplained in [23], the alignment of the measurement setupis an important issue in correctly evaluating the excitedmodes of the corrugated waveguide. The total mechanicalerror induced by the scanner alignment in the measure-ment plane parallel to the waveguide aperture is estimatedto be less than 0.15o in both the x and y directions. Theoffset error of the receiver waveguide with respect to thecenter of the waveguide aperture is estimated to be lessthan 0.5 mm in both the x and y directions. Measure-ments were done at several distances between 10 cm to70 cm from the waveguide aperture. The measured inten-sity patterns are shown in Figure (5). The accuracy ofthe measurements can be seen from the diffraction ringsobserved down to -45 dB around the main lobe. The equa-tions derived in section-III are then used to calculate themode contents of the excited modes in the waveguide us-
6
Y−
ax
is [
cm
]
−10
−5
0
5
10
−40
−30
−20
−10
0
X−axis [cm]
Y−
ax
is [
cm
]
−10 −5 0 5 10
−10
−5
0
5
10
−40
−30
−20
−10
0(d) (e) (f)
(a) (b) (c)
Fig. 5. Measured field intensities after the waveguide aperture at(a) 22.8 cm, (b) 30.8 cm, (c) 39.5 cm, (d) 48.2 cm, (e) 56.8 cm and(f) 65.5 cm.
ing these intensity profiles. Mode content retrieval whichutilizes the phase retrieval method [23] is also employedto calculate the mode contents from the measured inten-sity patterns. Table-I shows the comparison of the twomethods, mode contents retrieved using the irradiance mo-ment method vs. the mode content retrieval method usingphase retrieval technique. A very good agreement can beseen between the two methods and the results are in goodagreement within an error bar of less than ±0.4%.
TABLE I
Comparison of the experimental mode contents of 63.5 mm
corrugated waveguide (170 GHz) calculated using the
irradiance moment method vs. the phase retrieval method.
Calculated from Calculated from
Mode irradiance moments retrieval method
HE11 94.9 95.0
LPo11 2.3 1.9
LPe11 1.9 1.7
LP02 0.3 0.2
LP03 0.3 0.1
LP21 0.2 0.1
Others 0.1 1.0
VI. Results from 19 mm corrugated waveguideat 250 GHz
The proposed method is also applied to the measured in-tensity profiles from a 19 mm corrugated waveguide trans-mission line used at 250 GHz in a Dynamic Nuclear Polar-ization (DNP-NMR) experiment. The measurements wereperformed similarly to the 63.5 mm corrugated waveguideand are presented in [23]. The mode content results pre-sented in [23] are compared to the mode contents calcu-lated by the irradiance moment method. The results aresummarized in Table-II. Very good agreement, at aboutthe ±0.5% level, is seen in this case also between the twomethods.
TABLE II
Comparison of the mode contents of 19 mm corrugated
waveguide (250 GHz) calculated using the irradiance
moment method vs. the phase retrieval method.
Calculated from Calculated from
Mode irradiance moments retrieval method
(this paper) (Ref. [23])
HE11 94.6 94.2
LPo11 0.77 0.1
LPe11 0.27 0.3
LP02 3.75 4.0
LP03 0.08 0.3
LP21 0.53 0.4
Others 0.00 0.7
VII. Conclusions
The irradiance moment technique allows us to easily cal-culate the modal power of the dominant HOMs in a corru-gated waveguide without requiring a solution for the phaseusing the phase retrieval technique. The application of theirradiance moment method is based on the assumption thatthe power in the fundamental mode HE11 is large comparedto the other modes. The analytical investigation of the su-perposition of different modes breaks down the method tosimple linear equations which are easier to solve and makesthe proposed method simpler and faster compared to othermethods. Numerical benchmarking of the technique pre-dicts an error of less than ∼±0.7% in the retrieval of modalpower in the dominant modes. The results obtained basedon the experimentally measured intensity profiles, in a 63.5mm diameter ITER-relevant corrugated waveguide and a19 mm waveguide at 250 GHz used in a DNP-NMR ex-periment, demonstrate very good agreement between theresults obtained by the phase retrieval and mode contentretrieval techniques. The technique of mode content re-trieval using irradiance moments starts to deviate if thereis significant modal content of LP03 or other azimuthallysymmetric higher order modes. In the case of gyrotrons,the output beam is very well optimized to be a Gaussian-like beam by accurately designing the internal quasi-opticalhelical launcher and the set of beam shaping mirrors andfurther by using an external matching optics unit. There-fore the modes excited in the corrugated waveguide by thisGaussian-like beam are typically only the low index LPmnmodes, and, the power coupled into these modes by theGaussian-like beam is small. Due to this, the proposedmethod based on the irradiance moments can be very wellapplied in such a case.
VIII. Acknowledgments
The authors would like to thank E. Kowalski, E. Nanni,P. Woskov and J. X. Zhang of the Plasma Science and Fu-sion Center at MIT, Y. Oda of JAEA and T. Shimozuma ofthe National Institute of Fusion Sciences, Japan, F. Gan-dini of the ITER Organization, and G. Hanson of the OakRidge National Laboratory for helpful discussions. This re-
7
search was supported by the U.S. Department of Energy,Office of Fusion Energy Sciences, and by the U.S. ITERProject managed by Battelle/Oak Ridge National Labora-tory.
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