by...• Miniaturized-element frequency selective surfaces (MEFSS) are sub-wavelength unit cells...

1 Scott Massidda/APS/November, 2012

by

Scott Massidda

(Columbia University)

In collaboration with

William Capecchi1, Kenneth Hammond2, Francesco Volpe2

(University of Wisconsin-Madison) 1

(Columbia University) 2

Presented at the

54th Annual Meeting of the APS Division of Plasma Physics

Providence, RI

October 29 – November 2, 2012

Metamaterial Lens of Adjustable Frequency-Dependent Focus for Electron Cyclotron Emission in the DIII-D Tokamak

2

Abstract

Millimeter wave diagnostics of plasmas typically cover bands of several GHz

(reflectometry, scattering), tens of GHz (radiometry) or even hundreds of GHz

(Michelson interferometry), but their focus is optimized for a single frequency. For

other frequencies, the measuring volume is far from the beam waist. This results in

a loss of resolution in the poloidal direction, especially at higher poloidal mode

numbers (e.g., Alfvèn Eigenmodes). Ideally the beam should be focused at

different locations for different frequencies. Our recent numerical study suggests

that a zoned planar metamaterial lens can achieve this result in the 8-12GHz

band [W.J. Capecchi et al., Optics Express 20, 8761 (2012)]. Here we present the

design and full-wave simulations of a lens for possible use with Electron Cyclotron

Emission (ECE) at DIII-D in the 80-130GHz band, discuss the fabrication challenges

due to the required miniaturization, and present a system allowing the

adjustment of the focal points to accommodate changes in the magnetic field.

Because ECE at DIII-D undergoes one of the largest variations of optimal focal

length, similar metamaterial lenses can more easily be designed for other mm-

wave diagnostics and/or smaller devices.

Scott Massidda/APS/November, 2012

3

Planar Metamaterial Lens


• Miniaturized-element frequency selective surfaces (MEFSS) are sub-

wavelength unit cells that populate the lens aperture in a square grid

• An Nth order MEFSS is composed of N capacitive layers and is alternated by

N-1 inductive layers; it acts as an Nth-order coupled-resonator bandpass filter

• Each cell is dependent on physical parameters, D, g, h, and w by

for the capacitive layers and for the inductive layers

• A lens can be made by arranging groups of units cells with identical

parameters into zones, then varying zones as a function of lens radius

Unit cell (MEFSS)

5 zone array

aperture design

0

2ln sin

2eff

D gC

D

0 ln sin2 2

eff

D wL

D

4

Designing a Lens from Many Unit Cells


• The MEFSS is partitioned into a set of discrete

annular zones concentric with a perpendicular

axis. Each zone is associated with a certain

spatial phase shift

• Outer zones (i.e. those with larger annular radii)

have larger phase shifts than those near the

axis, such that an incident collimated beam of

light that passes through an MEFSS with this

zoned layout will undergo a transformation in

its radial phase distribution that will cause it to

focus a certain distance beyond the lens.

Many unit cells arranged in

of a 3rd order MEFSS Lens

5

Chromatic Aberration


Focal Length

• Higher frequencies focused closer to lens

• This is caused by the refractive index increasing with frequency

6

Modified Chromatic Aberration


Focal Length

• Metamaterial lenses allow exotic properties

• We can specify a desired function for our refractive index, , which allows

the lens to focus frequencies at desired focal lengths.

n f

7

Desired focal lengths


• B field and ECE frequency decrease with major radius in Tokamak,

• Current wave diagnostics are optimized for a single location (Left).

• Metamaterial lenses can focus at many locations (Right).

• The desired focal lengths for each frequency are given corresponding to

channels of the DIII-D ECE radiometer.

ce qB m

R

Current

diagnostic lens

130 GHz 80 GHz

30 cm

150 cm

ECE frequency

R

Metamaterial

lens

130 GHz 80 GHz

30 cm

150 cm

Red arrows represent

regions of poor focus

The Red and purple

dashed lines represent

different frequencies

8

Numerical Optimization of a Metamaterial Lens


• Owing to the high number of degrees of freedom (capacitor gap, g, or

inductor width, w for each layer), it is possible to numerically identify a

geometrical configuration that produces different focal lengths for different

frequencies, and maintains a good transmittance

• Our lens design has 10 capacitive layers and 9 inductive layers. A 10th order

MEFSS was needed to maintain a well-behaved, “flat-top” passband. The

trade off is a steeper cutoff, which causes a low transmittance for the outside

frequencies

• Our lens has 83 zones in order

to obtain a smooth phase-shift

profile

• Ultimately, by choosing 7

physical parameters for each

zone, we allowed for 581

physical parameters

Individual Zones

9

Derivation of Goal Functions


A unit cell will have a different phase response, , and transmittance, T ,

for each frequency depending upon its physical parameters g and w

The ability of the lens to focus is dependent only upon the relative phase

shift of each of the zones. Therefore, The phase shift for the inner most

zone can be chosen freely

The numerical optimization was carried out in three steps, with the goal of

finding the physical parameters 1 2 6, ...g g g , and w , that produce a phase

response, , closest to the desired phase response, i , and have a similar

transmittance, T . This is done for each benchmark frequency, if , and

zone, iz

10

Pool of Unit Cell Designs


• For each unit cell

design, the

transmittance and

phase response for

each benchmark

frequency is

calculated using full

wave simulations in

CST.

• The end result is a

pool of possible unit

cell designs

• These designs can be

arranged into zones

to form a lens

Phase responses and transmittance of various unit

cell designs for GHz. 83.5;129.5if

11

The Importance of the initial Offset


The ideal function, ( , )i f z , is determined by the required focal length, which

defines the difference in the applied phase shift for each zone from the

center zone

Any arbitrary phase shift can be used for the center zone, 0 ( )i f . This gives

six degrees of freedom: one for each of the six benchmark frequencies.

Any phase shift applied is identical to 2 , because it is the interference

pattern that has physical importance

This allows for many different “Ideal” functions; one of which is the easiest

for the lens to obtain.

0( , ) , ( )i i if z f r z f

12

-600 -500 -400 -300 -200 -100 00.000

0.083

0.167

0.250

0.333

0.417

0.500 Pool of Phase Responses for 92.5 GHz

Ave

rag

e T

ran

sm

itta

nce

Err

or

(ove

r a

ll fr

eq

ue

ncie

s)

Phase response (degrees)

The Passband Determines the Choice of the Offset


• Each point represents a

unit cell design

• The lens requires this

frequency to span a

phase response of about

350 degrees in order for it

to focus at the proper

place

• This span can begin

anywhere and can be

different for each

frequency; the blue

rectangle represents the

range, -100 to -450

degrees, chosen for the

lens for 92.5 GHz

13

The Initial Offset Can be Defined by g and w


Contour Plot of error for the all zones varying the offset function in 2

dimensions (g and w), before the 6 dimensional optimization of the offset

• The offset is defined by

the phase response of

zone 1.

• The phase response of

zone 1 is defined by the

parameters used to build

the unit cell that

populates that zone.

• A 2 dimensional

optimization can find a

good approximation for

the best offset 0.01 0.02 0.03 0.04

0.10

0.12

0.14

0.16

0.18

0.20 Error for All Zones

Ca

pa

cito

r G

ap

Inductor Width

0.50000.75001.0001.2501.5001.7502.0002.2502.500

14

Six dimensional optimization of offset


After an approximation of the offset was found with a 2 dimensional

optimization, a six dimensional optimization was done to find precisely the

most efficient offset, and therefore the most efficient ideal function, ( , )i f z

In previous efforts that tested the feasibility of a metamaterial lens, the initial

phase response profile was constrained to be linear, 0( )i f f , leaving

only two parameters, the offset of a particular frequency and the slope of

the initial phase response profile. This allowed for quicker optimizations

15

Comparison of 6 versus 2 dimensional offset function


Contour Plot of error for the 1st (inner-most) zone with 2 different “Offsets”

16

7 dimensional physical parameter optimization


The last step was to perform a seven dimensional optimization in CST using full-

wave simulations. For the final optimization, the initial constraint ,

1 2 6, ,...g g g g , was removed, and the parameters were varied in seven

dimensional space: 1 2 6, ,...g g g , and w

The exact physical parameters that best matched the ideal function were

found for every 5 zones, and the remaining zones were found by interpolating

the parameters.

17

Final Results of the Lens


Interference plots of the final lens design

83.5;129.5if

18

Final Parameters Determined by Optimization in CST


0 10 20 30 40 50 60 70 800

40

80

120

160

200

Siz

e in

m

Zone

Inside Gap Pre-optimization Gap

2nd Gap Pre-optimization Width

3rd Gap

4th Gap

Outside Gap

Inductor Width

19

Summary and Conclusions


• Metamateriel lenses have properties that allow them to have capabilities that

surpass those of traditional lenses

• By correctly choosing an efficient ideal function and subsequently choosing

the proper physical parameters, a lens can be constructed that meets the

goals required for the DIII-D ECE emission

• Because the requirements of phase shift and transmittance for DIII-D are

relatively difficult to attain, we can achieve most other applications of

millimeter wave diagnostics using the same techniques

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