An FDTD Optimization of a Circularly Polarized Reflectarray Unit Cell ´ Etienne GIRARD1 , Ronan MOULINET1, Rapha¨el GILLARD1 and Herv´e LEGAY2 1

IETR ( UMR CNRS ) INSA
Rennes, 20, avenue des buttes de cœsmes, CS 14315, 35043 Rennes, France, egirard,rmouline,raphael.gillard  @insa-rennes.fr 2 Alcatel Space Industrie, 26 avenue JF.Champollion, BP1187, 31037 Cedex, Toulouse, France , [email protected]

1

Introduction

This paper describes the numerical analysis of a reflectarray unit cell using the Finite Differences in Time Domain ( FDTD ). The studied reflectarray consists of printed radiating elements embedded in metallic cavities in order to reduce mutual couplings. When excited with a circular polarized incoming wave, each printed element is supposed to re-radiate a similar circulary polarized wave with a phase shift depending on its orientation in the cavity as in [1, 2]. The analysis and optimisation of such a structure requires a rigorous electromagnetic modelling. FDTD has been used as it can deal with complex 3D structures and provides wide bandwidth characterization in a single simulation. This paper presents the specific numerical procedure that was used to perform this study and its application to a K-band radiating cell. 2

Studied structure

The studied unit cell of the reflectarray is shown in figure 1. As explained earlier, the radiating element is located in a segment of metallic waveguide. This permits both to reduce coupling with adjacent cells and to minimize the effect of the incoming wave’s incidence. The input of the waveguide is covered with a dielectric matching layer that optimizes the transition from the incident radiated wave to the incident propagating wave [3] and will not be discussed here. As the structure is supposed to deal with circular polarization, the waveguide should be able to propagate two orthogonal degenerate modes. At that time, experiences have been carried out with both square and circular waveguides. In this paper, we focus on the analysis of the printed radiating element that is embedded in a waveguide. In the normal operating mode, it is simultaneously excited by two orthogonal propagating modes, having the same properties except for a phase quadrature. The numerical analysis should permit to optimize the geometry of the printed element in order to reflect the two modes with a common phase shift ( that determines the phase of the re-radiated wave ) while preserving the helicity of the incoming wave by providing an additionnal π phase shift to one of the modes. The studied radiating structure is constituted by 6 dipoles with 30 o angular shift ( see fig.1 ) . The final design will use MEMS switches to activate one dipole among the 6.

frag replacements

ΓXX ΓYY r´equence ( GHz ) Phase (o ) Γ ΓXX YY dB Axial ratio eight of dielectric ngth of the dipole EYrotation (1b) EYre f erence (1a) Z

Y

MEMS Switch

Y

X

X

Incoming wave a d

w2

Waveguide

L w1

Dielectric active dipole

h

( b ) rotated by -60°

( a ) along Y axis

εr

Figure 1: Unit cell

In such a configuration, the “active” dipole is supposed to reflect totally the component of the incident field that is parallel to it while the orthogonal component is reflected by the ground plane. Theoretically, the distance between the ground plane and the radiating element should be adjusted to λ4 for the reflected wave to have the same helicity as the incident one. The situation is much more complex in practice due to couplings between the active dipole and the passive ones. Therefore only an electromagnetic analysis can lead to the optimal geometry. 3

Analysis

The numerical analysis involves two steps. In the first step, only one of the two degenerate modes ( say mode TE01 for a rectangular waveguide ) is applied as the incident wave. Reflections cœfficients for both possible reflected modes ( Γ XX for reflection in TE01 and ΓY X for reflection in TE10 ) are computed. In the second step, the orthogonal mode ( TE 10 ) is applied and two new reflection cœfficients are defined ( ΓYY and ΓXY ). The global reflection for an incident field ( E inc ) is obtained as 

 

re f l



EX

EYre f l







ΓXX





ΓXY





ΓY X



EXre f l re f l EY 





ΓYY

ΓXX  ΓY X 

(1)

EYinc

For a RHCP incoming wave propagating with respect to 



EXinc

j ΓXY j ΓYY

z, E inc 



1 j 

and yields to :

(2)

The conditions to be satisfied for a correct behaviour of the cell ( say a phase shift φ and a

circular polarization similar for both the reflected and incident waves ) are : 

ΓXX  ΓY X 

j ΓXY j ΓYY

exp  j φ  j exp  j φ 





(3)

In the simpler case where no phase shift is required, ( φ  0 ) and the metallic element is symetric with respect to ( OX ) or ( OY ), the cross reflection cœfficients are null ( ΓY X  0 and ΓXY  0 ), and (3) reduces to : ΓXX ΓYY 



1 1

Γ ΓXX YY dB Axial ratio Variable Height of dielectric Variable Length of the dipole EYrotation (1b) EYre f erence (1a)

(4)

PSfrag replacements

The computation of the reflexion cœfficients is performed using the FDTD.

PSfrag replacements ΓXX

ΓYY ΓXX dB Axial ratio Variable Height of dielectric Variable Length of the dipole EYrotation (1b) EYre f erence (1a)



The Total field - Scattered field [4] formulation is used to excite the waveguide whose upper part is matched using a 10 cells thick Perfectly Matched Layer ( PML ).

ΓYY

The determination of the incident field requires an additionnal simulation where both extremities of the waveguide are matched with PML. ΓXX

ΓYY

70

Phase (o )

Phase (o )

80 60 50 40 5.5

5.6

5.7

5.8

Fr´equence ( GHz ) HFSS

FDTD

5.9

250 240 230 220 210 200 190 5.5

5.6

5.7

5.8

5.9

Fr´equence ( GHz ) HFSS

FDTD

Figure 2: L = 13.34 mm, a = 34.8 mm, h = 6.5 mm, d = 2.94 mm, w 1 = 2.28 mm, w2 = 0.32 mm 4 4.1

Simulations Optimization process

As an illustration, figure 2 presents the frequency response of a cell that was optimized at 5.7 GHz. It can be seen that the phase difference is close to 180 o at 5.7 GHz while the agreement with HFSS simulation is quite good ( 10 o ). Figure 3 illustrates the optimization process for a 18.55 GHz operation. Both the substrate height and dipole length can be used to tune the phase difference. 4.2

Rotation and phase shift

Figure 4(a) shows the obtained axial ratio for the chosen geometry ( see fig. 1 (a) ) which is less than 0.5 dB on a 1.5 GHz frequency band. Figure 4(b) presents the phase of the circularly polarized reflected wave ( LHCP in this example ) for active dipoles corresponding to case 1(a) and 1(b). It can be seen that a -60 o rotation of the dipole results in a -120 o phase shift as expected from [1] .



ΓXX

200 195

2.2

2.4

2.6

2.8

h ( mm )

(a) L = 4.9 mm

180 175 170

183.2

165 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5

3

L ( mm )

(b) h = 2.16 mm

Figure 3: a = 10.7 mm , d= 0.9 mm, w1 = 0.7 mm , w2 = 0.1 mm

Phase (o )

0.4

dB

0.2 0.1 17.8 18 18.2 18.4 18.6 18.8 19 19.2

Fr´equence ( GHz ) (a)

ΓXX ΓYY

ΓXX ΓYY

Axial ratio 0.3

PSfrag replacements

2

PSfrag replacements

2.16

185

ΓYY ΓXX dB Axial ratio Variable Height of dielectric Variable Length of the dipole

Phase (o )

190

Phase (o ) ΓYY ΓXX

Variable Height of dielectric Variable Length of the dipole E rotation (1b) Y re f erence EY (1a)

Phase (o )

ΓYY

frag replacements

ΓXX

ΓXX ΓYY r´equence ( GHz )

dB Axial ratio eight of dielectric ngth of the dipole EYrotation (1b) EYre f erence (1a)

frag replacements

ΓXX ΓYY r´equence ( GHz )

dB Axial ratio eight of dielectric ngth of the dipole EYrotation (1b) EYre f erence (1a) ΓYY 260 240 220 200 180 168 160 140 120 100 80 1.6 1.8

160 re f erence 140 EY (1a) 120 100 -120 80 60 40 20 EYrotation (1b) 0 -20 -40 17.6 17.8 18 18.2 18.4 18.6 18.8 19 19.2 19.4

Fr´equence ( GHz ) (b)

Figure 4: L = 4.1 mm , h = 2.16 mm , a = 10.7 mm , d = 0.9 mm ,w 1 = 0.7 mm,w2 = 0.1 mm

5

Conclusion

In this paper, a FDTD modelling has been proposed to study a reflectarray unit cell. A two steps procedure has been used to compute its reflection properties when excited with a circularly polarized incident wave. This method has been applied to study the effect of all different geometrical parameters and the ability to control the phase of the reflected wave. Results are in good agreement with HFSS simulations. References [1] J. Huang and R. Pogorzelski, “A ka-band refllectarray with elements having variable rotation,” IEEE Trans. on Antennas Propag., vol. 46, pp. 650–655, May 1998. [2] K. C. R.D. Janor, X.D. Wu, “Design and performance of a microstrip reflectarray antenna,” IEEE Trans. on Antennas Propag., vol. 43, September 1995. [3] G. H. Knittel, “Wide-angle impedance matching of phased-array antennas, a survey of theory and practice,” in Proceedings of the 1970 phased array antenna symposium, pp. 157–171. [4] A. Taflove, Computational Electrodynamics, The FDTD Method, ch. 6.5, p. 111. Artech House.