Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014 Article ID 812461 10 pageshttpdxdoiorg1011552014812461

Research ArticleDevelopment of Ray Tracing Algorithms for Scanning Plane andTransverse Plane Analysis for Satellite Multibeam Application

N H Abd Rahman12 M T Islam1 N Misran1 Y Yamada3 and N Michishita3

1 Department of Electrical Electronic and Systems Engineering Faculty of Engineering and Built EnvironmentUniversiti Kebangsaan Malaysia (UKM) 43600 Bangi Malaysia

2 Faculty of Electrical Engineering Universiti Teknologi MARA 40450 Shah Alam Selangor Malaysia3 National Defense Academy 1-10-20 Hashirimizu Yokosuka 239-8686 Japan

Correspondence should be addressed to N H Abd Rahman huda2811gmailcom

Received 18 November 2013 Accepted 18 March 2014

Academic Editor Rezaul Azim

Copyright copy 2014 N H Abd Rahman et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Reflector antennas have been widely used in many areas In the implementation of parabolic reflector antenna for broadcastingsatellite applications it is essential for the spacecraft antenna to provide precise contoured beam to effectively serve the requiredregion For this purpose combinations of more than one beam are required Therefore a tool utilizing ray tracing method isdeveloped to calculate precise off-axis beams for multibeam antenna system In the multibeam system each beam will be fedfrom different feed positions to allow the main beam to be radiated at the exact direction on the coverage area Thus detailedstudy on caustics of a parabolic reflector antenna is performed and presented in this paper which is to investigate the behaviour ofthe rays and its relation to various antenna parameters In order to produce accurate data for the analysis the caustic behavioursare investigated in two distinctive modes scanning plane and transverse plane This paper presents the detailed discussions onthe derivation of the ray tracing algorithms the establishment of the equations of caustic loci and the verification of the methodthrough calculation of radiation pattern

1 Introduction

Advances in wireless communications have introducedtremendous demands in the antenna technology [1 2] Due tothe significant raise in the number of geostationary satellitesthat provide high data rate services the demands for satelliteantenna system with multiple spot beams are also increasingIn satellite broadcasting application fine contoured beamdesign is very critical to ensure uniform gain to the desiredarea and to decrease the radiation level in the unwantedareas rapidly Thus to produce these contoured beamscombination of more than one beam is employed [3ndash5]Antenna structures for various multibeam scenarios havebeen discussed in [6]

Many reflector shaping methods and techniques havebeen identified as suitable to solve this issue For examplein [7] physical optics (PO) technique has been employed togenerate contoured beam from a shaped reflector antennas

The shaping of the reflector is carried out by using thirdparty commercial software called TICRA POS To fulfill thehigh gain requirement the feed is optimized and displacedlaterally in the focal plane however the detailed analysis onhow the lateral feed displacement relates to the beam shift isnot performed

Apart from the PO methods many studies on the focalregion of parabolic reflectors have been conducted Previ-ously a reflector shaping technique based on the aperturediffraction method has been studied [8] The technique hasbeen employed in [9] to estimate feed positions for twodistinctive surfaces resulting in the best feed location forelevation and azimuth plane patterns Based on the scanningof two planes the best focal surfaces which are expressedin two-dimensional coordinate have been used to determinethe antenna feed positions However the relationship of thecaustic displacement to the parameter 119891119863 is not clear

2 International Journal of Antennas and Propagation

Table 1 Study parameters

Antenna parameters Numerical valuesFrequency (GHz) 114GHz (120582 = 263mm)Antenna diameter (mm) 119863 = 20120582 = 5263mm

Focal length (mm)1198911 = 526mm (119891119863 = 1)1198912 = 789mm (119891119863 = 15)1198913 = 1578mm (119891119863 = 3)

Incident beam direction (∘) 120579in = minus35∘ to 35∘

D

x

y

z

fO

P120579

120579

120601

120601

= (Px Py Pz)

120588

Figure 1 Antenna configuration

In this paper a design tool called ray tracing method isdeveloped The tool is designed to measure the caustics ofa single parabolic reflector antenna system when radiatedfrom various incident beam directions Therefore the sameconcept is used to determine the best feed positions for thesatellite multibeam system In this method caustic pointsare determined based on the focal region generated bythe convergence of rays observed in the scanning planeand transverse plane of the antenna system The scanningplane and transverse plane analysis have been carried outpreviously for dielectric lens antenna [10] and the methodhas been very useful in determining feed positions Thecalculation of rays andparabolic surface points are performedon the basis of mathematical and physical optics algorithmsTo investigate the behaviour of caustics the ray tracingprogram is performed for various antenna configurationsand beam directions The results are recorded and the dataare interpolated Based on the data a set of equations torepresent the loci of caustics are derived These equations arevery useful to determine the best feed position especiallyfor multibeam applicationThe reliability and accuracy of theequations are verified in this paper

2 Ray Tracing Concept

21 Antenna and Feed Type The antenna system comprisesof one large parabolic reflector and one feeding elementThe fundamental parabolic reflector configuration and thecoordinate system are shown in Figure 1 The reflector issymmetrical around the 119911-axis while 119909-axis and 119910-axisindicate the radial direction of the reflector The origin ofthe coordinate system is expressed by 119874 Under a focusedoperation a feed antenna is placed at the focal point 119874 withdistance 119891 away from the reflector

During operation rays emitted from the feed will reachthe parabolic surface at a point on reflector surface denoted

by119875120579120601 that has vector components of119875119909119875119910119875119911120601 correspondsto the angle around the 119911-axis and 120579 indicates the angle fromthe feed point to the119875120579120601The calculation of reflector in119909- and119911-coordinate component is based on the standard equationof a parabolic surface [11]

119911 = 11990924119891 minus 119891 (1)

The distance from the feed to the reflector surface120588 can beexpressed in polar coordinates as follows [12]

120588 = 21198911 + cos 120579 (2)

22 Design Flow The flowchart expression of the developedray tracing program is shown in Figure 2 The program isdeveloped onMATLAB simulation tool Both scanning planeand transverse plane have the same fundamental concepthowever the calculation algorithms are different due to thevariation in plane wave configurations

The analysis is performed based on the receiving modecondition as explained in [13] Initially antenna parameterssuch as focal length (119891) diameter (119863) and frequency aredetermined based on the spacecraft requirements and lim-itations Other parameters such as reflector angle (120579) anddistance to reflector rim (120588) are calculated on the basis ofmathematical equations physical optic (PO) conditions andtrigonometry equations explained in the next section In thefocal region ray tracing all of the incident rays coming fromthe desired beams will produce multiple reflections at thereflector surfaceThe intersection of the rays will converge ona plane or point and will form a set of traces and curves thatare observed and studied in this paperThe convergence pointor plane is called a caustic [8 12] This position becomes thebest feed point for the beam scanning angle 120579in The processis repeated for many sets of incident angles and antennaconfigurations

3 Ray Tracing Algorithm Overview andParametric Setup

31 Receiving Mode Ray Tracing The ray tracing algorithmsare derived based on the receivingmode condition illustratedin Figure 3 The illumination is in a form of plane wavetransmitted from a fixed direction and tilted with respect tothe reflector axis

32 Scanning Plane The illustration of scanning plane ina single reflector system is shown in Figure 4 For thecalculation of caustic the incident plane is displaced fromthe axis perpendicular to the reflector by an angle 120579in andthe caustic position is measured The incident rays exist ina vertical line that is contained in the 119909119911-plane

In the scanning plane the reflector points 119875(120579) will bedetermined in terms of 120588 and 120579 where 120588 is the distancemeasured from feed point (119865) to the reflector surface and 120579is the angle from the origin to the reflector surface To beginwith the 119909-coordinate of reflector surface (119875119903119909) is set between

International Journal of Antennas and Propagation 3

Input antenna parameters

Set beam direction

Determine reflector angle (θ)

Calculate reflector

Determine the points for incident plane waves

Express rays from incident plane to reflector surface

Express rays from reflector surface to focal

region

Determine caustic point

Determine the relevant

(120579in )

surface (Prx Pry Prz) vectors ib n rb

(Pinx Piny Pinz)

F998400

Figure 2 Flow chart of focal region ray tracing program

Parabolic reflector

x

Plane wavezO

n

n

Focal region

120579

120579

in

120588

ib

rb

1205850

120585i

Figure 3 Illustration of focal region in the radiation mode raytracing

x

Locus of caustic Incident plane

Caustic

z

y

Incident rays

Prx Pry Prz)

F998400

F

= (PinPin x Piny Pinz)

P(120579) = (

minus120579in

Figure 4 Illustration of scanning plane in a single reflector system

the range of its minimum and maximum 1199091 = (minus1198632)to (1198632) By solving the simultaneous equations of (2) andits relation to 120579 from Figure 4 the expression of 120579 can beobtained as follows

120579 = cosminus1 (41198912 minus 1199092141198912 + 11990921) (3)

Thus by taking into account (3) and (1) the coordinate of119875(120579) in 119909- 119910- and 119911-direction is defined as follows

119875119903119909 = 1199091119875119903119910 = 0119875119903119911 = 120588 sin2 120579

4119891 minus 119891(4)

The incident point known as 119875in(119875in119909 119875in119910 119875in119911) is cal-culated based on the incident plane wave directed to thereflector with 120579in offset from the centre position Thus ingeneral by considering the incident beam the 119909- and 119911-component of incident plane can be expressed in tangentequation as follows

sin 120579incos 120579in = 119909119903 minus 119909in119911119903 minus 119911in = tan 120579in (5)

The line equation of the incident plane 119909in where 119897119900 repre-sents the length of the feed point 119865 to the incident point isgenerally expressed as follows

119909in = minuscot 120579in (119911in minus 119897119900cos 120579in) (6)

Therefore by solving the simultaneous equations of (5) and(6) the coordinate of 119875in can be calculated as follows

119875in119909 = 119909in119875in119910 = 0

119875in119911 = 119875119903119911 sin2 120579in minus 119875119903119909 sin 120579in cos 120579in + 119897119900 cos 120579in(7)

For the scanning plane analysis the coordinate 119875in119910 remainszero as only the 120579in direction is taken into account and onlyvertical scanning is performedThen the unit incident vector119894119887 from a known beam direction (119875in) to reflector surface (119875119903)is expressed by the following equation where 119894119887119909 119894119887119910 and 119894119887119911indicate the 119909 119910 and 119911 components of the vector 119894119887

119894119887 (119894119887119909 119894119887119910 119894119887119911) =997888997888997888rarr119875in119875119903100381610038161003816100381610038161003816997888997888997888rarr119875in119875119903100381610038161003816100381610038161003816

= [119875in119909 minus 119875119903119909 119875in119910 minus 119875119903119910 119875in119911 minus 119875119903119911]100381610038161003816100381610038161003816997888997888997888rarr119875in119875119903100381610038161003816100381610038161003816

(8)

4 International Journal of Antennas and Propagation

z

y

Incident plane

Incident raysLocus of caustic

x

CausticPrx Pry Prz)

F998400

F

= (PinPin x Piny Pinz)

P(120579) = (

minus120579inymax =D

2

ymin =minusD

2

Figure 5 Illustration of transverse plane in a single reflector system

The next step is to calculate the unit reflected vector119903119887=[119903119887119909 119903119887119910 119903119887119911] To solve that firstly the normal unit vectoron the parabolic reflector 119899119887= [119899119887119909 119899119887119910 119899119887119911] is derived asfollows

119899119887 = (minus sin(1205792) 0 cos(120579

2)) (9)

Then by applying the condition that all vectors exist on thesame plane the simultaneous equations of the vector dotproduct and cross product are solved [7] The vectors 119894119887 and119899119887 are expressed in terms of 1205850 as follows where 1205850 representsthe incident angle at the reflector surface

119899119887119909119894119887119909 + 119899119887119910119894119887119910 + 119899119887119911119894119887119911 = cos 1205850 (10)

The equation is expanded as follows assuming that theincidence angle 1205850 is equal to the reflected angle 120585119894 measuredfrom the normal vector 119899119887

119899119887119910119894119887119911 minus 119899119887119911119894119887119910 = 119899119887119910119903119887119911 minus 119899119887119911119903119887119910119899119887119911119894119887119909 minus 119899119887119909119894119887119911 = 119899119887119911119903119887119911 minus 119899119887119909119903119887119910119899119887119909119894119887119910 minus 119899119887119910119894119887119909 = 119899119887119909119903119887119910 minus 119899119887119910119903119887119909

(11)

The left hand side of (11) is given by known values thereforethe equations become constant and are expressed in terms of119886119887119909 119886119887119910 and 119886119887119911 respectively Finally by taking into accountall the relationships defined above the unknowns of 119903119887119909 119903119887119910and 119903119887119911 are solved

119903119887119909 = minus119899119909 cos 1205850 + 119899119887119911119886119887119910 minus 119899119887119910119886119887119911119903119887119910 = minus119899119910 cos 1205850 + 119899119887119909119886119887119911 minus 119899119887119911119886119887119909119903119887119911 = minus119899119911 cos 1205850 + 119899119887119910119886119887119909 minus 119899119887119909119886119887119910

(12)

In order to calculate the caustic points all rays reflectedfrom focal surface to the focal region will be expressed anddisplayed The focal region consists of a set of focal points

119865119887 = [119865119887119909 119865119887119910 119865119887119911] The equations of the 119865119887 defined in 119909- 119910-and 119911-coordinates are derived as follows

119865119887119909 = 119875119903119909 + 120588119903119887119909119865119887119910 = 119875119903119910 + 120588119903119887119910119865119887119911 = 119875119903119911 + 120588119903119887119911

(13)

The focal points are plotted and the most converged area orpoint is determined as caustic and will be used throughoutthe study for the calculation of feed position

33 Transverse Plane Theconfiguration of parabolic antennawith the illustration of transverse plane is shown in Figure 5From the diagram it shows that the incident rays exist ina plane perpendicular to the previous scanning plane Theincident rayswill cross the119910-axis before being reflected by thereflector surface hence the119910-coordinates remain unchangedthroughout the procedure

As explained in scanning plane the reflector points119875(120579)will be determined in terms of 120588 120579 and 120601 where 120601 is definedas the azimuth angle measured from the centre of reflector asillustrated in Figure 1Thus the119875(120579) in119909-119910- and 119911-directionis determined by the following equations

119875119903119909 = 120588 sin 120579 cos120601119875119903119910 = 120588 sin 120579 sin120601119875119903119911 = minus 120588 cos 120579

(14)

However in this case the values of 120579 and 120601 are unknownand will be determined by solving various equations Intransverse plane 119875119903119910 is fixed to 1199101 which ranges from minus1198632to 1198632 Based on the condition of incident rays in Figure 5the following expression is derived

tan 120579in = minus119875119903119909119875119903119911 =sin 120579119898 cos120601

cos 120579119898 (15)

By substituting 120588 to the equation of 119875119903119910 in (14) the followingequation is obtained

21198911 + cos 120579119898 sin 120579119898 sin120601 = 1199101 (16)

International Journal of Antennas and Propagation 5

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1Scanning plane-reflection mode caustics

z

x

fD = 1

(a)

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1

z

x

fD = 15

Scanning plane-reflection mode caustics

(b)

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1

z

x

Direction of caustic locus S (x z)

120579in = minus35∘120579in = minus15∘ 120579in = 05∘

120579in = 35∘120579in = 15∘

Scanning plane-reflection mode caustics

fD = 3

(c)

Figure 6 Results of scanning plane ray tracing for various incoming beam directions (a) 119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

In order to derive the value of 120579 and 120601 (15) and (16)will be solved simultaneously By using the technique calledldquocompleting the squarerdquo the quadratic equations can besolved and hence the values of 120579 and 120601 are obtained Thus

the exact parabolic surface points as mentioned in (14) canbe determined On the other hand the plane of incident rays

6 International Journal of Antennas and Propagation

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 1

(a)

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 15

(b)

minus020

02

Transverse plane-reflection mode caustics

y

minus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 3

Direction of caustic locus

120579in = minus35∘

120579in = minus15∘120579in = 05∘

120579in = 35∘

120579in = 15∘

S (x z)

(c)

Figure 7 Results of transverse plane ray tracing for various incoming beam directions (a) 119891119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

119875in (119875in119909 119875in119910 119875in119911) is calculated by solving the position of theray in 119911-direction

119875in119911 = 119875119903119911sin2120579in minus 119875119903119910 sin 120579in cos 120579in + 119897119900 cos 120579in (17)

Since the rays are rotated along the 119910-axis thus similar to the119875119903119910 the incident points also have the same values

119875in119910 = 119875119903119910 (18)

The calculation of the 119875in119909 is based on the same principleadopted in scanning plane and thus

119875in119909 = minuscot 120579in (119875in119911 minus 119897119900cos 120579in) (19)

The reflected vectors are calculated by deriving the normalvectors as in (10) and the prediction of the feed position canbe performed by using the same concept as done in scanningplane

4 Results and Discussion

41 Parametric Setup Ray tracing program for scanning andtransverse plane is performed for all major points on thereflectorThe analysis is carried out for a set of incident angles120579in to interpolate the locus of caustic and to observe thecaustic behaviour Critical antenna parameter such as 119891119863is varied to evaluate the effects towards the caustics To varythe 119891119863 the reflector diameter is kept constant at 5263mmwhilst only the focal length 119891 is changed The parameters ofstudy are summarized in Table 1

42 Displacement of Caustics The results of focal regionray tracing for scanning plane is shown in Figure 6 In thisanalysis the focal region of the parabolic reflector is analyzedfor five different incident angles which are set within therange of 120579in = minus35∘ to 35∘ The results are observed in119909119911-plane a two-dimensional graph that best represents the

International Journal of Antennas and Propagation 7

minus18minus16minus14minus12minus10minus08minus06minus04minus02

00minus08 minus06 minus04 minus02 00 02 04 06 08

Scanning plane fD = 1Scanning plane fD = 15Scanning plane fD = 3

Δx

Δz120579in = minus20∘

f= 0789m

f= 1578mParabolic reflector withdiameter D = 0526m

Figure 8 Scanning plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

minus18

minus18minus16minus16

minus14 minus14

minus15 15minus05 05

minus12minus12

minus10

minus10

minus08

minus08

minus06

minus06

minus04

minus04

minus02

02

02

minus02

minus02

00minus16 minus12 minus08 minus04 00

0

04 08 12 16

Transverse plane fD = 1Transverse plane fD = 15Transverse plane fD = 3

z

yx

f= 1578mm

f= 0789mm

Δx

Δz

120579in = minus20∘

Figure 9 Transverse plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

configuration of the scanning plane mode Based on thegraphs locus of caustic can be interpolated as the incidentdirection is varied The graphs are drawn on the samescale thus the changes in antenna configuration and causticbehaviour can be demonstrated

From the results it can be observed that as the incidentbeam is increased the caustics moved further away fromthe origin As seen from the zoomed diagrams the qualityof focusing also changed which means that the caustics areless converged at higher 120579in On the contrary the causticsare more converged and easier to determine when the119891119863 is increased Therefore it is expected that at higher119891119863 configuration the calculation of caustic points is moreaccurate and thus provide more reliable results The causticpoints for scanning plane are measured in terms of 119909- and 119911-axis displacement or denoted as Δ119909119904 and Δ119911119904 respectively As

minus05

minus04

minus03

minus02

minus01

00

01

minus10 minus08 minus06 minus04 minus02 00 02 04 06 08 10

Scanning plane fD = 1 Transverse plane fD = 1Scanning plane fD = 15 Transverse plane fD = 15Scanning plane fD = 3 Transverse plane fD = 3S(x z) for scanning plane S(x z) for transverse plane

Δzf

Scanning plane

Transverse plane

Δxf

Figure 10 Comparison of two-dimensional caustic displacementwith the equation lines of caustic loci 119878(119909 119911)

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 11 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 15 configuration

seen from Figure 6(c) a line is drawn to interpolate the locusof causticThe line of caustic locus is represented by a variable119878(119909 119911) which represents the distance between the centre ofparabolic reflector to the caustic points

Figure 7 shows the results of transverse plane ray tracingDue to the complexity of the transverse plane the behavioursof the transverse plane caustics are best observed in three-dimensional axes The axes are rotated several times duringmeasurement in order to calculate the 119909- and 119911-axis displace-ment also known as Δ119909119905 and Δ119911119905 respectively From thegraphs it can be seen that the caustics are less converged athigher 120579 in However similar to scanning plane the effectsare less when the 119891119863 is increased The caustic locus is

8 International Journal of Antennas and Propagation

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 12 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 15 configuration

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 13 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 3 configuration

interpolated and the distance 119878(119909 119911) as shown in Figure 7(c)is calculated and will be used in the next analysis

43 Equation of Caustic Locus The results of caustic dis-placements are plotted and the trend is observed Specificexamples of the caustic points with the comparison betweentwo different reflector configurations (119891119863 = 15 and 3)are presented in Figures 8 and 9 for scanning plane andtransverse plane respectively The convergence of parallelincident rays with 120579in = minus20∘ is shown in both diagrams The

0 5 10 15 20 25 30minus55minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus50

Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 14 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 3 configuration

caustics of other incident angles within the range of plusmn35∘ areplotted In transverse plane a clearer view of the convergenceof rays is shown in the zoomed image

Based on the results of caustic displacements the trajec-tory of the caustics can be approximated as follows

(a) in the scanning plane

119878 (119909 119911) = 119891 cos 120579in (20)

(b) in the transverse plane

119878 (119909 119911) = 119891cos 120579in (21)

Hence the locus of the caustics in the scanning planeis a circular arc with a curvature that is smaller by cos(120579in)Meanwhile for the transverse plane the trajectory is also acircular arc but with a curvature that is bigger by cos(120579in)Thus if an antenna system is to be designed to form a beam atthe desired direction given by 120579in the feed will be positionedon the curvature given by (20) and (21) on the scanningplane and the transverse plane respectivelyThe accuracy andvalidity of the loci equations can be shown in Figure 10 In thisdiagram the normalized caustic displacement data obtainedfrom both scanning plane and transverse plane analysis areplotted and lines joining the points are drawn to interpolatethe locus The lines of equations for caustic loci are alsoplotted according to the incident beamdirection Both resultsare compared and based on the plots it can be observedthat the lines of (20) and (21) almost matched with thecaustic displacement data of scanning and transverse planerespectively Therefore the general equations for caustic locican be used as a guide to preliminary locate the feed positionsof parabolic reflector antenna especially for the design ofmultibeam or shaped beam system

International Journal of Antennas and Propagation 9

44 Radiation Characteristics To ensure the accuracy of themethod the results of caustic obtained in the previous sectionare verified through the calculation of the radiation patternThe relative radiation pattern is calculated by using the fol-lowing equations where 120585(119909 119910) represents the aperture phasedistribution obtained through comparison of ray length froma feed to all aperture points [11]

119864119903 = intlfloorint 119890(119895119896119903 sin 120579 cos(ΦminusΦ1015840)+119895120577(119909119910))119889Φ1015840rfloor 119903119889119903where 119903 = 120588 sin 120579

(22)

Several examples based on the incident direction of 120579in =minus20∘ are shown here in order to demonstrate the behaviourof the radiation pattern Figure 11 and Figure 12 show theradiation characteristics of a parabolic reflector with 119891119863 =15 In these configurations the feed locations 119865(119909 119910 119911) areset to be located at two different positions Firstly the feed isset at the caustic point (Δ119909119904 Δ119911119904) or (Δ119909119905 Δ119911119905) that has beenobtained in the previous ray tracing program for scanningplane and transverse plane respectively Then the resultis compared to the radiation pattern measured when thefeed is placed at the position determined through the locusequation (20) and (21) The results of both scanning planeand transverse plane are shown

From Figure 11 themaximum gain is measured at around20∘ offset from the origin which is the expected result sincethe feed location was previously set at the minus 20∘ caustic pointHowever the plot given by (20) gives better representation ofthe exact beam shiftMeanwhile as for transverse planemoredeteriorate patterns are observed

For both scanning and transverse planes by comparingthe caustic displacement results with the ldquobluerdquo line it seemslike the locus equations for both scanning and transverse pro-duced more accurate beams with the highest gain obtainedat the direction closer to the desired beam direction In thiscase the equations of caustic loci become very useful indetermining the accurate position of the feed However asexpected the analysis conducted in scanning mode providesmore accurate results as compared to the transversemode Toinvestigate further similar exercise is carried out for119891119863 = 3configuration

Similar to 119891119863 = 15 it can be observed from Figures13 and 14 that the equations of caustic loci bring the beamdirection closer to the desired 120579in = minus20∘ direction althoughit seems like in transverse plane the result deviates further ascompared to scanning plane As shown by scanning plane inFigure 13 the radiation patterns for both feed configurationsoverlapped each other due to the similarity of both valuesThe highest gain is also measured at the same point Theagreement of the radiation gain values obtained at the twodifferent feed configurations validates the equation of causticloci derived by author in the previous section Therefore theaccuracy of the ray tracing algorithms for scanning plane andtransverse plane is ensured

5 Conclusion

A set of algorithms developed for the scanning plane andtransverse plane ray tracing program are presented Theresults obtained from the focal region ray tracing are dis-cussed and based on the caustic displacement results usefulequations representing the caustic loci are derived Theimportance of the equations are shown which could beused in designing a multibeam antenna application Off-feedcaustic displacements obtained in ray tracing method arecompared with the approximate equation of locus throughcalculation of radiation pattern The results of beam shiftshow good agreement between these two values and thusthe accuracy of the ray tracing method and the reliability ofthe locus equation for scanning plane and transverse planeanalysis are proven

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Teknologi MARAfor continuous support towards this research project

References

[1] R Azim M T Islam N Misran S W Cheung and YYamada ldquoPlanar UWB antenna with multi-slotted groundplanerdquo Microwave and Optical Technology Letters vol 53 no5 pp 966ndash968 2011

[2] L Liu S W Cheung R Azim and M T Islam ldquoA com-pact circular-ring antenna for ultra-wideband applicationsrdquoMicrowave and Optical Technology Letters vol 53 no 10 pp2283ndash2288 2011

[3] K Shogen S Tanaka and S Nakazawa ldquoStudies on the onboardAntennas for 21 GHz band broadcasting satellites and futurestudy itemsrdquo IEICE Transactions on Communications vol J94-B no 9 pp 1014ndash1024 2011

[4] G Toso C Mangenot and P Angeletti ldquoRecent advanceson space multibeam antennas based on a single aperturerdquoin Proceedings of the European Conference on Antennas andPropagation pp 454ndash458 April 2013

[5] S Yun M Uhm J Choi and I Yom ldquoMultibeam reflectorantenna fed by few elements for Ka-band communicationsatelliterdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 1ndash2 July 2012

[6] M Schneider C Hartwanger and H Wolf ldquoAntennas formultiple spot beam satellitesrdquo CEAS Space Journal vol 2 no1ndash4 pp 59ndash66 2011

[7] M Mahajan R Joti K Sood and S B Sharman ldquoA methodof generating simultaneous contoured and pencil beams fromsingle shaped reflector antennardquo IEEETransactions onAntennasand Propagation vol 61 no 10 pp 5297ndash5301 2013

[8] R E Collin and F J Zucker Antenna Theory Part 2 McGraw-Hill New York NY USA 1969

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012

International Journal of Antennas and Propagation 3

Input antenna parameters

Set beam direction

Determine reflector angle (θ)

Calculate reflector

Determine the points for incident plane waves

Express rays from incident plane to reflector surface

Express rays from reflector surface to focal

region

Determine caustic point

Determine the relevant

(120579in )

surface (Prx Pry Prz) vectors ib n rb

(Pinx Piny Pinz)

F998400

Figure 2 Flow chart of focal region ray tracing program

Parabolic reflector

x

Plane wavezO

n

n

Focal region

120579

120579

in

120588

ib

rb

1205850

120585i

Figure 3 Illustration of focal region in the radiation mode raytracing

x

Locus of caustic Incident plane

Caustic

z

y

Incident rays

Prx Pry Prz)

F998400

F

= (PinPin x Piny Pinz)

P(120579) = (

minus120579in

Figure 4 Illustration of scanning plane in a single reflector system

the range of its minimum and maximum 1199091 = (minus1198632)to (1198632) By solving the simultaneous equations of (2) andits relation to 120579 from Figure 4 the expression of 120579 can beobtained as follows

120579 = cosminus1 (41198912 minus 1199092141198912 + 11990921) (3)

Thus by taking into account (3) and (1) the coordinate of119875(120579) in 119909- 119910- and 119911-direction is defined as follows

119875119903119909 = 1199091119875119903119910 = 0119875119903119911 = 120588 sin2 120579

4119891 minus 119891(4)

The incident point known as 119875in(119875in119909 119875in119910 119875in119911) is cal-culated based on the incident plane wave directed to thereflector with 120579in offset from the centre position Thus ingeneral by considering the incident beam the 119909- and 119911-component of incident plane can be expressed in tangentequation as follows

sin 120579incos 120579in = 119909119903 minus 119909in119911119903 minus 119911in = tan 120579in (5)

The line equation of the incident plane 119909in where 119897119900 repre-sents the length of the feed point 119865 to the incident point isgenerally expressed as follows

119909in = minuscot 120579in (119911in minus 119897119900cos 120579in) (6)

Therefore by solving the simultaneous equations of (5) and(6) the coordinate of 119875in can be calculated as follows

119875in119909 = 119909in119875in119910 = 0

119875in119911 = 119875119903119911 sin2 120579in minus 119875119903119909 sin 120579in cos 120579in + 119897119900 cos 120579in(7)

For the scanning plane analysis the coordinate 119875in119910 remainszero as only the 120579in direction is taken into account and onlyvertical scanning is performedThen the unit incident vector119894119887 from a known beam direction (119875in) to reflector surface (119875119903)is expressed by the following equation where 119894119887119909 119894119887119910 and 119894119887119911indicate the 119909 119910 and 119911 components of the vector 119894119887

119894119887 (119894119887119909 119894119887119910 119894119887119911) =997888997888997888rarr119875in119875119903100381610038161003816100381610038161003816997888997888997888rarr119875in119875119903100381610038161003816100381610038161003816

= [119875in119909 minus 119875119903119909 119875in119910 minus 119875119903119910 119875in119911 minus 119875119903119911]100381610038161003816100381610038161003816997888997888997888rarr119875in119875119903100381610038161003816100381610038161003816

(8)

4 International Journal of Antennas and Propagation

z

y

Incident plane

Incident raysLocus of caustic

x

CausticPrx Pry Prz)

F998400

F

= (PinPin x Piny Pinz)

P(120579) = (

minus120579inymax =D

2

ymin =minusD

2

Figure 5 Illustration of transverse plane in a single reflector system

The next step is to calculate the unit reflected vector119903119887=[119903119887119909 119903119887119910 119903119887119911] To solve that firstly the normal unit vectoron the parabolic reflector 119899119887= [119899119887119909 119899119887119910 119899119887119911] is derived asfollows

119899119887 = (minus sin(1205792) 0 cos(120579

2)) (9)

Then by applying the condition that all vectors exist on thesame plane the simultaneous equations of the vector dotproduct and cross product are solved [7] The vectors 119894119887 and119899119887 are expressed in terms of 1205850 as follows where 1205850 representsthe incident angle at the reflector surface

119899119887119909119894119887119909 + 119899119887119910119894119887119910 + 119899119887119911119894119887119911 = cos 1205850 (10)

The equation is expanded as follows assuming that theincidence angle 1205850 is equal to the reflected angle 120585119894 measuredfrom the normal vector 119899119887

119899119887119910119894119887119911 minus 119899119887119911119894119887119910 = 119899119887119910119903119887119911 minus 119899119887119911119903119887119910119899119887119911119894119887119909 minus 119899119887119909119894119887119911 = 119899119887119911119903119887119911 minus 119899119887119909119903119887119910119899119887119909119894119887119910 minus 119899119887119910119894119887119909 = 119899119887119909119903119887119910 minus 119899119887119910119903119887119909

(11)

The left hand side of (11) is given by known values thereforethe equations become constant and are expressed in terms of119886119887119909 119886119887119910 and 119886119887119911 respectively Finally by taking into accountall the relationships defined above the unknowns of 119903119887119909 119903119887119910and 119903119887119911 are solved

119903119887119909 = minus119899119909 cos 1205850 + 119899119887119911119886119887119910 minus 119899119887119910119886119887119911119903119887119910 = minus119899119910 cos 1205850 + 119899119887119909119886119887119911 minus 119899119887119911119886119887119909119903119887119911 = minus119899119911 cos 1205850 + 119899119887119910119886119887119909 minus 119899119887119909119886119887119910

(12)

In order to calculate the caustic points all rays reflectedfrom focal surface to the focal region will be expressed anddisplayed The focal region consists of a set of focal points

119865119887 = [119865119887119909 119865119887119910 119865119887119911] The equations of the 119865119887 defined in 119909- 119910-and 119911-coordinates are derived as follows

119865119887119909 = 119875119903119909 + 120588119903119887119909119865119887119910 = 119875119903119910 + 120588119903119887119910119865119887119911 = 119875119903119911 + 120588119903119887119911

(13)

The focal points are plotted and the most converged area orpoint is determined as caustic and will be used throughoutthe study for the calculation of feed position

33 Transverse Plane Theconfiguration of parabolic antennawith the illustration of transverse plane is shown in Figure 5From the diagram it shows that the incident rays exist ina plane perpendicular to the previous scanning plane Theincident rayswill cross the119910-axis before being reflected by thereflector surface hence the119910-coordinates remain unchangedthroughout the procedure

As explained in scanning plane the reflector points119875(120579)will be determined in terms of 120588 120579 and 120601 where 120601 is definedas the azimuth angle measured from the centre of reflector asillustrated in Figure 1Thus the119875(120579) in119909-119910- and 119911-directionis determined by the following equations

119875119903119909 = 120588 sin 120579 cos120601119875119903119910 = 120588 sin 120579 sin120601119875119903119911 = minus 120588 cos 120579

(14)

However in this case the values of 120579 and 120601 are unknownand will be determined by solving various equations Intransverse plane 119875119903119910 is fixed to 1199101 which ranges from minus1198632to 1198632 Based on the condition of incident rays in Figure 5the following expression is derived

tan 120579in = minus119875119903119909119875119903119911 =sin 120579119898 cos120601

cos 120579119898 (15)

By substituting 120588 to the equation of 119875119903119910 in (14) the followingequation is obtained

21198911 + cos 120579119898 sin 120579119898 sin120601 = 1199101 (16)

International Journal of Antennas and Propagation 5

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1Scanning plane-reflection mode caustics

z

x

fD = 1

(a)

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1

z

x

fD = 15

Scanning plane-reflection mode caustics

(b)

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1

z

x

Direction of caustic locus S (x z)

120579in = minus35∘120579in = minus15∘ 120579in = 05∘

120579in = 35∘120579in = 15∘

Scanning plane-reflection mode caustics

fD = 3

(c)

Figure 6 Results of scanning plane ray tracing for various incoming beam directions (a) 119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

In order to derive the value of 120579 and 120601 (15) and (16)will be solved simultaneously By using the technique calledldquocompleting the squarerdquo the quadratic equations can besolved and hence the values of 120579 and 120601 are obtained Thus

the exact parabolic surface points as mentioned in (14) canbe determined On the other hand the plane of incident rays

6 International Journal of Antennas and Propagation

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 1

(a)

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 15

(b)

minus020

02

Transverse plane-reflection mode caustics

y

minus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 3

Direction of caustic locus

120579in = minus35∘

120579in = minus15∘120579in = 05∘

120579in = 35∘

120579in = 15∘

S (x z)

(c)

Figure 7 Results of transverse plane ray tracing for various incoming beam directions (a) 119891119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

119875in (119875in119909 119875in119910 119875in119911) is calculated by solving the position of theray in 119911-direction

119875in119911 = 119875119903119911sin2120579in minus 119875119903119910 sin 120579in cos 120579in + 119897119900 cos 120579in (17)

Since the rays are rotated along the 119910-axis thus similar to the119875119903119910 the incident points also have the same values

119875in119910 = 119875119903119910 (18)

The calculation of the 119875in119909 is based on the same principleadopted in scanning plane and thus

119875in119909 = minuscot 120579in (119875in119911 minus 119897119900cos 120579in) (19)

The reflected vectors are calculated by deriving the normalvectors as in (10) and the prediction of the feed position canbe performed by using the same concept as done in scanningplane

4 Results and Discussion

41 Parametric Setup Ray tracing program for scanning andtransverse plane is performed for all major points on thereflectorThe analysis is carried out for a set of incident angles120579in to interpolate the locus of caustic and to observe thecaustic behaviour Critical antenna parameter such as 119891119863is varied to evaluate the effects towards the caustics To varythe 119891119863 the reflector diameter is kept constant at 5263mmwhilst only the focal length 119891 is changed The parameters ofstudy are summarized in Table 1

42 Displacement of Caustics The results of focal regionray tracing for scanning plane is shown in Figure 6 In thisanalysis the focal region of the parabolic reflector is analyzedfor five different incident angles which are set within therange of 120579in = minus35∘ to 35∘ The results are observed in119909119911-plane a two-dimensional graph that best represents the

International Journal of Antennas and Propagation 7

minus18minus16minus14minus12minus10minus08minus06minus04minus02

00minus08 minus06 minus04 minus02 00 02 04 06 08

Scanning plane fD = 1Scanning plane fD = 15Scanning plane fD = 3

Δx

Δz120579in = minus20∘

f= 0789m

f= 1578mParabolic reflector withdiameter D = 0526m

Figure 8 Scanning plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

minus18

minus18minus16minus16

minus14 minus14

minus15 15minus05 05

minus12minus12

minus10

minus10

minus08

minus08

minus06

minus06

minus04

minus04

minus02

02

02

minus02

minus02

00minus16 minus12 minus08 minus04 00

0

04 08 12 16

Transverse plane fD = 1Transverse plane fD = 15Transverse plane fD = 3

z

yx

f= 1578mm

f= 0789mm

Δx

Δz

120579in = minus20∘

Figure 9 Transverse plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

configuration of the scanning plane mode Based on thegraphs locus of caustic can be interpolated as the incidentdirection is varied The graphs are drawn on the samescale thus the changes in antenna configuration and causticbehaviour can be demonstrated

From the results it can be observed that as the incidentbeam is increased the caustics moved further away fromthe origin As seen from the zoomed diagrams the qualityof focusing also changed which means that the caustics areless converged at higher 120579in On the contrary the causticsare more converged and easier to determine when the119891119863 is increased Therefore it is expected that at higher119891119863 configuration the calculation of caustic points is moreaccurate and thus provide more reliable results The causticpoints for scanning plane are measured in terms of 119909- and 119911-axis displacement or denoted as Δ119909119904 and Δ119911119904 respectively As

minus05

minus04

minus03

minus02

minus01

00

01

minus10 minus08 minus06 minus04 minus02 00 02 04 06 08 10

Scanning plane fD = 1 Transverse plane fD = 1Scanning plane fD = 15 Transverse plane fD = 15Scanning plane fD = 3 Transverse plane fD = 3S(x z) for scanning plane S(x z) for transverse plane

Δzf

Scanning plane

Transverse plane

Δxf

Figure 10 Comparison of two-dimensional caustic displacementwith the equation lines of caustic loci 119878(119909 119911)

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 11 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 15 configuration

seen from Figure 6(c) a line is drawn to interpolate the locusof causticThe line of caustic locus is represented by a variable119878(119909 119911) which represents the distance between the centre ofparabolic reflector to the caustic points

Figure 7 shows the results of transverse plane ray tracingDue to the complexity of the transverse plane the behavioursof the transverse plane caustics are best observed in three-dimensional axes The axes are rotated several times duringmeasurement in order to calculate the 119909- and 119911-axis displace-ment also known as Δ119909119905 and Δ119911119905 respectively From thegraphs it can be seen that the caustics are less converged athigher 120579 in However similar to scanning plane the effectsare less when the 119891119863 is increased The caustic locus is

8 International Journal of Antennas and Propagation

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 12 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 15 configuration

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 13 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 3 configuration

interpolated and the distance 119878(119909 119911) as shown in Figure 7(c)is calculated and will be used in the next analysis

43 Equation of Caustic Locus The results of caustic dis-placements are plotted and the trend is observed Specificexamples of the caustic points with the comparison betweentwo different reflector configurations (119891119863 = 15 and 3)are presented in Figures 8 and 9 for scanning plane andtransverse plane respectively The convergence of parallelincident rays with 120579in = minus20∘ is shown in both diagrams The

0 5 10 15 20 25 30minus55minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus50

Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 14 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 3 configuration

caustics of other incident angles within the range of plusmn35∘ areplotted In transverse plane a clearer view of the convergenceof rays is shown in the zoomed image

Based on the results of caustic displacements the trajec-tory of the caustics can be approximated as follows

(a) in the scanning plane

119878 (119909 119911) = 119891 cos 120579in (20)

(b) in the transverse plane

119878 (119909 119911) = 119891cos 120579in (21)

Hence the locus of the caustics in the scanning planeis a circular arc with a curvature that is smaller by cos(120579in)Meanwhile for the transverse plane the trajectory is also acircular arc but with a curvature that is bigger by cos(120579in)Thus if an antenna system is to be designed to form a beam atthe desired direction given by 120579in the feed will be positionedon the curvature given by (20) and (21) on the scanningplane and the transverse plane respectivelyThe accuracy andvalidity of the loci equations can be shown in Figure 10 In thisdiagram the normalized caustic displacement data obtainedfrom both scanning plane and transverse plane analysis areplotted and lines joining the points are drawn to interpolatethe locus The lines of equations for caustic loci are alsoplotted according to the incident beamdirection Both resultsare compared and based on the plots it can be observedthat the lines of (20) and (21) almost matched with thecaustic displacement data of scanning and transverse planerespectively Therefore the general equations for caustic locican be used as a guide to preliminary locate the feed positionsof parabolic reflector antenna especially for the design ofmultibeam or shaped beam system

International Journal of Antennas and Propagation 9

44 Radiation Characteristics To ensure the accuracy of themethod the results of caustic obtained in the previous sectionare verified through the calculation of the radiation patternThe relative radiation pattern is calculated by using the fol-lowing equations where 120585(119909 119910) represents the aperture phasedistribution obtained through comparison of ray length froma feed to all aperture points [11]

119864119903 = intlfloorint 119890(119895119896119903 sin 120579 cos(ΦminusΦ1015840)+119895120577(119909119910))119889Φ1015840rfloor 119903119889119903where 119903 = 120588 sin 120579

(22)

Several examples based on the incident direction of 120579in =minus20∘ are shown here in order to demonstrate the behaviourof the radiation pattern Figure 11 and Figure 12 show theradiation characteristics of a parabolic reflector with 119891119863 =15 In these configurations the feed locations 119865(119909 119910 119911) areset to be located at two different positions Firstly the feed isset at the caustic point (Δ119909119904 Δ119911119904) or (Δ119909119905 Δ119911119905) that has beenobtained in the previous ray tracing program for scanningplane and transverse plane respectively Then the resultis compared to the radiation pattern measured when thefeed is placed at the position determined through the locusequation (20) and (21) The results of both scanning planeand transverse plane are shown

From Figure 11 themaximum gain is measured at around20∘ offset from the origin which is the expected result sincethe feed location was previously set at the minus 20∘ caustic pointHowever the plot given by (20) gives better representation ofthe exact beam shiftMeanwhile as for transverse planemoredeteriorate patterns are observed

For both scanning and transverse planes by comparingthe caustic displacement results with the ldquobluerdquo line it seemslike the locus equations for both scanning and transverse pro-duced more accurate beams with the highest gain obtainedat the direction closer to the desired beam direction In thiscase the equations of caustic loci become very useful indetermining the accurate position of the feed However asexpected the analysis conducted in scanning mode providesmore accurate results as compared to the transversemode Toinvestigate further similar exercise is carried out for119891119863 = 3configuration

Similar to 119891119863 = 15 it can be observed from Figures13 and 14 that the equations of caustic loci bring the beamdirection closer to the desired 120579in = minus20∘ direction althoughit seems like in transverse plane the result deviates further ascompared to scanning plane As shown by scanning plane inFigure 13 the radiation patterns for both feed configurationsoverlapped each other due to the similarity of both valuesThe highest gain is also measured at the same point Theagreement of the radiation gain values obtained at the twodifferent feed configurations validates the equation of causticloci derived by author in the previous section Therefore theaccuracy of the ray tracing algorithms for scanning plane andtransverse plane is ensured

5 Conclusion

A set of algorithms developed for the scanning plane andtransverse plane ray tracing program are presented Theresults obtained from the focal region ray tracing are dis-cussed and based on the caustic displacement results usefulequations representing the caustic loci are derived Theimportance of the equations are shown which could beused in designing a multibeam antenna application Off-feedcaustic displacements obtained in ray tracing method arecompared with the approximate equation of locus throughcalculation of radiation pattern The results of beam shiftshow good agreement between these two values and thusthe accuracy of the ray tracing method and the reliability ofthe locus equation for scanning plane and transverse planeanalysis are proven

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Teknologi MARAfor continuous support towards this research project

References

[1] R Azim M T Islam N Misran S W Cheung and YYamada ldquoPlanar UWB antenna with multi-slotted groundplanerdquo Microwave and Optical Technology Letters vol 53 no5 pp 966ndash968 2011

[2] L Liu S W Cheung R Azim and M T Islam ldquoA com-pact circular-ring antenna for ultra-wideband applicationsrdquoMicrowave and Optical Technology Letters vol 53 no 10 pp2283ndash2288 2011

[3] K Shogen S Tanaka and S Nakazawa ldquoStudies on the onboardAntennas for 21 GHz band broadcasting satellites and futurestudy itemsrdquo IEICE Transactions on Communications vol J94-B no 9 pp 1014ndash1024 2011

[4] G Toso C Mangenot and P Angeletti ldquoRecent advanceson space multibeam antennas based on a single aperturerdquoin Proceedings of the European Conference on Antennas andPropagation pp 454ndash458 April 2013

[5] S Yun M Uhm J Choi and I Yom ldquoMultibeam reflectorantenna fed by few elements for Ka-band communicationsatelliterdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 1ndash2 July 2012

[6] M Schneider C Hartwanger and H Wolf ldquoAntennas formultiple spot beam satellitesrdquo CEAS Space Journal vol 2 no1ndash4 pp 59ndash66 2011

[7] M Mahajan R Joti K Sood and S B Sharman ldquoA methodof generating simultaneous contoured and pencil beams fromsingle shaped reflector antennardquo IEEETransactions onAntennasand Propagation vol 61 no 10 pp 5297ndash5301 2013

[8] R E Collin and F J Zucker Antenna Theory Part 2 McGraw-Hill New York NY USA 1969

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012

4 International Journal of Antennas and Propagation

z

y

Incident plane

Incident raysLocus of caustic

x

CausticPrx Pry Prz)

F998400

F

= (PinPin x Piny Pinz)

P(120579) = (

minus120579inymax =D

2

ymin =minusD

2

Figure 5 Illustration of transverse plane in a single reflector system

The next step is to calculate the unit reflected vector119903119887=[119903119887119909 119903119887119910 119903119887119911] To solve that firstly the normal unit vectoron the parabolic reflector 119899119887= [119899119887119909 119899119887119910 119899119887119911] is derived asfollows

119899119887 = (minus sin(1205792) 0 cos(120579

2)) (9)

Then by applying the condition that all vectors exist on thesame plane the simultaneous equations of the vector dotproduct and cross product are solved [7] The vectors 119894119887 and119899119887 are expressed in terms of 1205850 as follows where 1205850 representsthe incident angle at the reflector surface

119899119887119909119894119887119909 + 119899119887119910119894119887119910 + 119899119887119911119894119887119911 = cos 1205850 (10)

The equation is expanded as follows assuming that theincidence angle 1205850 is equal to the reflected angle 120585119894 measuredfrom the normal vector 119899119887

119899119887119910119894119887119911 minus 119899119887119911119894119887119910 = 119899119887119910119903119887119911 minus 119899119887119911119903119887119910119899119887119911119894119887119909 minus 119899119887119909119894119887119911 = 119899119887119911119903119887119911 minus 119899119887119909119903119887119910119899119887119909119894119887119910 minus 119899119887119910119894119887119909 = 119899119887119909119903119887119910 minus 119899119887119910119903119887119909

(11)

The left hand side of (11) is given by known values thereforethe equations become constant and are expressed in terms of119886119887119909 119886119887119910 and 119886119887119911 respectively Finally by taking into accountall the relationships defined above the unknowns of 119903119887119909 119903119887119910and 119903119887119911 are solved

119903119887119909 = minus119899119909 cos 1205850 + 119899119887119911119886119887119910 minus 119899119887119910119886119887119911119903119887119910 = minus119899119910 cos 1205850 + 119899119887119909119886119887119911 minus 119899119887119911119886119887119909119903119887119911 = minus119899119911 cos 1205850 + 119899119887119910119886119887119909 minus 119899119887119909119886119887119910

(12)

In order to calculate the caustic points all rays reflectedfrom focal surface to the focal region will be expressed anddisplayed The focal region consists of a set of focal points

119865119887 = [119865119887119909 119865119887119910 119865119887119911] The equations of the 119865119887 defined in 119909- 119910-and 119911-coordinates are derived as follows

119865119887119909 = 119875119903119909 + 120588119903119887119909119865119887119910 = 119875119903119910 + 120588119903119887119910119865119887119911 = 119875119903119911 + 120588119903119887119911

(13)

The focal points are plotted and the most converged area orpoint is determined as caustic and will be used throughoutthe study for the calculation of feed position

33 Transverse Plane Theconfiguration of parabolic antennawith the illustration of transverse plane is shown in Figure 5From the diagram it shows that the incident rays exist ina plane perpendicular to the previous scanning plane Theincident rayswill cross the119910-axis before being reflected by thereflector surface hence the119910-coordinates remain unchangedthroughout the procedure

As explained in scanning plane the reflector points119875(120579)will be determined in terms of 120588 120579 and 120601 where 120601 is definedas the azimuth angle measured from the centre of reflector asillustrated in Figure 1Thus the119875(120579) in119909-119910- and 119911-directionis determined by the following equations

119875119903119909 = 120588 sin 120579 cos120601119875119903119910 = 120588 sin 120579 sin120601119875119903119911 = minus 120588 cos 120579

(14)

However in this case the values of 120579 and 120601 are unknownand will be determined by solving various equations Intransverse plane 119875119903119910 is fixed to 1199101 which ranges from minus1198632to 1198632 Based on the condition of incident rays in Figure 5the following expression is derived

tan 120579in = minus119875119903119909119875119903119911 =sin 120579119898 cos120601

cos 120579119898 (15)

By substituting 120588 to the equation of 119875119903119910 in (14) the followingequation is obtained

21198911 + cos 120579119898 sin 120579119898 sin120601 = 1199101 (16)

International Journal of Antennas and Propagation 5

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1Scanning plane-reflection mode caustics

z

x

fD = 1

(a)

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1

z

x

fD = 15

Scanning plane-reflection mode caustics

(b)

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1

z

x

Direction of caustic locus S (x z)

120579in = minus35∘120579in = minus15∘ 120579in = 05∘

120579in = 35∘120579in = 15∘

Scanning plane-reflection mode caustics

fD = 3

(c)

Figure 6 Results of scanning plane ray tracing for various incoming beam directions (a) 119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

In order to derive the value of 120579 and 120601 (15) and (16)will be solved simultaneously By using the technique calledldquocompleting the squarerdquo the quadratic equations can besolved and hence the values of 120579 and 120601 are obtained Thus

the exact parabolic surface points as mentioned in (14) canbe determined On the other hand the plane of incident rays

6 International Journal of Antennas and Propagation

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 1

(a)

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 15

(b)

minus020

02

Transverse plane-reflection mode caustics

y

minus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 3

Direction of caustic locus

120579in = minus35∘

120579in = minus15∘120579in = 05∘

120579in = 35∘

120579in = 15∘

S (x z)

(c)

Figure 7 Results of transverse plane ray tracing for various incoming beam directions (a) 119891119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

119875in (119875in119909 119875in119910 119875in119911) is calculated by solving the position of theray in 119911-direction

119875in119911 = 119875119903119911sin2120579in minus 119875119903119910 sin 120579in cos 120579in + 119897119900 cos 120579in (17)

Since the rays are rotated along the 119910-axis thus similar to the119875119903119910 the incident points also have the same values

119875in119910 = 119875119903119910 (18)

The calculation of the 119875in119909 is based on the same principleadopted in scanning plane and thus

119875in119909 = minuscot 120579in (119875in119911 minus 119897119900cos 120579in) (19)

The reflected vectors are calculated by deriving the normalvectors as in (10) and the prediction of the feed position canbe performed by using the same concept as done in scanningplane

4 Results and Discussion

41 Parametric Setup Ray tracing program for scanning andtransverse plane is performed for all major points on thereflectorThe analysis is carried out for a set of incident angles120579in to interpolate the locus of caustic and to observe thecaustic behaviour Critical antenna parameter such as 119891119863is varied to evaluate the effects towards the caustics To varythe 119891119863 the reflector diameter is kept constant at 5263mmwhilst only the focal length 119891 is changed The parameters ofstudy are summarized in Table 1

42 Displacement of Caustics The results of focal regionray tracing for scanning plane is shown in Figure 6 In thisanalysis the focal region of the parabolic reflector is analyzedfor five different incident angles which are set within therange of 120579in = minus35∘ to 35∘ The results are observed in119909119911-plane a two-dimensional graph that best represents the

International Journal of Antennas and Propagation 7

minus18minus16minus14minus12minus10minus08minus06minus04minus02

00minus08 minus06 minus04 minus02 00 02 04 06 08

Scanning plane fD = 1Scanning plane fD = 15Scanning plane fD = 3

Δx

Δz120579in = minus20∘

f= 0789m

f= 1578mParabolic reflector withdiameter D = 0526m

Figure 8 Scanning plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

minus18

minus18minus16minus16

minus14 minus14

minus15 15minus05 05

minus12minus12

minus10

minus10

minus08

minus08

minus06

minus06

minus04

minus04

minus02

02

02

minus02

minus02

00minus16 minus12 minus08 minus04 00

0

04 08 12 16

Transverse plane fD = 1Transverse plane fD = 15Transverse plane fD = 3

z

yx

f= 1578mm

f= 0789mm

Δx

Δz

120579in = minus20∘

Figure 9 Transverse plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

configuration of the scanning plane mode Based on thegraphs locus of caustic can be interpolated as the incidentdirection is varied The graphs are drawn on the samescale thus the changes in antenna configuration and causticbehaviour can be demonstrated

From the results it can be observed that as the incidentbeam is increased the caustics moved further away fromthe origin As seen from the zoomed diagrams the qualityof focusing also changed which means that the caustics areless converged at higher 120579in On the contrary the causticsare more converged and easier to determine when the119891119863 is increased Therefore it is expected that at higher119891119863 configuration the calculation of caustic points is moreaccurate and thus provide more reliable results The causticpoints for scanning plane are measured in terms of 119909- and 119911-axis displacement or denoted as Δ119909119904 and Δ119911119904 respectively As

minus05

minus04

minus03

minus02

minus01

00

01

minus10 minus08 minus06 minus04 minus02 00 02 04 06 08 10

Scanning plane fD = 1 Transverse plane fD = 1Scanning plane fD = 15 Transverse plane fD = 15Scanning plane fD = 3 Transverse plane fD = 3S(x z) for scanning plane S(x z) for transverse plane

Δzf

Scanning plane

Transverse plane

Δxf

Figure 10 Comparison of two-dimensional caustic displacementwith the equation lines of caustic loci 119878(119909 119911)

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 11 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 15 configuration

seen from Figure 6(c) a line is drawn to interpolate the locusof causticThe line of caustic locus is represented by a variable119878(119909 119911) which represents the distance between the centre ofparabolic reflector to the caustic points

Figure 7 shows the results of transverse plane ray tracingDue to the complexity of the transverse plane the behavioursof the transverse plane caustics are best observed in three-dimensional axes The axes are rotated several times duringmeasurement in order to calculate the 119909- and 119911-axis displace-ment also known as Δ119909119905 and Δ119911119905 respectively From thegraphs it can be seen that the caustics are less converged athigher 120579 in However similar to scanning plane the effectsare less when the 119891119863 is increased The caustic locus is

8 International Journal of Antennas and Propagation

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 12 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 15 configuration

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 13 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 3 configuration

interpolated and the distance 119878(119909 119911) as shown in Figure 7(c)is calculated and will be used in the next analysis

43 Equation of Caustic Locus The results of caustic dis-placements are plotted and the trend is observed Specificexamples of the caustic points with the comparison betweentwo different reflector configurations (119891119863 = 15 and 3)are presented in Figures 8 and 9 for scanning plane andtransverse plane respectively The convergence of parallelincident rays with 120579in = minus20∘ is shown in both diagrams The

0 5 10 15 20 25 30minus55minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus50

Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 14 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 3 configuration

caustics of other incident angles within the range of plusmn35∘ areplotted In transverse plane a clearer view of the convergenceof rays is shown in the zoomed image

Based on the results of caustic displacements the trajec-tory of the caustics can be approximated as follows

(a) in the scanning plane

119878 (119909 119911) = 119891 cos 120579in (20)

(b) in the transverse plane

119878 (119909 119911) = 119891cos 120579in (21)

Hence the locus of the caustics in the scanning planeis a circular arc with a curvature that is smaller by cos(120579in)Meanwhile for the transverse plane the trajectory is also acircular arc but with a curvature that is bigger by cos(120579in)Thus if an antenna system is to be designed to form a beam atthe desired direction given by 120579in the feed will be positionedon the curvature given by (20) and (21) on the scanningplane and the transverse plane respectivelyThe accuracy andvalidity of the loci equations can be shown in Figure 10 In thisdiagram the normalized caustic displacement data obtainedfrom both scanning plane and transverse plane analysis areplotted and lines joining the points are drawn to interpolatethe locus The lines of equations for caustic loci are alsoplotted according to the incident beamdirection Both resultsare compared and based on the plots it can be observedthat the lines of (20) and (21) almost matched with thecaustic displacement data of scanning and transverse planerespectively Therefore the general equations for caustic locican be used as a guide to preliminary locate the feed positionsof parabolic reflector antenna especially for the design ofmultibeam or shaped beam system

International Journal of Antennas and Propagation 9

44 Radiation Characteristics To ensure the accuracy of themethod the results of caustic obtained in the previous sectionare verified through the calculation of the radiation patternThe relative radiation pattern is calculated by using the fol-lowing equations where 120585(119909 119910) represents the aperture phasedistribution obtained through comparison of ray length froma feed to all aperture points [11]

119864119903 = intlfloorint 119890(119895119896119903 sin 120579 cos(ΦminusΦ1015840)+119895120577(119909119910))119889Φ1015840rfloor 119903119889119903where 119903 = 120588 sin 120579

(22)

Several examples based on the incident direction of 120579in =minus20∘ are shown here in order to demonstrate the behaviourof the radiation pattern Figure 11 and Figure 12 show theradiation characteristics of a parabolic reflector with 119891119863 =15 In these configurations the feed locations 119865(119909 119910 119911) areset to be located at two different positions Firstly the feed isset at the caustic point (Δ119909119904 Δ119911119904) or (Δ119909119905 Δ119911119905) that has beenobtained in the previous ray tracing program for scanningplane and transverse plane respectively Then the resultis compared to the radiation pattern measured when thefeed is placed at the position determined through the locusequation (20) and (21) The results of both scanning planeand transverse plane are shown

From Figure 11 themaximum gain is measured at around20∘ offset from the origin which is the expected result sincethe feed location was previously set at the minus 20∘ caustic pointHowever the plot given by (20) gives better representation ofthe exact beam shiftMeanwhile as for transverse planemoredeteriorate patterns are observed

For both scanning and transverse planes by comparingthe caustic displacement results with the ldquobluerdquo line it seemslike the locus equations for both scanning and transverse pro-duced more accurate beams with the highest gain obtainedat the direction closer to the desired beam direction In thiscase the equations of caustic loci become very useful indetermining the accurate position of the feed However asexpected the analysis conducted in scanning mode providesmore accurate results as compared to the transversemode Toinvestigate further similar exercise is carried out for119891119863 = 3configuration

Similar to 119891119863 = 15 it can be observed from Figures13 and 14 that the equations of caustic loci bring the beamdirection closer to the desired 120579in = minus20∘ direction althoughit seems like in transverse plane the result deviates further ascompared to scanning plane As shown by scanning plane inFigure 13 the radiation patterns for both feed configurationsoverlapped each other due to the similarity of both valuesThe highest gain is also measured at the same point Theagreement of the radiation gain values obtained at the twodifferent feed configurations validates the equation of causticloci derived by author in the previous section Therefore theaccuracy of the ray tracing algorithms for scanning plane andtransverse plane is ensured

5 Conclusion

A set of algorithms developed for the scanning plane andtransverse plane ray tracing program are presented Theresults obtained from the focal region ray tracing are dis-cussed and based on the caustic displacement results usefulequations representing the caustic loci are derived Theimportance of the equations are shown which could beused in designing a multibeam antenna application Off-feedcaustic displacements obtained in ray tracing method arecompared with the approximate equation of locus throughcalculation of radiation pattern The results of beam shiftshow good agreement between these two values and thusthe accuracy of the ray tracing method and the reliability ofthe locus equation for scanning plane and transverse planeanalysis are proven

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Teknologi MARAfor continuous support towards this research project

References

[1] R Azim M T Islam N Misran S W Cheung and YYamada ldquoPlanar UWB antenna with multi-slotted groundplanerdquo Microwave and Optical Technology Letters vol 53 no5 pp 966ndash968 2011

[2] L Liu S W Cheung R Azim and M T Islam ldquoA com-pact circular-ring antenna for ultra-wideband applicationsrdquoMicrowave and Optical Technology Letters vol 53 no 10 pp2283ndash2288 2011

[3] K Shogen S Tanaka and S Nakazawa ldquoStudies on the onboardAntennas for 21 GHz band broadcasting satellites and futurestudy itemsrdquo IEICE Transactions on Communications vol J94-B no 9 pp 1014ndash1024 2011

[4] G Toso C Mangenot and P Angeletti ldquoRecent advanceson space multibeam antennas based on a single aperturerdquoin Proceedings of the European Conference on Antennas andPropagation pp 454ndash458 April 2013

[5] S Yun M Uhm J Choi and I Yom ldquoMultibeam reflectorantenna fed by few elements for Ka-band communicationsatelliterdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 1ndash2 July 2012

[6] M Schneider C Hartwanger and H Wolf ldquoAntennas formultiple spot beam satellitesrdquo CEAS Space Journal vol 2 no1ndash4 pp 59ndash66 2011

[7] M Mahajan R Joti K Sood and S B Sharman ldquoA methodof generating simultaneous contoured and pencil beams fromsingle shaped reflector antennardquo IEEETransactions onAntennasand Propagation vol 61 no 10 pp 5297ndash5301 2013

[8] R E Collin and F J Zucker Antenna Theory Part 2 McGraw-Hill New York NY USA 1969

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012

International Journal of Antennas and Propagation 5

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1Scanning plane-reflection mode caustics

z

x

fD = 1

(a)

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1

z

x

fD = 15

Scanning plane-reflection mode caustics

(b)

minus15 minus1 minus05 0 1

minus15

minus1

minus05

0

05

05 15

1

z

x

Direction of caustic locus S (x z)

120579in = minus35∘120579in = minus15∘ 120579in = 05∘

120579in = 35∘120579in = 15∘

Scanning plane-reflection mode caustics

fD = 3

(c)

Figure 6 Results of scanning plane ray tracing for various incoming beam directions (a) 119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

In order to derive the value of 120579 and 120601 (15) and (16)will be solved simultaneously By using the technique calledldquocompleting the squarerdquo the quadratic equations can besolved and hence the values of 120579 and 120601 are obtained Thus

the exact parabolic surface points as mentioned in (14) canbe determined On the other hand the plane of incident rays

6 International Journal of Antennas and Propagation

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 1

(a)

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 15

(b)

minus020

02

Transverse plane-reflection mode caustics

y

minus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 3

Direction of caustic locus

120579in = minus35∘

120579in = minus15∘120579in = 05∘

120579in = 35∘

120579in = 15∘

S (x z)

(c)

Figure 7 Results of transverse plane ray tracing for various incoming beam directions (a) 119891119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

119875in (119875in119909 119875in119910 119875in119911) is calculated by solving the position of theray in 119911-direction

119875in119911 = 119875119903119911sin2120579in minus 119875119903119910 sin 120579in cos 120579in + 119897119900 cos 120579in (17)

Since the rays are rotated along the 119910-axis thus similar to the119875119903119910 the incident points also have the same values

119875in119910 = 119875119903119910 (18)

The calculation of the 119875in119909 is based on the same principleadopted in scanning plane and thus

119875in119909 = minuscot 120579in (119875in119911 minus 119897119900cos 120579in) (19)

The reflected vectors are calculated by deriving the normalvectors as in (10) and the prediction of the feed position canbe performed by using the same concept as done in scanningplane

4 Results and Discussion

41 Parametric Setup Ray tracing program for scanning andtransverse plane is performed for all major points on thereflectorThe analysis is carried out for a set of incident angles120579in to interpolate the locus of caustic and to observe thecaustic behaviour Critical antenna parameter such as 119891119863is varied to evaluate the effects towards the caustics To varythe 119891119863 the reflector diameter is kept constant at 5263mmwhilst only the focal length 119891 is changed The parameters ofstudy are summarized in Table 1

42 Displacement of Caustics The results of focal regionray tracing for scanning plane is shown in Figure 6 In thisanalysis the focal region of the parabolic reflector is analyzedfor five different incident angles which are set within therange of 120579in = minus35∘ to 35∘ The results are observed in119909119911-plane a two-dimensional graph that best represents the

International Journal of Antennas and Propagation 7

minus18minus16minus14minus12minus10minus08minus06minus04minus02

00minus08 minus06 minus04 minus02 00 02 04 06 08

Scanning plane fD = 1Scanning plane fD = 15Scanning plane fD = 3

Δx

Δz120579in = minus20∘

f= 0789m

f= 1578mParabolic reflector withdiameter D = 0526m

Figure 8 Scanning plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

minus18

minus18minus16minus16

minus14 minus14

minus15 15minus05 05

minus12minus12

minus10

minus10

minus08

minus08

minus06

minus06

minus04

minus04

minus02

02

02

minus02

minus02

00minus16 minus12 minus08 minus04 00

0

04 08 12 16

Transverse plane fD = 1Transverse plane fD = 15Transverse plane fD = 3

z

yx

f= 1578mm

f= 0789mm

Δx

Δz

120579in = minus20∘

Figure 9 Transverse plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

configuration of the scanning plane mode Based on thegraphs locus of caustic can be interpolated as the incidentdirection is varied The graphs are drawn on the samescale thus the changes in antenna configuration and causticbehaviour can be demonstrated

From the results it can be observed that as the incidentbeam is increased the caustics moved further away fromthe origin As seen from the zoomed diagrams the qualityof focusing also changed which means that the caustics areless converged at higher 120579in On the contrary the causticsare more converged and easier to determine when the119891119863 is increased Therefore it is expected that at higher119891119863 configuration the calculation of caustic points is moreaccurate and thus provide more reliable results The causticpoints for scanning plane are measured in terms of 119909- and 119911-axis displacement or denoted as Δ119909119904 and Δ119911119904 respectively As

minus05

minus04

minus03

minus02

minus01

00

01

minus10 minus08 minus06 minus04 minus02 00 02 04 06 08 10

Scanning plane fD = 1 Transverse plane fD = 1Scanning plane fD = 15 Transverse plane fD = 15Scanning plane fD = 3 Transverse plane fD = 3S(x z) for scanning plane S(x z) for transverse plane

Δzf

Scanning plane

Transverse plane

Δxf

Figure 10 Comparison of two-dimensional caustic displacementwith the equation lines of caustic loci 119878(119909 119911)

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 11 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 15 configuration

seen from Figure 6(c) a line is drawn to interpolate the locusof causticThe line of caustic locus is represented by a variable119878(119909 119911) which represents the distance between the centre ofparabolic reflector to the caustic points

Figure 7 shows the results of transverse plane ray tracingDue to the complexity of the transverse plane the behavioursof the transverse plane caustics are best observed in three-dimensional axes The axes are rotated several times duringmeasurement in order to calculate the 119909- and 119911-axis displace-ment also known as Δ119909119905 and Δ119911119905 respectively From thegraphs it can be seen that the caustics are less converged athigher 120579 in However similar to scanning plane the effectsare less when the 119891119863 is increased The caustic locus is

8 International Journal of Antennas and Propagation

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 12 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 15 configuration

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 13 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 3 configuration

interpolated and the distance 119878(119909 119911) as shown in Figure 7(c)is calculated and will be used in the next analysis

43 Equation of Caustic Locus The results of caustic dis-placements are plotted and the trend is observed Specificexamples of the caustic points with the comparison betweentwo different reflector configurations (119891119863 = 15 and 3)are presented in Figures 8 and 9 for scanning plane andtransverse plane respectively The convergence of parallelincident rays with 120579in = minus20∘ is shown in both diagrams The

0 5 10 15 20 25 30minus55minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus50

Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 14 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 3 configuration

caustics of other incident angles within the range of plusmn35∘ areplotted In transverse plane a clearer view of the convergenceof rays is shown in the zoomed image

Based on the results of caustic displacements the trajec-tory of the caustics can be approximated as follows

(a) in the scanning plane

119878 (119909 119911) = 119891 cos 120579in (20)

(b) in the transverse plane

119878 (119909 119911) = 119891cos 120579in (21)

Hence the locus of the caustics in the scanning planeis a circular arc with a curvature that is smaller by cos(120579in)Meanwhile for the transverse plane the trajectory is also acircular arc but with a curvature that is bigger by cos(120579in)Thus if an antenna system is to be designed to form a beam atthe desired direction given by 120579in the feed will be positionedon the curvature given by (20) and (21) on the scanningplane and the transverse plane respectivelyThe accuracy andvalidity of the loci equations can be shown in Figure 10 In thisdiagram the normalized caustic displacement data obtainedfrom both scanning plane and transverse plane analysis areplotted and lines joining the points are drawn to interpolatethe locus The lines of equations for caustic loci are alsoplotted according to the incident beamdirection Both resultsare compared and based on the plots it can be observedthat the lines of (20) and (21) almost matched with thecaustic displacement data of scanning and transverse planerespectively Therefore the general equations for caustic locican be used as a guide to preliminary locate the feed positionsof parabolic reflector antenna especially for the design ofmultibeam or shaped beam system

International Journal of Antennas and Propagation 9

44 Radiation Characteristics To ensure the accuracy of themethod the results of caustic obtained in the previous sectionare verified through the calculation of the radiation patternThe relative radiation pattern is calculated by using the fol-lowing equations where 120585(119909 119910) represents the aperture phasedistribution obtained through comparison of ray length froma feed to all aperture points [11]

119864119903 = intlfloorint 119890(119895119896119903 sin 120579 cos(ΦminusΦ1015840)+119895120577(119909119910))119889Φ1015840rfloor 119903119889119903where 119903 = 120588 sin 120579

(22)

Several examples based on the incident direction of 120579in =minus20∘ are shown here in order to demonstrate the behaviourof the radiation pattern Figure 11 and Figure 12 show theradiation characteristics of a parabolic reflector with 119891119863 =15 In these configurations the feed locations 119865(119909 119910 119911) areset to be located at two different positions Firstly the feed isset at the caustic point (Δ119909119904 Δ119911119904) or (Δ119909119905 Δ119911119905) that has beenobtained in the previous ray tracing program for scanningplane and transverse plane respectively Then the resultis compared to the radiation pattern measured when thefeed is placed at the position determined through the locusequation (20) and (21) The results of both scanning planeand transverse plane are shown

From Figure 11 themaximum gain is measured at around20∘ offset from the origin which is the expected result sincethe feed location was previously set at the minus 20∘ caustic pointHowever the plot given by (20) gives better representation ofthe exact beam shiftMeanwhile as for transverse planemoredeteriorate patterns are observed

For both scanning and transverse planes by comparingthe caustic displacement results with the ldquobluerdquo line it seemslike the locus equations for both scanning and transverse pro-duced more accurate beams with the highest gain obtainedat the direction closer to the desired beam direction In thiscase the equations of caustic loci become very useful indetermining the accurate position of the feed However asexpected the analysis conducted in scanning mode providesmore accurate results as compared to the transversemode Toinvestigate further similar exercise is carried out for119891119863 = 3configuration

Similar to 119891119863 = 15 it can be observed from Figures13 and 14 that the equations of caustic loci bring the beamdirection closer to the desired 120579in = minus20∘ direction althoughit seems like in transverse plane the result deviates further ascompared to scanning plane As shown by scanning plane inFigure 13 the radiation patterns for both feed configurationsoverlapped each other due to the similarity of both valuesThe highest gain is also measured at the same point Theagreement of the radiation gain values obtained at the twodifferent feed configurations validates the equation of causticloci derived by author in the previous section Therefore theaccuracy of the ray tracing algorithms for scanning plane andtransverse plane is ensured

5 Conclusion

A set of algorithms developed for the scanning plane andtransverse plane ray tracing program are presented Theresults obtained from the focal region ray tracing are dis-cussed and based on the caustic displacement results usefulequations representing the caustic loci are derived Theimportance of the equations are shown which could beused in designing a multibeam antenna application Off-feedcaustic displacements obtained in ray tracing method arecompared with the approximate equation of locus throughcalculation of radiation pattern The results of beam shiftshow good agreement between these two values and thusthe accuracy of the ray tracing method and the reliability ofthe locus equation for scanning plane and transverse planeanalysis are proven

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Teknologi MARAfor continuous support towards this research project

References

[1] R Azim M T Islam N Misran S W Cheung and YYamada ldquoPlanar UWB antenna with multi-slotted groundplanerdquo Microwave and Optical Technology Letters vol 53 no5 pp 966ndash968 2011

[2] L Liu S W Cheung R Azim and M T Islam ldquoA com-pact circular-ring antenna for ultra-wideband applicationsrdquoMicrowave and Optical Technology Letters vol 53 no 10 pp2283ndash2288 2011

[3] K Shogen S Tanaka and S Nakazawa ldquoStudies on the onboardAntennas for 21 GHz band broadcasting satellites and futurestudy itemsrdquo IEICE Transactions on Communications vol J94-B no 9 pp 1014ndash1024 2011

[4] G Toso C Mangenot and P Angeletti ldquoRecent advanceson space multibeam antennas based on a single aperturerdquoin Proceedings of the European Conference on Antennas andPropagation pp 454ndash458 April 2013

[5] S Yun M Uhm J Choi and I Yom ldquoMultibeam reflectorantenna fed by few elements for Ka-band communicationsatelliterdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 1ndash2 July 2012

[6] M Schneider C Hartwanger and H Wolf ldquoAntennas formultiple spot beam satellitesrdquo CEAS Space Journal vol 2 no1ndash4 pp 59ndash66 2011

[7] M Mahajan R Joti K Sood and S B Sharman ldquoA methodof generating simultaneous contoured and pencil beams fromsingle shaped reflector antennardquo IEEETransactions onAntennasand Propagation vol 61 no 10 pp 5297ndash5301 2013

[8] R E Collin and F J Zucker Antenna Theory Part 2 McGraw-Hill New York NY USA 1969

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012

6 International Journal of Antennas and Propagation

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 1

(a)

minus020

02

Transverse plane-reflection mode caustics

yminus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 15

(b)

minus020

02

Transverse plane-reflection mode caustics

y

minus15 minus1 minus05 0

0

1

minus15

minus1

minus05

05

05 15

1

z

x

fD = 3

Direction of caustic locus

120579in = minus35∘

120579in = minus15∘120579in = 05∘

120579in = 35∘

120579in = 15∘

S (x z)

(c)

Figure 7 Results of transverse plane ray tracing for various incoming beam directions (a) 119891119863 = 1 (b) 119891119863 = 15 and (c) 119891119863 = 3

119875in (119875in119909 119875in119910 119875in119911) is calculated by solving the position of theray in 119911-direction

119875in119911 = 119875119903119911sin2120579in minus 119875119903119910 sin 120579in cos 120579in + 119897119900 cos 120579in (17)

Since the rays are rotated along the 119910-axis thus similar to the119875119903119910 the incident points also have the same values

119875in119910 = 119875119903119910 (18)

The calculation of the 119875in119909 is based on the same principleadopted in scanning plane and thus

119875in119909 = minuscot 120579in (119875in119911 minus 119897119900cos 120579in) (19)

The reflected vectors are calculated by deriving the normalvectors as in (10) and the prediction of the feed position canbe performed by using the same concept as done in scanningplane

4 Results and Discussion

41 Parametric Setup Ray tracing program for scanning andtransverse plane is performed for all major points on thereflectorThe analysis is carried out for a set of incident angles120579in to interpolate the locus of caustic and to observe thecaustic behaviour Critical antenna parameter such as 119891119863is varied to evaluate the effects towards the caustics To varythe 119891119863 the reflector diameter is kept constant at 5263mmwhilst only the focal length 119891 is changed The parameters ofstudy are summarized in Table 1

42 Displacement of Caustics The results of focal regionray tracing for scanning plane is shown in Figure 6 In thisanalysis the focal region of the parabolic reflector is analyzedfor five different incident angles which are set within therange of 120579in = minus35∘ to 35∘ The results are observed in119909119911-plane a two-dimensional graph that best represents the

International Journal of Antennas and Propagation 7

minus18minus16minus14minus12minus10minus08minus06minus04minus02

00minus08 minus06 minus04 minus02 00 02 04 06 08

Scanning plane fD = 1Scanning plane fD = 15Scanning plane fD = 3

Δx

Δz120579in = minus20∘

f= 0789m

f= 1578mParabolic reflector withdiameter D = 0526m

Figure 8 Scanning plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

minus18

minus18minus16minus16

minus14 minus14

minus15 15minus05 05

minus12minus12

minus10

minus10

minus08

minus08

minus06

minus06

minus04

minus04

minus02

02

02

minus02

minus02

00minus16 minus12 minus08 minus04 00

0

04 08 12 16

Transverse plane fD = 1Transverse plane fD = 15Transverse plane fD = 3

z

yx

f= 1578mm

f= 0789mm

Δx

Δz

120579in = minus20∘

Figure 9 Transverse plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

configuration of the scanning plane mode Based on thegraphs locus of caustic can be interpolated as the incidentdirection is varied The graphs are drawn on the samescale thus the changes in antenna configuration and causticbehaviour can be demonstrated

From the results it can be observed that as the incidentbeam is increased the caustics moved further away fromthe origin As seen from the zoomed diagrams the qualityof focusing also changed which means that the caustics areless converged at higher 120579in On the contrary the causticsare more converged and easier to determine when the119891119863 is increased Therefore it is expected that at higher119891119863 configuration the calculation of caustic points is moreaccurate and thus provide more reliable results The causticpoints for scanning plane are measured in terms of 119909- and 119911-axis displacement or denoted as Δ119909119904 and Δ119911119904 respectively As

minus05

minus04

minus03

minus02

minus01

00

01

minus10 minus08 minus06 minus04 minus02 00 02 04 06 08 10

Scanning plane fD = 1 Transverse plane fD = 1Scanning plane fD = 15 Transverse plane fD = 15Scanning plane fD = 3 Transverse plane fD = 3S(x z) for scanning plane S(x z) for transverse plane

Δzf

Scanning plane

Transverse plane

Δxf

Figure 10 Comparison of two-dimensional caustic displacementwith the equation lines of caustic loci 119878(119909 119911)

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 11 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 15 configuration

seen from Figure 6(c) a line is drawn to interpolate the locusof causticThe line of caustic locus is represented by a variable119878(119909 119911) which represents the distance between the centre ofparabolic reflector to the caustic points

Figure 7 shows the results of transverse plane ray tracingDue to the complexity of the transverse plane the behavioursof the transverse plane caustics are best observed in three-dimensional axes The axes are rotated several times duringmeasurement in order to calculate the 119909- and 119911-axis displace-ment also known as Δ119909119905 and Δ119911119905 respectively From thegraphs it can be seen that the caustics are less converged athigher 120579 in However similar to scanning plane the effectsare less when the 119891119863 is increased The caustic locus is

8 International Journal of Antennas and Propagation

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 12 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 15 configuration

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 13 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 3 configuration

interpolated and the distance 119878(119909 119911) as shown in Figure 7(c)is calculated and will be used in the next analysis

43 Equation of Caustic Locus The results of caustic dis-placements are plotted and the trend is observed Specificexamples of the caustic points with the comparison betweentwo different reflector configurations (119891119863 = 15 and 3)are presented in Figures 8 and 9 for scanning plane andtransverse plane respectively The convergence of parallelincident rays with 120579in = minus20∘ is shown in both diagrams The

0 5 10 15 20 25 30minus55minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus50

Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 14 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 3 configuration

caustics of other incident angles within the range of plusmn35∘ areplotted In transverse plane a clearer view of the convergenceof rays is shown in the zoomed image

Based on the results of caustic displacements the trajec-tory of the caustics can be approximated as follows

(a) in the scanning plane

119878 (119909 119911) = 119891 cos 120579in (20)

(b) in the transverse plane

119878 (119909 119911) = 119891cos 120579in (21)

Hence the locus of the caustics in the scanning planeis a circular arc with a curvature that is smaller by cos(120579in)Meanwhile for the transverse plane the trajectory is also acircular arc but with a curvature that is bigger by cos(120579in)Thus if an antenna system is to be designed to form a beam atthe desired direction given by 120579in the feed will be positionedon the curvature given by (20) and (21) on the scanningplane and the transverse plane respectivelyThe accuracy andvalidity of the loci equations can be shown in Figure 10 In thisdiagram the normalized caustic displacement data obtainedfrom both scanning plane and transverse plane analysis areplotted and lines joining the points are drawn to interpolatethe locus The lines of equations for caustic loci are alsoplotted according to the incident beamdirection Both resultsare compared and based on the plots it can be observedthat the lines of (20) and (21) almost matched with thecaustic displacement data of scanning and transverse planerespectively Therefore the general equations for caustic locican be used as a guide to preliminary locate the feed positionsof parabolic reflector antenna especially for the design ofmultibeam or shaped beam system

International Journal of Antennas and Propagation 9

44 Radiation Characteristics To ensure the accuracy of themethod the results of caustic obtained in the previous sectionare verified through the calculation of the radiation patternThe relative radiation pattern is calculated by using the fol-lowing equations where 120585(119909 119910) represents the aperture phasedistribution obtained through comparison of ray length froma feed to all aperture points [11]

119864119903 = intlfloorint 119890(119895119896119903 sin 120579 cos(ΦminusΦ1015840)+119895120577(119909119910))119889Φ1015840rfloor 119903119889119903where 119903 = 120588 sin 120579

(22)

Several examples based on the incident direction of 120579in =minus20∘ are shown here in order to demonstrate the behaviourof the radiation pattern Figure 11 and Figure 12 show theradiation characteristics of a parabolic reflector with 119891119863 =15 In these configurations the feed locations 119865(119909 119910 119911) areset to be located at two different positions Firstly the feed isset at the caustic point (Δ119909119904 Δ119911119904) or (Δ119909119905 Δ119911119905) that has beenobtained in the previous ray tracing program for scanningplane and transverse plane respectively Then the resultis compared to the radiation pattern measured when thefeed is placed at the position determined through the locusequation (20) and (21) The results of both scanning planeand transverse plane are shown

From Figure 11 themaximum gain is measured at around20∘ offset from the origin which is the expected result sincethe feed location was previously set at the minus 20∘ caustic pointHowever the plot given by (20) gives better representation ofthe exact beam shiftMeanwhile as for transverse planemoredeteriorate patterns are observed

For both scanning and transverse planes by comparingthe caustic displacement results with the ldquobluerdquo line it seemslike the locus equations for both scanning and transverse pro-duced more accurate beams with the highest gain obtainedat the direction closer to the desired beam direction In thiscase the equations of caustic loci become very useful indetermining the accurate position of the feed However asexpected the analysis conducted in scanning mode providesmore accurate results as compared to the transversemode Toinvestigate further similar exercise is carried out for119891119863 = 3configuration

Similar to 119891119863 = 15 it can be observed from Figures13 and 14 that the equations of caustic loci bring the beamdirection closer to the desired 120579in = minus20∘ direction althoughit seems like in transverse plane the result deviates further ascompared to scanning plane As shown by scanning plane inFigure 13 the radiation patterns for both feed configurationsoverlapped each other due to the similarity of both valuesThe highest gain is also measured at the same point Theagreement of the radiation gain values obtained at the twodifferent feed configurations validates the equation of causticloci derived by author in the previous section Therefore theaccuracy of the ray tracing algorithms for scanning plane andtransverse plane is ensured

5 Conclusion

A set of algorithms developed for the scanning plane andtransverse plane ray tracing program are presented Theresults obtained from the focal region ray tracing are dis-cussed and based on the caustic displacement results usefulequations representing the caustic loci are derived Theimportance of the equations are shown which could beused in designing a multibeam antenna application Off-feedcaustic displacements obtained in ray tracing method arecompared with the approximate equation of locus throughcalculation of radiation pattern The results of beam shiftshow good agreement between these two values and thusthe accuracy of the ray tracing method and the reliability ofthe locus equation for scanning plane and transverse planeanalysis are proven

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Teknologi MARAfor continuous support towards this research project

References

[1] R Azim M T Islam N Misran S W Cheung and YYamada ldquoPlanar UWB antenna with multi-slotted groundplanerdquo Microwave and Optical Technology Letters vol 53 no5 pp 966ndash968 2011

[2] L Liu S W Cheung R Azim and M T Islam ldquoA com-pact circular-ring antenna for ultra-wideband applicationsrdquoMicrowave and Optical Technology Letters vol 53 no 10 pp2283ndash2288 2011

[3] K Shogen S Tanaka and S Nakazawa ldquoStudies on the onboardAntennas for 21 GHz band broadcasting satellites and futurestudy itemsrdquo IEICE Transactions on Communications vol J94-B no 9 pp 1014ndash1024 2011

[4] G Toso C Mangenot and P Angeletti ldquoRecent advanceson space multibeam antennas based on a single aperturerdquoin Proceedings of the European Conference on Antennas andPropagation pp 454ndash458 April 2013

[5] S Yun M Uhm J Choi and I Yom ldquoMultibeam reflectorantenna fed by few elements for Ka-band communicationsatelliterdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 1ndash2 July 2012

[6] M Schneider C Hartwanger and H Wolf ldquoAntennas formultiple spot beam satellitesrdquo CEAS Space Journal vol 2 no1ndash4 pp 59ndash66 2011

[7] M Mahajan R Joti K Sood and S B Sharman ldquoA methodof generating simultaneous contoured and pencil beams fromsingle shaped reflector antennardquo IEEETransactions onAntennasand Propagation vol 61 no 10 pp 5297ndash5301 2013

[8] R E Collin and F J Zucker Antenna Theory Part 2 McGraw-Hill New York NY USA 1969

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012

International Journal of Antennas and Propagation 7

minus18minus16minus14minus12minus10minus08minus06minus04minus02

00minus08 minus06 minus04 minus02 00 02 04 06 08

Scanning plane fD = 1Scanning plane fD = 15Scanning plane fD = 3

Δx

Δz120579in = minus20∘

f= 0789m

f= 1578mParabolic reflector withdiameter D = 0526m

Figure 8 Scanning plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

minus18

minus18minus16minus16

minus14 minus14

minus15 15minus05 05

minus12minus12

minus10

minus10

minus08

minus08

minus06

minus06

minus04

minus04

minus02

02

02

minus02

minus02

00minus16 minus12 minus08 minus04 00

0

04 08 12 16

Transverse plane fD = 1Transverse plane fD = 15Transverse plane fD = 3

z

yx

f= 1578mm

f= 0789mm

Δx

Δz

120579in = minus20∘

Figure 9 Transverse plane caustic in 119909119911-plane 120579in = minus20∘ with119891119863 = 15 and 3

configuration of the scanning plane mode Based on thegraphs locus of caustic can be interpolated as the incidentdirection is varied The graphs are drawn on the samescale thus the changes in antenna configuration and causticbehaviour can be demonstrated

From the results it can be observed that as the incidentbeam is increased the caustics moved further away fromthe origin As seen from the zoomed diagrams the qualityof focusing also changed which means that the caustics areless converged at higher 120579in On the contrary the causticsare more converged and easier to determine when the119891119863 is increased Therefore it is expected that at higher119891119863 configuration the calculation of caustic points is moreaccurate and thus provide more reliable results The causticpoints for scanning plane are measured in terms of 119909- and 119911-axis displacement or denoted as Δ119909119904 and Δ119911119904 respectively As

minus05

minus04

minus03

minus02

minus01

00

01

minus10 minus08 minus06 minus04 minus02 00 02 04 06 08 10

Scanning plane fD = 1 Transverse plane fD = 1Scanning plane fD = 15 Transverse plane fD = 15Scanning plane fD = 3 Transverse plane fD = 3S(x z) for scanning plane S(x z) for transverse plane

Δzf

Scanning plane

Transverse plane

Δxf

Figure 10 Comparison of two-dimensional caustic displacementwith the equation lines of caustic loci 119878(119909 119911)

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 11 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 15 configuration

seen from Figure 6(c) a line is drawn to interpolate the locusof causticThe line of caustic locus is represented by a variable119878(119909 119911) which represents the distance between the centre ofparabolic reflector to the caustic points

Figure 7 shows the results of transverse plane ray tracingDue to the complexity of the transverse plane the behavioursof the transverse plane caustics are best observed in three-dimensional axes The axes are rotated several times duringmeasurement in order to calculate the 119909- and 119911-axis displace-ment also known as Δ119909119905 and Δ119911119905 respectively From thegraphs it can be seen that the caustics are less converged athigher 120579 in However similar to scanning plane the effectsare less when the 119891119863 is increased The caustic locus is

8 International Journal of Antennas and Propagation

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 12 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 15 configuration

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 13 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 3 configuration

interpolated and the distance 119878(119909 119911) as shown in Figure 7(c)is calculated and will be used in the next analysis

43 Equation of Caustic Locus The results of caustic dis-placements are plotted and the trend is observed Specificexamples of the caustic points with the comparison betweentwo different reflector configurations (119891119863 = 15 and 3)are presented in Figures 8 and 9 for scanning plane andtransverse plane respectively The convergence of parallelincident rays with 120579in = minus20∘ is shown in both diagrams The

0 5 10 15 20 25 30minus55minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus50

Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 14 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 3 configuration

caustics of other incident angles within the range of plusmn35∘ areplotted In transverse plane a clearer view of the convergenceof rays is shown in the zoomed image

Based on the results of caustic displacements the trajec-tory of the caustics can be approximated as follows

(a) in the scanning plane

119878 (119909 119911) = 119891 cos 120579in (20)

(b) in the transverse plane

119878 (119909 119911) = 119891cos 120579in (21)

Hence the locus of the caustics in the scanning planeis a circular arc with a curvature that is smaller by cos(120579in)Meanwhile for the transverse plane the trajectory is also acircular arc but with a curvature that is bigger by cos(120579in)Thus if an antenna system is to be designed to form a beam atthe desired direction given by 120579in the feed will be positionedon the curvature given by (20) and (21) on the scanningplane and the transverse plane respectivelyThe accuracy andvalidity of the loci equations can be shown in Figure 10 In thisdiagram the normalized caustic displacement data obtainedfrom both scanning plane and transverse plane analysis areplotted and lines joining the points are drawn to interpolatethe locus The lines of equations for caustic loci are alsoplotted according to the incident beamdirection Both resultsare compared and based on the plots it can be observedthat the lines of (20) and (21) almost matched with thecaustic displacement data of scanning and transverse planerespectively Therefore the general equations for caustic locican be used as a guide to preliminary locate the feed positionsof parabolic reflector antenna especially for the design ofmultibeam or shaped beam system

International Journal of Antennas and Propagation 9

44 Radiation Characteristics To ensure the accuracy of themethod the results of caustic obtained in the previous sectionare verified through the calculation of the radiation patternThe relative radiation pattern is calculated by using the fol-lowing equations where 120585(119909 119910) represents the aperture phasedistribution obtained through comparison of ray length froma feed to all aperture points [11]

119864119903 = intlfloorint 119890(119895119896119903 sin 120579 cos(ΦminusΦ1015840)+119895120577(119909119910))119889Φ1015840rfloor 119903119889119903where 119903 = 120588 sin 120579

(22)

Several examples based on the incident direction of 120579in =minus20∘ are shown here in order to demonstrate the behaviourof the radiation pattern Figure 11 and Figure 12 show theradiation characteristics of a parabolic reflector with 119891119863 =15 In these configurations the feed locations 119865(119909 119910 119911) areset to be located at two different positions Firstly the feed isset at the caustic point (Δ119909119904 Δ119911119904) or (Δ119909119905 Δ119911119905) that has beenobtained in the previous ray tracing program for scanningplane and transverse plane respectively Then the resultis compared to the radiation pattern measured when thefeed is placed at the position determined through the locusequation (20) and (21) The results of both scanning planeand transverse plane are shown

From Figure 11 themaximum gain is measured at around20∘ offset from the origin which is the expected result sincethe feed location was previously set at the minus 20∘ caustic pointHowever the plot given by (20) gives better representation ofthe exact beam shiftMeanwhile as for transverse planemoredeteriorate patterns are observed

For both scanning and transverse planes by comparingthe caustic displacement results with the ldquobluerdquo line it seemslike the locus equations for both scanning and transverse pro-duced more accurate beams with the highest gain obtainedat the direction closer to the desired beam direction In thiscase the equations of caustic loci become very useful indetermining the accurate position of the feed However asexpected the analysis conducted in scanning mode providesmore accurate results as compared to the transversemode Toinvestigate further similar exercise is carried out for119891119863 = 3configuration

Similar to 119891119863 = 15 it can be observed from Figures13 and 14 that the equations of caustic loci bring the beamdirection closer to the desired 120579in = minus20∘ direction althoughit seems like in transverse plane the result deviates further ascompared to scanning plane As shown by scanning plane inFigure 13 the radiation patterns for both feed configurationsoverlapped each other due to the similarity of both valuesThe highest gain is also measured at the same point Theagreement of the radiation gain values obtained at the twodifferent feed configurations validates the equation of causticloci derived by author in the previous section Therefore theaccuracy of the ray tracing algorithms for scanning plane andtransverse plane is ensured

5 Conclusion

A set of algorithms developed for the scanning plane andtransverse plane ray tracing program are presented Theresults obtained from the focal region ray tracing are dis-cussed and based on the caustic displacement results usefulequations representing the caustic loci are derived Theimportance of the equations are shown which could beused in designing a multibeam antenna application Off-feedcaustic displacements obtained in ray tracing method arecompared with the approximate equation of locus throughcalculation of radiation pattern The results of beam shiftshow good agreement between these two values and thusthe accuracy of the ray tracing method and the reliability ofthe locus equation for scanning plane and transverse planeanalysis are proven

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Teknologi MARAfor continuous support towards this research project

References

[1] R Azim M T Islam N Misran S W Cheung and YYamada ldquoPlanar UWB antenna with multi-slotted groundplanerdquo Microwave and Optical Technology Letters vol 53 no5 pp 966ndash968 2011

[2] L Liu S W Cheung R Azim and M T Islam ldquoA com-pact circular-ring antenna for ultra-wideband applicationsrdquoMicrowave and Optical Technology Letters vol 53 no 10 pp2283ndash2288 2011

[3] K Shogen S Tanaka and S Nakazawa ldquoStudies on the onboardAntennas for 21 GHz band broadcasting satellites and futurestudy itemsrdquo IEICE Transactions on Communications vol J94-B no 9 pp 1014ndash1024 2011

[4] G Toso C Mangenot and P Angeletti ldquoRecent advanceson space multibeam antennas based on a single aperturerdquoin Proceedings of the European Conference on Antennas andPropagation pp 454ndash458 April 2013

[5] S Yun M Uhm J Choi and I Yom ldquoMultibeam reflectorantenna fed by few elements for Ka-band communicationsatelliterdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 1ndash2 July 2012

[6] M Schneider C Hartwanger and H Wolf ldquoAntennas formultiple spot beam satellitesrdquo CEAS Space Journal vol 2 no1ndash4 pp 59ndash66 2011

[7] M Mahajan R Joti K Sood and S B Sharman ldquoA methodof generating simultaneous contoured and pencil beams fromsingle shaped reflector antennardquo IEEETransactions onAntennasand Propagation vol 61 no 10 pp 5297ndash5301 2013

[8] R E Collin and F J Zucker Antenna Theory Part 2 McGraw-Hill New York NY USA 1969

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012

8 International Journal of Antennas and Propagation

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 12 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 15 configuration

0 5 10 15 20 25 30minus55

minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (20)Caustic displacement (Δxs Δzs)

Figure 13 Comparison of relative radiation patterns using scanningplane caustic and locus for 119891119863 = 3 configuration

interpolated and the distance 119878(119909 119911) as shown in Figure 7(c)is calculated and will be used in the next analysis

43 Equation of Caustic Locus The results of caustic dis-placements are plotted and the trend is observed Specificexamples of the caustic points with the comparison betweentwo different reflector configurations (119891119863 = 15 and 3)are presented in Figures 8 and 9 for scanning plane andtransverse plane respectively The convergence of parallelincident rays with 120579in = minus20∘ is shown in both diagrams The

0 5 10 15 20 25 30minus55minus50

minus45

minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus50

Relative radiation pattern

Rad

Pat

(dB

)

120579 (deg)

Feed position given by Eq (21)Caustic displacement (Δxt Δzt)

Figure 14 Comparison of relative radiation patterns using trans-verse plane caustic and locus for 119891119863 = 3 configuration

caustics of other incident angles within the range of plusmn35∘ areplotted In transverse plane a clearer view of the convergenceof rays is shown in the zoomed image

Based on the results of caustic displacements the trajec-tory of the caustics can be approximated as follows

(a) in the scanning plane

119878 (119909 119911) = 119891 cos 120579in (20)

(b) in the transverse plane

119878 (119909 119911) = 119891cos 120579in (21)

Hence the locus of the caustics in the scanning planeis a circular arc with a curvature that is smaller by cos(120579in)Meanwhile for the transverse plane the trajectory is also acircular arc but with a curvature that is bigger by cos(120579in)Thus if an antenna system is to be designed to form a beam atthe desired direction given by 120579in the feed will be positionedon the curvature given by (20) and (21) on the scanningplane and the transverse plane respectivelyThe accuracy andvalidity of the loci equations can be shown in Figure 10 In thisdiagram the normalized caustic displacement data obtainedfrom both scanning plane and transverse plane analysis areplotted and lines joining the points are drawn to interpolatethe locus The lines of equations for caustic loci are alsoplotted according to the incident beamdirection Both resultsare compared and based on the plots it can be observedthat the lines of (20) and (21) almost matched with thecaustic displacement data of scanning and transverse planerespectively Therefore the general equations for caustic locican be used as a guide to preliminary locate the feed positionsof parabolic reflector antenna especially for the design ofmultibeam or shaped beam system

International Journal of Antennas and Propagation 9

44 Radiation Characteristics To ensure the accuracy of themethod the results of caustic obtained in the previous sectionare verified through the calculation of the radiation patternThe relative radiation pattern is calculated by using the fol-lowing equations where 120585(119909 119910) represents the aperture phasedistribution obtained through comparison of ray length froma feed to all aperture points [11]

119864119903 = intlfloorint 119890(119895119896119903 sin 120579 cos(ΦminusΦ1015840)+119895120577(119909119910))119889Φ1015840rfloor 119903119889119903where 119903 = 120588 sin 120579

(22)

Several examples based on the incident direction of 120579in =minus20∘ are shown here in order to demonstrate the behaviourof the radiation pattern Figure 11 and Figure 12 show theradiation characteristics of a parabolic reflector with 119891119863 =15 In these configurations the feed locations 119865(119909 119910 119911) areset to be located at two different positions Firstly the feed isset at the caustic point (Δ119909119904 Δ119911119904) or (Δ119909119905 Δ119911119905) that has beenobtained in the previous ray tracing program for scanningplane and transverse plane respectively Then the resultis compared to the radiation pattern measured when thefeed is placed at the position determined through the locusequation (20) and (21) The results of both scanning planeand transverse plane are shown

From Figure 11 themaximum gain is measured at around20∘ offset from the origin which is the expected result sincethe feed location was previously set at the minus 20∘ caustic pointHowever the plot given by (20) gives better representation ofthe exact beam shiftMeanwhile as for transverse planemoredeteriorate patterns are observed

For both scanning and transverse planes by comparingthe caustic displacement results with the ldquobluerdquo line it seemslike the locus equations for both scanning and transverse pro-duced more accurate beams with the highest gain obtainedat the direction closer to the desired beam direction In thiscase the equations of caustic loci become very useful indetermining the accurate position of the feed However asexpected the analysis conducted in scanning mode providesmore accurate results as compared to the transversemode Toinvestigate further similar exercise is carried out for119891119863 = 3configuration

Similar to 119891119863 = 15 it can be observed from Figures13 and 14 that the equations of caustic loci bring the beamdirection closer to the desired 120579in = minus20∘ direction althoughit seems like in transverse plane the result deviates further ascompared to scanning plane As shown by scanning plane inFigure 13 the radiation patterns for both feed configurationsoverlapped each other due to the similarity of both valuesThe highest gain is also measured at the same point Theagreement of the radiation gain values obtained at the twodifferent feed configurations validates the equation of causticloci derived by author in the previous section Therefore theaccuracy of the ray tracing algorithms for scanning plane andtransverse plane is ensured

5 Conclusion

A set of algorithms developed for the scanning plane andtransverse plane ray tracing program are presented Theresults obtained from the focal region ray tracing are dis-cussed and based on the caustic displacement results usefulequations representing the caustic loci are derived Theimportance of the equations are shown which could beused in designing a multibeam antenna application Off-feedcaustic displacements obtained in ray tracing method arecompared with the approximate equation of locus throughcalculation of radiation pattern The results of beam shiftshow good agreement between these two values and thusthe accuracy of the ray tracing method and the reliability ofthe locus equation for scanning plane and transverse planeanalysis are proven

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Teknologi MARAfor continuous support towards this research project

References

[1] R Azim M T Islam N Misran S W Cheung and YYamada ldquoPlanar UWB antenna with multi-slotted groundplanerdquo Microwave and Optical Technology Letters vol 53 no5 pp 966ndash968 2011

[2] L Liu S W Cheung R Azim and M T Islam ldquoA com-pact circular-ring antenna for ultra-wideband applicationsrdquoMicrowave and Optical Technology Letters vol 53 no 10 pp2283ndash2288 2011

[3] K Shogen S Tanaka and S Nakazawa ldquoStudies on the onboardAntennas for 21 GHz band broadcasting satellites and futurestudy itemsrdquo IEICE Transactions on Communications vol J94-B no 9 pp 1014ndash1024 2011

[4] G Toso C Mangenot and P Angeletti ldquoRecent advanceson space multibeam antennas based on a single aperturerdquoin Proceedings of the European Conference on Antennas andPropagation pp 454ndash458 April 2013

[5] S Yun M Uhm J Choi and I Yom ldquoMultibeam reflectorantenna fed by few elements for Ka-band communicationsatelliterdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 1ndash2 July 2012

[6] M Schneider C Hartwanger and H Wolf ldquoAntennas formultiple spot beam satellitesrdquo CEAS Space Journal vol 2 no1ndash4 pp 59ndash66 2011

[7] M Mahajan R Joti K Sood and S B Sharman ldquoA methodof generating simultaneous contoured and pencil beams fromsingle shaped reflector antennardquo IEEETransactions onAntennasand Propagation vol 61 no 10 pp 5297ndash5301 2013

[8] R E Collin and F J Zucker Antenna Theory Part 2 McGraw-Hill New York NY USA 1969

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012

International Journal of Antennas and Propagation 9

44 Radiation Characteristics To ensure the accuracy of themethod the results of caustic obtained in the previous sectionare verified through the calculation of the radiation patternThe relative radiation pattern is calculated by using the fol-lowing equations where 120585(119909 119910) represents the aperture phasedistribution obtained through comparison of ray length froma feed to all aperture points [11]

119864119903 = intlfloorint 119890(119895119896119903 sin 120579 cos(ΦminusΦ1015840)+119895120577(119909119910))119889Φ1015840rfloor 119903119889119903where 119903 = 120588 sin 120579

(22)

Several examples based on the incident direction of 120579in =minus20∘ are shown here in order to demonstrate the behaviourof the radiation pattern Figure 11 and Figure 12 show theradiation characteristics of a parabolic reflector with 119891119863 =15 In these configurations the feed locations 119865(119909 119910 119911) areset to be located at two different positions Firstly the feed isset at the caustic point (Δ119909119904 Δ119911119904) or (Δ119909119905 Δ119911119905) that has beenobtained in the previous ray tracing program for scanningplane and transverse plane respectively Then the resultis compared to the radiation pattern measured when thefeed is placed at the position determined through the locusequation (20) and (21) The results of both scanning planeand transverse plane are shown

From Figure 11 themaximum gain is measured at around20∘ offset from the origin which is the expected result sincethe feed location was previously set at the minus 20∘ caustic pointHowever the plot given by (20) gives better representation ofthe exact beam shiftMeanwhile as for transverse planemoredeteriorate patterns are observed

For both scanning and transverse planes by comparingthe caustic displacement results with the ldquobluerdquo line it seemslike the locus equations for both scanning and transverse pro-duced more accurate beams with the highest gain obtainedat the direction closer to the desired beam direction In thiscase the equations of caustic loci become very useful indetermining the accurate position of the feed However asexpected the analysis conducted in scanning mode providesmore accurate results as compared to the transversemode Toinvestigate further similar exercise is carried out for119891119863 = 3configuration

Similar to 119891119863 = 15 it can be observed from Figures13 and 14 that the equations of caustic loci bring the beamdirection closer to the desired 120579in = minus20∘ direction althoughit seems like in transverse plane the result deviates further ascompared to scanning plane As shown by scanning plane inFigure 13 the radiation patterns for both feed configurationsoverlapped each other due to the similarity of both valuesThe highest gain is also measured at the same point Theagreement of the radiation gain values obtained at the twodifferent feed configurations validates the equation of causticloci derived by author in the previous section Therefore theaccuracy of the ray tracing algorithms for scanning plane andtransverse plane is ensured

5 Conclusion

A set of algorithms developed for the scanning plane andtransverse plane ray tracing program are presented Theresults obtained from the focal region ray tracing are dis-cussed and based on the caustic displacement results usefulequations representing the caustic loci are derived Theimportance of the equations are shown which could beused in designing a multibeam antenna application Off-feedcaustic displacements obtained in ray tracing method arecompared with the approximate equation of locus throughcalculation of radiation pattern The results of beam shiftshow good agreement between these two values and thusthe accuracy of the ray tracing method and the reliability ofthe locus equation for scanning plane and transverse planeanalysis are proven

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors would like to thank Universiti Teknologi MARAfor continuous support towards this research project

References

[1] R Azim M T Islam N Misran S W Cheung and YYamada ldquoPlanar UWB antenna with multi-slotted groundplanerdquo Microwave and Optical Technology Letters vol 53 no5 pp 966ndash968 2011

[2] L Liu S W Cheung R Azim and M T Islam ldquoA com-pact circular-ring antenna for ultra-wideband applicationsrdquoMicrowave and Optical Technology Letters vol 53 no 10 pp2283ndash2288 2011

[3] K Shogen S Tanaka and S Nakazawa ldquoStudies on the onboardAntennas for 21 GHz band broadcasting satellites and futurestudy itemsrdquo IEICE Transactions on Communications vol J94-B no 9 pp 1014ndash1024 2011

[4] G Toso C Mangenot and P Angeletti ldquoRecent advanceson space multibeam antennas based on a single aperturerdquoin Proceedings of the European Conference on Antennas andPropagation pp 454ndash458 April 2013

[5] S Yun M Uhm J Choi and I Yom ldquoMultibeam reflectorantenna fed by few elements for Ka-band communicationsatelliterdquo in Proceedings of the IEEE Antennas and PropagationSociety International Symposium pp 1ndash2 July 2012

[6] M Schneider C Hartwanger and H Wolf ldquoAntennas formultiple spot beam satellitesrdquo CEAS Space Journal vol 2 no1ndash4 pp 59ndash66 2011

[7] M Mahajan R Joti K Sood and S B Sharman ldquoA methodof generating simultaneous contoured and pencil beams fromsingle shaped reflector antennardquo IEEETransactions onAntennasand Propagation vol 61 no 10 pp 5297ndash5301 2013

[8] R E Collin and F J Zucker Antenna Theory Part 2 McGraw-Hill New York NY USA 1969

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012

10 International Journal of Antennas and Propagation

[9] C J Sletten and R A Shore ldquoFocal Surfaces of offset dual-reflector antennasrdquo IEE Proceedings H Microwaves Optics andAntennas vol 129 no 3 pp 109ndash115 1982

[10] Y Yamada and S Sasaki ldquoEstimations of radiation character-istics of an Off-focus feed shaped dielectric lens antenna by aray tracing methodrdquo in Proceedings of the IEEE Antennas andPropagation Society International Symposium pp 398ndash401 SanAntonio Tex USA June 2002

[11] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons New York NY USA 3rd edition 2005

[12] W L Stutzman and G A Thiele Antenna Theory and DesignJohn Wiley and Sons New York NY USA 2nd edition 1998

[13] N Abd Rahman M Islam N Misran Y Yamada and NMichishita ldquoDesign of a aatellite antenna for Malaysia beamsby ray tracing methodrdquo in Proceedings of the InternationalSymposium on Antennas and Propagation (ISAP rsquo12) pp 1385ndash1388 October 2012