design trades for rotman lenses

464 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 39, NO. 4, APRIL 1991

Design Trades for Rotman Lenses R. C. Hansen, Fellow, IEEE

Abstract-The foundation of a satisfactory Rotman lens design is geometric. The effects of the seven design parameters (focal angle, focal ratio, beam angle ratio, maximum beam angle, beam port curve ellipticity, array element spacing, and focal length/ A) on the shape, and on the geometric phase and amplitude errors of a Rotman lens are described. The advantage of beam port shaping to reduce phase error, and of pointing port horns at the opposite apex (instead of normal to the curve) to reduce off-axis beam amplitude asymmetries, are shown numerically. A design procedure for selecting these parameters is given, and a new calculation of lens gain is presented.

I. INTRODUCTION ULTIPLE beam antennas have proved useful for various M applications such as ECM, and the Rotman lens is often

used. Design of these lenses must involve both geometric trades and mutual coupling effects between the lens ports. The latter is relatively difficult to control, but the former is crucial to the realization of an efficient and compact lens. Thus a careful geometric optics design should be accomplished first; then ad- justments must be made to reduce mutual coupling effects. This paper describes the geometric design trades.

The Rotman lens has six basic design parameters: focal angle a , focal ratio 0, beam angle to ray angle ratio y , maximum beam angle $,, focal length f l , and array element spacing d. The last two are in wavelengths, and y is a ratio of sines. A seventh design parameter allows the beam port arc to be elliptical instead of circular. Since the design equations are implicit and transcendental, with only one sequence of solution, the interplay of design parameters is difficult to discern. In this paper a series of lens plots is used to show the effects of each parameter. Geometric phase and amplitude errors over the element port arc vary primarily with a and 0, and with an implicit parameter which is the normalized element port arc height. Representative plots show how these errors depend upon the parameters.

For lenses where the beam port arc and feed port arc are identical, resulting in a completely symmetric lens, the design equations are greatly simplified [8]. However, these lenses are seldom used because their design options are much more con- strained.

A new calculation of lens gain is presented, with the lens connected to an array of isotropic elements. Port spillover, and phase and amplitude errors are included in the gain calculation, but not impedance mismatches due to mutual coupling. Finally, a design procedure is outlined.

11. LENS PARAMETERS The lens equations equate path lengths from the foci to the

array elements; see [7] or [6] for a derivation of these. Using the nomenclature of Fig. 1, it is convenient to normalize all dimensions by the principal focal length f This is also the lens width

Manuscript received December 4, 1989; revised September 11, 1990. The author is at P.O. Box 570215, Tarzana, CA 91357. IEEE Log Number 9041791.

focus

y 3

y-

x

Fig. 1. Ray geometry.

at the center. The focal angle a is subtended by the upper and lower foci at the center of the element port curve. It is assumed here that the foci are symmetrically disposed about the axis, and that the lens is also symmetric. Then the parameter /3 is the ratio' of upper (and lower) focal length f 2 to f l :

P = f 2 l f l

Clearly the lens width in wavelengths, f l /A, is another parameter. Now the angle of the beam radiated by the array is $, and if one of the off-axis foci is excited, then the ratio of lens ray angle a to array beam angle $ is y:

sin $

y=G An indirect parameter of utility is {, which relates the distance y, of any point on the array from the axis, to fl. This parameter controls the portion of the phase and amplitude error curves that the lens experiences. It is expressed:

Y3Y (-= -

f l (3)

Note that the line lengths w of Fig. 1 are an integral and essential part of the lens. The maximum beam angle, $,, is an important parameter, as is the array element spacing in wave-

' Note that this ratio 0 is the inverse of the ratio g used by Rotman, and by McGrath [4]. As it is convenient to normalize all dimensions by f, , the ratio of f2 / f l is more appropriate.

OO18-926X/91/04OO-0464$01 .OO 0 1991 IEEE

HANSEN: DESIGN TRADES FOR ROTMAN LENSES

1.0

. 8

ZETA

.6

. 4

. 2

n .80 .84 . d 8 . 9 2 . 9 6

BETA

Fig. 2. Upper limit on parameter zeta.

lengths d/X. The {,,,ax that corresponds is given by

(NE - 1)yd

2fl lmax = (4)

where NE is the number of elements in the linear array,’ since y,, = (NE - 1)/2. An upper limit on { occurs when the tangent to the element port curve is vertical; this also gives w = 0. This value of [ is given by

l w = o = S ( 5 )

Fig. 2 gives this limiting value versus P, for several values of a. Since the useful range of { is roughly from 0.5 to 0.8, a range of P appropriate for a given a may be inferred.

The geometric lens equation is a quadratic in the line length w that connects an element port to the corresponding array element:

a(w/f1)’ + bw/fl + c = 0 (6)

where the coefficients involve the parameters, a, 0, and y :

( 1 - 0)’ l2 ( 1 - PC)’ P2

a = l - _ _ (7)

2 p 2(1 - 0) 12S2(1 - P ) P 1 -PC ( 1 -PC)’

(8) b = - 2 + - + - -

and C = cosa, S = sina. Usually the number of beams and number of elements, the

maximum beam angle and element spacing, are specified from

465

Fig. 3 . Effect of focal angle. f i = 0.9: y = 1 . 1 , psim = 50, f, = 4 WV, d = 0.5 WV.

the system requirements. Thus, the task is to select the optimum a, P , ~9 and fl /A.

111. EFFECT OF PARAMETERS ON LENS SHAPE AND PORT POSITIONS

The lens shape is important, both in conserving space and in reducing loss. For example, a wide lens tends to have path lengths that are more nearly equal, and allows the beam port curve and the element port curve to have different heights, and even different curvatures. As in Fig. 1 , the width is along the lens axis. Wide lenses have large spillover loss, and higher transmission line loss. A compact lens tends to minimize spillover losses; roughly equal port curve heights now become important, to avoid severe asymmetric amplitude tapers and large phase errors. Curvatures of the two port curves may be different; use of array element spacing greater than half-wavelength allows more beam ports than element ports to be used. For this case, the beam port curve may be more curved, and the element port curve flatter.

The effects of the seven parameters will be shown through a series of charts. Six charts are shown in Figs. 3-8. Beam ports, which are ticked, are on the left. Element ports, also ticked, are on the right. Foci are indicated by asterisks. The focal length is normalized to unity, so that each tick mark on the axis (and on


Fig. 4. Effect of focal ratio. (Y = 40, y = 1 . 1 , psim = 50, f, = 4 WV, d = 0.5 WV.

the ordinate) is 0.05. From the ordinate scale the element positions may be inferred. Each lens curve is extended past the outermost port by half the width of that port. These examples have nine beam ports, and 11 element ports, and of course 11 elements in an equally spaced linear array.

With all other variables fixed, increasing a opens the beam port curve, and closes the element port curve. Port positions are roughly unchanged. But the outer foci locations change markedly as expected. The three lens plots of Fig. 3 illustrate these effects. It can be seen that a value CY can be selected that roughly equalizes the heights of the two curves. Of course, CY must be selected in conjunction with the other variables, to minimize phase errors over the aperture. The outer foci should be com- fortably inside the beam port curve.

Increasing @ has an effect similar to increasing CY; the beam port curve opens, and the element port curve closes. Fig. 4 contains three lens plots to show this. Again, port positions are roughly unchanged. Also the focal locations change relatively little. Again, a value of @ can be selected that roughly equalizes the curve heights.

There are pairs of a and @ that produce closely the same lens shape, and port positions. However, the foci vary with a , and the connecting lines (from element ports to elements) are different. Table I shows three CY - @ lens pairs that have common lens curves and ports, all for y = 1.1, $,,, = 50, f l / A = 4, and d = X/2. One may thus infer that the phase error over the

1.0

Fig. 5. Effect of angle ratio. CY = 40, 0 = 0.9, psim = 50, f, = 4 WV, d = 0.5 WV.

aperture for each beam will be different, depending upon CY.

This will be shown in the next section. For any set of the other four parameters, there are probably some CY - @ pairs that behave similarly.

Increasing y leaves both lens curves unchanged, but the beam ports are moved closer together, while the element ports are spread apart. A three-lens set in Fig. 5 shows this trend. Although the foci remain fixed, the ends of the curves change, so that the relative position of the foci changes. y also can affect the relative heights of the two curves.

Values of y here are one or greater, as the cases used are all for large beam angles. When the beam cluster subtends a more modest angle, e.g., 30°, values of y < 1 are appropriate as they allow a “fat” or curved lens.

When $, is changed, only the beam port spacings change. Increasing $, spreads the beam ports and extends the port curve, so that this parameter helps produce a lens with roughly equal heights of beam and element port curves. The three lens plots of Fig. 6 depict this behavior.

Element spacing is critical as it controls the appearance of grating lobes [ 2 ] . For a maximum beam angle of $m, the spacing that just admits a grating lobe is

d / h = 1/(2 + sin $,)

In general, spacings are kept below this value.


91

461

VALUES OF d/X SHOWN VALUES OF PSI-MAX SHOWN i

Fig. 6. Effect of maximum beam angle. cy = 35, p = 0.92, y = 1.1, f, = 4 WV, d = 0.5 WV.

When d is changed, only the element port spacings and the extent of the port curve change, analogous to $,,, changing beam port spacing. Fig. 7 uses two lens plots to show this.

Increasing the lens focal length (width) in general increases the separation between the end ports as well. But changing f l / A also changes all spacings, as the lens equations are normalized by fi. Thus as shown in the two lens plots of Fig. 8 changing fl / A also changes the element port arc and element port spacings. The minimum value of f l is smaller for Rotman lenses than for other types of lenses [9].

Next, phase and amplitude errors will be examined.

Iv. EFFECT OF PARAMETERS ON PHASE AND AMPLITUDE ERRORS

Aperture errors depend upon CY and 0, and upon eccentricity, but only indirectly on the other parameters. Thus the most insight results from plotting phase and amplitude errors versus the normalized parameter (; see (3). Since phase errors are zero at angles corresponding to the three foci, a satisfactory approach uses one beam position midway between the central and edge foci, and a second beam position beyond the edge focus. Ampli- tude errors occur at all beam ports, so more cases are needed to display amplitude error behavior.

Fig. 7 . Effect of array element spacing, 01 = 40, 6 = 0.88, y = 1.1, psim = 50, f, = 4 WV.

A . Phase Errors Figs. 9- 12 show phase error versus ( for lenses with CY of 30

and 40 deg., for the two beam positions. Note that to get phase error, the values from the figures are to be multiplied by f i /A . For the midfoci beams, the phase error is small, except for very large lenses. Phase errors for the wider angle beams are still modest, and will not be important except for large lenses, or designs with { > 0.75. In general, the phase errors increase as CY is increased, for all beam positions.

Although Rotman and Turner indicated an optimum value of 0, which was 2/(2 + CY*), examination of Figs. 9-12 show that an optimum 0 exists only for one range of ( and one ray angle. Best values of 0 are different for between foci rays and rays outside the foci. Since the latter usually have larger phase error, the designer could optimize, but the value would vary with both ray angle and with lm,. This old value also gives poor lens shapes when the number of beam and element ports are roughly equal.

The results of using an elliptical beam port curve are shown in Fig. 13, where the phase errors are equalized at ( = -10.7 for a ray angle of 45". Note that the elliptical beam port curve is a simple way of realizing the optimum curve at Katagi et al. [3]. This gives a 13% reduction in phase error; the phase errors for the midfoci ray at 17.5" become slightly more asymmetric, but


Fig. 8. Effect of focal length. (Y = 40, f l = 0.9, y = 1.1, psim = 50, d = 0.5 WV.

TABLE I ff-0 PAIRS

Number 1 OL = 30 p = .94 2 (Y = 35 f l = .92 3 (Y = 40 f l = .90

are still well below the 45" ray errors. Note that the ellipticity of -0.3 only changes the principal radius by 596, so amplitudes are essentially unchanged. The beam port ellipse major axis is along the lens axis for this ellipticity.

B. Amplitude Errors Amplitude errors are calculated using beam port and element

port horn patterns of sinc x u , where horn widths are all set to a nominal X/2. Each port horn has its axis normal to the port curve. Fig. 14 shows amplitude error, normalized to 0 dB for the axial ray, for a lens with a = 30, /3 = .94. Curves for ray angles of O n , 15", and 45" are given. Similarly, Fig. 15 is for a lens with a = 40, @ = 0.9, for ray angles of O", 20", and 50". These examples are two of the a - /3 pairs of Table I, and thus the amplitude errors are similar. As expected, for wide ray

angles the near and far ends of the element port curve experience modest amplitude changes. Compared to the amplitude taper needed to produce 25 dB sidelobes, these amplitude errors are small. Actual lenses may have port widths larger than X/2, so the amplitude tapers can be expected to increase, especially for edge beams.

The asymmetry of amplitude for the off-axis beams can be reduced by pointing each port horn at the opposite apex, instead of normal to the port curve [ 5 ] . For example, a nine-beam, 11-element lens with a! = 40, /3 = 0.9, y = 1.1, $, = 50, d = 0.51 and f = 4X has amplitude taper for the outside beam as shown in Table 11. Also shown is the taper for apex pointed horns. Use of apex pointing produces appreciable improvement. Gain is slightly improved.

V. CALCULATION OF LENS GAIN

Element and beam port spillover, phase and amplitude errors, port impedance mismatches, and transmission line loss all con- tribute to reducing lens gain. Note that, as in the case of a horn feeding a reflector antenna, there is no feed horn spreading loss, due to the equal path property through the foci. The small inequality of other paths is subsummed in the path phase and amplitude errors. Gain will be calculated here based on port

HANSEN: DESIGN TRADES FOR ROTMAN LENSES 469

i.OL i

ZETA

Fig. 9. Phase error between foci. Rotman lens, (Y = 30, ray angle = 15.

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 ZETA

Fig. 10. Phase error beyond foci. Rotman lens, a! = 30, ray angle = 45

8

spillover and on aperture errors. Since the amplitude error calculation includes both beam port and element port horn patterns, spillover is included [lo]. The phase and amplitude errors at the element ports are transferred to an array of isotropic elements. Then the problem reduces to that of calculating gain of a symmetric linear array of isotropic elements with complex coefficients. This is readily done [2]:

(11) IxAn12 G =

CCA,A,*sinc(n - m ) 2 a d / X '

Actual gain is then that of (1 1) multiplied by the element gain, times the impedance mismatch factors.

Variation of gain with parameters is very small. For example, for a typical small lens only 0.2 dB change occurs from the center beam to the edge beam. And the gain values are roughly independent of a, 0, y etc. With a larger array the gain increases just as expected. Using the same nine-beam, ll-element example of the previous section, gain ranges from 10.2 to 10.4 dB; the latter value, for the center beam, is within 0.1 dB of the gain for a similar uniformly excited array. For this lens the [ range is kO.756.


-0.a -0.6 -0 .4 -0.2 0.0 0.2 0 .4 0.6 ZETA

Fig. 11. Phase error between foci. Rotman lens, 01 = 40, ray angle = 20.

2.0 //

1.5-

-0.5

-O.:L

BETA = . 9

Fig. 12. Phase error between foci. Rotman lens, 01 = 45, ray angle = 22.5,

Effects of feed horn spillover and internal lens reflections can be reduced by either employing dummy (terminated) feed horns adjacent to the edge horns, or through the use of absorber between the ends of the beam port arc and the element port arc [5] . Port impedance mismatches are outside the scope of this paper.

VI. A DESIGN PROCEDURE

System requirements usually specify the frequency range, the number of beams and the angular coverage, and either the

beamwidth or adjacent beam crossover level. From these, a suitable combination of number of elements and d/X may be inferred. The design process starts by the selection of a center frequency, at which all dimensions are computed. Then, a starting value of fl / A is selected, to keep lmx well below 0.8. The focal length will be somewhat less than the array length. Next, using the guidelines of Sections 111 and IV, CY, p, and y are selected, to: locate the outer beam port a modest amount past the outer focus; produce beam port and element port arcs of comparable heights; and yield acceptable phase and amplitude errors at each port. Achieving this may require adjustment of


7 - E = O

47 1

J 0 2 0 4 0 6 0 8 -0 6 -0 4 -0 2 0 0

4 0 ~ 1 ~ ~ 1 " ' ' I ' ~ ' ' ' ' ~ ~ ~ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ~ " ' ' ' ' ' ' ~ ~ ' ' ' ' ' ' ' ' ' ~ ' ' ' ' ' ' ' 1 1 ~ ' ' ' ' ' ' ' ' ZETA

-0 E

Fig. 13. Effect of ellipticity. Rotman lens, a = 35, 0 = 0.92.

f, / A or d /A, and of a, 0, and y. Use of an elliptical beam port arc is usually not warranted except for large lenses.

When a satisfactory design is realized at the center frequency, phase and amplitude errors at each port are calculated at representative frequencies, to assess performance over the frequency range. And of course the actual beam and element port horn widths are used. At this stage, calculation of a beam rosette (a set of beam patterns) at each frequency is appropriate. Some compromise and iterative adjustment of parameters may be necessary to obtain good wide-band results, and to best accom- modate mutual coupling effects.

Although the lens width fl is less than the array length, the lens height is always greater than the array length. Lens dimensions are reduced by the square root of effective dielectric constant for either stripline or microstrip implementation. See [l] for examples.

VII. CONCLUSION Guidelines have been given on how the seven Rotman lens

parameters affect lens performance, and on how to select values for them, based on geometrical optics. Such a design must be tempered by mutual impedance considerations.

412 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. VOL. 39, NO. 4, APRIL 1 9 9 1

\ RAY ANGLE = 0

-0.8 -0 .6 -0.4 -0.2 0.0 0.2 0 . 4 0.6 -6.0

ZETA

Fig. 15. Amplitude errors. Rotman lens, 01 = 40, (3 = 0.9.

TABLE I1 BEAM ONE AMPLITUDE TAPER; f, = 4 h

Axes Axes Element Normal to Arc through Apex Number (dB) (dB)

1 2 3 4 5 6 7 8 9

10 1 1

- 8.49 -7.03 -5.50 -4.29 -3.33 -2.58 -2.04 - 1.70 - 1.57 - 1.68 - 2.08

- 2.26 - 1.84 - 1.46 - 1.31 - 1.37 - 1.63 -2.08 -2.73 -3 .60 -4.71 - 5.95

REFERENCES D. H. Archer, “Lens-fed multiple beam arrays,” Microwave J., vol. 27, pp. 171-195, Sept. 1984. R. C. Hansen, “Linear arrays,” in Handbook of Antenna Design, vol. 2, A. W. Rudge et al., Eds. U.K.: Inst. Elec. Eng./Peregrinus, 1983, ch. 9. T. Katagi et al., “An improved design method of Rotman lens antennas,” IEEE Trans. Antennas Propagat., vol. AP-32, pp. 524-527, May 1984. D. T. McGrath, “Contrained lenses,” in Reflector and Lens Antennas, C. J. Sletten, Ed. Dedham, MA: Artech House, 1988, ch. 6. L. Musa and M. S. Smith, “Microstrip port design and sidewall absorption for printed Rotman lenses,” Proc. Inst. Elec. Eng., vol. 136, pt. H, pp. 53-58, Feb. 1989. D. M. Pozar, Antenna Design Using Personal Computers. Dedham, MA: Artech House, 1985, sec. 4.6. W. Rotman and R. F. Turner, “Wide-angle microwave lens for line source applications,” IEEE Trans. Antennas Propagat.,

J. P. Shelton, “Focusing characteristics of symmetrically con- figured bootlace lenses,” ZEEE Trans. Antennas Propagat.,

M. S. Smith, “Design considerations for Ruze and Rotman lenses,” Radio Electron. Eng., vol. 52, pp. 181-197, Apr. 1982. M. S. Smith and A. K. S. Fong, “Amplitude performance of Ruze and Rotman lenses,” Radio Electron. Eng., vol. 5 3 , pp. 329-336, Sept. 1983.

vol. AP-11, pp. 623-632, NOV. 1963.

vol. AP-26, pp. 513-518, July 1978.

3

R. C. Hansen (S’47-A’49-M’55-SM’56- F’62) received the B.S.E.E. degree from the Missouri School of Mines and the Ph.D. degree from the University of Illinois in 1955.

From 1949 to 1955 he worked in the Antenna Laboratory of the University of Illinois on fer- rite loops, streamlined airborne antennas, and DF and homing systems. He was Section Head in the Microwave Laboratory of Hughes Air- craft Co., working on surface wave antennas, slot arrays, near fields and radio power transfer,

electronic scanning and steerable arrays and dynamic antennas. In 1960 he became a Senior Staff Member in the Telecommunications Labora- tory of STL, Inc., engaged in communication satellite telemetry, track- ing, and command. He was Associate Director of Satellite Control responsible for converting the Air Force satellite control network into a realtime computer-to-computer network. From 1964 to 1966, he formed and was Director of the Test Mission Analysis Office, responsible for computer programs for the planning and control of classified Air Force Satellites. From 1966 to 1967, he was Operations Group Director of the Manned Orbiting Laboratory Systems Engineering Office of the Aerospace Corp., with responsibilities for: flight crew training, simula- tor, the software system for mission profiles, and mission control center equipment and displays.

Dr. Hansen has been involved in many professional activities includ- ing: Chairman of U.S. Commission VI of URSI (1967-1969), Chair- man of the 1958 WESCON Technical Program Committee, Chairman of IEEE Antennas and Propagation Society (1963-1964, and 1980), Chairman of Standards Subcommittee 2.5 which revised Methods of Testing Antennas, Editor of GAP Newsletter (1961- 1963), member GAP AdCom (1959-1974), member, IEEE Publications Board (1972 and 1974), and Director IEEE, 1975. He is a Fellow of the Institution of Electrical Engineers, is a registered Professional Engineer in California and England, and is a member of the American Physical Society, Tau Beta Pi, Sigma Xi, Eta Kappa Nu, and Phi Kappa Phi. He was awarded an honorary Doctor of Engineering degree by the University of Mis- souri-Rolla in 1975. The University of Illinois Electrical Engineering Department gave him a Distinguished Alumnus award in 1981, and the College of Engineering awarded him a Distinguished Alumnus Service Medal in 1986. He has written over 100 papers on electromagnetics, has been an Associate Editor of Microwave Journal since 1960, was Associate Editor of Radio Science (1967- l969), Associate Editor of Microwave Engineer’s Handbook (1971), Editor of Microwave Scanning Antennas, vol. I (1964), vols. I1 and 111 (1966), Editor of Sign@cant Phased Array Papers (1973), Geometric Theory of Diffraction (1981), and Moment Methods in Antennas and Scatter- ing (1990).

design trades for rotman lenses

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