[1] A continuum solution for the phase evolution of a coupled oscillator array with coupling delay, when one oscillator is step detuned in time, is derived from the discrete model solution previously published. From this Green's function a differential equation is derived for the Laplace transform of the phase dynamics. The solution of this differential equation is expressed as a sum of the eigenfunctions of the differential operator for a finite length array with one oscillator detuned. The dynamic behavior of the aperture phase for beam steering is obtained by superposing two of the Green's function solutions corresponding to antisymmetric excitation at each end of the array. The far zone beam steering behavior is obtained by summing the elemental radiated signals. Generalization to planar arrays appears to be straightforward. An alternative more flexible formulation suitable for early times is also presented.

[2] Recently, Pogorzelski [2008b] described a generalization of the linearized discrete model of a linear array of oscillators coupled to nearest neighbors that was designed to account for the presence of significant delay in the coupling between the oscillators. The analysis also showed how such delays impact the so-called continuum version of the linearized theory in which the oscillator phase variation across the array is represented by a continuous function satisfying a partial differential equation, second order in the variable along the array and first order in time, describing the phase dynamics of the array as a diffusion-like process [Pogorzelski, 2008a]. As described some time ago by Liao and York [1993], the oscillator arrays treated in those previous analyses may be used to provide properly phased excitations for the elements of a linear phased array antenna. The primary advantage of such an arrangement is that the resulting far-zone radiated beam may be steered merely by antisymmetrically detuning the end oscillators of the array only. Thus, the beam steering control system is vastly simpler than that of a conventional phased array. Here, the previous discrete and continuum formulations are applied in the analysis of such a coupled oscillator fed phased array in the presence of significant coupling delay and the result is used to illustrate the impact of such coupling delay on the dynamics of the steered far-zone beam.

[3] The approach will be similar to that of Pogorzelski et al. [1999]. First, the previous formulations will be used to derive a second-order differential equation for the temporal Laplace transform of the phase distribution across the array. Being an equation of Sturm-Liouville type, the Green's function is be immediately expressed as a sum of the eigenfunctions of the differential operator. The Green's function, in turn, is used to obtain the phase distribution for antisymmetric detuning of the end oscillators in terms of the Laplace variable. Finally, inverse Laplace transformation yields the temporal behavior of the phase at each array element which may then be used to obtain the temporal behavior of the far-zone beam by summation of the elemental signals in the usual manner.

2. Solution for the Phase Dynamics

[4] We begin with the continuum solution adapted from the discrete model for the temporal Laplace transform of the phase distribution across an infinite array as given by equation (1) of Pogorzelski [2008a], repeated here for convenience as equation (1), which expresses the phase of the oscillator at x in response to detuning of the oscillator at y.

where d is the coupling delay measured in inverse locking ranges (ILRs) for these mutually injection locked oscillators. The temporal variable, τ, corresponding to the Laplace variable s is also measured in these units. Recall that in the continuum model, the oscillators are located at integer values of x and y. Δ_{tune}(y) is the Laplace transform of the temporal detuning function applied to the oscillator at position y. Thus, (1) is the Green's function for the infinite array.

[5] For x not equal to y, it is easily shown that f satisfies the homogeneous differential equation,

Noting from (1) that the derivative of f with respect to x is discontinuous across x = y, it becomes clear that the second derivative of f with respect to x will contain a Dirac delta function dependence on x and this delta function will be located at x = y. Thus, for all x, f satisfies the inhomogeneous differential equation,

The Green's function for a finite array of N + 1 oscillators can now be written in terms of the eigenfunctions of the differential operator on the left side of (3). Determination of these eigenfunctions requires that we make use of the boundary conditions at the ends of the array. The reflection coefficient at the ends of the array was provided in Appendix A of Pogorzelski [2008b]. However, the corresponding eigenfunctions would be quite complicated due to the s dependence of the reflection coefficient rendering inverse Laplace transformation difficult. Here, instead, we assume that a half-length coupling line is connected to each of the end oscillators of the finite array. With this small modification, the reflection coefficient becomes unity and the normalized eigenfunctions are simply,

Whereupon, the Green's function for the finite array may be written in the form,

where,

and,

Now, taking

representing unit amplitude (one inverse locking range) step function temporal dependence of the detuning, one may write the solution of (3) in the form,

As will be discussed again later, by linearity, the detuning, (8), may be scaled by any desired factor thus scaling the solution for the phase, (9), by the same factor.

[6] Inverse Laplace transformation is now merely a matter of summing the residues at the poles of (9) in the complex s plane. These poles are located where,

or,

for integer values of ℓ. These equations can be solved for the pole locations. The result is,

where W is the Lambert W function defined by,

The locations of these poles are shown in Figure 1, wherein the black dots correspond to odd values of ℓ and gray dots correspond to even values of ℓ.

[7] The overall time constant of the array may now be approximated analytically by evaluating the real part of the pole closest to the origin using (12). This may be accomplished by expanding the W function in a Taylor series about 2de^{2d}, or near ℓ = 0. That is,

But, z is the argument of the W function, 2de^{2d} cos(), so that,

[8] The pole at the origin (ℓ = 0) together with the 1/s from the step detuning lead to a double pole that accounts for the ensemble frequency shift (linear time dependence of the phase) when one oscillator is detuned. Because the poles of even ℓ do not contribute under antisymmetric excitation, the pole with odd ℓ closest to but not at the origin determines the array beam steering time constant. It corresponds to ℓ = 1 so it lies on the real axis at,

Therefore, the time constant of the array is,

or just (2d + 1) times the time constant of a similar array without time delay. (Compare with Pogorzelski et al. [1999, equation (30)].) This remarkably simple result will be shown to give very accurate estimates of the behavior of arrays with coupling delay in terms of the behavior of those without.

3. Beam Steering Dynamics

[9] As mentioned above, beam steering is achieved by antisymmetric detuning of the end oscillators of the array. Thus, the phase distribution for beam steering may be expressed as a superposition of two solutions of the form (9), one with y = N/2 and one with y = −N/2. Under such circumstances, the cosine terms cancel and one is left with,

[10] The inverse Laplace transform may now be evaluated by summing the residues at the poles found earlier (see equation (12)) and shown as black dots in Figure 1. Note that the apparent branch cuts due to the square root in the denominator are canceled by those of the arc secant function in the numerator leaving only the poles. The residues are computed by expanding the denominators in powers of (s − s_{m,2n+1}); that is, letting,

and expanding in the Taylor series,

where,

we find that the residue sum yields the inverse Laplace transform for the time dependence of the aperture phase in the form,

As noted above, for this antisymmetric case only the poles with odd index, 2n + 1, contribute. Moreover, since x is an integer at each oscillator, the sum on n need only extend from n = 0 to n = (N/2) − 1 to obtain the oscillator phase dynamics. This is because, for integer values of x, the terms for n = 0 through n = (N/2) − 1 are equal to the terms for n = (N/2) + 1 through n = N, the order merely being reversed. The terms for n values higher than N only affect the result between the oscillators and so are not relevant. Figure 2 shows a graph of (25) for each oscillator of a 21 element array with d = 2 inverse locking ranges. Interestingly, a close look at (21) reveals that the Bromwich contour may in fact be closed in the left half plane for −d < τ < ∞ although some oscillatory behavior may be encountered for negative time due to the truncation of the series. To illustrate this, (25) was derived by closing the contour in the left half plane beginning when τ = −d/4 which leads to the appearance of the ed/4 term. Figure 3 shows this result over a longer time duration and displays the steady state condition of equal phase increments between oscillators; that is, a linear aperture phase distribution. Figures 4 and 5 provide a different perspective on the data plotted in Figures 2 and 3. In Figures 4 and 5 one may view the time evolution of the aperture phase distribution all at once. Finally, Figure 6 shows the same data plotted over a time interval of 250 inverse locking ranges. Figure 6 is provided for comparison with Figure 6 of Pogorzelski et al. [1999] which shows similar data for a 21 element array with no coupling delay plotted over a time interval of 50 inverse locking ranges. Note that our Figure 6 and that of Pogorzelski et al. [1999] are nearly identical showing that the 2d + 1 slowing of the array response due to coupling delay is a quite accurate formula as in this case of delay equal to 2 inverse locking ranges the slowing factor is 5 which is, of course, just the 250/50 ratio of timescales.

[11] Having obtained (25) for the time dependence of the aperture phase distribution for this 21 element, half wavelength spaced, array with coupling delay, we are now in a position to compute the far zone radiated beam as a function of time during the beam steering transient ensuing upon application of detuning to the end oscillators. The result is shown in Figure 7. Here we see the sidelobe effects of the temporary phase aberration in the aperture during the transient as well as the smooth motion of the beam from the initial angle normal to the array to the final steady state position at 18.56 degrees from normal. This expected steady state condition can be obtained from (21) via the final value theorem. That is, we multiply by s and take the limit as s approaches zero resulting in,

this function is odd and periodic in x with period 2(N + 1) and satisfies the Neumann boundary conditions at x = −(N + 1)/2 and x = (N − 1)/2. For half-wavelength element spacing and interoscillator phase difference, Δϕ = 1, the beam angle from normal is then given by,

Finally, it should be noted that the interoscillator phase difference of one radian stresses the accuracy of this linearized formulation since the linearization is performed by approximating the sine of this phase difference by its argument leading to the second partial derivative with respect to x in the differential equation, (3). This rather large steady state interoscillator phase difference was chosen so that the results could be directly compared with those in the work of Pogorzelski et al. [1999]. As mentioned earlier, the present phase results can easily be scaled to a phase slope reduced by a factor C by multiplying (8) by C. For example, if C = 0.5, the detuning represented by (8) will be 0.5 ILRs, the steady state interoscillator phase difference will become 0.5 radians, and the sine of 0.5 is 0.479, an error of only about 4%. Of course, using (27), the corresponding steady state beam angle will then be 9.16 degrees from normal instead of 18.56 degrees.

4. An Alternative Representation for Early Time

[12] Using the representation (25) the number of residues required for good accuracy depends upon the time at which the expression is to be evaluated. For large values of time, very few residues are required as only the poles closest to the origin in the s plane play a significant role. In fact, as discussed above, the overall dynamic response time of the array depends only upon the residue at the pole closest to the origin and this led to the 2d + 1 formula for the time constant modification relative to that of an array with no time delay. However, for small values of time, many residues are required. For the above examples, m ranged from −15 to 15; i.e., 31 residues per n value. Using the discrete formulation presented by Pogorzelski [2008b], one may obtain the results for early times with fewer terms. That is, beginning with Pogorzelski [2008b, equation (22)] for the Laplace transform of the phase of oscillator i under detuning of oscillator j, repeated here as,

where R = 1,

and the exponents, n_{<} and n_{>}, on Q are the lesser and greater of i and j, respectively, one may show that the corresponding discrete formulation result for the beam steering phase in the above example is,

Expanding this expression in powers of b and analytically performing the inverse Laplace transform as in the work of Pogorzelski [2008b] yields the time dependence of the phase of each oscillator. The highest power of b required in the expansion is determined by the number of delay times over which the transient response is desired so for early times very few terms are required. In the present example, 20 terms were included in the expansion for each x. (This included higher powers of b than needed for some x values but was convenient from a programming point of view.) The resulting phase behavior is shown in Figure 8. This is to be compared with Figure 2 obtained via the eigenfunction expansion requiring more terms for these early time values. The results are, of course, identical except for some invisibly small oscillations near zero time in the eigenfunction results due to truncation of the residue series. The power series results have no truncation error and are thus exact over the plotted time interval.

[13] Finally, it is noted that, although somewhat limited to early times for practical reasons, the power series approach does have more flexibility in terms of boundary conditions. That is, if instead of terminating the array in half length coupling lines one terminates it without such lines, the eigenfunction expansion becomes unwieldy but the power series expansion merely requires that instead of R = 1 one use [see Pogorzelski, 2008b].

The power series expansion proceeds as before, and the resulting early time dependence of the oscillator phases is shown in Figure 9 and may be compared with Figure 8 where half length terminations were used. In observing the phase of the end oscillators of the array one may note that, whereas with the terminations, the effects of reflections appear two inverse locking ranges or one delay time after τ = 0 corresponding to one round trip between the end oscillator and the open end of the terminating line, without the terminations these effects do not appear until four inverse locking ranges accounting for the 2d roundtrip delay between the end oscillator and the next one inward in the array.

5. Concluding Summary

[14] By means of the continuum solution for the phase distribution over the aperture of a coupled oscillator based phased array with coupling delay derived from the discrete model solution, we have derived a differential equation for the phase dynamics. Being a system of Sturm-Liouville type, the Green's function for this differential equation has been expressed as a sum of products of the eigenfunctions in the Laplace domain. The inverse Laplace transform of this Green's function was computed as a residue series yielding the dynamic phase evolution when one oscillator is detuned. Beam steering is accomplished by detuning the two end oscillators and, as such, the phase dynamics for beam steering may be expressed as a superposition of two detuning responses. This yields a solution which substantiates the approximation that coupling delay results in a slowing of the transient response of the array by a factor of (2d + 1) where d is the coupling delay in inverse locking ranges.

[15] The theory presented here is formulated for a one-dimensional array. However, it appears that following Pogorzelski [2001], one may generalize this formulation to two-dimensional arrays by taking advantage of the separability of the partial differential equation,

for a planar array. Moreover, it appears that this formalism may be similarly applied for arrays with triangular or hexagonal coupling lattices following Pogorzelski [2004]. However, generalization of the alternative formulation for early times to two dimensions appears to be formidable if indeed possible.

Acknowledgments

[16] The work reported here was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration (NASA). Partial support from the JPL Mars Technology Program, the Interplanetary Network Directorate Technology Program, and the JPL Advanced Concepts Office of the NASA Office of Space Exploration are gratefully acknowledged. Thanks are due Vahraz Jamnejad of JPL for helpful theoretical discussions during the early stages of the present work and R. A. York of the University of California, Santa Barbara, for his early insight (circa 1998) concerning the Sturm-Liouville nature of the differential equations and the concomitant simple inverse Laplace transform of the Green's function without coupling delay. Particular thanks are due Jason Martinez of Wolfram Research for his help with the plots. Much of the computation and graphing associated with this work was done using Mathematica® by Wolfram Research. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not constitute or imply its endorsement by the United States Government or the Jet Propulsion Laboratory, California Institute of Technology.