3. K. J . Weingarten, M. J. Rodwell, and D . M. Bloom, “Picosecond Sampling of GaAs Integrated Circuits,” IEEE J . Quantum Elec- tron., Vol. QE-24, No. 2, Feb. 1988, pp. 198-220.

4. H. W. Yen, M. K. Bamoski, R. G. Hansperger, and R. T. Melville, “Switching to GaAs IMPA’IT Diode Oscillator by Optical Illu- mination,”Appl. Phys. Lett., Vol. 31, No. 2, July 1977, pp. 120- 122.

5. R. A. Keihl, “Behavior and Dynamics of Optically Controlled TRAPATT Oscillators,” ZEEE Trans. Electron Devices, Vol. ED- 25, No. 6, June 1978, pp. 703-710.

6. D. S. McGregor, C. S. Park, M. H. Weichold, H. F. Taylor, and K. Chang, “Optically Excited Microwave Ring Resonators in Ga- llium Arsenide,” Microwave Opt. Technol. Lett., Vol. 2, No. 5, May 1989, pp. 159-162.

7. G. K. Gopalakrishnan, B. W. Fairchild, C. L. Yeh, C. S. Park, K. Chang, M. H. Weichold, and H. F. Taylor, “Microwave Per- formance of Nonlinear Optoelectronic Microstrip Ring Resona- tor,” Electron. Lett., Vol. 27, No. 2, Jan. 1991, pp. 121-123.

8. G. K. Gopalakrishnan, B. W. Fairchild, C. L. Yeh, C. S. Park, K. Chang, M. H. Weichold, and H. F. Taylor, “Experimental Investigation of Microwave-Optoelectronic Interactions in a Microstrip Ring Resonator,” IEEE Trans. Microwave Theory Tech., Vol. MTT-39, No. 12, Dec. 1991, pp. 2052-2060.

Received 8-9-93

Microwave and Optical Technology Letters, 712, 45-48 0 1994 John Wiley & Sons, Inc. CCC 0895-2477J94

matrix partitioning scheme [l] is routinely used to accomplish this purpose; however, when the attachments are large com- pared with the initial structure, it may not be efficient com- pared with an LU decomposition of the entire body. Although intermediate solutions may be stored in the form of LU de- composed matrices, each step of this scheme results in an explicit inverse of the matrix, as can be seen in Eqs. (4), cited from [l]. For quasiseparable cases, the add-on approach [2, 31 has been recently proposed for efficient comutations of additions to basic bodies. Other related schemes are the cas- caded mode matching scheme and the Sherman-Morrison- Woodbury algorithm [4]. In this work-, a recursive LU de- composition algorithm, using the partitioning approach, is described. It is based on the algebraic operations of the well- known Crout algorithm, except in a different order, which adds the recursive feature to the algorithm. Other efficient numerical features are thus retained, including the ability to incorporate it into any existing moment-method code. A de- tailed operation count and examples demonstrate the effi- ciency of the method in the process of adding different at- tachments to an initial scatterer known by its LU decomposition.

2. FORMULATION Assume a moment-method (MM) matrix 2 defined as

z = [f ;I. In Eq. (l), Q is a submatrix, for which either LU decom- position or inverse are assumed known. Q is the matrix per- taining to all the internal interactions within a subset of the basis functions representing a portion of the scatterer, de-

A RECURSIVE LU DECOMPOSITION ALGORITHM Yigal Twig and Raphael Kastner Department of Electrical Engineering, Physical Electronics Tel Aviv Universitv

noted as scatterer No. 1. Upon adding an additional sub- scatterer No. 2, whose internal interactions are described by the matrix d, the mutual interactions between the two sub-

Tel Aviv 69978, Is’rael

KEY TERMS Recursive algorithms, LU decomposition, integral equations, mo- ment method

scatterers are expressed in the matrices b and c (in a Galerkin formulation, b = cT because of symmetry). Assuming that body No. 1 has N , unknowns and body No. 2 is represented by N, unknowns, the matrices are of the following orders:

ABSTRACT A recursive LU decomposition scheme extends the capabilities of

The inverse of 2 is then nonrecursive algorithms for the reduction of computation time when geometrical modifications are made to a precomputed Xcatterer. Sig- nificant savings are achieved for cases of many different attach- ments, or when the initial scatterer is amenable to a particular efji- cient solution. 0 1994 John Wiley & Sons, Inc.

(3)

1. INTRODUCTION Recursive methods for scattering problems are best suited for cases where a number of different attachments are to be made to a common known scatterer. This process makes use of available information on the initial known scatterer. Many problems fall into this category; examples include bodies of missiles or aircraft, which can be augmented with fins, wings, blade antennas, and pods, as well as dielectric coatings. In many cases, the platform is designed iteratively, based on a certain initial structure with a number of possible attachments. It then becomes crucially important to be able to analyze the many variations without the need to analyze the entire scat- terer over again, in order to assess the impact of the different additions on the RCS characteristics of the entire body. The

where [l]

This scheme could be attractive for a case such as a cylinder with fins, where body No. 1 is the cylinder and the fins are body No. 2. Q-l is efficiently computed using the FFT, taking advantage of the circulant property of Q. The operation count shows that the dominant factor is of the order of N , e , stand- ing for the computation of E as in Eq. (4a). This operation

48 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS I Vol. 7, No. 2, February 5 1994

count should be compared with the case of inversion of the full matrix of order N, + N , using LU decomposition and N, + N , back substitutions, i.e., the order of (N , + N,)3. Assuming N , << N,, the ratio between the two operation counts is of the order of N,IN,. In practice, however, there is some additional overhead associated with matrix partition- ing; hence the advantage is not fully realized for small ma- trices. This ideal ratio can be regarded as an asymptotic value for large matrices. The partitioning idea can be incorporated into the direct evolution of the LU decomposition. This de- composition, defined in the standard way as

is traditionally computed in an efficient (and space-saving) manner by methods such as the Crout algorithm [4,5], where lii = 1. The decomposed matrix, defined by

rull u12 U13 U 1 4 1

is filled by the Crout algorithm column by column (see Figure l), in a way which is indeed gradual. Once a column has been filled, it is not changed in the subsequent operations, except for interchange row-wise permutation under the diagonal ele- ment for the sake of pivoting. A different order of the same algebraic operations, suggested here, converts this scheme into a recursive one by generating the decomposed matrix in an upper-left-blocks order. For a matrix

L

as in (l), assume that the partial LU decomposition Q = LU is known. One can make use of this information to decompose the matrix Z as follows:

LU b L -'b '= [ c d ] = [ c i - l :I[: d - cU-lL- lb

This is a block LU decomposition, unless N , = 1, in which case it is a true LU decomposition. Equation (7) can be used

as a replacement for Eqs. (3) and (4). In this equation, one partial forward substitution and one partial back substitution need to be performed, for solving the two triangular systems of equations as expressed in the terms L-'b and cU-l , re- spectively. In order to obtain true LU decomposition, for cases other than N , = 1, one needs first to evaluate the decomposition

such that the total decomposition becomes

LU b U L-'b = [ c d ] = [ c i - l i ' ] [ O U' ] (9)

The operation count for Eq. (7) or (9) is much smaller than the one for matrix partitioning. In fact, if at each stage we augment Q with one row and one column at a time, all the algebraic operations involved in the Crout algorithm are ex- ecuted, except at a different order; hence the total number of operations for the augmented matrix is the same. The upper-left square blocks filling process is depicted in Figure 2, and it can be contrasted with Figure 1. The process ex- pressed by Eqs. (7) and (9) is recursive; if it is stopped at any intermediate stage, the partial LU decomposition has been obtained for a subset of the basis functions representing a subscatterer, and the process can be continued from this point. In contrast, the Crout algorithm is not recursive and cannot be continued from this point, because the decomposed matrix does not include the remainders of the columns of the aug- mented matrix underneath the initial decomposed matrix; therefore the Crout process has to be restarted for the entire problem.

Equations (7) and (9) can provide saving in computation time if the information on a portion of the scatterer is avail- able. Assuming that the initial scatterer comprises N, un- knowns and the complete scatterer has N, + N , unknowns, the total number of operations would be of the order of

( (N, + Nw)3 - N : ) / ( N c + N,)3. (10)

If N, >> N,, this ratio reduces to 3N,lN,. When N , becomes very large, the ratio approaches unity and the advantage is obviously lost.

The Crout algorithm also includes partial pivoting, which may be essential for the stability of the process. In the Crout algorithm, row permutation is performed under the diagonal

Figure 1 algorithm

Filling a decomposed matrix (6) by columns in the Crout Figure 2 Filling the decomposed matrix by upper-left blocks according to Eqs. (7) and (9)

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS I Vol. 7, No. 2, February 5 1994 49

element of the current column, i.e., for rows which contain l's that have already been computed and u's which are yet to be changed. The upper portion of the columns which have been determined so far are not changed, because they are already in the LU decomposed form. For this reason, partial pivoting can take place equally well in the case of Eq. (7), where the upper left block has been determined as the LU decomposition of Q . This is true for the case of block LU decomposition. The bottom rows of the remainder of the matrix, represented by the rectangular matrix c, can be in- terchanged in accordance with the selection of the pivot at each stage, and the computation of CU-' modified accordingly. The remainder of the rows, represented by the rectangular matrix b , does not take part in partial pivoting, hence the computation of L- lb will not be affected by it. If a true LU decomposition is performed by the addition of one basis function at a time, then the order by which the sub- problems are computed overrides pivoting; therefore it should constitute a stable order of additions from a physical point of view. In addition, column exchange to the right of the upper- left block may also be considered for a possible introduction of full pivoting.

The method is also well suited for cases where an initial portion of the scatterer is a special case which can be easily computed. Examples include circular cylinders or bodies of revolution (BORs), where the axial symmetry enables short- cuts in the computations using the FFT. BORs form the basis for many practical three-dimensional cases, and are therefore of great interest. Flat plates, generating block Toeplitz ma- trices, and certain long bodies that are amenable to two- dimensional approximation also fall into this category. If the inverse Q-l is known, it may be used for the efficient com- putation of the LU decomposition of Q. In this way, the advantages are incorporated into the recursive computation of the LU decomposition. This point is subject to further investigation.

3. RESULTS The finned cylinder, shown in Figure 3, is a model repre- senting many practical arrangements where the initial cylinder is augmented with a number of alternative fin configurations. The circular cylinder is composed of N , = 320 basis functions, which require 10,922,560 multiply-add operations for the LU decomposition. It is assumed that this initial decomposition has been performed, and it can serve as a starting point to several applications of Eq. (7). The actual operation count is

a / Einc

-----t ' ' +

I I I ' -h 10

I -10 -5.09 0 5.09

Figure 3 A finned cylinder with 5.09h radius. The circular cylinder is the initial structure, augmented by various fin arrangements

TABLE 1 Actual Numbers of Operations for Various Attachments to an initial Cylinder of 320 Basis Functions and Initial 10,922,560 Operations

Additional Total No. N , N , Operations Operations Percentage

1 320 32 3,615,392 14,537,952 24.9% 2 352 32 4,336,288 18,874,240 23.0%

4 370 50 7,811,650 24,695,860 31.6% 5 320 80 10,410,640 21,333,200 48.8% 6 320 180 30,743,940 41,666,500 73.8%

3 320 50 5,961,650 16,884,210 35.3%

listed in Table 1. In rows 1 and 2 of Table 1, the cylinder is augmented first with one fin of Nwl = 32 basis functions, and then with an additional fin of another Nw2 = 32 basis func- tions. The additions of the first and the second fins require 3,615,392 and 4,336,288 operations, respectively. Comparison can be made with the total numbers of 14,537,952 and 18,874,240 operations needed for decomposing the entire matrices of the order of 352 and 484 for each of the two cases, showing that the additions of the NW1 and Nw2 basis functions are attained with 24.9% and 23.0% of the total time. Sim- ilarly, for adding NW1 = 50 and Nw2 = 50 in rows 3 and 4 we have 5,961,650 and 7,811,650 operations, respectively, re- sulting in 35.3% and 31.6% out of a total of 16,884,210 and 24,695,860 operations for each of the entire problems.

If two fins with 40 basis functions each, totaling N , = 80, are to be added to the same initial cylindrical structure with N , = 320, as in row 5, the number of operations involved is 10,410,640, or 48.8% of the total 21,333,200 operations needed for the decomposition of the entire problem. For the addition of four fins with a total of N , = 180 basis functions (row 6), 30,743,940 operations are needed, or 73.8% out of the total 41,666,500 that would have been carried out if the entire problem were to be solved as a whole. This actual operation count is in accordance with (10). One can see that indeed the advantages are more significant as "INc is smaller, as alluded to in conjunction with (10).

Each of the examples described here has also been worked out as a whole with a standard library solver, and the current distributions have been found to be identical. At this point, pivoting has not been incorporated into the process, and no effects have been observed with regard to stability. The cur- rent distributions for the example of NW1 = Nw2 = 50 are shown in Figure 4 for 45" TM incidence on the doubly finned cylinder of Figure 3. The fin facing the incoming wave, marked as No. 1 in Figure 3, carries the current shown in Figure 4(a), the opposite fin No. 2 has the current shown in Figure 4(b), and the current on the cylinder with N , = 320 is depicted in Figure 4(c). These distributions are in accordance with the independent moment-method computation.

4. CONCLUSIONS The recursive LU decomposition approach is best suited for cases where a number of different attachments are to be made to a common known scatterer. For these cases, the total computational cost with a given known subproblem should be significantly reduced compared with a direct computation of the LU decomposition of the entire matrix. No special geometries are assumed, and no assumptions on small changes as the shape is modified are included. In fact, the addition of just one basis function can produce substantial changes in the current distribution, as in the case of an open cavity evolving

50 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS I Vol. 7, No. 2, February 5 1994

mavelength mavelength

PHI@EGREES) Figure 4 Current distribution on the finned cylinder of Figure 3, subject to 45” TM incidence. (a) Current of illuminated fin No. 1, (L) current on shadowed fin No. 2, (c) current on cylinder

into a closed one. The solution at each stage is in accordance with the rigorous MM solution. The algorithm can be in- tegrated into existing moment-method packages, because the matrix element computation is done independently of the algorithm, and the subsequent back and forward substitution processes are not affected by this modification.

At present, partial or full pivoting has not been incorpo- rated into the scheme, and its effects on stability are a subject of further investigation. Another topic under consideration is determining the LU decomposition of structures for which the inverse of the matrix is rapidly obtainable. In the examples of Section 3, the circular cylinder has been used as the initial structure; however, no advantage has been taken of the cir- culant matrix and the ease by which the matrix inverse can

be obtained using the FFT. Shortcuts such as this can con- tribute substantially to the efficiency of spatial decomposition of mixed structures with a simple initial structure augmented with more general additions. Although the principle is well established and the method applies to any size of problem modeled by an integral equation, larger problems will be addressed as better computational facilities become available.

REFERENCES 1. C. Lanczos, Applied Analysis, Prentice Hall, New York, 1956, pp.

2. R. Kastner, “A Recursive “Add-on” Method for the Anaiysis of Scattering from Planar Structures,” In Proc. 1987 IEEE AP-S

141-143.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 7, No. 2, February 5 1994 51

lnternational Symposium, Blacksburg, VA, June 15-19,1987, Vol.

3. R. Kastner, “An ‘Add-on’ Method for the Analysis of Large Planar Structures,” lEEE Trans. Antennas Propagat., Vol. AP- 37, March 1989, pp. 353-361.

4. A. S. Householder, The Theory of Matrices in Numerical Analysis, Dover Publications, New York, 1964.

5. A. Jennings, Matrix Computation for Engineers and Scientists, Wiley, New York, 1977.

1, pp. 276-279.

Received 8-30-93

Microwave and Optical Technology Letters, 712, 48-52 0 1994 John Wiley & Sons, Inc. CCC 08952477194

ON THE EXISTENCE CONDITION FOR BlSTABlLlTY IN A ONE-COUPLER OPTICAL FIBER RING RESONATOR BY USING DEGENERATE TWO-WAVE MIXING Y. H. Ja Telecorn Australia Research Laboratories PO Box 249 Clayton, Victoria 3168, Australia

KEY TERMS Optical bistability, optical fiber ring resonators, degenerate two-wave mixing

ABSTRACT The existence condition for bistability, B2G > 1, in a one-coupler single-mode optical fiber ring resonator using two different all-fiber schemes of degenerate two-wave mixing (DTWM) is studied in de- tail. It is shown that the existence condition for bistability is in gen- eral a necessary condition only, not a sufJicient condition for the oc- currence of the DTWM bistability. 0 I994 John Wiley & Sons, Inc.

1. INTRODUCTION Recently we studied theoretically a new type of bistability in a two-coupler [l] and a one-coupler [2] optical fiber ring res- onator by using degenerate two-wave mixing (DTWM). The bistability is caused by the dependence of the DTWM (power) gain on the intensity ratio of the signal to the pump beam and the inherent positive feedback in the fiber ring [2].

One obvious difference between the DTWM bistability and other types of optical bistabilities caused by self-focusing or self-trapping [3] and optical Kerr effect [4, 51 is the use of an additional beam, that is, the pump beam. Use of the pump beam results in the special property that the DTWM bistability can occur at any intensity level of the input, because it depends only on the intensity ratio of the input to pump beam [l]. Such a property is different from many other types of optical bistability [3-51, where the input intensity must be larger than a certain threshold level. However, the DTWM bistability does have an existence condition, B2G > 1 [see Eq. (l)], which is material dependent [ l , 21 but not intensity dependent.

In this Letter, we present a general theoretical study of the existence condition for the DTWM bistability when two different all-fiber schemes [I, 6,7] are used to implement the DTWM in a one-coupler fiber ring resonator. Figure 1 shows their schematic diagrams. In Scheme A [Figure l(a)], the

I, pump

Figure 1 A single-mode fiber ring resonator with one coupler using two different all-fiber schemes of the degenerate two-wave mixing, (a) Scheme A, and (b) Scheme B

entire fiber ring is made with a photorefractive fiber (p.f.), and the pump beam also travels along the fiber core, but in an opposite direction to the signal beam. In Scheme B [Figure l(b)], a short strand of a p.f. with length of about a few millimeters is spliced into a fiber ring made with a conven- tional (silica or fluoride) fiber and the pump beam illuminating the p.f. from the side. The pump beam makes a small angle with the axis of the p.f. [for clarity, in Figure l(b) the angle and the length of the p.f. strand are greatly exaggerated] and it does not propagate inside the fiber. In both cases, a re- flection phase grating is formed inside the p.f. due to the photorefractive effect [7]. Energy transfer can take place be- tween the mutually coherent signal and pump beams, and we assume that energy is always transferred from the pump beam to the signal beam. Thus, the signal beam can be directly amplified.

2. THEORY In Scheme A of the DTWM as shown in Figure l(a) the pump beam is launched into the p.f. through a circulator or a second fiber coupler. The DTWM bistability may occur only when [I , 21

B2G > 1, (1)

where

B = a t F K , ( 2 ) G = exp(I’) = exp(yd). (3)

In the above equations, a = is the amplitude trans- mission coefficient of the coupler, and yc is the excess loss of the coupler, K is the intensity coupling coefficient of the cou- pler, t = exp[ - (ad + 01 is the amplitude transmission coef- ficient of the p.f., a is the amplitude attenuation coefficient of the p.f., d is the length of the ring made with the p.f., y and r are the effective coupling coefficient and the coupling strength of the p.f., respectively, ,$is the splicing loss, B2 can be regarded as the round trip intensity transmission coefficient of the fiber ring, and G is the maximum value of the DTWM (power) gain g [l].

One can see that for a given B, it is easier to satisfy con- dition (1) with a larger G. This means a stronger DTWM process and larger amplification of the signal beam. Note that condition (1) is similar to B2G = 1, the threshold gain con- dition for a fiber ring oscillator [6], where no input beam is needed [Ii = 0 in Figure l(a)] and the phase change e of the fiber ring is required to be 2mv with m = 1, 2, . . , , N. Although condition (1) requires a larger G, oscillation will

52 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS I Vol. 7, No. 2, February 5 1994