Sumanth Kaushik Vol. 15, No. 4 /April 1998 /J. Opt. Soc. Am. A 1009

Vector Fresnel equations and Airy formula forone-dimensional multilayer and

surface-relief gratings: reply to comment

Sumanth Kaushik

Sandia National Laboratories, Albuquerque, New Mexico 87185-1405

Received July 21, 1997; accepted October 26, 1997; revised manuscript received November 20, 1997

In a recent paper [J. Opt. Soc. Am. A 14, 596 (1997)], S. Kaushik describes a modal theory of diffraction inwhich a number of features from scalar optics are generalized. The paper describes an S-matrix propagationalgorithm that is characterized as being new and an improvement over earlier work. In a response to thispaper, L. Li [J. Opt. Soc. Am. A. 15, 1006 (1998)] disputes this claim and claims that the algorithm is wellknown and presents no significant improvement over earlier work. These criticisms are addressed in thisreply. Specifically, it is shown that unlike earlier work cited by Li, the method described by Kaushik is agenuine S-matrix method since energy balance is automatically guaranteed for dielectric gratings, irrespectiveof truncation order. Further, it is shown that the algorithm is easier to relate to scalar optics and is compu-tationally more efficient than the specific algorithms cited by Li. © 1998 Optical Society of America[S0740-3232(98)01104-1]

OCIS codes: 050.0050, 050.2770, 230.4170, 350.5550.

In a recent paper of mine that appeared in the Journal ofthe Optical Society of America A1 I use a modal theory todescribe a formulation of vector diffraction that is a gen-eralization of results well known in scalar optics. In afollow-up paper2 L. Li disputes the novelty of one of theprincipal results claimed in Ref. 1. Li argues that thematrix propagation algorithm (referred to as an S-matrixpropagation algorithm in Ref. 1) is well known to re-searchers (in particular, Refs. 3–7) and that it offers nonew physical insight or intuition as claimed in Ref. 1.The aim of this paper is to address these criticisms.

The starting point is to clearly define the algorithm andhighlight the features that are novel. The term ‘‘matrixpropagation algorithm’’ used in Ref. 1 refers not only toEqs. (16) but also to the specific procedure of discretizinga surface-relief grating into M individual lamellar grat-ings and to the specific matrices that define Eqs. (16).There are principally three novel features of the algo-rithm. First, the algorithm as described in Ref. 1 auto-matically satisfies the energy balance for dielectric grat-ings, even for truncated matrices. Second, every featureof the propagation algorithm, including the individualmatrices that compose the propagation algorithm, can bereadily traced to a corresponding entity in a scalar theory.Finally, the algorithm in its most straightforward imple-mentation is more computationally efficient. The latterfeature, however, was not the focus of Ref. 1 and was notemphasized but will be elaborated briefly in this paper.

At a most basic level, the algorithms in Refs. 4, 6, and 7are not identical to Eqs. (16) of Ref. 1. For example, inEqs. (12) of Ref. 6 the propagation equation for Rdu re-quires two different r matrices (rdu and rud),

Rdu~ j ! 5 r̃du

~ j ! 1 t̃dd~ j !@1 2 Rdu

~ j21 !r̃ud~ j !#21Rdu

~ j21 !t̃uu~ j ! ,

whereas the corresponding equation in Ref. 1 requires asingle g i matrix. This is because the propagation algo-

0740-3232/98/041009-03$15.00 ©

rithm in Ref. 1 is expressed in terms of layer transmissionand reflection matrices (g i and ti) instead of the interfacematrices (r and t matrices in Refs. 6 and 7 and t11 , t12

in Ref. 4). By expressing the propagation algorithm interms of layer matrices and generalizing the work of Bot-ten et al.8 to multilayer gratings, the algorithm in Ref. 1is shown to rigorously conserve energy even for truncatedmatrices. On the other hand, for the algorithm describedin Refs. 6 and 7, no immediate statements on energy bal-ance can be made, even for infinite truncation order. Infact, in Ref. 7 Li explicitly states that his S-matrix propa-gation algorithm does not satisfy energy balance.

The usefulness of this feature appears to be a subject ofcontroversy in Ref. 2. Li argues that ‘‘satisfying energytheorem is not an intrinsic property of the matrix propa-gation algorithm’’ (p. 1008). He points out that energybalance cannot be used as a criterion for metallic grat-ings. Furthermore, he argues that violation of energybalance has been used by other researchers as an indica-tor of convergence and that therefore it is not necessarilyunappealing that an algorithm does not conserve energybalance automatically.

Although it is certainly true that researchers have usedthe energy-balance criterion to quantify the effect of trun-cation, it should be noted that the earlier researchers donot refer to the matrices arising in the computation as Smatrices.8,9 Furthermore, to my knowledge, no re-searcher has justified this procedure on the basis of physi-cal arguments. It should be remembered that the T(transition), R (reactance), and S (scattering) matriceshave a rich history in atomic, nuclear, and molecular scat-tering theory, with literature dating back to the early1920’s.10,11 In the physics community there is a well-established terminology, and these three matrices havewell-understood properties. One of the most important

1998 Optical Society of America

1010 J. Opt. Soc. Am. A/Vol. 15, No. 4 /April 1998 Sumanth Kaushik

properties of the S and R matrices is that for lossless sys-tems they are unitary. Since the S matrix is unitary, thecomplex poles of the matrix can be readily interpreted asthe excitation spectrum, with the imaginary part repre-senting the resonance linewidths. A similar interpreta-tion is much more difficult to obtain by use of a nonuni-tary S matrix. Therefore even when exact represen-tation of the S matrix is not possible (e.g., by use of atruncated basis set), there are well-known procedures forobtaining an approximate S matrix that is unitary.10

In linear network and transmission-line theory, thesematrices also play a prominent role. Like atomic andnuclear scattering processes, the S matrices appearing innetwork analysis are also unitary for lossless systems.For lossy linear networks and transmission lines, S ma-trices are not required to be unitary; however, they retainother important properties.12 For a linear system com-posed of several finite-dimensional systems, there arewell-established propagation algorithms, in which the Smatrix for the larger system is obtained from the S ma-trices of the smaller systems. In Ref. 2 Li expands onthis point. However, an important point that is missed isthat for a lossless system the S matrix (as per establishedterminology) for the combined system must also be uni-tary. It is true that a matrix propagation algorithm is analgebraic method for connecting the characteristic matri-ces of the different layers. However, to justify calling analgorithm an S-matrix algorithm, it is only reasonable toexpect the algorithm to preserve the properties of the Smatrix at each propagation step. Since unitarity is animportant property of the S matrix (for a lossless system),it seems only natural to insist that the algorithm main-tain this property where applicable.

It should be noted that the R-matrix algorithm de-scribed by Li in Ref. 7 correctly retains the unitarity(though this is never mathematically proven in Refs. 6and 7). The unitary property of the R-matrix propaga-tion algorithm has been stressed in earlier literature.13

However, this feature appears to have been lost in theS-matrix formalism of Li in Refs. 6 and 7, and no reason-able justification for this is provided. In fact, on the basisof conventionally accepted terminology, the algorithmpresented in Refs. 6 and 7 is not a true S-matrix theory.On the other hand, the algorithm shown in Ref. 1 retainsall the important features of an S-matrix theory.

Having shown that the algorithm in Ref. 1 differs fromthe algorithm presented in Refs. 6 and 7, I next addressthe question of whether the new algorithm offers any ad-ditional insight, or physical intuition, into grating compu-tation. Clearly, the use of the S matrix in optical thin-film work is well established. In Ref. 2 Li comments thatRef. 1 does not adequately acknowledge earlier referencessuch as Refs. 3–7. However, it should be noted thatmany of these references address only the physics of opti-cal thin films (e.g., Refs. 3 and 5). If we consider thatthese methods are found in introductory optics texts (e.g.,see Ref. 14), it is not clear that references for this estab-lished formalism are required. Li opines that the princi-pal physics of the S-matrix algorithm is its description ofthe optical interference arising from multiple reflections.Although the S matrix certainly captures this physics, sodo the T and R matrices. In scattering theory the S ma-

trix is important not because it incorporates multiplescattering (which it does) but because, unlike the T and Rmatrices, it characterizes the input response of a physicalsystem. As described earlier in this reply, the eigenval-ues and eigenvectors are the excitation frequencies andnormal modes of the system. This is the fundamentalphysical content of the S matrix.

Tamir and co-workers have used the S matrix exten-sively in their analysis of grating-assisted couplers.15,16

Guided modes appear as poles in the S matrix. Theseauthors also develop a matrix propagation algorithm formultilayer gratings. This algorithm appears to be simi-lar to that presented in Ref. 1 and, on the basis of the de-scription provided, appears to be a true S-matrix theory.In fact, the Airy and Fresnel formulas of Ref. 1 are pre-sented in terms of equivalent transmission-lineformulas.15,16 Therefore the methods developed in thesereferences are directly applicable to the formalism devel-oped in Ref. 1. Since the matrices obtained by the propa-gation algorithm in Refs. 6 and 7 do not preserve the im-portant properties of the S matrix, it is not clear that thisalgorithm is similarly general in its applicability.

From the viewpoint also of scalar optics, the propaga-tion algorithm in Refs. 6 and 7 is inadequate. In scalaroptics, reciprocity would imply that rdu 5 2rud . Thiswould erroneously imply that Eqs. (12) in Ref. 6 could befurther simplified, which is clearly false. Thus intuitionfrom scalar optics is seen to break down. There is nosuch ambiguity in the formulation developed in Ref. 1since, for a single lamellar slab, there is only a single setof transmission and reflection matrices. Of course, invector diffraction the order in which the matrices appearin Eqs. (16) is important. However, the order can bereadily intuited by expanding 1 /(1 2 gG) in series.

At a larger level, it is certainly appealing that everymatrix in Eqs. (16) of Ref. 1 can be identified with scalarformulas such as the Fresnel equations and the Airy for-mulas. Reciprocity relations are generalized in a simplemanner. Results for both TE and TM polarization areprovided in terms of well-understood scalar formulas.Additionally, the methods outlined in Ref. 1 are readilyapplicable to coupled-wave analysis as well as to classicalmodal analysis. It is hard to gain similar insights fromEqs. (4) in Ref. 6 and Eqs. (15) in Ref. 17.

Finally, I will briefly comment on the relative numeri-cal efficiencies of the two algorithms. It should be notedthat nowhere in the abstract is an argument for improvednumerical efficiency mentioned. Li somehow arrives atthis meaning despite the clear statement in the introduc-tion that ‘‘the purpose here is not to dwell too much onimplementation details’’ (Ref. 1, p. 597). This notwith-standing, as Li correctly points out, Ref. 1 claims that thepropagation algorithm requires fewer matrix operationsthan that presented in Ref. 6. This difference is not dueto a misunderstanding of the matrix propagation algo-rithm as claimed in Ref. 2.

Indeed, the number of matrix operations in Eqs. (16) isthe same as the number of matrix operations in Eqs. (12)of Ref. 6. However, the matrices in Ref. 1 are layer ma-trices, whereas the matrices appearing in Ref. 6 are inter-face matrices; and for gratings with many layers, thereare roughly twice as many interfaces as there are layers.

Sumanth Kaushik Vol. 15, No. 4 /April 1998 /J. Opt. Soc. Am. A 1011

Therefore the propagation algorithm has to be appliedtwice as often in Refs. 6 and 7. Of course, in theS-matrix algorithm of Ref. 1, the Airy and Fresnel formu-las have to be applied at each layer to obtain the layermatrices, and this results in additional matrix operations.Based on an operation count, the most straightforwardimplementation of Eqs. (16) of Ref. 1 requires five matrixmultiplications and one matrix division. In TE polariza-tion the Airy and Fresnel formulas require nine matrixmultiplications and three matrix divisions, resulting in atotal of fourteen matrix multiplications and four matrixdivisions for propagation through an entire layer. In TMpolarization two additional matrix multiplications are re-quired for the Fresnel formulas.

In contrast, the algorithm as presented by Eqs. (12) ofRef. 6 requires nine matrix multiplications and three ma-trix divisions at each interface. Five multiplications andone division arise from the propagation algorithm, andthe remaining four multiplications and two divisions arerequired for assembling the interface r and t matrices asgiven in Eqs. (4) and (15) of Refs. 6 and 17, respectively.Thus the algorithm in Ref. 6 takes a total of eighteen ma-trix multiplications and six matrix divisions. This isclearly more computationally intensive than the algo-rithm in Ref. 1 and was duly noted in Sec. 4 of Ref. 1.

To summarize, I have identified the features of theS-matrix propagation algorithm in Ref. 1 that differ fromearlier work, most notably that of Li in Refs. 6 and 7. Ihave shown that the work in Ref. 1 is a true S-matrixtheory, whereas the work in Refs. 6 and 7 is not, since thepropagated matrices do not have the required propertiesof an S matrix. Furthermore, I have shown that theS-matrix algorithm in Ref. 1 is indeed more computation-ally efficient (in its most direct implementation) becausethe algorithm is expressed in terms of layer matricesrather than interface matrices. Additionally, I haveshown why the physics implied by the algorithm is moretransparent than that of earlier work since every featureof the algorithm, including the physics, has a correspond-ing equivalent in a scalar theory.

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