Methods for 3-D vector microcavity problems involving a planar dielectric mirror

Transcription

1 Methos for 3-D vector microcavity problems involving a planar ielectric mirror Davi H. Foster an Jens U. Nöcel Department of Physics University of Oregon Eugene, OR Publishe in Optics Communications 234, We evelop an emonstrate two numerical methos for solving the class of open cavity problems which involve a curve, cylinrically symmetric conucting mirror facing a planar ielectric stac. Such ome-shape cavities are useful ue to their tight focusing of light onto the flat surface. The first metho uses the Bessel wave basis. From this metho evolves a two-basis metho, which ultimately uses a multipole basis. Each metho is evelope for both the scalar fiel an the electromagnetic vector fiel an explicit en user formulas are given. All of these methos characterize the arbitrary ielectric stac mirror entirely by its 2 2 transfer matrices for s- an p-polarization. We explain both theoretical an practical limitations to our metho. Non-trivial emonstrations are gi ven, incluing one of a stacinuce effect the mixing of near-egenerate Laguerre-Gaussian moes that may persist arbitrarily far into the paraxial limit. Cavities as large as 5λ are treate, far exceeing any vectorial solutions previously reporte. Contents 1. Introuction 3 2. Overview of the Moel an Notation 5 1

2 3. Plane Wave Bases an the Bessel Wave Metho The Fiel Expansion in the Simple Plane Wave Bases Scalar basis Vector basis The Fiel Expansion in the Bessel Wave Bases Scalar basis Vector basis The Linear System of Equations for the PWB The planar mirror M1 bounary equations The curve mirror M2 bounary equations The see equation Solution of Ay=b Calculating the fiel from y Multipole Bases an the Two-Basis Metho The Scalar Multipole Basis The M1 Equations in the Scalar MB Variant Variant The Linear System of Equations in the Scalar MB Calculating the Fiel in the Layers with the Scalar MB The Vector Multipole Basis The M1 Equations in the Vector MB The M1 equations for s-polarization The M1 equations for p-polarization Variant Variant The Linear System of Equations in the Vector MB Calculating the Fiel in the Layers with the Vector MB Demonstrations an Comparisons The V Moe: A Stac Effect Persistent Stac-Inuce Mixing of Near-Degenerate Laguerre-Gaussian Moe Pairs Paraxial Theory for Vector Fiels A emonstration of persistent mixing Moes with m N irs an l max : Comparing the Two Primary Methos Almost-Real MB Coefficients Conclusions 42 2

3 A. Further Explanations an Limitations of the Moel 44 A.1. Exclusion of High-Angle Plane Waves A.2. The Hat Brim A.2.1. The infinitesimal hat brim A.2.2. The infinite hat brim an the 1-D half-plane cavity B. Negative m Moes an Sine an Cosine Moes 46 B.1. Plotting with -m B.2. Plotting cosine an sine moes C. Stacs use in Section Introuction There is currently consierable interest in the nature of electromagnetic vector moes both for free space propagation [1, 2, 3, 4] an in cavity resonators [5]. In particular, recent avances in fabrication technology have given rise to optical cavities which cannot be moele by effectively two-imensional, scalar or pseuo-vectorial wave equations [6]. The resulting moes may exhibit non-paraxial structure an nontrivial polarization, but this ae complexity also gives rise to esirable effects; an example from free-space optics is the observation of enhance focusing for raially polarize beams [4]. This effect is shown here to arise in cavities as well, among a rich variety of other moes that epen on the three-imensional geometry. The main goal of this paper is to present a set of numerical techniques aapte to a realistic cavity esign as escribe below. Much wor involving optical cavity resonators utilizes mirrors that are compose of thin layers of ielectric material. These ielectric stac mirrors offer both high reflectivity an a low ratio of loss to transmission, which are esirable in many applications. The simplest moel of such cavities treats the mirrors as perfect conuctors E tangential =, H normal =. In applications involving paraxial moes an highly reflective mirrors, it is often acceptable to use this treatment. The mature theory of Gaussian moes c.f. Siegman [7] is applicable for this class of cavity resonator. When an application requires going beyon the paraxial approximation to escribe the optical moes of interest, the problem becomes significantly more involve an moeling ielectric stac mirrors as conucting mirrors may become a poor approximation. Also, one is often intereste in the fiel insie the ielectric stac an for this reason must inclue the stac structure into the problem. In this paper we present a group of improve methos for resonators with a cylinrically symmetric curve mirror facing a planar mirror. The paraxial conition is not necessary for these methos; tightly focuse moes can be stuie. Furthermore, the true vector electromagnetic fiel is use, rather than a scalar fiel approximation. The planar mirror is treate as infinite an is characterize by its polarization-epenent reflection functions r s θ inc an r p θ inc for plane waves with incient angle θ inc. Our moel encompasses both cavities for which the planar mirror is an arbitrary ielectric 3

4 stac, an cavities for which the planar mirror is a simple mirror conucting with r s/p θ inc = 1 or free with r s/p θ inc = +1. The opposing curve mirror is always treate as a conuctor. It shoul be note that, for most moes which are highly focuse at the planar mirror, moes which are liely to be of interest in applications, this limitation can be expecte to cause little error because the local wave fronts at the curve mirror are mostly perpenicular to its surface. Of course, for applications in which the curve mirror is inee conucting, our moel is very well suite. On the other han, the correct treatment of the planar mirror can be a great improvement over the simplest moel. Applications with both ielectric an conucting curve mirrors have been an are currently being use experimentally [5, 8, 9]. The methos escribe here belong to a class of methos which we refer to as basis expansion methos. In basis expansion methos, a complete, orthogonal basis such as the basis of electromagnetic plane waves is chosen. Each basis function itself obeys Maxwell s equations. The equations that etermine the correct value of the basis coefficients are bounary equations, resulting from matching appropriate fiels at ielectric interfaces, setting appropriate fiel components to zero at conuctor-ielectric interfaces, an setting certain fiels to be zero at the origin or infinity. In the usual application of this metho, each homogenous ielectric region is allocate its own set of basis coefficients. Our methos use a single set of basis coefficients; the matching between ielectric layers is hanle by the 2 2 transfer matrices of the stac. As ielectric stacs have nonzero transmission, optical cavities with this type of mirror are necessarily open, or lossy. The methos escribe here eal with the openness ue to the stac an the solutions are quasimoes, with iscrete, isolate complex wavenumbers which enote both the optimal riving frequency an the resonance with 1. While the ielectric mirror is partially responsible for the openness of our moel system, the openness is not primarily responsible for moe pattern changes resulting from replacing a ielectric stac mirror with a simple mirror. The phase shifts of plane waves reflecte off a ielectric stac can vary with incient angle, an it is this variation which can cause significant changes in the moes, even though reflectivities may be greater than.99. Generally speaing, the eviation of r s/p θ inc from 1 is not as important as the eviation of argr s/p θ inc from, say, argr s/p. We evelop two general methos, the two-basis metho an the Bessel wave metho. The scalar fiel versions of both methos are also evelope an are iscusse first, acting as peagogical stepping stones to the vector fiel versions. The Bessel wave metho uses the Bessel wave basis which is the cylinrically symmetric version of the plane wave basis. This metho is escribe in Section 3. The two-basis metho ultimately uses the vector or scalar multipole basis. The multipole basis has an avantage in that it is the eigenbasis of a conucting hemisphere, the canonical ome-shape cavity. The unusual aspect of 1 For many moes, there is also loss ue to lateral escape from the sies of the cavity. While our moel intrinsically incorporates the openness ue to lateral escape in the calculation of the fiels by simply not closing the curve mirror surface, or extening its ege into the ielectric stac, this loss is not inclue in the calculate resonance with or quality factor, Q. Because a single set of basis vectors is use to escribe the fiel in the half-plane above the planar mirror, this entire half-plane is the cavity as far as the calculation of resonance with is concerne. 4

5 the two-basis metho is the intermeiate use of the Bessel wave basis. The two-basis metho is evelope in Section 4. We have implemente both methos an have use them as numerical checs against each other. Various emonstrations an comparisons are given in Section 5. Our implementations of all methos are programme in C++, use the GSL, LAPACK, SLATEC, an PGPlot numerical libraries, an run on a Macintosh G4 with OS X. Limitations of our moel an methos are iscusse in Appenix A. Appenix B iscusses plotting moes that are associate with linear polarization an Appenix C specifies ielectric stacs that are use in Section Overview of the Moel an Notation Figure 1: The cavity moel. A iagram of the moel is shown in Figure 1. The conucting surface is inicate by the heavy line. The annular portion of this surface extening horizontally from the ome ege will be referre to as the hat brim. The ome is cylinrically symmetric with maximum height z = R an ege height z = z e. The shape of the ome is arbitrary, but in our emonstrations the ome will be a part of an origin-centere sphere of raius R s = R unless otherwise specifie. The region surrouning the curve mirror will be referre to as layer. The ielectric interface between layer an layer 1 has height z = z 1. The last layer of the ielectric stac is layer N an the exit layer is calle layer X. The epiction of the stac layers in the figure suggests a esign in which the stac consists of some layers of experimental interest perhaps containing quantum wells, ots, or other structures [5, 8] at the top of the stac where the fiel intensity is high, an a highly reflective perioic structure below. 5

6 At the heart of the proceure to solve for the quasimoes is an overetermine, complex linear system of equations, Ay = b. The column vector y is mae up of the coefficients of eigenmoes in some basis B. The fiel in layer is given by expansion in B using these coefficients. For a given wavenumber,, a solution vector y = y best can be foun so that A y b 2 is minimize with respect to y. Dips in the graph of the resiual quantity, r A y best b, versus signify the locations of the isolate eigenvalues of theoretically r shoul become at the eigenvalues. The solution vector y best at one of these eigenvalues escribes a quasimoe. The system of equations is mae up of three parts as shown below: M1 equations, M2 equations an an arbitrary amplitue or see equation. [ ] [ M1. ] A y = [ M2 y =. 1 ]. s. eqn. 1 Henceforth M1 refers to the planar mirror an M2 to the curve mirror. The M1 bounary conition for a plane wave basis is expresse simply in terms of the 2 2 stac transfer matrices T s θ inc an T p θ inc, as suggeste by the M1 region enclose by the ashe line in Fig. 1. In the scalar an vector multipole bases, a sort of conversion to plane waves is require as an intermeiate step. The ashe vectors in the figure incoming from the bottom of the stac represent plane waves that are given zero amplitue, in orer to efine a quasimoe problem rather than a scattering problem. The plane waves enote by the soli vectors have nonzero amplitue. The M2 bounary conition is implemente as follows. A number of locations on the curve mirror are chosen the X mars in Fig. 1. The with of the hat brim is w b as shown. An infinitesimal hat brim w b λ is introuce to give the ome a iffractive ege. An infinite hat brim can be introuce theoretically an can mae the moel more easily unerstanable in certain respects. More about the moel in relation to the hat brim is iscusse in Appenix A.2. The M2 equations are the equations in basis B setting the appropriate fiels at these locations to zero. For a problem not possessing cylinrical symmetry, these locations woul be points. The simplification ue to this symmetry, however, allows these locations to be entire rings about the z axis, specifie by a single parameter such as the ρ coorinate. Finally, the see equation sets some combination of basis coefficients equal to one an is the only equation with a nonzero value on the right han sie b. The cylinrical symmetry of the bounary conitions allows one to always fin solutions which have a φ epenence of expımφ, where m is an integer. This in turn leas to a imensionally reuce version of the plane wave basis calle the Bessel wave basis, in which each basis function is a superposition of all the plane waves with the same wavevector polar angle, θ. The weight function of the superposition is proportional to expımφ. We will refer to the non-reuce basis as the simple plane wave basis. The 6

7 unaorne phrase plane wave basis PWB; same abbreviation for plural will refer henceforth to either or both of the Bessel wave an simple plane wave bases. When using the scalar or vector multipole basis MB; same abbreviation for plural, cylinrical symmetry allows the problem to be solve separately for each quantum number m of interest 2. The imensional reuction in this case amounts to the removal of a summation over m in the basis expansion. The refractive inex in region layer q is enote ñ q. Layers are also enote with an upper subscript in parenthesis: E q means the electric vector fiel in layer q. Sometimes fs is use as a value of q, meaning in free space e.g. ñ fs, whether or not any of the layers in the moel we are consiering actually are free space. We note here that ñ fs = 1 only in cavity type I iscusse below. The symbol, where not in a super/subscript an not bol nor having any super/subscripts, always refers to what may be calle the wavenumber in free space, although it will have an imaginary part if M1 is a ielectric mirror. An imaginary part in wavenumber, refractive inex, an/or frequency is often introuce as it is in this problem to turn open cavity problems into eigenvalue problems. The efinition of is as follows. Define q 2 as the constant of separation use to separate space an time equations from the wave equation for layer q: 2 X q = ñ2 q c 2 2 X q t 2. 2 Here X q may be a vector or scalar fiel. In a few steps, the selection of a global monochromatic time epenence exp ıωt reveals that the ratio q /ñ q is inepenent of q. Then is efine as fs, so that q = n q where n q ñ q /ñ fs. In the moel, the inex ratios n q are assume to be real. The single plane wave solution to 2 has the form X q = C q e ıq x e ıωt where C q is a constant vector or scalar an the complex wave vector q is given by q q Ω q vector of the plane wave, specifie by θ q = n q Ω q with Ω q being the unit irection an φ. The generally complex frequency is given by ω = c q /ñ q. At a refractive interface, the angle θ q changes as given by Snell s law. To unerstan the meaning of a complex, it is helpful to realize that the spatial epenence of the quasimoes are ientical 3 in the following two physical cavities: I : ñ fs = 1, ñ q = n q, II : ñ fs = Υ, ñ q = Υn q, Υ C. 3 2 The moes with low m are liely to be of practical interest since they have the simplest transverse polarization structure. The m = 1 family of vector eigenmoes is exceptional because the proportionality of E ρ an E φ to expımφ means that moes for m = 1 have no average transverse electric fiel, even instantaneously. It is straightforwar to show that ReE x φ = ReE y φ = if m = 1. Thus a uniformly polarize, focuse beam centere on the cavity axis can not couple to cavity moes with m 1! 3 There is the minor ifference for the magnetic fiel first note in Eqn. 12. Since the iscrepancy is a constant factor multiplying H, however, this ifference is not necessarily part of the spatial epenence. 7

8 The shape an size of each ielectric an conucting region are the same for cavities I an II. Cavity I is compose of of conuctors an zero-gain regions of real refractive inex. Cavity II is constructe by taing cavity I an multiplying the refractive inex of each region, incluing free space, by an arbitrary complex number Υ. The congruence of the spatial quasimoes follows from separating the variables in 2. The values of, q, an n q for a given quasimoe are the same in cavities I an II. The frequency in cavity II is ω II = ω I /Υ. If we henceforth consier only the specific cavity II for which Υ = 1 ıg where g is tune to be the ratio Im/Re for a given quasimoe, we see that ω II = cre = Reω I. While in cavity I the quasimoe ecays in time, in cavity II the quasimoe is a steay state because the gain exactly offsets the loss. Either of the two cavity types may be imagine to be the case in our treatment. The only ifference is the existence of the ecay factor expcimt for cavity I. The inequality Im always turns up for an eigenvalue problem with conucting an/or ielectric interface bounary conitions. We note that the relation of to the free space wavelength always real of a plane wave is = 1 ıg2π/λ. The quality factor, Q, of the quasimoe is Re/2 Im = 1/2g. We note that Snell s law, ñ q sin θ q have a cavity of type I or II because the quantity 1 ıg, if present, ivies out. One = ñ q+1 sin θ q+1, is inepenent of whether we of the limitations of our metho is the omission of evanescent waves in layer an in layers where n q n see Appenix A.1. Snell s law may cause θ q to become complex for layers with inex ratios n q less than n. In this case sin θ q > 1 an cos θ q = ısgncos θ [sin2 θ q 1] 1/2. In most cases the symbols ψ, E, an H stan for complex-value fiels. The time epenence is exp ıωt an it is usually suppresse. Physical fiels are obtaine by multiplying by the time epenence an then taing the real part. Throughout this paper, the common functions enote by Y lm, Pl m, P l, J n, j l, an n l are efine as they are in the boo by Jacson [1]. In the implementation, c an the relate constants ɛ, µ, an Z are all unity, an they will usually be roppe in our treatment. We also assume non-magnetic materials so that µ q = µ = Plane Wave Bases an the Bessel Wave Metho Although this major section escribes the Bessel wave metho, much of what is iscusse here is applicable to the two-basis metho with little alteration. The iscussion in Section 4 is greatly shortene ue to this overlap of concepts an proceures The Fiel Expansion in the Simple Plane Wave Bases Scalar basis A single scalar plane wave in layer q has the form ψ = C exp ı q x ıωt. For a general monochromatic fiel, an ω are fixe an the fiel can be expresse ue to 8

9 the completeness of the scalar PWB uniquely ue to the orthogonality of the scalar PWB as a sum over plane waves in ifferent irections. In our treatment however, we omit plane waves in layer q which woul only exist as evanescent waves when refracte into layer. The expansion for the fiel in layer q is ψ q x = 2π φ π θ sinθ q ψ eıq x. 4 ψ q Here the basis expansion coefficients are the continuous coefficients in the integral, an iscrete coefficients in implementation. The above expansion effectively propagates each plane wave existing in the cavity own whether forwar or bacwar into the stac q layers, an as up all of their contributions. In orer to express the ψ in terms of ψ, it is first necessary to separate the coefficients with z > from those with z < an write the above expansion as ψ q = 2π ψq u φ π/2 e ıq u x + α sinα q ψ eıq x. 5 The u an refer to the plane waves going upwar or ownwar, e.i. is the expansion q coefficient ψ for z q > or z >, since sgn z q = sgn z q an ψ taes the q place of ψ for z <. The wavevector q in cylinrical coorinates is ρ q, q φ, q z, for which the following relationships hol: This leas to where q ρ q z = q sin θ q = n sin θ = n sin α = ρ, = q cos θ q = n q sgncos θ cos αq 2 = n q sgncos θ n 1 sin α, n q q φ = φ inep. of q. 6 ψ q = 2π ψq u φ π/2 e ıϕz + α sinα eıϕρ ψ q u ψ q e ıϕz, 7 ϕ ρ ρn sinα cosφ φ, ϕ z zn q cos α q. 8 9

10 From stanar theory regaring plane waves an layere meia[11], one can calculate the 2 2 complex transfer matrix, T s q, that obeys the following equation ψq ψ e ıϕz ψ u q = T q e ıϕz s ψ u. 9 Defining the column sums + β s q T q s,12 + T q s,22 an + γ s q write the expansion of the scalar wave in the layers as ψ q = 2π φ π/2 e ıϕρ +β s q α sinα ψ u + + γ q s ψ T q s,11 + T q s,21 allows us to. 1 The reason for the notation with the subscripts + an s will become apparent in the vector iscussion. The s refers to s-polarization. The variables in the simple PWB for scalar fiels are the complex ψ u an ψ the superscript will often be roppe. Next we consier the simple PWB for vector fiels Vector basis We assume that for our purposes a general monochromatic electromagnetic fiel can be expresse uniquely as a sum of vector electromagnetic plane waves. For every given frequency an wavevector irection Ω there are two orthogonally polarize plane waves as oppose to a single plane wave in the scalar case. Instea of a single coefficient ψ for each spatial irection we nee two, S an P, which we can efine as follows. S is the amplitue of the vector plane wave propagating in irection Ω which has its electric fiel polarize in the x-y plane E z =. Thus, this plane wave is an s-wave with regar to the planar mirror. P is the amplitue of the p-wave, the vector plane wave in irection Ω which has its electric fiel polarize in the plane of incience E φ =. The coefficients S an P will be separate into S u, S, Pu, an P. To specify the polarization of the fiels we will use unit vectors enote by ɛ. The unit vector ɛ q s, enotes the irection of the electric fiel associate with the plane wave with wavevector q an s-polarization. We tae the irection of the unit vectors to be: ɛ q s, = ˆφ = ˆx sin φ ŷ cos φ, ɛ q p, q = ˆθ sgncos θq = ˆρ sgncos θ cos θq ẑsgncos θ sin θq. 11 In this phase convention use by Yeh[11], the projections of the ɛ q p, vectors for the incient an reflecte waves onto the x-y plane are equal. The other common phase convention has these projections being in opposite irections. 1

11 The entire electric an magnetic fiel can be broen up into two parts: E q = E s q + E p q an H q = H s q + H p q where H s q is the fiel with magnetic s-polarization H z = an H p q is the fiel with magnetic p-polarization H φ =. We can now write own the most compact expansion of the vector fiel. E q s = E q p = H q s H q p Ω Ω = ñ q = ñ q S q ɛq eıq x s,, P q ɛq eıq x p,, Ω Ω S q P q ɛq p, ɛq s,sgncos θ eıq x, sgncos θ eıq x. 12 The factors of ñ q in the H equations come from the physical relation of H to E for a plane wave. Note that ñ q is ifferent for cavity types I an II as given in Eqn. 3. Separating up an own coefficients yiels S q u/ E q s = E q p = 2π Sq u 2π [ φ ɛ q s, e ıϕz + φ π/2 ˆρ cosα q + ẑ sinα q π/2 α S q e ıϕz, α sinα sinα P q e ıϕz + u q P u e ıϕz + eıϕρ eıϕρ P q e ıϕz P q e ıϕz ]. 13 These expressions explicitly use coorinate vectors only where necessary ue to a epenence of the ɛ vectors on the sign of cos θ. The expressions for Hq are omitte for brevity. q To relate an P u/ to S u/ an P u/ we can use the transfer matrices: T s for s-polarize light an T p for p-polarize light. The transfer matrix use for the scalar fiel in Eqn. 9 is the same matrix we will use here for s-polarization. These matrices perform the following transformations We efine Sq e ıϕz S u q e ıϕz P q e ıϕz P u q e ıϕz ±β q s ±γ q s = T q s = T q p S S u P P u T q s,12 ± T q s,22, T q s,21 ± T q s,11,,

12 Note + β q s an + γ q s ±β p q T q p,12 ± T p,22, q ±γ q p T q p,21 ± T q p, are efine as before. The β an γ quantities are functions of z an z 1 an not of ρ or φ. They are functions of an α but not of φ. Now the fiel expansions become E q s = E q p = H q s H q p 2π 2π φ ɛ q s, +β s q S u π/2 α + + γ q s sinα, S π/2 φ α sinα [ ˆρ cosα q +β p q P u + ẑ sinα q β p q P u = ñ q 2π β q p φ ɛ q s, P u π/2 α P γ q p eıϕρ eıϕρ + + γ q p γ q p P sinα, P ], eıϕρ 2π π/2 = ñ q φ α sinα eıϕρ [ ˆρ cosα q β s q S u + γ s q ẑ sinα q +β s q S u + + γ s q S S ]. 16 The variables in the simple PWB for vector fiels are the complex the superscript will often be roppe. P S u, S, P u, an 3.2. The Fiel Expansion in the Bessel Wave Bases Scalar basis We have alreay assume a time epenence of exp ıωt. As mentione in the Overview, a cylinrically symmetric set of bounary conitions allows us to assume an azimuthal epenence of expımφ with m being an integer. Consier the expansion 4. We wish ψ q to fin the conitions on expımφ. The general Fourier series expansion of which cause the entire epenence of ψx on φ to be ψ q is ψ q θ, φ = n f q n θ eınφ

13 We can then write 4 as where ψ q = n π θ 2π The last integral is of the solve form 2π sinθ eı ϕz f q n θ φ e ıρn sinθ cosφ φ e ınφ, 18 ϕ z zn q cos θ q. 19 e ıy cosφ φ e ınφ φ = 2πı n J n ye ınφ, 2 where J n enotes the regular Bessel function of orer n n can be negative. This yiels ψ q = 2π n ı n e ınφ π e ı ϕz J n ρn sin θ θ sinθ q f n θ. 21 In orer to have only expımφ epenence on φ, the integral on the right han sie must be zero for n m. Because f n q θ cannot be a function of z or ρ, the only way to q have this for all z an ρ is to pic f n = for n m. Thus ψ θ, φ = f m q θ eımφ. At this point we efine the symbol ψ q to mean the coefficient f m q. The cylinrically symmetric expansion is where ψ q = ξ π θ sinθ eı ϕz J m ρn sin θ This is an expansion in scalar Bessel waves, efine to be ψq θ, 22 ξ 2πı m e ımφ. 23 ξ expızn q cos θ q J mρn sin θ, 24 with {ψ q θ } being the set of coefficients. Each Bessel wave is a set of simple plane waves with fixe polar angle but having the full range to 2π of azimuthal angles, φ. The weight factors of the plane waves are proportional to expımφ. The final cylinrically symmetric scalar expansion with up an own separate is π/2 ψ q x = ξ α sinα J mρn sin α +β s q ψ u + + γ s q ψ. 25 The ψ u an ψ are the complex variables in the Bessel wave metho for scalar fiels; they mae up the solution vector y in 1. 13

14 Vector basis For cylinrically symmetric bounary conitions, the φ epenence of E ρ, E z, E φ, H ρ, H z, an H φ can be taen for a single moe to be expımφ. Consier oing the φ integrations in 12 or 16. The unit vectors ɛ q s/p, an ˆρ epen on φ. The z components o not epen on φ so we will loo at these first. There is no contribution to the z component of the electric fiel from E s q nor is there any contribution to the z component of the magnetic fiel from H s q. We efine zp q x E q ẑ = E p q ẑ, H z P q x H q ẑ = H p q ẑ. 26 Requiring that these quantities have an expımφ epenence prouces results similar to the scalar case. Defining S q eımφ P q eımφ q S, P q, 27 an using 16, we have the final, useful expansions for z P q an H z P q : π/2 α zp q = ξ sinα sinαq J m ρn sin α π/2 H z P q = ξñ q α sinα J m ρn sin α β p q P u γ p q P sinαq +β s q S u + + γ s q S,. 28 To eal with the transverse part of the electromagnetic fiel it is helpful to use quantities relate to circular polarization. We efine Inverting 29 yiels ±S q ±ıe q s = e ±ıφ ±ıe q s ˆx E q s ŷ ˆρ E q s H ±S q ±ıh s q ˆx H s q ŷ = e ±ıφ ±ıh q s ±P q E q p = e ±ıφ E q p ˆx ± ıe q p ˆρ H q s ŷ ˆρ ± ıe q p H ±P q H p q ˆx ± ıh p q ŷ E q s E q s = e ±ıφ H q p ˆφ, ˆφ, ˆφ, ˆρ ± ıh q p ˆφ. 29 ˆρ = ı 2 +S q e ıφ S q e ıφ, ˆφ = 1 2 +S q e ıφ + S q e ıφ, 14

15 E q p ˆρ = 1 2 +P q e ıφ + P q e ıφ, E q p ˆφ = ı 2 +P q e ıφ P q e ıφ, 3 with the magnetic fiel quantities having similar relations. Now we use 12 an 11 with 29. The resulting electric fiel quantities are ±S q = ±P q = π θ 2π π θ 2π sinθ eı ϕz φ e ±ıφ e ıρn sinθ cosφ φ Sq, sinθ cosθq sgncos θ eı ϕz φ e ±ıφ e ıρn sinθ cosφ φ q P. 31 It is the exp±ıφ factors in the integrans here that motivate the efinitions of ± S/P 29. We see that the substitution of 27 into 31 results in φ integrals of the form 2. Performing this substitution, oing the integrals, an separating the up an own parts gives the final expansions: where ±S q = ξ ± π/2 α sinα J m±1ρn sin α +β s q S u + + γ s q S, π/2 ±P q = ξ ± α sinα J m±1ρn sin α cosα q +β p q P u + + γ p q P, π/2 H ±S q = ξ ± ñ q π/2 H ±P q = ξ ± ñ q α sinα J m±1ρn sin α β p q P u γ p q P cosα q β q, α sinα J m±1ρn sin α s S u + γ s s S, 32 ξ ± 2πı m±1 e ım±1φ. 33 At this point one can quicly verify, using 3 an the above equations, that E ρ, E φ, H ρ, an H φ o inee have a φ-epenence of expımφ. 15

16 The S u, S, P u, an P are the complex variables in the Bessel wave metho an mae up the solution vector y in 1. They are essentially coefficients of electromagnetic Bessel waves, although we nee not explicitly combine 32, 3, 28, an 26 to obtain an explicit expression for the E an H vector Bessel waves as we i for the scalar case The Linear System of Equations for the PWB Until now the PWB coefficients have been treate as continuous, when in practice they must be chosen iscrete. Let us eep in min this iscrete nature in the following subsections. We enote by N irs the number of irections α we choose. Thus there are 2N irs coefficient variables for a scalar problem an 4N irs coefficient variables for a vector problem. The istribution of the α on [, π/2] nee not be uniform, an the effect of istribution choice will be briefly mentione later The planar mirror M1 bounary equations The M1 equations planar mirror bounary conition equations in the PWB are very simple. In fact, because of this simplicity, the M1 equations in the MB are basically a transformation to an from the PWB with the M1 equations for the PWB sanwiche between. The reflection of a plane wave off of a layere potential is a well nown problem. For the purpose of etermining the fiel in layer, the entire ielectric stac is characterize by the complex r s an r p coefficients acting at the first surface of the stac. For the scalar case with the layer layer 1 interface at z 1 =, the bounary conition is just ψ u = r s α ψ where r s α is the stac reflection function. Since Bessel waves are linear superpositions of many plane waves with the same θ parameter, this equations is true for Bessel waves: ψ u = r s α ψ. 34 For a conucting mirror, set r s = 1 an for a free mirror set r s = 1. If the interface is at a general height z 1, then the same r s function is use at this surface yieling or where ψ u e ız 1n cos α ψ = r s α u r s α ψ e ız 1n cos α, ψ =, 35 r s/p α r s/pα e ı2z 1n cos α

17 The equation is given for both s- an p-polarization since we will shortly be using the r p quantities. The quantities r s/p α are inepenent of z an z 1. When M2 is a q=layer X ielectric stac mirror, the r s/p α are etermine by T accoring to s/p r s/p α = T X X s/p,21 /T s/p, For the vector case the equation for the s-polarize plane waves is E u ɛ s,u = r s α E ɛ s,, 38 where E u/ is the total electric fiel of the two plane waves going in the irection specifie by α, φ, an u or. Shifting to our current notation an to Bessel waves, the equation becomes S u r s α S =. 39 For p-polarization there is an arbitrary conventional sign. chosen in 11 the equation is In our phase convention E u ɛ p,u = r p α E ɛ p,, 4 or The vector ɛ p,u/ P u r p α P =. 41 is ɛ p, with being force into the up/own version of itself. For a simple mirror, set r p = r s = { 1 for conucting, +1 for free} an use 36 instea of 37 to etermine r s/p. Sometimes we will use cos α as the explicit argument to r s/p or r s/p instea of α. Equation 35 or Eqns. 39 an 41, given for each iscrete α, form the M1 equations The curve mirror M2 bounary equations As mentione in Section 2, the M2 equations come from setting the appropriate fiel components equal to zero at some number of locations on the curve mirror an the hat brim. If we chose iniviual points on the two imensional surface, the φ-epenence factor, expımφ, coul be ivie away. Thus picing locations with the same ρ an z but ifferent φ yiels ientical bounary equations. Therefore we simply set φ = an t = an pic equations by incrementing a single parameter such as ρ on the one imensional curve given by the intersection of the conucting mirror an the x-z plane. The number of locations, N M2 loc, etermines the number of M2 bounary equations. All of the locations are taen to lie in layer. We obtain the M2 equations by, in effect, oing the α integrals in 25 or in 28 an 32. Before maing the integrals iscrete, the istribution of representative irections 17

18 must be chosen. If the interval α between successive irections is not constant, the integral over α must be transforme to an integral over a new variable, x, where x is constant. Such a transformation will generate a new integration factor. At this point the integral is turne into a sum accoring to: b a j, x b a/n. Choosing the irection istribution to be uniform in α requires no change in integration factor an yiels a summation factor of π/2n irs. Using 25 an 23, the M2 equations for the scalar problem are sinα j π 2π +β s 2N irs Nirs j=1 z=z ψ uj + + γ s [ J m ρ n sin α j ] z=z ψ j =. 42 An equation is ae to the linear system for each chosen location specifie by ρ, z. All phase factors have been ivie out of 42 but the scale factor π 2 /N irs has been ept for representative weighting. Of course there is also an effective weight prouce by the istribution of the evaluation locations on the conucting mirror. In our implementation, we choose equal steps of θ to cover the ome an equal steps of ρ to cover the hat brim see Fig. 1. For the vector problem there are three equations associate with each location: E φ =, E =, an H =. Here the subscript refers to the irection that is both tangential to the M2 surface an perpenicular to ˆφ. From 3 the E φ = equation is 1 2 +S + S + ı + P P =. 43 The expansions for ± S/P in terms of the unnowns S/P u/ in equation 32 must now be use, along with the ientical integral-to-sum conversion use in the scalar problem 42. It is probably not beneficial to wor out the long form of this bounary equation, as its computer implementation can be one with substitutions. The E = equation epens on the shape of the mirror. If η is the angle that the outwar-oriente surface normal maes with the z axis, then E is given by where E z = z P an, using 3, E ρ = 1 2 E = E ρ cos η E z sin η, 44 ı S + S + + P + P. 45 Again equation 32 an the integral-to-sum conversion must be use to obtain the explicit row equation. The H = equation is obtaine by oing the same type of substitutions with H = H ρ sin η + H z cos η

19 Here H z = H z P an H ρ = 1 2 For locations on the hat brim, η = The see equation ı H S H +S + H +P + H P. 47 All the M1 an M2 equations have no constant term. Thus the best numerical solution will be the trivial solution y best =. To prevent from being a solution, an equation with a constant term must be ae. One simple type of equation sets a single variable equal to 1, for instance S uj=5 = 1. Another simple type sets the sum of all of the coefficients equal to 1. A more complicate type sets the fiel or a fiel component at a certain point in space equal to a constant. No one type of see equation is always best Solution of Ay=b As epicte in 1, the matrix A is mae up of the left han sies of the M1, M2, an see equations. For the scalar case there are 2N irs columns an N irs + N M2 loc + 1 rows. For the vector case there are 4N irs columns an 2N irs + 3N M2 loc + 1 rows. The value of N M2 loc is pice so that A has several times as many rows as columns. A value of is pice an the overetermine system of equations is solve as well as possible by a linear least squares metho. The best such methos rely on a technique nown as singular value ecomposition [12]. Our implementation relies on the function zgels of the LAPACK fortran library. To fin the eigenvalues of, the imaginary part of is set to zero an the real part of is scanne. As mentione in the Overview, this results in ips in the value of r. Using Brent s metho [12] for minimization, the minimum of the ip is foun. The real part of is now fixe an Brent s metho is use again to fin the best imaginary part of. Then Brent s metho may again be use on the real part of. By this alternating metho, the complex eigenvalue of is foun, along with the eigenmoe, y. In practice Brent s metho nee only be use two to four times per scan ip to get an accurate complex. We usually normalize each row of A to 1 so that the normalize error per equation in the system can be expecte to be aroun n r /[ y number of rows in A 1/2 ]. n is one inicator of the accuracy of the solution Calculating the fiel from y Once y is calculate for a quasimoe, the values of the fiel in any layer can foun by using the expansion 25 for the scalar fiel an equations 32, 3, 28, an 26 for the vector fiel. Of course the integrals over α must be mae iscrete as iscusse previously. Appenix B explains more regaring the plotting of the fiels. 19

20 4. Multipole Bases an the Two-Basis Metho As mentione in the Introuction, the two-basis metho ultimately uses the scalar or vector MB. The MB is the eigenbasis for the close, conucting hemisphere or sphere. Both the vector an scalar multipole bases have nown forms which we will use but not erive. The basis functions alreay possess an azimuthal epenence of expımφ an the imensional reuction ue to cylinrical symmetry is accomplishe by picing a value for m instea of summing over basis functions with many m. The metho of stepping along a one-parameter location curve to obtain the M2 equations is the same as for the PWB. Explicit formulas in the MB of course will be completely ifferent an will be given in this major section. The methos of solution to the linear system of equations are the same as for the PWB. The evelopment of the M1 equations in the MB, however, requires consierable wor. After the system of equations has been solve, using the resulting solution vector y to calculate/plot the fiels in layers other than layer also requires significant wor. We use the term two-basis metho because of the role of plane waves in these two calculations. Figure 2 represents the linear system of equations of the two primary methos an how they are relate. S u, S, P u, P a l, b l PWB M1 eqns. r s/p MB M1 eqns. Bessel wave metho two-basis metho PWB M2 eqns. MB M2 eqns. see eqn. see eqn. Figure 2: Diagram for the two primary methos. The close loops suggest the selfconsistency or constructive interference of the quasimoe solutions. Grey regions inicate intersection between PWB an MB methos. Size roughly inicates the post-basis-erivation wor require to get the equations. The variable coefficients for the vector problem are shown. 2

21 4.1. The Scalar Multipole Basis The scalar MB functions we use are the ψ lm = j l n ry lm θ, φ where j l enotes the spherical Bessel function of the first in an Y lm is the spherical harmonic function. The scalar MB functions, lie the scalar PWB functions, satisfy the wave equation. We assume the fiel in layer, in a region large enough to encompass the cavity, can be expane uniquely in terms of the scalar MB functions. Using the cylinrical symmetry of the cavity to solve the problem separately for each value of m, we expan the fiel in the cavity as ψ x = l max l= m c l j l n ry lm θ, φ. 48 The expansion coefficients, c l, are complex. One shoul never nee to choose l max much larger than Ren r max where r max is the maximum raial extent of the ome not the hat brim. Semiclassically, the maximum angular momentum a sphere or hemisphere of raius R s can support for a given is Ren R s, which correspons to a whisperinggallery moe. The scalar MB functions are the exact eigenfunctions of the problem of a hypothetical spherical conuctor specifie by r = R s = R, with eigenvalues given by the zeros of j l n R. The basis functions for which l + m is o are the eigenfunctions of the close hemispherical conuctor. This is because Y lm has parity 1 l+m in cos θ 49 an thus is zero at θ = π/2. It can also be shown, using the power series expansion of P l x given in Ref. [13], that Y lm π/2, φ is nonzero if l + m is even The M1 Equations in the Scalar MB The expansion of a plane wave in terms of the monochromatic scalar MB functions is [1]: e ı x = 4π l= m= l l ı l Ylmθ, φj l ry lm θ, φ. 5 The inverse of this relation, the expansion of a scalar MB function in terms of monochromatic plane waves, is ı l j l ry lm θ, φ = Ω 4π Y lmθ, φ e ı x. 51 This equation is easy to verify by inserting 5 into the right han sie. The use of Y lm = 1 m Yl, m an the orthogonality relation for the Y lm leas irectly to the left han sie. The expansion 51, applie to layer = n, is the founation for this section an its counterpart for vector fiels. 21

22 Using 48 an 51 yiels the scalar fiel expansion for a given m: ψ = Ω e ı x ı l c l 4π Y lmθ, φ. 52 l= m The quantity in the curve bracets is ψ, the simple plane wave coefficient compare to Eqn. 4 with q set to. We can now use 35. For z >, ψ u = ψ an ψ = ψ, where in cylinrical coorinates = ρ, φ, z. For z <, ψ = ψ an ψ u = ψ. Using 49, one gets ψ = l c ly lm θ, φ 1 l+m ı l /4π. We can now erive an equation that expresses 35 an hols for all Ω l= m c l ı l 4π Y lmθ, φ At this point there are two ways to procee Variant 1 : { } 1 r s, l + m is even sgncos θ 1 + r =. 53 s, l + m is o One way to construct the M1 portion of the matrix A is to pic some number N M1 irs of iscrete irections, θ, an use 53 for each irection the φ epenence ivies out. Inspection of the equation reveals that it is even in cos θ ; thus only polar angles in the omain [, π/2] are neee. Using α as before to enote this reuce omain, the M1 equations in this variant become: l max l= m c l ı l 4π Y lmα, 1 1 l+m r s cos α e ı2z 1n cos α =. 54 We note for later comparison that for each α we nee only evaluate a single associate Legenre function insie the Y lm, because the recursive calculation technique for computing Pl m max x also computes Pl m x for m l l max. Since each step of this recursive calculation involves a constant number of floating point operations, we may say that the complexity associate with the Pl m calculations for each α is Olmax. 2 Generally N M1 irs l max so the Pl m complexity is Olmax 3 an the overall number of evaluations of r s is Ol max Variant 2 Rather than picing iscrete irections to turn 53 into many equations, we coul project the entire left han sie onto the spherical harmonic basis, {Y l m }, that is, 22

23 integrate 53 against Y l m Ω in Ω. Due to the uniqueness of the basis expansion, the integral for each l, m pair must be zero. The integrals for m m are zero an can be neglecte. Thus the number of equations generate is N l = l max m + 1. We set l max = l max. Since 53 is even in cos θ, integrating against Y gives if l m l + m is o, halving the number of equations. When l + m is even the integration must be one numerically. The most analytically simplifie version of the coefficients of c l in the M1 equation corresponing to l for l + m even is: where M1 l,l = ζ 1 P m l xp m x 1 1 l+m r l s x x, 55 ζ ıl l m! l 1/2 m! 2l + 12l π l + m! l + m! This comes from using the properties of Y lm uner a sign change of m, using the efinition of Y lm, oing the φ integral, an halving the omain of the even integral over x = cos α. Noting that the number of unnowns in the solution vector y is N l = l max m + 1, the number of complex coefficients that must be calculate is about Nl 2/2. Noting that M1 l,l = ı l l M1 l,l when l + m an l + m are both even, the calculation is reuce to about 3Nl 2 /4 real-value integrations. While this variant of the metho is in some ways the most elegant, the numerical integrals are extremely computationally intensive. Our implementation use an aaptive Gaussian quarature function, gsl_integration_qag of the GSL. Aaptive algorithms choose a ifferent set of evaluation points for each integration an achieve a prescribe accuracy; in a sense the integral is one in a continuous rather than a iscrete manner. For an aaptive algorithm, the complexity 4 associate with the Pl m calculation is Olmax, 3+ν where ν 1. The number of evaluations of r s is Olmax. 2+ν In practice, variant 2 is much slower than variant 1 even for cavities as small as R/λ 5 with simple mirrors. Checing between the two variants has generally shown very goo agreement The Linear System of Equations in the Scalar MB The M1 equations have been given in the previous section. Using 48 yiels an M2 equation l max l= m c l j l n r Y lm θ, =, 57 4 The exponent ν is efine so that the number of integration points that must be sample for the most complicate integrals where l l l max to maintain a constant accuracy goes lie lmax. ν This efinition may not be rigorous if ν is a function of l max. Since Pl x has Ol zeros an the Pl m for m > are even more complicate, it is reasonably certain that ν 1. 23

24 for each location r, θ. The iscussions from Section 3.3 regaring the M2 equations, the see equation, an the metho of solution to Ay = b apply here. The number of columns in A is N l an the number of rows is N M1 irs + N M2 loc + 1 for variant 1 or about N l /2 + N M2 loc + 1 for variant Calculating the Fiel in the Layers with the Scalar MB To calculate the complex fiel anywhere in layer once a quasimoe solution y has been foun, Eqn. 48 can be use irectly. To calculate the fiel in the layers below the cavity q > where the expansion oes not apply, the Bessel waves must be use, along with the T s q matrix that propagates them. Performing the φ integration in 52, using 49, an then comparing with 9 an 25 with q = in both of these yiels the Bessel wave coefficients: ψ u ψ = l = l c l ı l 4π Y lmα,, c l ı l 4π Y lmα, 1l+m. 58 Now 25 can be use with q > yieling ψ q x = ım 2 eımφ π/2 J m ρn sin α +β q s α sinα [ c l ı l Y lm α, + 1 l+m +γ q s l ]. 59 This is a costly numerical integration to o with an aaptive algorithm. For every sample α, the quantities T s, Pl m max an J m must be calculate once. For isplaying large regions of the fiel in the stac, it is sufficiently accurate to simply convert the solve MB solution vector y into a PWB solution vector with N irs l max by means of a separate program using 58, an then use the iscrete form of 25 to plot The Vector Multipole Basis We expan the electromagnetic fiel in layer at least in a finite region surrouning the cavity in the vector multipole basis using spherical Bessel functions of the first in j l. The multipole basis uses the vector spherical harmonics VSH; same abbreviation for singular, which are given by[14] M lm x = j l n r x Y lm θ, φ, N lm x = 1 n M lm. 6 24

25 The VSH are not efine for l =. The nature of the electromagnetic multipole expansion is evelope in section 9.7 of the boo by Jacson[1] using somewhat ifferent terminology. The multipole expansion of the electromagnetic fiels is E x = H x = ñ l max l=l min a l N lm + ıb l M lm, l max l=l min ıa l M lm + b l N lm, 61 where cylinrical symmetry has been invoe to remove the sum over m, an l min = max1, m. The a l an b l are complex coefficients an there are N l = l max l min + 1 of each of them. The a l coefficients correspon to electric multipoles an the b l coefficients correspon to magnetic multipoles. The explicit forms of the VSH are M lm x = ˆθ ım sin θ j ln ry lm θ, φ + ˆφ j l n r θ Y lmθ, φ, ll + 1 N lm x = ˆr n r j ln ry lm θ, φ + ˆθ 1 n r r rj ln r θ Y lmθ, φ + ˆφ ım n r sin θ 4.6. The M1 Equations in the Vector MB r rj ln r Y lm θ, φ. 62 The goal of the calculation here is to transform the M1 relations, 38 an 4, into equations such that each ot prouct is written in the form l a lf a + b l f b. The resulting two equations will be the vector analogues of 53 an will hol for all Ω, leaing to two variants of solution metho in the same manner as before. First we must Fourier expan E as we expane ψ to get 52. The only Fourier relation we have is 51 which expans the quantity j l Y lm x. We must therefore brea M lm into terms containing this quantity. Here we can mae use of the orbital angular momentum operator L = ıx. Using the laer operators, L ± = L x ± ıl y, the VSH M lm = ıj l LY lm can be shown to be where [ M lm = ıj l ˆx 1 + lm 2 Y lm+1 + lm Y lm 1 + ŷ ı ] + lm 2 Y lm+1 + lm Y lm 1 + ẑ mylm, 63 ± lm l ml ± m

26 Using 51 multiple times yiels M lm = with Ω M lm e ı x, 65 where M lm = ˆx ıl 8π ıe+ lm + ŷ ıl 8π 1e lm + ẑ ıl 4π my lmω, 66 e ± lm + lm Y lm+1ω ± lm Y lm 1Ω. 67 We can avoi the tas of expaning N lm into j l Y lm terms by just using N lm = M lm = Ω ı n n M lm e ı x. 68 Performing the cross prouct an substituting into 61 using yiels E x x = E y x = E z x = { Ω e ı x Ω e ı x l { { Ω e ı x l l ı l [a l 4π [a l ı l 4π ı l 4π [ a l ı 2 m sin θ sin φ Y lm ı 2 m sin θ cos φ Y lm 1 2 sin θ cos φ e lm cos θ e lm cos θ e + lm sin θ sin φ e + lm 1 + b l 2 lm ]} e+, ı ]} + b l 2 e lm, + b l 69 my lm ]}, where Y lm is unerstoo to mean Y lm Ω. The quantities insie the curly braces in 69 will be enote by Ẽx, Ẽ y, an Ẽz, where the superscript on has been roppe. We will also nee Ẽ where = ρ, φ, z as before. Again using 49 we fin Ẽ x = l Ẽ y = l Ẽ z = l ı l 4π 1l+m ı l 4π 1l+m ı l 4π 1l+m [a l [a l [ a l ı 2 m sin θ sin φ Y lm ı 2 m sin θ cos φ Y lm 1 2 sin θ cos φ e lm cos θ e lm cos θ e + lm 1 ] + b l 2 e+ lm, ı + b l 2 lm ] e, sin θ sin φ e + lm ] + b l my lm. 7 26

27 The M1 equations for s-polarization We will now efine f lm an g lm by Ẽ ɛ s, l ı l 4π a lg lm + b l f lm. 71 Performing the ot prouct using 11 yiels f lm = ı + lm 2 Y lm+1e ıφ lm Y lm 1e ıφ, g lm = 1 cos θ + lm 2 Y lm+1e ıφ + lm Y lm 1e ıφ m sin θ Y lm = 1 2l + 1 l ml m 1Yl 1,m+1e ıφ 2 2l 1 + l + ml + m 1Y l 1,m 1 e ıφ. 72 The simplification in the last step has been one using recursion relations for the P m l. Performing the ot prouct for Ẽ yiels Ẽ ɛ s, = l ı l 4π 1l+m a l g lm b l f lm. 73 The M1 relation from 38 is Ẽ ɛ s, = r s Ẽ ɛ s, for z > an Ẽ ɛ s, = r s Ẽ ɛ s, for z <. These lea to an M1 equation that hols for all Ω : l max ıl 4π l=l min { } al s 1 g lm + b l s 2 f lm, l + m even =, 74 a l s 2 g lm + b l s 1 f lm, l + m o where s 1 1 r s cos θ an s 2 sgncos θ 1 + r s cos θ The M1 equations for p-polarization The M1 equation for p-polarization is obtaine the same way as above. The calculation is simplifie by temporarily switching phase conventions. We efine the unit vector ɛ p, sgncos θ ɛ p, using 11 so that ɛ p, = ˆx cos θ cos φ + ŷ cos θ sin φ ẑ sin θ, ɛ p, = ˆx cos θ cos φ ŷ cos θ sin φ ẑ sin θ. 75 The M1 relation is now Ẽ ɛ p, = r p Ẽ ɛ p, for z > an Ẽ ɛ p, = r p Ẽ ɛ p, for z <, where r p has not change in any way. Performing the ot 27

28 proucts, we fin that Ẽ ɛ p, = l Ẽ ɛ p, = l ı l 4π a lf lm + b l g lm, ı l 4π 1l+m a l f lm + b l g lm. 76 This leas to the M1 equation for p-polarization which hols for all Ω : l max ıl 4π l=l min { } al p 2 f lm + b l p 1 g lm, l + m even =, 77 a l p 1 f lm + b l p 2 g lm, l + m o where p r p cos θ an p 2 sgncos θ 1 r p cos θ Variant 1 Variant 1 for the vector case procees exactly as it oes for the scalar case. The M1 equations 74 an 77 are seen to be even in cos θ, allowing us to pic irections only in the omain [, π/2]. Thus for each iscrete irection α we get one equation in the M1 portion of A for each polarization. No moification of 74 or 77 is neee, except that we can tae φ = an cos θ = cos α. The complexity of calculating the M1 portion of A is of the same orer in l max as it is for the scalar MB. However, numerous constant factors mae the computation time an orer of magnitue longer, as there are more Pl m functions to compute an r p must be compute in aition to r s Variant 2 In this minor section, we assume that m see Appenix B for plotting negative m moes from positive m solutions. We follow the same proceure use in variant 2 for the scalar fiel. Integrating 74 an 77 against Y l m Ω yiels zero if either m m or l + m is o. For m = m an l + m even, the M1 equation associate with each l in { m, m + 1,..., l max } for s-polarization is l max 1 1 [a 2 ζ l 1 1 l+m r s l=l min 1 + b l ı l+m r s P m+1 l P m+1 l 1 m 1 x + l + ml + m 1Pl 1 x P m l x x ] x l + ml m + 1P m 1 l x P m x x =, l 78 28

29 an the M1 equation for p-polarization is l max 1 [ 1 2 ζ a l ı 1 1 l+m r p P m+1 l l=l min 1 + b l l+m r p P m+1 m 1 l 1 x + l + ml + m 1Pl 1 x x l+ml m+1p m 1 l x P m P m l x x l x x ] =, where P n µ = for n > µ. Note that if m = the equations for l = are inclue. We implemente this by calculating an array of each of the following types of integrals: 1 P m+1 1 P m+1 µ Pν m x, µ Pν m r s x, µ Pν m r p x, 1 P m+1 1 P m 1 1 P m 1 µ Pν m x, µ Pν m r s x, µ Pν m r p x. 1 P m 1 The integrals in the first row can be calculate ahea of time an store in a ata file. The other integrals must be calculate each time the matrix A is calculate. The complexity of the calculation is of the same orer in l max as for variant 2 in the scalar case. These integrals are very numerically intensive an in practice variant 2 taes much longer than variant 1. We have foun goo agreement between the two variants, inicating that variant 1 can be use all or most of the time The Linear System of Equations in the Vector MB The M2 equations are foun in the usual way, choosing locations ρ, θ with φ = an constructing the equations E φ =, E = E ρ cos η E z sin η =, an H = H ρ sin η + H z cos η =. The fiels E an H are foun by substituting 62 into 61. The erivatives in 62 are an, for m, r rj ln r = n rj l 1 n r lj l n r, 8 θ Y lmθ, φ = 1 2 where P l+1 l [ 2l + 1l m! 4πl + m! ] 1/2 P m+1 l l + ml m + 1P m 1 l e ımφ, 81 =. For negative m use Y l, m = 1 m Ylm. The number of columns in A is 2N l an the number of rows is 2N M1 irs + 3N M2 loc + 1 for variant 1 an about N l + 3N M2 loc + 1 for variant 2. 29

30 4.8. Calculating the Fiel in the Layers with the Vector MB To calculate the fiel in layer, the expansion 61 is use irectly, with 62, 8, an 81 being use to obtain the explicit form. As for the scalar case, there is no irect expansion for the layers q > in the MB, an we must use the Bessel wave basis. Using 71, 73, an 76, the conversion is seen to be l max S u = ıl 4π a lg lm + b l f lm l=l min S = ı l 4π 1l+m a l g lm b l f lm l P u = ı l 4π a lf lm + b l g lm l P = ı l 4π 1l+m a l f lm + b l g lm l θ =α θ =α,φ =, θ =α,φ =,φ =, θ =α,φ =,. 82 This is the vector case analogue of Eqn. 58. At this point the Bessel wave coefficients S u, etc., are continuous functions of α, not iscrete as they were in the Bessel wave metho. These continuous coefficients are substitute into the integrals 32, an finally 3 is use to give the vector fiels in the layers. As in the scalar case, however, oing the integrals with an aaptive algorithm is impractical if one is plotting the vector fiels in a sizable region. In this case, it is best to mae the Bessel wave coefficients iscrete an use a simple integration metho, or, as was suggeste in the scalar case, to pic a N irs an create a PWB solution vector that can then be plotte via a PWB plotting routine. 5. Demonstrations an Comparisons In this section we will emonstrate, but not thoroughly analyze, several results that we have obtaine using our moel an methos. The moes shown here all have n <.2 an o not contain large amounts of high-angle plane waves, which in principle can cause problems see Appenix A.1. The authors expect to publish a separate paper focusing on the moes themselves. In this section we will also compare our implementations of the two-basis metho an the Bessel wave metho. Before reaing this section it may be helpful to loo at Appenix B The V Moe: A Stac Effect As mentione several times in the Introuction, the correct treatment of a ielectric stac can be essential for getting results that are even qualitatively correct 5. One of 5 It is not always essential however, see Note Ae in Proof. 3

31 Figure 3: Contrast-enhance plot of ReE x in the y-z plane for a V moe m = 1, = ı.19. M2 is spherical an origin-centere; R = 4; z 1 =.3; z e = 3; M1 = stac II; s = 8.16; N s = 2. In this an other sie view plots, the fiel outsie the cavity has been set to zero. Figure 4: An x-y cross section of the moe in Fig. 3 at z = Units are in µm as usual. For m = +1, the forwar time evolution is simply the counterclocwise rotation of the entire vector fiel. If m = +1 an m = 1 are mixe to create a cosine moe, the arrow irections remain fixe an their lengths simply oscillate sinusoially in time so that the moe is associate with x polarization. The inset shows the location of the cross section in the view of Fig. 3. The fiel is plotte only in layer. 31

32 the most remarable ifferences that we have observe when switching from simple to ielectric M1 mirrors occurs right where someone looing for highly-focuse, rivable moes woul be intereste: the wiening behavior of the funamental Gaussian moe the moe. A Gaussian moe of a near-hemispherical cavity will become more an more focuse wie at the curve mirror an narrow at the flat mirror as z 1 is ecrease from a starting value for which the Gaussian moe is paraxial. As the moe becomes more focuse, of course, the paraxial approximation becomes less vali an at some point Gaussian theory no longer applies. We have foun, using realistic Al 1 x Ga x As AlAs stac moels stacs I an II escribe in Appenix C, that the moe splits into two parts, so that in the sie view a V shape is forme. Figure 3 shows a split moe an Fig. 4 shows the physical transverse electric fiel, ReE T ReE x ˆx + ReE y ŷ, for this moe near the focal region. Figure 5 shows the values of ReR as z 1 is change from.3 to about.63. Figure 6 shows a zoome view of the fiel at the focus of the moe at z 1 =.63, where it is qualitatively a moe. If a conucting mirror is use, the central cone simply grows wier an wier, but oes not split. The V moe is preominately s-polarize an appears in the scalar problem as well. Thus it appears that the V moe is a result of the non-constancy of argr s α for a ielectric stac. We note here that following the moe as z 1 is change is an imperfect process: it is possible that the following proceure may sip over narrow anticrossings that are ifficult to resolve. However, the character of the moe is maintaine through such anticrossings. The apparent splitting of the central cone/lobe for higher orer Gaussian moes has also been observe. In the next section we loo at some higher orer Gaussian moes, focusing not on what occurs at the breaing of the paraxial conition, but on what is allowe an observe for moes that are very paraxial Persistent Stac-Inuce Mixing of Near-Degenerate Laguerre-Gaussian Moe Pairs As mentione in the previous section, we o observe the funamental Gaussian moe for paraxial cavities with both ielectric an conucting planar mirrors. The situation becomes more interesting when we loo at higher orer Gaussian moes. It is true that, as moes become increasingly paraxial, Gaussian theory must apply. However, the way in which it applies allows the esign of M1 to play a significant role. Here we will emonstrate the ability of a ielectric stac stac II to mix near-egenerate pairs preicte by Gaussian theory into new, more complicate near-egenerate pairs of moes. We must provie consierable bacgroun to put these moes into context. Our iscussion applies to moes with paraxial geometry, moes for which the paraxial parameter h λ/πw = w /z R is much less than 1. Here w is the waist raius an z R is the Rayleigh range, as use in stanar Gaussian theory. In this section, the solutions we are referring to are always the +m or m moes iscusse in Appenix B, an never mixtures of these, such as the cosine an sine moes iscusse in the same section. 32

33 Figure 5: Following R as z 1 is varie. The points of the two graphs correspon one-toone, with the parameter z 1 increasing from left to right in both graphs. The peas in resonance with have not yet been analyze Paraxial Theory for Vector Fiels From Gaussian theory, using the Laguerre-Gaussian LG basis, we expect that the transverse electric fiel of any moe in the paraxial limit is expressible in the form: E T = N [ j= A + j 1ı + A j 1 ı ] LG 2j N minj,n j x. 83 Here N is the orer of the moe. The A ± j are complex coefficients an 1 ı an 1 ı are the Jones vectors for right an left circular polarization, respectively. The explicit forms of the normalize LG 2j N minj,n j functions are given in Ref. [15] as ulg nm with the substitutions n N j an m j. The important aspects of the LG 2j N minj,n j functions are that the φ-epenence is expı2j Nφ, an the ρ-epenence inclues the factor L 2j N minj,n j 2ρ2 /wz 2 where L l p is a generalize Laguerre function an wz is the beam raius. In the paraxial limit the vector eigenmoes of the same orer become 33

34 Figure 6: The fiel in layers X for the moe at z 1 =.63. The inset shows the entire moe. The V nature of the moe has been lost as it has become more paraxial an more lie the funamental Gaussian. ReE T everywhere lies nearly in a single irection at any instant here it is in the x irection. The x-z plane is shown here, although the plots of E x in any plane containing the z axis are very similar. Here = ı.41. egenerate. There are 2N + 1 inepenent vector eigenmoes present in the expansion 83. Inepenent from the iscussion above, E T = E ρ ˆρ + E φ ˆφ can be written as E T = E ρ + ıe φ e ıφ }{{} ı 2 E ρ ıe φ e ıφ. 84 }{{} ı expımφ expımφ Since the E ρ an E φ fiels have a sole φ-epenence of expımφ, comparing with 83 reveals that, for a given m, at most two terms in 83 are present: the A an A + j + j terms where j ± = N + m ± 1/2. Explicitly, E ρ ± ıe φ /2 A LG m±1 j [N+minm±1, m 1]/2. 85 ± If the solution 84 has both terms nonzero for almost all x, the constraints j N force N to have a value given by N = m +1+2ν, ν =, 1, 2,.... However, if only right left circular polarization is present in the solution, N = m 1 is also allowe, provie that m +1 m -1. So, given a paraxial numerical solution with its m value, we 34

35 Table 1: Vector Laguerre-Gaussian moes. can etermine which orers the solutions can belong to. The reverse proceure, picing N an etermining which values of m are allowe an how many inepenent vector solutions are associate with each m, can be one by stepping j in 83 an comparing with 84 or 85. The results for the first four orers are summarize in Table 1. The LG 1ı an LG 1 ı moes are the funamental Gaussian moes. Since the cavity moes are not perfectly paraxial, the 2N + 1-fol egenerate moes from the Gaussian theory are broen into N + 1 separate egenerate pairs. The truly egenerate pairs for m of course consist of a +m an a m moe which are relate by reflection see Appenix B. The pairings are shown in the last column of Table 1. The ashe boxes in Table 1 enclose pairs of moes which may be mixe in a single solution for fixe m Eqn. 84. Only the mixable m = moes are exactly egenerate an may be arbitrarily mixe. For the other mixable pairs, the egree of mixing will be fixe by the cavity, in particular by the structure of M1. We now iscuss our results regaring the mixing of the two moes with N = 2 an m = A emonstration of persistent mixing Figure 7 shows the cross sections of two near-egenerate m = 1 moes foun for a conucting cavity. Here z 1 = z e = an M2 is spherical with raius R s = 7 but is 35

36 A B Figure 7: 8 8 µm cross sections of moes of a conucting cavity near maximum amplitue z =.25. A = an B = The inset plots show ReE x in the x-z plane with horizontal an vertical tic mars every 1 µm. centere at z = 59.5 instea of the origin, so that R = 1.5. The paraxial parameter h is at least as small as.1 see the sie views in the inset plots. Moe A correspons very well to the pure LG 1ı 1 moe, while moe B correspons very well to the pure LG 2 1 ı moe. These are mixable moes, as inicate in Table 1, an the conucting cavity has chosen essentially zero mixing. Note that moe B woul have zero overlap with an incient funamental Gaussian beam centere on the z axis, while moe A woul have nonzero overlap. Pictures C an D in Figure 8 are cross sections of near-egenerate m = 1 moes for a cavity with stac II. To show that the transverse part of these moes are mixtures of A an B, we have ae an subtracte the solution vectors y A an y B to create the new non-solution vectors: y C = y A + ηy B, y D = y A.225ηy B, 86 an have plotte these fiels C an D using = A + B /2. The scaling factor η, here.7, is unphysical an relate to the effect of the see equation on overall amplitue. Comparing C an D with C an D shows that we have reconstructe the moe cross sections quite well up to a constant factor. Moes C an D are not pure Hermite-Gaussian HG moes, but their resemblance to the HG 2 an HG 2 moes is not an accient. Moe conversion formulas in Ref. [15] show that LG 2 is mae of the HG 2, HG 11, an HG 2 moes while LG 1 contains only HG 2 an HG 2. Note that both moes C an D woul couple to a centere funamental Gaussian beam. An interesting property of the mixe moes C an D is that their general forms appear to be persistent as h is varie, provie that the paraxial approximation remains sufficiently applicable. Furthermore, we o not have to be particularly careful about 36

37 C D C' D' Figure 8: Cross sections of true moes C an D of a cavity with stac II near maximum amplitue z =.5, an constructe moes C an D near their maximum amplitue z =.25. The constructe cross sections are nearly ientical to the true cross section. C = ı.5415, D = ı.5516, an the constructe moes are plotte with C /D = , the average of A an B. The inset plots show ReEx in the x-z plane. The stac parameters are Ns = 22 an s = setting λs to correspon very closely to 2π/Re. The moes shown in Figure 9 are not nearly as paraxial as the C an D moes. Here s = 8.16 which is not close to the values of the moes. Furthermore, stac I was use which is missing the spacer layer of stac II. Nevertheless the moes of Fig. 9 bear a strong resemblance to the more carefully pice C an D moes Moes with m 6= 1 The m = moes may be circularly polarize, such as the paraxial m = moes of Table 1. The more interesting polarization, perhaps, is the m = analogue of x an y polarization: raial an azimuthal polarization. The moe shown in Fig. 1 is raially polarize, although it is impossible to tell this from the plots because the left an right circularly polarize moes are raial ientical to the plot at t = an azimuthal 37

38 Figure 9: Cross sections of neighboring, poorly paraxial moes near maximum amplitue z = M2 is spherical an origin-centere with R = 1. M1 is stac I with N s = 2, s = 8.16, an z 1 = z e = 1.. The moe on the left has = ı.2491 an the moe on the right has = ı lie a vortex at ωt = π/2. The raial or azimuthal polarizations are easier to obtain numerically than the right or left polarizations because raially azimuthally polarize moes have all of the b l a l MB coefficients equal to zero an thus can be selecte by the see equation see Appenix B. Figure 11 shows a m = 2 moe that correspons to the LG1 1 1ı moe from Gaussian theory. We also have complete the N = 2 near-egenerate family by fining the LG 2 1ı m = 3 moe for both the cavity use in Fig. 7 = A B an the cavity use in Fig. 8 = ı.5466 C D. The cross sections of this moe for the two cavities, which are not shown to conserve space, appear ientical, as preicte by the absence of a m = +3 near-egenerate partner for this moe Table N irs an l max : Comparing the Two Primary Methos The authors of this paper first implemente the more complicate two-basis metho, believing that many more PWB coefficients than MB coefficients woul be require to expan ome-shape cavity fiels. When the Bessel wave metho was implemente as a chec, it was iscovere that the PWB wors surprisingly well, with usable values of N irs being of the same orer of magnitue as usable values of l max. The choice of see equation, N irs or l max, an other parameters can both affect the epth an narrowness of the ips in the graph of r versus Re. This maes it ifficult to perform a comprehensive an conclusive comparison of the two methos. One practical problem with the Bessel wave metho is that its performance is some- 38

39 E x E z Figure 1: Tightly focuse, raially polarize m = moes of a conucting cavity. The focusing has cause E z to be greater than the E ρ. The spot size of the ominant E z fiel for such moes can be surprisingly small, as has recently been emonstrate experimentally by Dorn et al.[4] for a focuse beam with no cavity. Parameters of the non-inset cavity an moe are R = 1, z 1 =.5, an = λ =.867. The cross section is at z =.3. The insets show the moe after it was followe to the perfectly hemispherical z 1 = cavity shape; here the moe has an even smaller central spot an E z E ρ. We expect a hemispherical moe to have only a single nonzero MB coefficient an this was verifie by the solution: a 1 = 1. an a l < for l 1. The hemispherical moe has = λ =.87. times quite sensitive to the value of N irs. In both methos, setting the number of basis function too high results in a failure to solve for y best ; the program acts as if Ay = b were uneretermine, even though the number of equations is several times the number of unnowns. When increasing l max or N irs towar its problematic value, the r values for all ramatically ecrease, sharply lowering the contrast neee to locate the eigenvalues of. The solutions that are foun begin to attempt to set the fiel to zero in the entire cavity region, with some regions of layer outsie the cavity having fiel intensities that are orers of magnitue larger than the fiel insie the cavity. In our experience this problem has not occurre in the two-basis metho for l max near the semiclassical limit of l, ReR, an thus has never been a practical annoyance. On the other han, the problem can occur at surprisingly low values of N irs. Yet taing N irs to be too low can often cause solutions to simply not be foun: ips in the graph of r vs. Re can simply isappear. The winow of goo values of N irs can be at least as narrow as N irs /1. The winow of goo l max values for the MB seems to be much wier for many moes, it is sufficient to tae l max to be half of ReR or less. In this respect, the two-basis metho is easier to use than the Bessel wave metho. 39

40 Figure 11: A m = 2 moe for a conucting cavity = M2 is spherical an origin-centere with R = 1. M1 is a conuctor; z 1 = 1.. The inset shows ReE x in the x-z plane. On the other han, there were situations when a scan of r versus Re with the Bessel wave metho reveale a moe that was sippe in the same scan with the twobasis metho, ue to the narrowness of the ip feature. To the best of our recollection, we have always been able to fin a moe by both methos if we have mae an effort to search for it. A irect comparison of the methos by looing at the solution plots rarely reveals fiel value ifferences greater than 1% of the maximum value when the moes are restricte to those that o not have a large high-angle component. The eigenvalues of locate by the two methos are usually quite close. Here we show one case in which there is a small but visible ifference between the moe plots for the Bessel wave metho an the two-basis metho. Figure 12 shows E x in the x-z plane for a moe in a cavity with n = 1. an n X =.5 an no stac layers at all. This situation is meant to moel a ielectric-fille ome cavity surroune by air. Our program assumes that layer is free space n = 1, so we have set n X < n to achieve this effect. Here M2 is spherical an centere at the origin with R = 1 an z e = z 1 =.5. The reason for the isagreement between the two methos for this case is not nown. This was the only low finesse cavity we have trie, as well as being the only cavity with a high inex of refraction in layer. Another unusual property of this moe solution, which loos similar to a funamental Gaussian, is that its 4

41 Figure 12: Top: Solution obtaine with Bessel wave metho. = ı.5857; n = , N irs = 3; α istribution is uniform in sin α, not α. Bottom: Solution obtaine with two-basis metho. = ı.59; n = ; l max = 86. The fiel in layer X is not plotte. electric fiel in the x-y plane actually spirals: its instantaneous linear irection rotates with z as well as with t. The other moes we show throughout the emonstrations section give extremely goo agreement between the two primary methos. In our implementation, computation time to set up an solve Ay = b is of the same orer for the Bessel wave metho an the two-basis metho with variant 1. For the large cavity shown in Fig. 3, the Bessel wave metho too about 1 s with N irs = 1 an about 6 locations chosen on M2. Variant 1 with l max = 2, 6 M2 locations, an 6 uniformly space α, too about 15 s. For comparison, Variant 2 too about 2 hours achieving a 1 6 relative accuracy for each α integral Almost-Real MB Coefficients When M1 is a conuctor we fin that the imaginary parts of a l an b l are essentially zero. Furthermore, we often fin that for ielectric M1 mirrors, the imaginary parts of a l an b l are one or more orers of magnitue smaller than their real parts. We show one way in which this tenency simplifies the interpretation of moe polarization in Appenix B. There is a an argument that suggests that a l an b l shoul be real for a conucting cavity. A conucting cavity has a real eigenvalues,, an real reflection functions, 41