Effective mode volumes and Purcell factors for leaky optical cavities

Effectiv e mo de v olumes and Purc ell factors for leaky opt ic al ca vities Philip T røst Kristensen, 1, 2 Cole V an Vlack, 2 and Stephen Hughes 2 1 DTU F otonik, T e chnic al University of De nmark, D K -2800 Kgs. Lyngby, Denmark 2 Dep ar tment of Physics, Que en ’s Uni versity, Ontario, Canada K7L 3N6 (Dated: Nov em ber 27, 2024) W e sh o w that for optical cavities with any finite dissipation, th e term “cavit y mode” should b e understo od as a solution to the Helmholtz eq uation with outgoing wa v e b oundary conditions. This c hoice of b ound ary condition renders the problem non-Hermitian, and w e demonstrate that the common d efinition of an effective mo de volume is ambiguous and not app licable. Instead, we prop ose an alternativ e effective mode volume whic h can b e easily ev aluated based on the mode calculation meth ods typically applied in the literature. This corrected mo de volume is d irectly applicable t o a muc h wider range of physical systems, allowi ng one to compute th e Purcell effect and other interes ting optical phenomena in a rigorous and unambiguous w a y . Optical micro cavities are in herently dissipative and a re t ypically characterized by a quality factor, or Q -v alue, describing the relative ener gy loss p er cycle as well as an effectiv e mo de volume, V eff , which g iv es a measure of the spa tia l confinement of the electroma gnetic field in the cavity . Cavities with hig h Q -v alues and sma ll mo de volumes provide enhanced light-matter interaction and are of fundamen tal as well as tec hnologica l interest [1, 2]. Effective mo de volumes are ubiquitous in physics and connect to a wide ra nge of phenomena, including sens- ing [3], switchin g [4], ca vity quantum e le ctro dynamics (QED) [5], circuit-Q ED [6 ], and optomechanics [7]. As a striking exa mple of the use o f mo de volumes, an emitter in an optical cavity will exp erienc e a medium-enhanced radiation ra te relative to that in a homogeneo us medium given by the so - called Purcell factor [8] F P = 3 4 π 2  λ c n c  3  Q V eff  , (1) where λ c is the free space wa velength, and n c is the ma- terial r efractive index at the field antinode r c . Purcell factors a re widely used in quantum optics as a figure o f merit for sing le photon sources [9]. Mo de volumes are often attr ibuted to the physically app ealing idea o f a single cavit y mo de. How ev er, in spite of the fact that cavity mo des ar e widely used in the lit- erature, ther e seems to b e a disturbing lack of a pr ecise definition, and their mathematical prop erties therefo r e remain unsp ecified. The lack of a definition is ev ide nc e d in pa rt by the div erse nomenclature a t use (“res o nance”, “leaky mo de ” or “quasi mo de”), suggesting that the dis- sipative na ture of cavity mo des s omehow makes them different fr o m other mo des, but an explicit distinction is rarely made. It app ear s that cavit y modes a re widely b e- lieved to share the prop erties of Hermitian eig en vectors, although a mo de with a finite lifetime is incompa tible with the solution space o f Hermitian eigenv alue pro b- lems. Strictly , o nly in the limit of la rge (infinite) Q will the mo des in optical cavities app ear as the solutions to Hermitian eigenv alue problems. F or finite Q , the lack of hermiticity effectively renders expressions s uc h as Eq. (1) ambiguous, since the volume V eff cannot b e inferred from the usua l inner pro duct of Hermitian systems. In particu- lar, if ǫ r ( r ) describ es the rela tive permittivity distribution and ˜ f c ( r ) is the cavit y mo de, then a direct application of the c ommon nor mal mo de pr escription V N eff = Z V ǫ r ( r ) | f c ( r ) | 2 ǫ r ( r c ) | f c ( r c ) | 2 d r , (2) cannot b e exp ected to pr ovide the co r rect mo de v olume. In many practical ca lc ulations, the leaky cavity mo de is found with the finite-difference time-do main (FDTD) metho d by launching a short pulse and monitoring the resonant field tha t leaks from the cavit y a t a rate se t by the Q -v a lue [10]. Figure 1 shows a sketch o f an exa mple cavit y along with mo de profiles ca lc ula ted with FDTD [11]. F or this cavit y mo de the int egral in Eq. (2) diverges as a function of the integration volume V , and indeed this is formally the case for all cavities with a finite Q . F o r very hig h- Q c avities, how ev er, the divergence is slow and may no t be discernable in pr actice due to numerical accuracy , but the formal divergence still r enders E q. (2) questionable. 0 0.2 0.4 0.6 0.8 1 −2 0 2 −2 0 2 −2 0 2 −1 1 x/a z / a y /a FIG. 1. Sketc h of a photonic crystal made from a triangular lattice of air holes (lattice constan t a ) in a membrane of high refractiv e index . A defect cavit y is formed by the omission of a single hole. Right: Ab solute v alue of th e y -p olarized ca vity mod e in the p lanes z = 0 (t op ) and y = 0 (b ottom). In this L e tter w e arg ue that the ter m “cavit y mo de” should b e understo o d a s a solution to the Helmholtz 2 equation with outgoing w av e b oundary conditions. This definition render s the cavity mo des ident ical to the quasi- normal mo de s o f Lee et al. [12] which have complex res- onance frequencies (with a negative imaginary part, as exp ected for dissipative mo des) and ex hibit a n inher ent exp onent ial divergence at la rge distances. W e illustrate directly how this definition complies with t ypical c alcula- tions of cavity modes using FDTD and we elucidate how the cavit y mo des fro m FDTD calculatio ns also show an exp onent ial divergence. Quasinormal mo des hav e pr o p- erties that a re different from the so lutions to Hermitian eigenv alue problems and therefore imp ortant results such as or thogonality and completeness of the solutions ca n- not b e taken for granted. Nevertheless, an inner pro duct and a corres po nding orthog onality rela tion c a n b e de- fined, and the qua sinormal modes can b e used as a basis for ex pansion o f the electromagnetic Gree n’s tensor in certain r egions [12]. This enable s a precise and unam- biguous des cription of light -matter interaction in general leaky optical cavities, including an expression for the ef- fective mo de volume and thus the Purcell fac tor. It is illustrative to start by highlighting the differ- ences b etw een tw o t yp es of modes that can b e asso ci- ated with fields in optical ca vities. The electr ic field in general non-mag netic materials satisfy the wa v e equa- tion with time-harmo nic so lutions of the form E ( r , t ) = E ( r , ω ) exp {− iω t } . The p osition-dep endent electric field E ( r ) s o lves the vector Helmholtz equation ∇ × ∇ × E ( r , ω ) − k 2 0 ǫ r ( r ) E ( r , ω ) = 0 , (3) where k 0 = ω /c . T ogether with a suita ble set of b ound- ary conditions, Eq. (3) provides a generalized eigenv alue equation. W e will use the ter m normal mo d e to deno te a so lution to Eq. (3) with any set of b oundary condi- tions that renders the problem Her mitian. In this c a se we denote the vector eigenfunctions and co rresp onding real eigenfrequencie s as f µ ( r ) and ω µ , res pectively . The normal mo de s are normalized as h f µ | f λ i = Z V ǫ r ( r ) f ∗ µ ( r ) · f λ ( r ) d r = δ µ,λ , (4) where the integral is over the volume defined by the bo undary conditions. In many applications , the limit V → ∞ is taken, in which case the sp ectrum of eigenv al- ues b ecomes contin uous. W e will use the term quasinor- mal mo des for s olutions to Eq. (3) with outgo ing w av e bo undary conditions (the Sommerfeld radiation condi- tion [13]). This c hoice of b oundar y condition renders the eigenv alue problem non- Hermitian with a discrete sp ectrum, and we denote the vector eig enfunctions with a tilde a s ˜ f µ ( r ). The corresp onding e ig enfrequencies, ˜ ω µ = ˜ ω R µ + i ˜ ω I µ , are in genera l co mplex with ˜ ω I µ < 0, and it follows from Eq. (3) that, c o n trary to the Hermitian case, ˜ f µ ( r ) and ˜ f ∗ µ ( r ) are not eigenv ectors c o rresp onding to the same eigenv alue. The quasinormal modes may b e normalized a s [12] hh ˜ f µ | ˜ f λ ii = lim V →∞ Z V ǫ r ( r ) ˜ f µ ( r ) · ˜ f λ ( r ) d r + i √ ǫ r ˜ ω µ + ˜ ω λ Z ∂ V ˜ f µ ( r ) · ˜ f λ ( r ) d r = δ µ,λ , (5) where ∂ V denotes the bo rder of the v olume V . The limit V → ∞ is ca lculated by increasing the volume to obtain conv ergence. F or the systems that we inv estigate in this Letter, the conv ergence is rema r k a bly fa st. F or very low- Q cavities, howev er, the conv ergence is nontrivial due to the exp onential divergence of the qua sinormal mo des which ma y cause the inner pro duct to os cillate around the prop er v alue a s a function of calculation domain size. In addition to the mo des of the cavit y it is conv enien t to intro duce the ele ctromagnetic Green’s tensor throug h ∇ × ∇ × G ( r , r ′ , ω ) − k 2 0 ǫ r ( r ) G ( r , r ′ , ω ) = I δ ( r − r ′ ) , (6 ) sub ject to the Sommerfeld radia tion co ndition. The Green’s tensor provides the pro per framework for cal- culating ligh t emission and sca tter ing in gener al dielec- tric s tr uctures. In general, the decay rate Γ α ( r , ω ) of a dipo le emitter with orientation e α may b e enhanced or suppressed as compared to the ra te Γ B in a homogeneous medium. The relative r a te may b e expressed as Γ α ( r , ω ) Γ B ( ω ) = Im { e α G ( r , r , ω ) e α } Im { e α G B ( r , r , ω ) e α } , (7) where G B ( r , r ′ , ω ) is the Green’s tenso r in a ho mogeneous background medium with ǫ r ( r ) = ǫ B [14]. In certain re- gions, s uc h as inside the sc attering regio n, the transverse part of the Green’s tensor may b e ex pa nded through [12] G T ( r , r ′ , ω ) = c 2 X µ ˜ f µ ( r ) ˜ f µ ( r ′ ) 2 ˜ ω µ ( ˜ ω µ − ω ) . (8) The implicit assumption behind the notio n of a cavit y mo de is that one term µ = c dominates the expansio n in Eq. (8) and hence that the Green’s tensor ca n be ap- proximated by this ter m only . With this assumption, and noting that Im { G ( r , r , ω ) } = Im { G T ( r , r , ω ) } , one can use E qs. (7) and (8) with ω = ω c = ˜ ω R c to recov er Eq. (1 ) with a c o rr e cte d effective mo de volume given a s V Q eff = 1 n 2 c | v Q | 2 v R Q , v Q = hh ˜ f c | ˜ f c ii ˜ f 2 c ( r c ) , (9) where v Q = v R Q + iv I Q is complex in general. This pre- scription provides a direct and unambiguous way of cal- culating the effective mo de volume. The q uasinormal mode s ca n b e calculated analytically for sufficiently simple structur es, but for g eneral struc- tures the outgo ing wa v e b oundar y conditions a re not im- mediately compatible with standar d n umerical solution metho ds. One o ptio n is to r ewrite E q. (3) as ∇ × ∇ × E ( r , ω ) − k 2 0 ǫ B E ( r , ω ) = k 2 0 ∆ ǫ ( r ) E ( r , ω ) , (10) 3 where ∆ ǫ = ǫ r ( r ) − ǫ B , and calculate the quasino r mal mo des from a F r edholm type integral equation, E ( r , ω ) =  ω c  2 Z V G B ( r , r ′ , ω ) ∆ ε ( r ′ ) E ( r ′ , ω )d r ′ , (1 1) which manifestly resp ects the outg o ing wa ve boundar y conditions. Another option is to use FDTD to calcula te the qua sinormal mo de as the reso nant field that is ex- cited b y an initial sho rt input pulse. W e ha ve used b oth Eq. (1 1) and FDTD to calcula te the qua sinormal mo des in different exa mple cavities in tw o and three dimensions. F o r our example b elow, we solve Eq . (11) using the ex- pansion technique of Ref. [15] with an a dditional iter ation of k 0 to make the solution self-consistent. In addition, we per form FDTD calc ula tions using p erfectly matched la y- ers (PMLs ) [16] to enforce the outgoing wa ve b ounday conditions. W e first co ns ider a 2 D finite-sized hexagonal crystal- lite of high- index ro ds in a ir w ith a single mis s ing ro d in the center. The ro ds have rela tiv e p ermittivity ǫ r =11 . 4 and ra dius R =0 . 15 a , wher e a is the la ttice constant. W e fo cus on transverse magnetic (TM) p olariz a tion in which the electric field is in the directio n of the ro ds. In the limit of infinite size, the photonic cry stal e xhibits a pho- tonic ba nd gap [17], and the Q o f the cavit y therefor e depe nds on the size of the structure which we may char- acterize by the num ber of r o d la y ers, N . F or N =1 a nd N = 2, Fig . 2 shows the supp orted c avity mo des with fre - quencies ˜ ω c a/ 2 π c =0 . 4259 − 0 . 0 135 i ( Q = − ˜ ω R / 2 ˜ ω I ≈ 16) and ˜ ω c a/ 2 π c = 0 . 421 8 − 0 . 001 3 i ( Q ≈ 163). In the ca se of N = 3 (not shown), the structure supp o rts a cavit y mo de with frequency ˜ ω c a/ 2 π c = 0 . 4 2 16 − 0 . 000 1 i ( Q ≈ 1 576). As exp ected, the quasino rmal modes are co ncen trated in the center o f the cavit y and seem to fall off with in- creasing distance to the crystallite. At large distances, how ev er, the qua s inormal mo des (b y definition) behav e as outgoing wa ves of the form ˜ f ( r ) ∝ ex p( ik 0 r ) / √ r (2D) and ˜ f ( r ) ∝ exp( ik 0 r ) /r (3D), a nd since k 0 = k R + ik I with k I < 0, they diverge exp onentially a s r → ∞ . F o r the case of N =1, the top panel in Fig . 2 illustra tes di- rectly that the so lutions to Eq . (11) are iden tical to those obtained fro m FDTD. In particular, bo th solutio n meth- o ds pick up the divergence in the field at la rge distances. Figure 3 shows, a s a function of the s iz e of the calcu- lation domain, the corr ected effective mode volume in Eq. (9) alo ng with the common definition in Eq. (2) with f c ( r ) = ˜ f c ( r ). F r o m the figure it is clea r that whereas V Q eff conv erges quic kly to the limiting v a lue s , V N eff seems to increase with the size of the domain. The linea r diver- gence in V N eff with the size of the normalizatio n do main was also noted in Ref. [18] and derives from the fact that the field does not go to zero a t p ositions outside the crys- tallite, cf. Fig. 2. At m uch larg er V , the field, and he nc e V N eff , diverges exp onentially . F or incr easing Q, the linear divergence with domain size be comes less and less pro- nounced, sugges ting that the tw o forma lisms conv erge in 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 (a) | ˜ f ( x ) | / | ˜ f (0) | 0 10 20 30 40 0.2 0.4 0.6 0.8 1 x/a | ˜ f ( x ) | / | ˜ f (0) | 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 (b) x/a | ˜ f ( x ) | / | ˜ f (0) | −2 0 2 −2 −1 0 1 2 0.2 0.4 0.6 0.8 x/a y /a FIG. 2. (a): Absolute val ue along th e x - axis of the quasinor- mal mode in the 2D crystallite for t he case of N =1. Blue solid line shows the solution to Eq. (11) , and b lac k circles sho w the calculation using FDTD. Inset show s long distance b ehavior on a logarithmic scale. (b): Absolute v alue along the x -axis of the qu asinormal mod e for the case of N = 2 with the inset sho wing th e distribution in the x y -plane. Grey shaded areas indicate the high-index rod s. the limit of infinite Q a s exp ected. Next, for a practical 3D example we co nsider a pho- tonic c r ystal mem brane ( ǫ r =12) o f thic kness h = 0 . 5 a and hole radius r = 0 . 275 a . A sing le air ho le is omit- ted to create a cavit y , and Fig. 1 shows the supp orted cavit y mo de with fre q uency ˜ ω a/ 2 π c =0 . 290 4 − 0 . 00 04 i ( Q ≈ 362). F rom the re s ults in tw o dimensions we know that the qua sinormal mo des c a n b e directly calcula ted using FDTD with PMLs. The field in Fig. 1 was cal- culated in the sa me way , a nd we therefore a rgue tha t it is indeed a quasinor mal mo de and that we should use Eq. (9) rather than Eq. (2) to ca lculate the effective mo de volume. Fig. 4 shows b oth V Q eff and V N eff as a function of calculation domain size. At the qua sinormal mode fr e- quency , the pho tonic band ga p preven ts in-plane propa- gation, and ther efore the only wa y for the field to leak out of the cavit y is in the z -dir e ction. This mea ns that b oth V Q eff and V N eff conv erge quickly as a function of width a nd depth of the calculation domain and we focus only on the v a riation in the effective mo de v olumes with the height of the calcula tion domain. As in t w o dimensio ns, the data sho ws a fast convergence o f V Q eff , while V N eff clearly diverges, confirming that Eq. (2) is not applicable. 4 0 2 4 6 8 10 0.17 0.18 0.19 0.2 0.21 R/a V eff / ( λ c /n c ) 3 FIG. 3. Effective mode volumes V N eff (thick lines) and V Q eff (thin lines) for N = 1 (red dash-d otted), N = 2 (green dashed) and N = 3 ( blue solid) as a function of radius R of the calculation domain. Circles indicate reference mo d e vol umes V tot eff from indep endent Green’s ten sor calculations [15], and grey d ashed areas sho w the rod cross sections along the x -axis. 0 1 2 3 4 5 6 0.31 0.32 0.33 0.34 L z /a V eff / ( λ c /n c ) 3 FIG. 4. Effective mo de volume V N eff (red dashed) and V Q eff (blue solid) for the ca vit y in Fig . 1 as a function of h eigh t of the calculation domain. Circles indicate reference mo d e vol umes V tot eff derived fr om indep endent Green’s tensor cal- culations [10] with estimated error bars at differen t domain heights. Gray dashed area shows the extend of the m embrane. Finally , we compare the calculated mo de volumes to in- depe ndent calcula tio ns using the Green’s tens o r [10, 15]. Substituting F P = Γ c ( r , ω c ) / Γ B ( ω c ) in the expressio n for the Pur cell factor [19] defines an effective mo de v olume V tot eff . F o r eac h of the c avities, V tot eff is indicated with a circle in Fig . 3. The maximum estimated a bsolute error in these calcula tions is less than 0.0 003. The observ able discrepancies for N = 1 and N =2 stem from the single mo de approximation and indicates the limited v alidity of the Purcell factor. In Fig. 4, V tot eff was calculated with FDTD as the r espo nse to an input dipole source at three different domain sizes a nd with estimated error bar s as indicated. These independent calculations confirm that Eq. (9) not only is unambiguous, but also leads to the correct v alue within the single mo de approximation. In conclusio n, we have shown that the term “cavit y mo de” should be under sto od a s a so- called quasinor- mal mo de, defined as a solution to the Helmholtz equa- tion with outgoing wa ve b oundary co nditions. This can hav e pro found co nsequences, since this choice o f bo und- ary co nditions r e nders the differential equation problem non-Hermitian so that common r esults from Hermitian eigenv alue analysis do not apply . In particular, the quasi- normal mo des hav e complex frequencies and exhibit an inherent divergence at long distances which makes the calculation of a n effective mode volume non trivial. In- tro ducing an inner pro duct that carefully accounts for the long distance behavior, it is p ossible to nor malize the quasinorma l mo des and to define an effective mo de vol- ume in a direct and unambiguous wa y . In practical cal- culations, this corre c ted mo de volume can b e obtained in a straightforw ard way using exac tly the same cavit y mo des that are typically computed for use in mo de vol- ume calculations. This w ork was supp orted b y NSERC and The Danish Council for Indep endent Resea rch (FTP 10-093 651). [1] R. K. Chang and A. J. Campil lo, Optic al Pr o c esse s in Micr o c avities (W orld S cien tific, 1996). [2] K. J. V ahala, N ature 424 , 839 (2003). [3] M. Loncar, A. Scherer, and Y. Qiu, Applied Physics Let- ters 82 , 4648 (2003). [4] C. Husk o, A. D. Rossi, S. Com brie, Q. V. T ran, F. Raineri, and C. W. W ong, Applied Physics Letters 94 , 021111 (2009). [5] H. Carmic hael, Statistic al Metho ds in Quantum Optics 1 (Springer, 2008). [6] A. W allraff, D. 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