DOI 10.1007/s11082-006-0018-5Optical and Quantum Electronics (2006) 38:71–81 © Springer 2006

Transient transformation of whispering gallery

resonator modes due to time variations in dielectric

permittivity

n. s a k h n e n k o1,2,∗, t . m . b e n s o n1, p . s e w e l l1 a n da . n e r u k h21George Green Institute for Electromagnetics Research, University of Nottingham, University Park,Nottingham NG7 2RD, UK2Kharkov National University of Radio Electronics, 14 Lenin Ave., Kharkov 61166, Ukraine(∗author for correspondence: E-mail: n sakhnenko@yahoo.com)

Received 29 September 2005; accepted 31 January 2006

Abstract. This paper considers the switching effects on whispering gallery modes in circular dielectricresonators when their material is subject to abrupt time changes in permittivity. Both the transientresponse and the new steady-state regime are described in detail by means of an analytical approach.

Key words: time-domain analysis, time-varying media, whispering gallery modes (WGM) resonators

1. Introduction

Whispering Gallery Mode, WGM, resonators or micro-cavities, have beenthe subject of significant interest in recent years as they exhibit propertiesideal for a wide range of potential applications. These include: optical filter-ing (Little et al. 1997), bio-sensing (Blair and Chen 2001; Boyd and Heebner2001) and low-threshold micro-lasers (Fujita et al. 1999; Baba et al. 2003).Moreover, arrays of such resonators not only provide more complex opticalfiltering, but have also led to the observation of the so-called slow and storedlight phenomena (Heebner et al. 2002).

In addition to the uses of passive resonators, the fabrication of opticallyactive and tuneable devices is of great importance for all-optical data pro-cessing and switching purposes and, generally, active micro-cavities signifi-cant extend the functionality of many devices and enhance their applicationrange. Micro-cavity tuning can be achieved by changing the material refrac-tive index which can be practically realised by varying the input signal in anon-linear micro-cavity (Blom et al. 1997); by voltage control (Savchenkovet al. 2005) or else by free carrier plasma injection (Djordjiev et al. 2002).

To date, much of the theoretical work focused upon micro-resonatorshas concentrated upon prediction of their frequency domain properties,although for applications such as optical memory (Hill et al. 2004) and

72 N. SAKHNENKO ET AL.

optical switching, accurate time domain characterisation is also clearlyessential. In order to gain understanding and insight into the fundamen-tal processes occurring in micro-cavities, we investigate, in a rigorous math-ematical manner, the transient response of a WGM of the resonator to anabrupt change in refractive index. Except for this abrupt change, the reso-nator materials are otherwise considered to be linear.

It is well known that a monochromatic electromagnetic field incident ona temporally invariant object is scattered with changes in its wavenumberand field pattern but that its frequency remains unaltered. In contrast, tem-poral variations of the permittivity of an unbounded medium transform thefrequency of an existing monochromatic field, but both the wavenumberand instantaneous field pattern are conserved (Morgenthaler 1958; Felsenand Whitman 1971). In a dielectric resonator, the WGM modes are definedby a complex interaction of the field with both the material and the bound-aries of the cavity. If the permittivity of the cavity is changed in time it isnot immediately apparent which characteristics of an existing WGM trans-form and which ones remain unchanged. Furthermore, the details of thefield evolution during the transient period have yet to be described in detailand specifically. It is clearly interesting for many applications to understandthe processes that determine the duration of this transient before a stablesteady-state regime is reached. A rigorous formulation will be derived inthis work that allows such questions to be addressed.

2. Mathematical formulation

We consider a circular dielectric resonator of radius ρ0 and dielectric per-mittivity ε1 embedded in a uniform dielectric medium of infinite extent andof permittivity ε. Both media are considered to be non-magnetic and to becharacterised by a simple scalar permeability µ= 1. To describe the fields,the polar system of coordinates ρ,ϕ, centred on the resonator, is intro-duced. Before switching the permittivity, an initial field exists in the reso-nator which is an E-polarised WGM of the resonator described by

E0 =Ek,m =⎧⎨

⎩

bkJk(ω0

v1ρ) cos kϕ, ρ <ρ0

H(2)k (

ω0

vρ) cos kϕ, ρ >ρ0

bk =H

(2)k (

ω0

vρ0)

Jk(ω0

v1ρ0)

(1)

where k, the number of angular field maxima, and m, the number of radialfield variations, classify the particular mode and ω0 is the corresponding ei-genfrequency which is a solution of the equation

Jk

(ω0

v1ρ0

)

H′(2)k

(ω0

vρ0

)− v

v1J ′

k

(ω0

v1ρ0

)

H(2)k

(ω0

vρ0

)=0 (2)

TRANSIENT TRANSFORMATION OF WHISPERING GALLERY RESONATOR MODES 73

v1 = c (ε1)−0.5 and v = c (ε)−0.5 are the phase velocities in the cavity and the

surrounding medium, respectively; c is the velocity of light in a vacuum.The associated time dependence for this initial field may be expressed as

eiω0(t−t∗)�(t − t∗), where t∗ is the moment of instantaneous creation of themode, where �(..) is the Heaviside unit function and in our considerationt∗ <0.

At zero moment of time the dielectric permittivity inside the resonatorchanges in value from ε1 to ε2 in response to an external source which is illus-trated schematically in Fig. 1. We now investigate the mechanisms that couplethe initial mode to those of the cavity with the new permittivity with partic-ular emphasis on the transient processes occurring in such a single dynamicresonator. The formulation of the problem in the above manner permits con-struction of an analytical solution that explains the interesting phenomenain detail. In general, there are two classes of approaches for solving suchtransient electromagnetic problems. One technique includes solving Maxwell’sequations directly in time domain by using a time-step algorithm, and is typ-ified by the finite difference time domain method. Such approaches, whilstflexible, are computationally intensive and, of more concern here, do notallow the key mechanisms at play to be succinctly extracted. In contrast, theapproach to be pursued here is to solve Maxwell’s equations analytically inthe frequency domain and then to recover the time domain electromagneticfield via an inverse frequency transform.

3. Solution

We start with wave equation satisfied by polarised fields

1c2

∂2

∂t2ε(t)E −�E =0 (3)

Here E represents the Ez component of the field which is perpendicular tothe plane of the resonator and � is the Laplacian operator.

1 0, Eε 2 1, Eε

ε ε

t < 0 t > 0

Fig. 1. Schematic diagram of the phenomenon.

74 N. SAKHNENKO ET AL.

�E =(

∂2

∂ρ2+ 1

ρ

∂

∂ρ+ 1

ρ2

∂2

∂ϕ2

)

E (4)

In (3) the derivatives are considered as a generalised derivative (Gelfandand Shilov 1964) that naturally takes into account the discontinuity of thematerial function.

To distinguish the field before and after the moment of switching the res-onator’s permittivity the following notation is adopted.

E(t)={

E0(t), t <0E1(t), t >0 (5)

We apply the Laplace transform (�) directly to the wave equation and,taking into account the initial conditions (continuity of electric displace-ment D at zero moment of time), obtain

ε2

c2

[

p2�E+1 (p)−p

ε1

ε2E+

0 (0)− ε1

ε2E

′+0 (0)

]

−��E+1 (p)=0

ε

c2

[p2�E−

1 (p)−pE−0 (0)−E

′−0 (0)

]−��E−

1 (p)=0 (6)

The superscripts + and −, respectively denote the electromagnetic fieldsinside and outside of the circular cavity region.

Here we adopt the evolutionary approach presented in (Nerukh et al.2001). In this approach, the complete transient solution is explicitly con-structed as a superposition of the waves reflecting from the cavity boundaries;the field at an instant just after the moment of switching being evaluated as ifthe transient medium were unbounded. Therefore, because of the finite speedof the electromagnetic waves, the influence of the cavity’s boundary appearsonly after a finite time delay from the moment of switching and this will bediscussed further below. The ‘unbounded’ field within the cavity just after themoment of switching is given by

�Eunb =bk cos kϕv2

2(p + iω0)

p2v21 +ω2

0ν22

Jk

(ω0

v1ρ

)

e−iω0t∗

(7)

and this expression can be easily inverted into the time domain to give

Eunb = v2

2v1

(v2 +v1

v1eiω1t + v2 −v1

v1e−iω1t

)

bk cos kϕJk

(ω0

v1ρ

)

e−iω0t∗

(8)

where ω1 = (v2/v1)ω0 is the new frequency present.As stated above, the complete field within the cavity consists of the

unbounded term (8) and additional contributions due to the boundary.

TRANSIENT TRANSFORMATION OF WHISPERING GALLERY RESONATOR MODES 75

Similarly, in the outer region the field comprises a superposition of the ini-tial field and contributions due to the presence of the boundary. Introduc-ing the terms due to the boundary reflections gives

�E+ =bk cos kϕv2

2(p + iω0)

p2v21 +ω2

0ν22

Jk(ω0

v1ρ)e−iω0t∗ + cos kϕBk(p)Ik(

p

v2ρ) (9)

�E− = cos kϕH(2)k (

ω0

vρ)

e−iω0t∗

p − iω0+ cos kϕCk(p)Kk(

p

vρ) (10)

which in the Laplace domain are expressed in terms of the modified Bes-sel functions Ik(..),Kk(..). Outside the cavity, the function Kk(..)guaranteessatisfaction of the Sommerfeld outgoing radiation condition at infinity andBk and Ck are unknown coefficients, to be determined, which are chosen sothat at the cylindrical boundary, ρ = ρ0, the tangential components of thefield are continuous, i.e.

E+z (ρ0)=E−

z (ρ0) (11)

H+ϕ (ρ0)=H−

ϕ (ρ0) (12)

�Hϕ = 1p

1µ0

∂

∂ρ�Ez (13)

From (9)–(13) it follows that

Bk(p)Ik

(p

v2ρ0

)

−Ck(p)Kk

(p

vρ0

)=bkJk

(ω

v1ρ0

)p2(v2

1 −v22)

(p − iω0)(v21p

2 +ω20v

22)

Bk(p)p

v2I ′k(

p

v2ρ0)−Ck(p)

p

vK ′

k(p

vρ0)=bk

ω

v1J ′

k(ω

v1ρ0)

p2(v21 −v2

2)

(p − iω0)(v21p

2 +ω20v

22)

(14)

which has a solution of the form

Bk(p)=bk

p2(v21 −v2

2)

(p − iω0)(v21p

2 +ω20v

22)

ω0v1

J ′k(

ω0v1

ρ0)Kk(p

vρ0)− p

vJk(

ω0v1

ρ0)K′k(

p

vρ0)

p

v2I ′k(

p

v2ρ0)Kk(

p

vρ0)− p

vIk(

p

v2ρ0)K

′k(

p

vρ0)

e−iω0t∗ (15)

Ck(p)=bk

p2(v21 −v2

2)

(p − iω0)(v21p

2 +ω20v

22)

ω0v1

J ′k(

ω0v1

ρ0)Ik(p

v2ρ0)− p

v2Jk(

ω0v1

ρ0)I′k(

p

v2ρ0)

p

v2I ′k(

p

v2ρ0)Kk(

p

vρ0)− p

vIk(

p

v2ρ0)K

′k(

p

vρ0)

e−iω0t∗ (16)

It is observed that these expressions possess both poles and a branch cutalong the negative real axis in the complex p-plane. Physically, the polescontribute to the steady-state solution while the branch point fields onlycontribute to the transient solution.

76 N. SAKHNENKO ET AL.

4. Transient analysis

The ‘early time’ response can be found using the following theoremconcerning the Laplace transform

limp→∞ p�f (p)= lim

t→0f (t) (17)

Using the asymptotic expansions for modified Bessel functions with largearguments

In(q)∼ 1√

2πqeq; Kn(q)∼

√π

2qe−q (18)

it follows that

p ·Bk(p)Ik

(p

v2ρ

)

∼bk

v21 −v2

2

v21

v2

v +v2

√ρ0

ρJk

(ω0

v1ρ0

)

e−p

ρ0−ρ

v2 , ρ <ρ0 (19)

p ·Ck(p)Kk

(p

vρ)

∼−bk

v21 −v2

2

v21

v

v +v2

√ρ0

ρJk

(ω0

v1ρ0

)

e−pρ−ρ0

v , ρ >ρ0

(20)

From (19) and (20) it is observed that, upon inversion to the timedomain, the expressions corresponding to Bk(p)Ik(

p

v2ρ) and Ck(p)Kk(

p

vρ)

will exhibit a time delay that can be expressed in terms of unit Heavisidefunctions; �(v2t + ρ − ρ0) and �(vt − ρ + ρ0) inside and outside of thecavity, respectively.

It is also noted that throughout space and time the transformed fieldpreserves the angular dependence of the initial field.

Therefore, to describe the ‘early time’ response just after the moment ofswitching we divide all space into ‘time-spatial zones’. In the ‘early time’regime for which t <

ρ0v2

, four such zones are observable which are denotedby the regions I–IV in Fig. 2. In the inner transient region, zone I , thefield corresponds to that of the unbounded value problem, (7), and exhibitsthe same wave number and shifted frequencies predicated from the abruptchange in the material parameters. It does not depend upon the bound-ary shape. In the outer region, zone IV, only the initial field (1) is present.Near the boundary region, zones II and III, transient waves appear thatcorrespond to total fields E+ and E− given in (9) and (10) which in thetime domain, are expressible in terms of a residue sum over all the singu-lar points and an integral along the branch cuts of (9) and (10). Examina-tion of (9) and (10) in more detail reveals that the singularities p = ±iω1

and p= iω0 in zones II and III do not contribute to the residue sum. It is

TRANSIENT TRANSFORMATION OF WHISPERING GALLERY RESONATOR MODES 77

v2v

IIIIIIIV

vt=ρ

0=ρ ρ

0 2v t= −ρ ρ

v2v

Fig. 2. Schematic diagram of ‘early time’ response for fixed moment of time.

also confirmed that there is a term within the transient response inside theresonator that provides immediate cancellation of the primary wave.

It can be seen that the boundaries between regions I and II and IIIand IV propagate in opposite directions from the boundary with veloci-ties v2 and v, respectively. Considering the fields inside the cavity, we definethe ‘late time’ transient response to begin after the moment t = ρ0

v2when

the moving wave front between I and II reaches the centre of the reso-nator. From this point onwards the field inside the cavity is representedin terms of the spectrum of the modes of the dynamic resonator in itsnew state.

In general, transient excitation of all modes occurs within the cavity butin practice, most of the modes have insignificantly small amplitudes, theexception being that which is most similar to the initial wave in terms offield profile and eigenfrequency. Regarding this mode to be a transforma-tion of the initial field, the overall effect of the abrupt change in refractiveindex is a transformed amplitude comparable to that of the initial field anda slight shift of the mode’s eigenfrequency.

For convenience dimensionless values are now introduced

n1 = v

v1; n2 = v

v2; w = ω

vρ0; R = ρ

ρ0; T = v

ρ0t (21)

Attention is now focussed upon evaluating the contribution of the branchcut to the transient field. Figure 3 shows the field outside the resonator dueto the integral along the branch cut. Here, the WGM E7,1 is considered asthe initial field in the resonator which is of refractive index n1 = 3.4 thatsubsequently abruptly changes to n2. The normalised moment of creationof this mode is at T ∗ = −5 and we observe the normalised transient fieldcomponent at a distance R = 2 from the centre of the cavity. It is seen inFig. 3 that the branch cut contributes a monotonically decreasing wave that

78 N. SAKHNENKO ET AL.

Fig. 3. The contribution of the branch cut to the field in the outer region (n1 = 3.4, R = 2, WGE7,1 isinitial field).

has maximum amplitude at the moment the transient wave arrives at theobservation point (T =1).

It can be seen that the amplitude of this component of the transientis rather small in comparison with amplitude of the field due to the resi-due sum at the same observation point which is shown in Fig. 4 for thecase when the refractive index changes to a value of n2 =3.3. Furthermore,Fig. 4 also shows the primary field, with frequency (w0 =2.89847+3.4458 ·10−5i), before the arrival of the transient field at time T = 1. Discontinu-ous field behaviour is evident in Figs. 3 and 4 at T = 1. However, thesetwo discontinuities cancel when Figs. 3 (dashed line) and 4 are combinedto give the total field. The contribution of the branch cut integrals to thetotal field inside the cavity is rather similar to that just shown, but is of aneven smaller amplitude.

To gain a clear picture of the transient process in zone I we now con-sider E20,1 as the initial field which is of short wavelength with respect tothe size of the cavity. The fields are now observed within the cavity at adistance of R =0.6 from the centre when the refractive index changes fromvalue n1 =3.4 to n2 =3.3. Before zero moment of time (i.e. that time whenthe permittivity jumps) the initial field of frequency w0 = 7.1406 + 1.878 ·10−14i is observable. As expected, the time jump in the dielectric permit-tivity causes a field discontinuity at zero moment of time. Within the timeinterval 0 <T < 1.2 oscillations of frequency w1 = (n1/n2)w0 are present aspredicted by the unbounded part of the solution (7). After T =1.2 when theinterface between regions I and II passes the observation point, the contri-bution of the residue sum is apparent.

TRANSIENT TRANSFORMATION OF WHISPERING GALLERY RESONATOR MODES 79

Fig. 4. The contribution of the residue sum to the field in the outer region (n1 = 3.4, n2 = 3.3, WGE7,1

is initial mode).

5. Steady-state analysis

The steady-state regime is regarded to exist for t >ρ0v2

and is characterisedby a superposition of the natural modes of the resonator in its new state.

To more clearly illuminate the phenomena associated with the changingfrequencies and amplitudes we consider relatively large changes in refrac-tive index. Although they are impractical they do reveal the key phenomenavery clearly. Figure 6 illustrates the electromagnetic field inside the resona-tor in the steady-state regime. Here the WGM E7,1 is chosen as the initialfield and the observation point is at a distance of R = 0.8 from the cavitycentre. Again regarding the transformation of the initial field into the mostsimilar WGM of the new structure it is seen that abruptly decreasing therefractive index leads to an increase in the frequency and the amplitude anda degradation of the Q factor (Fig. 6(a)). Figure 6(b) shows that increasingthe refractive index leads to the opposite effect.

Changing of the amplitude follows from initial conditions of electric dis-placement continuity at zero moment of time. The frequency shifting agreeswith the solution of the initial value problem (8).

The curves shown in Fig. 6 are evaluated using the 10 most significantmodes of the residue sum of (9), however, using just one (E7,1 in this case)gives essentially the same result, which confirms the above assertion thatthe mode of the same spatial shape as the initial field dominates; the othermodes that are excited have rather small amplitude (typically ≈0.1% of the

80 N. SAKHNENKO ET AL.

Fig. 5. The transient response inside the resonator (n1 =3.4, n2 =3.3, R =0.6, WGE20,1 is initial mode).

Fig. 6. Temporal evolution of the field inside the cavity (R =0.8, WGE7,1 is the initial field).

amplitude of the initial mode). Therefore it is concluded that time variationof the refractive index of the cavity does not change the near field pattern.

6. Conclusions

Transformation of the natural WGM of 2D circular resonators has beenstudied by means of an analytical formulation that clearly reveals the key

TRANSIENT TRANSFORMATION OF WHISPERING GALLERY RESONATOR MODES 81

mechanisms at play. It has been shown that at an early stage of the tran-sient process inside the resonator there exists a wave that corresponds tothe result for unbounded space. The influence of the resonator’s boundariessubsequently leads to the cancellation of this wave and the appearance ofthe natural modes of the new structure. Outside the resonator there existsa moving cylindrical front that propagates with velocity v within which theprimary field is cancelled and the field comprises the natural modes ofthe new structure. Within the resonator, the transient process extends overthe finite time interval 0 < t <

ρ0v2

before the appearance of the resonatormodes. In contrast, outside the resonator the transient extends to infinityin time and space.

The maximum WGM amplitude of the new structure is observed for thatmode with the same number of angular and radial field variations as theinitial mode. Therefore, it is concluded overall that changes of the permit-tivity of the material inside micro-cavities lead to a steady-state regime inwhich a change of wavenumber and frequency are observed but the fieldpattern is conserved.

Acknowledgement

The authors wish to thank the support of this project under a Royal Soci-ety/NATO Fellowship.

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