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Спецкурс Оптические микрорезонаторы. Лекция 14. Применение микрорезонаторов с модами шепчущей галереи. Лекция основана на обзоре В.С.Ильченко и А.Б.Мацко в IEEE J. of Sel. Topics in Quantum Electronics, v.12, No.1, 2006
М.Л.Городецкий 25 мая 2007 г.

В этой главе мы приведем обзор применений открытых диэлектрических резонаторов в фотонике Говоря о применениях в фотонике, мы понимаем под этим широким термином все применения, которые связаны с линейной, нелинейной, квантовой оптикой и прикладной оптикой и связанными с ними другими областями науки и технологии. Особое внимание уделяется микроволновой фотнике, когда диэлектрические резонаторы используются для получения и обработки СВЧ сигналов оптическими методами. Мы рассматриваем резонаторы, которые изготовлены из прозрачных материалов и имеют монолитную форму, как правило аксиально симметричного резонатора, моды в которых можно представить как лучи распространяющиеся внутри вблизи поверхности благодаря эффекту полного внутреннего отражения. Современные открытые диэлектрические оптические резонаторы могут иметь цилиндрическую, сферическую, сфероидальную, тороидальную форму и другие геометрии и топологии с различными граничными условиями. В целях некоторой общности мы далее везде используем термины резонаторы шепчущей галереи (РШГ) и моды шепчущей галереи (МШГ) для обозначения таких резонаторов и мод в них. Следует отметить, что, строго говоря, термин "шепчущие галереи"не вполне применим к таким квазиодномерным резонаторам, как интегральные кольцевые резонаторы, в которых кривизна образующей резонатора не играет определяющей роли в образовании пространственной структуры моды, и такие объекты можно часто описывать как петли, образованные направляющими. Однако при уменьшении размеров кольцевых резонаторов влияние внутренней образующей волновода постепенно уменбшается и мо-

1


ды переходят в моды шепчущей галереи, поэтому многие свойства кольцевых микрорезонаторов и МШГ весьма близки. В настоящее время исследование кольцевых интегральных структур также активно развивается (см., напр. [ ,

? ?, ?

]). Ниже мы останавливаемся на работах, в которых использу-

ются кольцевые микрорезонаторы лишь в случаях, которые представляют общий интерес.

1

Устройства с пассивными резонаторами ШГ

Unique sp ectral prop erties of WGMs that include narrow linewidth, tunability, and high stability under environment conditions, make WGRs attractive for numerous practical applications. In this section we review applications of passive WGRs for ltering, frequency stabilization and sensing. Photonic lters based on optical WGRs are currently among the most develop ed devices that involve WGMs. For optical telecommunication purp oses, the main task of the lters is to select channels in Wavelength Division Multiplexing (WDM) schemes. In this domain, where channel spacing is usually no less than 10 GHz, planar ring resonators with WGM and similar devices with are adequate.

Q<

I ? IH

5

Ultra-high-Q WGRs with MHz range resonance bandwidths oer a unique opp ortunity for creation of photonic microwave lters in which optical-domain selection is used for separating the RF channels imprinted as sidebands on a stable optical carrier. Imp ortant applications of WGRs are in metrology for optical and microwave frequency stabilization, where a long photon storage time helps to suppress phase and frequency deviation of oscillators. High Q and long recirculation of light in compact WGRs oer interesting new capabilities in sp ectroscopy and sensing, where the change in Q or resonance frequency of WGMs can serve as a measure of absorption in surrounding medium, or in a small (down to single molecule) quantity of dep osited substance on resonator surface. The resonator can also b e used for measurement of change in ambient parameters, such as temp erature, pressure, motion etc.

1.1

Оптические фильтры на основе отдельных резонаторах

The simplest resonator-based lter includes a WGR and an optical coupler, e.g. a prism coupler. Transmission of a mono chromatic electromagnetic wave of frequency

!

by an optical WGR in a single prism conguration may b e

characterized by the co ecient

T
where

a c i@! ! A c C C i@! ! A
0 0

;
, and

(1)

T

describ es the amplitude transmission,

, c

!0

are the absorption

and coupling linewidth, and resonance frequency of a mo de of the resonator

2


resp ectively (we assume that Condition

tral range). The p ower transmission



The lter describ ed by Eq. (1) is a stop-band lter b ecause it is characterized by the absorption resonance. A WGR with two, input and output, couplers is characterized by a transmission resonance. This is an example of a pass-band lter. The transmission and reection co ecients through the resonator are

a

c

j! !0 j is jT j2

much less than the cavity free sp ecthrough the resonator is Lorentzian.

corresp onds to critical coupling of the resonator [3, 4].

T
where

c a C i@ ! ! A c
0

;R

@ a iC!i@ ! A A ! ! c
0 0

;

(2)

T

and

R

describ e the amplitude transmission (light go es into one coupler

and exits the other coupler) and reection (light go es and exits the same coupler) resp ectively, and

c

)



is assumed for simplicity.

Single ring-shap ed WGR-based lter were studied in [5, 6, 7, 8, 9, 10], see also [11, 12] for a review. An all-optical passive four-p ort system including a fused silica microsphere and two tap ered b ers was used as a channel addingdropping device [13]. The lter resp onse of single-ring resonators with integrated semiconductor optical ampliers based on GaInAsP-InP is presented in [14]. A channel dropping lter based on a dielectric microsphere integrated to a silicon photo dio de was studied in [15]. Two dielectric waveguides that are evanescently coupled to a few micron size square or rectangular region of increased refractive index can serve as a very compact integrated optical microresonator, similar to a ring WGR. Applications of the device for ltering was discussed in [16]. Unfortunately, the Lorentzian lineshap e of the lter function asso ciated with a single microresonator represents a limitation for its application in many systems that require large sidemo de rejection, in addition to a narrow bandpass and a large tuning range.

1.2

Фильтры высокого порядка

Cascaded resonators, such as coupled optical b er resonators are widely used as optical and photonic lters [17, 18]. WGRs oer new p ossibilities for multip ole ltering b ecause of their small size, low losses, and integrability into optical networks. Multi-p ole lters based on cascaded integrated microring resonators fabricated with silica have b een demonstrated in compact and robust packages. The lters have der of
5

These lters provide passbands with at tops and sharp skirts, suitable for high p erformance applications, esp ecially in optical WDM. A second-order optical lter with a MHz bandwidth was realized with two coupled high-Q ( of germanate glass [24]. Tunable three-resonator lter made of LiNbO3 WGRs was demonstrated in [25]. The lter has the following distinctive features/advantages over other crosphere resonators, one of which was tunable. The tunable WGR was made

IH IHH IH IH
4

GHz bandwidths and corresp onding optical

Qs

on the or-

[6, 19, 20, 21, 22, 23], and are in fact commercially available.

IH

8

) mi-

3


WGM lters: i) agile tunability accompanied by a high-order lter function, ii) narrow linewidth ( three features makes this lter a unique device for a wide range of applications in optics. Since the microwave signals in photonic systems are sidebands of an optical carrier, these lters, in principle, can b e used at any microwave frequency, providing the same characteristics throughout the band, from 1 to 100 GHz, and higher. Multi-resonator lters have signicantly more sparse sp ectra as compared with a stand alone WGM resonator. This feature is due to the so called Vernier eect [26] and is similar to the feature observed in coupled b er-ring resonators [17, 18] which are noted for a rare sp ectrum. An ecient nesse of such multiresonator systems, intro duced as a ratio of frequency dierence b etween the transmission bands of the lter divided over the frequency width of a band, is very large, e.g. FSR of the lter rep orted in [25] exceeds a THz. Polymer double micro-ring lters with thermo-optic as well as electro-optic tuning were demonstrated and rep orted in [27]. It was noted, that one of several advantages of using coupled micro-rings for ltering is in the p ossibility of high tuning enhancement factor

PH

MHz), and iii) low b er-to-b er loss. Combination of the

M
where

I a I a

M

given by (3)

2

=a1 M

a1

and

a2

are the radii of the two rings. The tuning range of the double

micro-ring lter is

M

times the tuning range of a single ring [28]. Since

can b e

very large in double-WGR structures, the faster but much smaller electro-optic eect can b e used for tuning. It is imp ortant to note, that the the derivation of the parameter is valid for the resonators with such radius ratio, that less than the nesse of the resonators. Each ring resonator has a set of transmission sp ectra and the wavelength p erio d determined by its free sp ectral range

M

stays

f

F S R . The two rings have slightly

dierent radii (or eective indices). Therefore, the two sets of transmission p eak combs have small dierent p eak spacing. Wavelength tuning is achieved by aligning the p eaks in the two sets of combs with the adjustment of index in one or b oth ring resonators. The lter transmission discreetly jumps from transmission at wavelength ring by

f0

to

f

0

rings compared with voltage necessary to continuously shift the sp ectrum of the

C

f

F S R with

M

times less voltage applied to one of the

f

F SR .

A tuning enhancement factor of at wavelengths near

is 120 GHz/mW and for the electro-optic device is 120 GHz/12V. A tunable laser with a side-mo de suppression ratio greater than 30 dB was demonstrated using this lter and erbium-dop ed b er amplier gain. Thermal tuning over 35 nm was achieved [27]. One of apparent applications of the optical lters is in optical delay lines. Optical delay devices based on chains of coupled WGRs have b een studied in [29, 30]. It was shown that the Q-factor of the coupling-split mo des for a system of

I:SS



M

m [27]. The tuning rate for the thermo-optic device

a RH

in a double-ring lter was achieved

N

identical coupled resonators is greater than that of a single resonator in

4


the chain by a factor of was discussed in [32].

N

, and even more, in the case of optimum coupling [31].

Stopping light all optically with a chain of interacting tunable optical resonators Another concept of coupled resonator optical waveguides was develop ed in [33, 34]. In particular, such waveguides can b e realized in chains of coupled WGRs [35]. A numerical simulation of light propagation in micro cylinder coupled resonator waveguides was rep orted in [36]. The simulations show that light propagates slower in WGMs with bigger than smaller azimuthal numb ers. The light propagation by WGMs of the same azimuthal numb er have the same sp eed regardless of the size and the material of the resonators. Generally, WGR tunable lters allow shifting the sp ectrum of the resonators, however they do not provide linewidth tuning capabilities. Cascaded resonators were prop osed to b e used for real-time shaping of their mo dal structures [37, 38]. A key feature of the approach is that it p oints to a simple tuning of the frequency and the width of the lter transmission window, resulting in the tuning of the group delay of optical signals, a highly desirable feature for signal pro cessing applications. The transmission sp ectral window of the lter could b e curiously narrow. Theoretically, in the case of resonators without absorption, the width of the window can b e arbitrarily narrow [37], however, in reality, the minimum width of the resonance is determined by the material absorption. The physical principle of the lter op eration that results in the narrow sp ectral window has b een recognized in [6, 17, 18]. The existence of the window also has b een demonstrated exp erimentally [39].

1.3

Перестраиваемый фильтры

Tunability is a highly desirable prop erty of any application of resonators. Though WGRs are solid state devices and their tunability is not readily conceivable, it can b e, in fact, achieved by several metho ds. Mechanical trimming of WGMs with applied strain [40, 41, 42] and temp erature tuning [43, 44] have b een previously used. Though the mechanical as well as temp erature tuning ranges are relatively large, e.g. on the order of a few to several tens of nanometers for thermal tuning, these metho ds are not very convenient for many applications b ecause of small tuning sp eeds and low tuning accuracy. The tuning accuracy is esp ecially imp ortant for high-Q resonators with narrow lter bandwidth. An all-optical tunable lter design based on "discontinuity-assisted ring resonators"that do es not have the ab ove mentioned disadvantages has also b een prop osed theoretically [45], but, to our knowledge, no exp erimental implementation of the conguration has b een rep orted. A technique for WGM resonance tuning was demonstrated using microring resonators with a photosensitive coating. In that study, glass microrings were dipp ed in a p olymer coating material and were exp osed to UV light. This metho d pro duced resonators with relatively small Q (ab out induced absorption; but it still allowed large tunability of the optical resonance of the microring, enough for wavelength selective applications [39].

VHH

) b ecause of the p olymer-

5


A metho d for the trimming of p olymer optical microresonators was prop osed in [46]. The metho d is based on photobleaching CLD-1 chromophores. A maximum wavelength shift of 8.73 nm was observed at 1.55 value of



m. The resonators had

a 3-dB bandwidth of 0.12 nm, a free sp ectral range of 1.11 nm, an intrinsic Q Another approach for trimming the frequency of microresonators exploits the photosensitivity of the germanate silica glass. When exp osed to UV light, this material undergo es a small p ermanent change in structure that alters its index of refraction. In the case of a WGR, the spatially uniform change in the index of refraction results in a uniform translation of the resonant frequencies. Such a tunable resonator as well as a second-order optical lter based on two coupled resonators, one of which was tunable, was exp erimentally realized for optical high-Q ( Recently, fabrication of optical WGM resonators with lithium niobate [49] has led to the demonstration of a high-Q microwave lter with a linewidth of ab out 10 MHz and the fast electro-optic tuning with tuning range in excess of 10 GHz [50]. The b est tunability for LiNbO3 single resonator lter was by applying DC voltage of

$ P ? IH4

, and a nesse of

$ IH.

IH

8

) WGMs [24, 47, 48].

The frequency shift of the TE and TM mo des in a WGR may b e found from the theory of electro-optic eect [51]. For LiNbO3 WGR lter discussed in [50] we nd

ЃSH
n2 e

V to an electro de placed over the resonator.

ЃPH

GHz

where and

r

!0

13

the refractive indices of LiNbO3 ,

?!T E a a P ? P ? IH a IH =
14

!

0

P

r33 EZ ;

?!T M a

!0

n

P

2

o

r13 EZ ; r
33

(4)

Hz is the carrier frequency of the laser,

pm V are the electro-optic constants,

EZ

n

e

is the amplitude of the electric eld applied

a P:PV

and

n

a QI o a P:P

pm=V are

along the cavity axis. TM mo des were used in the exp eriment rep orted in [50] b ecause they have larger quality factors than the TE mo des. If the quality factor is not very imp ortant, it is b etter to use the TE mo des b ecause their electrooptic shifts are three times as large as those of TM mo des for the same values of the applied voltage. Theoretically,

ties and are related to the fundamental limitations of optical resonator-based high sp eed electro-optic mo dulators [52]. For example, the domain reversal in a congruent LiNbO3 crystal o ccurs at relative WGR frequency shift

?!T

E and

?

!T

M do not dep end on the resonator prop er-

which is well ab ove the observed shifts.

?!T

E

EZ 9 =!0 9 :

PH I T ? IH

kV/mm, which corresp onds to
3

and

?!

TM

9 S ? IH 4

,

Lithium niobate lters are convenient, however their linewidth is restricted by a few MHz b ecause of the residual absorption of the material. Crystalline WGRs could p ossess many order of magnitude narrower lines. For instance, optical lters with bandwidths of ab out ten kiloHertz using CaF2 WGM resonators was demonstrated in [53]. The CaF2 resonators have stable ultrahigh Q-factors compared with fused silica resonators, where Q degrades with time. Limited tuning of the CaF2 lters can b e realized with temp erature. The insertion loss of the lter was at

S

dB level.

6


To characterize the absolute tunability of an optical resonator based photonic lter it is useful to intro duce the ratio of the resonator free sp ectral range (FSR) and linear tunability range given by the host material. Tuning the lters do es not change the FSR in the rst approximation, but only shifts the comb of the optical mo des making it overlap with itself for each frequency shift prop ortional to the FSR. Hence, the lter can b e tuned at any prescrib ed single frequency if the linear tunability exceeds the FSR. The lack of selectivity in a single-resonator lter can b e comp ensated with application of coupled-resonator lters. If each resonator in the lter can b e tuned by its FSR, the whole lter can b e tuned at any frequency, while the sp ectrum of the multi-resonator lter can b e very rare due to the Vernier eect. Some photonic applications call for narrowband lters passing simultaneously b oth the carrier and sidebands. For example, this is imp ortant for generation of sp ectrally pure microwave signals in opto-electronic oscillators [54], where b eating of the optical sidebands and the carrier on a fast photo dio de generates microwaves. Tunability of the microwave frequency of the oscillator requires that the frequency dierence b etween the lter passbands change controllably. This prop erty is lacking in existing tunable lters where the entire lter sp ectrum shifts as a whole as the tuning voltage is applied. A critical comp onent of a novel miniature lter with electro-optically recongurable sp ectrum was recently rep orted. The lter is based on a WGR fabricated from a commercially available lithium niobate wafer having a sp ecially engineered domain structure [55].

1.4

Фильтры на резонаторах ШГ для стабилизации оптоэлектронных осцилляторов и лазеров

1.4.1

Фильтры ШГ в ОЭО

Generation of sp ectrally pure signals at

nications, radar, and navigation. The advent of high throughput optical communications links p oints to the prosp ects for networks op erating at data rates as high as GHz. Schemes for realizing this typ e of capability rely on sources capable of providing high frequency, low phase noise signals, without which error-free high data rate system would not b e p ossible. Similarly, high p erformance radar systems require low phase noise oscillators to allow detection of feeble signals from a dense background clutter. The opto-electronic oscillator (OEO) is a device that pro duces sp ectrally pure signals at many tens of GHz based on photonic techniques, and thus overcomes some of the inherent limitations of the conventional electronic devices [54, 56, 57, 58, 59, 60, 61]. The OEO is a generic architecture consisting of a laser as the source of light energy. The laser radiation propagates through a mo dulator and an optical energy storage element, such as an optical b er, b efore it is converted to the electrical energy with a fast photo dio de. The RF electrical signal at the output of the photo dio de is amplied and ltered, and then fed

I

to

IHH

GHz is required in commu-

ITH

Gb/s and consisting of multiples of channels separated by a few

7


back into the mo dulator, closing the lo op. If the total gain exceeds linear loses of the lo op, the system oscillates at the frequency determined by the lter. The use of optical storage elements allows for the realization of extremely high Q's and thus sp ectrally pure signals in optical oscillators, since the noise p erformance of an oscillator is determined by the energy storage time, or quality factor Q. In particular, a long b er delay leads to realization of micro-second storage times, corresp onding to Q's of ab out a million at frequency. This is a high value compared to conventional dielectric microwave cavities used in oscillators [62, 63]. The b er delay line also provides for wideband frequency op eration unhindered by the usual degradation of the oscillator Q with increasing frequency. Thus, sp ectrally pure signals at frequencies as high as demonstrated. In a generic OEO [54] the long b er delay line supp orts many microwave mo des imp osed on an optical wave. A narrow band electrical lter should b e inserted into the electronic segment of the OEO feedback lo op to achieve a stable single mo de op eration. The center frequency of this lter determines the op erational frequency of the OEO. While this approach yields the desired sp ectrally pure high frequency signals, physical size of the OEO is rather bulky b ecause of the kilometers of b er delay needed. Moreover, the long b er delay is very sensitive to the surrounding environment so the OEO do es not pro duce an output with high long term frequency accuracy and stability. The OEO is typically phase lo cked to a stable reference for long term stability. The prop erties of the OEO with a high-Q WGR in place of the electronic lter as well as the b er delay was studied in [64]. It was shown that the metho d allows one to cho ose virtually an arbitrary frequency of oscillation by tuning the resonator.

IH

GHz oscillation

RQ

GHz, limited only by the mo dulator and detector bandwidth, have b een

1.4.2

Резонаторы ШГ для стабилизации лазеров

In addition to the stabilization of the OEO, WGRs can b e used for laser stabilization. Optical feedback from a high-Q microsphere resonator was used to narrow the sp ectrum of a miniature high-coherent dio de laser, and a nearly halfpitch gradient-index lens served as a coupling element [65]. As was estimated from the variation in frequency-tuning range (chirp-reduction factor) the fast line width of the laser was reduced by more than three orders of magnitude. A mo dication of external optical feedback that includes a WGR was used to narrow the line of a dio de laser [66]. A WGM of a high-Q microsphere was excited by means of frustrated total internal reection while the feedback for optical lo cking of the laser was provided by the intracavity Rayleigh backscattering. A b eat note of the two laser dio des optically lo cked to a pair of orthogonally p olarized mo des of the same microresonator had the indicated a sp ectral width of 20 kHz, and the stability of mo del for the laser stabilization with a WGR was presented in [67]. Finally, instead of lo cking a laser to a WGM, the opp osite was realized in [68]. A WGM of a fused-silica microsphere was lo cked to a frequency-scanning

P ? IH

6

over averaging times of 10 s. A theoretical

8


laser. The resonance frequency was mo dulated by axial compression of the microsphere, and phase-sensitive detection of the b er-coupled optical throughput was used for lo cking. Such a system is particularly useful in WGR-based chemical sensors the following section is devoted to.

1.5

Спектроскопия и исследование химических и биологических агентов

Starting from liquid WGRs used for resonator-enhanced sp ectroscopy (see [69] for review), solid state WGRs were utilize to enhance the interaction b etween light and atoms/molecules. One of the rst exp eriments on the sub ject was realized in the frame of cavity-QED [70]. The radiative coupling of free atoms to the external evanescent eld of a WGM was detected. The coupling manifested itself as a narrow absorption line observed in the resonator transmission sp ectrum. It was proven that the evanescent eld of the high Q ( ume

b etween single photon in the WGM and single atom in the surrounding atomic vap or. An ultrasensitive sp ectrometer based on a stretched silica microsphere was prop osed in Refs. [42, 71]. The next stage in the sensor development was related to WGR-based biosensors [72, 73, 74]. Optical biosensors are typically transducers that detect the presence of molecules at a surface. They have several desirable features, particularly for the detection of biological molecules, that include: (i) high sensitivity (less than nanomoles); (ii) non-destructivity to the sample; (iii) high selectivity; and (iv) applicability to various substances. The transduction pro cesses in optical biosensors generally take place on a surface and can b e tailored to sense almost any kind of molecule, chemical and pre-biotic as well as biological. WGR sensors b elong to the evanescent wave sensors, which are among the most sensitive class of biosensors [75, 76]. An evanescent wave pro duced by the total reection of light within the waveguide interacts with analytes on the waveguide surface in the evanescent eld sensors. The evanescent wave protrudes ab ove the waveguide surface by the relative index of refraction of the waveguide and the sample medium) and samples only the analyte on the surface. Surface treatments such as antib o dies or oligonucleotide strands can provide sp ecicity for the analyte; the sensor then detects only those b ound to the surface. Transduction mechanisms for b ound analyte include uorescence, mass change in the evanescent region [77], and change in the index of refraction [78]. Typical sensitivity of evanescent wave biosensors based on b er optic sensors or planar waveguide sensors is in the range of nM to pM. The basic detection scheme that utilizes WGRs is that binding of molecules to the micro-resonator surface induces an optical change prop ortional to the amount of b ound molecules. The paradigm for this pro cess is a change in the cavity Q as the surface b ound molecules aect the photon storage time either through increased scattering or absorption. In eect, the analyte sp oils the Q and that change can b e measured.

(IH

8

cm ) fused silica microsphere enables velo city-selective interactions

3

S ? IH

7

) and small mo de vol-

$ IHH

nm (the actual distance dep ends on

9


Any protein will adhere to glass surface of a generic WGR, and hence rep olished spheres are entirely nonsp ecic. Two conditions must b e met for chemical mo dication of the microsphere surface: rst, the glass must b e coated with a comp ound that will minimize nonsp ecic binding. Second, an antib o dy or other protein with sensitivity to a particular ligand must b e linked to the sphere in such a way that b oth the protein's functionality and the sphere's Q are preserved. A thin lm of a material with thickness smaller than the WGM's evanescent eld will not signicantly alter the Q of the micro-resonator, thus, a thickness of its high Q.

$ IH IHH

nm can b e applied to the microsphere while retaining

A p ossibility of enhancement of the detection sensitivity of evanescent-wave optical biosensors was discussed in [79, 80, 81, 82, 83, 84]. It was shown that the resonant coupling of p ower into the WGR allows for ecient use of the long photon lifetimes of the high-Q WGMs to increase the interaction of the light and the particles under the study. This enhancement results in stronger uorescence and in changes of the resonator parameters. A sp ectroscopic technique for high-sensitivity, lab el-free DNA quantication were develop ed in [85]. It was demonstrated that a WGM excited in a micronsized silica sphere can b e used to detect and measure nucleic acids. The surface of the silica sphere is to b e chemically mo died with oligonucleotides. A rst-order p erturbation theory was develop ed for WGMs in a dielectric microsphere [86, 87]. The theory were applied to three sensor applications of the microsphere to prob e the medium in which the sphere is immersed: a refractiveindex detector, an adsorption sensor, and a refractive-index prole sensor. Biosensors based on the shift of WGMs in microspheres accompanying protein adsorption were describ ed by use of a p erturbation theory in [86]. For random spatial adsorption, theory predicts that the shift should b e inversely prop ortional to microsphere radius sity and excess p olarizability. Hybrid zinc oxide/silica micro disk lasers were utilized to sense volatile organic comp ounds, such as toluene and nitrob enzene [88]. Nonsp ecic adsorption of these organic molecules onto the WGR surface causes an increase in the disk refractive index, ultimately resulting in a red shift of the observed lasing wavelengths. Improvement of photonic WGM sensors using the Fano-resonant line shap e was prop osed in [89]. Polystyrene microring resonators were fabricated by the nanoimprinting technique, and the optical sp ectra were measured in glucose solutions of dierent concentrations. The shift in resonant wavelength and variation of the normalized transmitted intensity were linearly related to the concentration of the glucose solution. Application of WGRs in high-eld high-frequency electron magnetic resonance measurements was discussed in [90].

a

and prop ortional to protein surface den-

10


1.6

Механические сенсоры

High-Q WGMs result in increase of sensitivity of various mechanical exp eriments. For instance, WGMs could b e used for the measurement of strain in optical b ers [40]. A two-resonator sensor of small displacements that utilizes high Q and mechanical tunability of normal mo des in coupled optical WGRs was prop osed in [91]. An accelerometer utilizing high-Q WGRs was presented in [92]. Induced console displacements were monitored through changes in the resonance characteristics of a spherical optical cavity coupled to the exure. Instantaneous measurement sensitivity of b etter than o or of

The idea of usage of passive and active optical ring interferometers for detection of rotation was develop ed and implemented a couple o decades ago [93, 94, 95]. A miniature integrated WGM optical sensors for gyroscop e systems was recently prop osed [96]. It was predicted, that the sensor may p ossess high enough sensitivity even on a millimeter size scale. A passive WGM gyroscop e was discussed in [97]. The basic dierence of the gyroscop e compared with the existing ring resonator gyroscop es is in the usage of crystalline WGR instead of the usual ring resonator. The WGR-based gyroscop e is exp ected to have much less backscattering and p olarization rotation noises compared with conventional b er-based gyroscop es.

IHH



g were achieved.

I

mg at

PSH

Hz bandwidth, and a noise

1.7

Фундаментальная физика с пассивными резонаторами ШГ

WGRs oer interesting p ossibilities in b oth classical as well as quantum p oints of view. High Q-factors as well as small mo de volumes of WGMs result in a multitude of interesting and imp ortant phenomena. In this sections we discuss those phenomena related to "passive"WGMs, which do not lead to generation of light, leaving fundamental prop erties of WGM lasers and other active devices to another section.

1.7.1

Хаос

One of the fundamental problems is related to WGMs in an asymmetric WGR. It was shown, that departure from an axial symmetry results in the o ccurrence of chaotic b ehavior of light in the resonator. This has b een predicted to give rise to a universal, frequency-indep endent broadening of the WGRs and to highly anisotropic emission [98, 99, 100, 101, 102]. A solution of the problem which conrms these predictions but also reveals frequency-dep endent eects characteristic of quantum chaos was presented in [103]. It was shown that for small WGR deformations the lifetime is controlled by evanescent leakage, the optical analogue of quantum tunneling [104]. The problem of the directional emission from egg-shap ed asymmetric resonant cavities was discussed theoretically in [105].

11


The lifetime of light conned in a WGR can b e signicantly shortened by a pro cess known as "chaos-assisted tunneling"[106]. Surprisingly, even for large deformations some resonances were found to have longer lifetimes than predicted by the ray chaos mo del due to the phenomenon of "dynamical lo calization"[107]. Mo des of partially-stable WGRs were discussed in [108] using a theory, where exp onentially suppressed tunneling interaction b etween regular and chaotic mo des was considered as a p erturbation. It was shown that chaos-assisted tunneling can lead to splitting of regular WGMs in asymmetric optical resonances. A theory of inuence of the chaos-assisted tunneling on lifetimes and emission patterns of the optical mo des in generic microresonators was develop ed in [109] using approach presented in [110]. The rst exp eriment on chaos-assisted tunnelling in a two-dimensional annular billiard was rep orted in [111]. Highly directional emission from WGMs was demonstrated in deformed nonaxisymmetric fused-silica "microspheres"[112].

1.7.2

"Фотонные атомы"

Another fundamental area of application of WGRs is based on the ability of the resonators to mimic atomic prop erties. It was shown that WGMs can b e thought of as classical analogy of atomic orbitals [113]. It was p ointed out that WGM mo de numb ers corresp ond to angular, radial and the azimuthal quantum numb ers, resp ectively, the same as in the atomic physics. Such an approach resulted in intro ducing the term "photonic atoms"with resp ect to WGM resonators [114, 115]. "Photonic molecules", based on coupled WGRs, was studied in [116, 117].

1.7.3

Квантовая электродинамика резонаторов

There is a great activity in b oth theoretical and exp erimental investigations of cavity quantum electro dynamics [118, 119, 120, 121, 122] eects in WGRs. For instance, sp ontaneous emission pro cesses may b e either enhanced or inhibited in a cavity due to a mo dication of the density of electromagnetic states compared with the density in a free space [123, 124]. This eect was studied theoretically [125, 126, 127] as well as exp erimentally [128, 129] in WGRs. Metho ds for control of atomic quantum state in atoms coupled to singlemo de and multi-mo de cavities and microspheres were discussed in [130]. Those metho ds include excitation, decay control, lo cation-dep endent control of interference of decay channels, and decoherence control by "conditionally interfering parallel evolutions". Prop erties of atomic interaction with the eld of a high-Q cavity was studied in [131] using "pseudomo de"theory. It was shown that the theory can b e derived by applying the Fano diagonalization metho d to a system in which the atomic transitions are coupled to a discrete set of cavity "quasimo des". The cavity mo des decay into a continuum set of external "quasimo des". It was shown that each "pseudomo de"can b e identied with a discrete "quasimo de", which contains structure to the actual reservoir.

12


Ponderomotive interaction of an atom and a WGM was discussed in [132, 133, 134], see also [135, 136] for review. In particular, it was shown that the external elds of optical WGMs may b e used to conne atoms in stable orbits around a dielectric microsphere [132]. The b ound state structure and dynamics for the atom trap were investigated in [133]. The dynamics of the center-of-mass of an ultracold excited atomic oscillator in the vicinity of a dielectric microsphere was studied in [134]. The p onderomotive interaction of an atom and photons conned in a WGM can b e used for quantum nondemolition measurements. It was shown [137, 138, 139] that the dip ole force exp erienced by an atom in an o-resonant spatially inhomogeneous light eld is quantized by the discrete nature of the photon. Similar schemes to p erform quantum nondemolition detection of optical photons by observing the deection of a b eam of atoms ying close to an op en dielectric resonator were prop osed in the studies. The p onderomotive interaction of an electron, instead of an atom, and photons in a WGM was prop osed for a quantum nondemolition measurement of photon numb er (the photon numb er is dened as energy stored in the mo de divided by on the eect of quadratic scattering of electrons traveling along the resonator with a velo city close to the phase velo city of the wave in the resonator [140]. The measurement idea relies on the fact that an electron traveling along a bare dielectric waveguide (or surface of a WGR) at a velo city near the phase velo city, acquires a transverse momentum prop ortional to the photon energy of the light in the waveguide. It was noted that this momentum can b e measured [141]. The scattering eect was analyzed with consideration for the waveguide (and WGM) disp ersion, radiation friction and the spurious Cherenkov radiation. A radiative coupling of a nanoparticle/atom with a WGM was studied in [70, 142], see also [122, 143] for a review. The p ossibility of strong coupling b etween a photon conned in a WGM and an atom was analyzed in [144, 145, 146]. The resonant interaction of an atom with dip olar

"

h!0

, where

!0

is the frequency of the mo de). The technique is based

J

transition with the quantized eld in a dielectric spheres and spheroids was studied in [144]. The p ossibility of application of a micro disk WGR for the detection of a single trapp ed atom was studied in [147]. A measurements of cavity-QED eects for the radiative coupling of atoms in a dilute vap or to the external evanescent eld of a WGM was rep orted in [70]. Exp eriments on coupling of a single nano-emitter and WGMs were discussed in [142, 148]. A comp osite system consisting of a GaAs quantum well structure placed in the evanescent eld of a fused silica microsphere and evanescent coupling b etween excitons in the quantum well and WGMs of the comp osite system was demonstrated in [149]. A comp osite system consisting of CdSeZnS nano crystals and a fused-silica microsphere was demonstrated in [150]. The Q-factors of the system were of the order of the level of single quantum dots. Optical prop erties of conned photon states in an extremely small spher-

aH6J aI

angular-momentum

IH

8

, providing a mo del for investigating cavity QED and microlasers at

13


ical WGRs with sizes of

nantly excited by photons emitted from semiconductor nano crystals (the quantum dots) was studied in [151, 152], with particular fo cus on QED prop erties of WGRs containing CdSe quantum dots and quantum ro ds. Both glass and p olymer WGRs were characterized by spatially and temp orally resolved microphotoluminescence.

P


IH

(dubb ed as "photonic dots") reso-

2

Резонаторы ШГ с активными модами

Small volumes and high Q-factors of WGMs result in enhancement of nonlinear optical pro cesses. Due to this enhancement WGR based nonlinear-optic devices p ossess unique characteristics. For example, usage of WGMs allows realizing lasers and wave-mixing devices with micro-Watt thresholds. Narrow linewidth of WGMs results in narrow sp ectral characteristics of the lasers. In this section we review results of recent studies in the eld.

2.1

Непрерывная лазерная генерация на модах ШГ

Miniature lasers are among the most obvious applications of WGRs. High quality factor of the resonators leads to the reduced threshold of the lasing. The rst WGM lasers were realized in solid materials [153, 154, 155]. However, probably b ecause of the lack of input-output techniques for WGMs, the work was discontinued at that p oint. Next development of the WGM-based lasers was in liquid aerosols and individual liquid droplets [156, 157, 158, 159, 160, 161]. Finally, during the last decade the lasers based on a sole solid state WGRs were rediscovered, demonstrated exp erimentally, and intensively studied. In this sections we review recent results with WGR continuous wave lasers, leaving WGR Raman lasers for the following section.

2.1.1

Лазерная генерация в капилярах

WGR laser can b e realized in a cylindrical resonator. The simplest resonator of this kind is a capillary. The gain medium could reside inside the capillary, where WGMs are lo calized. For instance, laser emission from WGMs in a highly refractive dye-dop ed solvent owing in a normally illuminated silica capillary b er was demonstrated in [162]. The cylindrical WGM laser diers from the spherical droplet laser [156] in that it has an internal refractive index discontinuity. The light p enetrates into the active medium if the refractive index of the medium is higher than the one of the capillary material; e.g. no laser p eaks are observed when the refractive index of the solvent is less than that of silica [162]. An example of microring lasing using CdSe nano crystal quantum dots incorp orated into micro capillary tub es was demonstrated in [163]. The lasing in a capillary based on the evanescent eld coupling with the gain medium is also p ossible. The layered micro cavity was realized in [164, 165] by owing dye-dop ed ethanol through a thin-wall fused silica capillary tub e whose

14


refractive index was larger than that of the liquid. Lasing sp ectrum showed a strong mo de selection, and even nearly single constructive interference p eaks, due to the interferential coupling of WGMs at the inner b oundary. Various mo de orders, which are not allowed in the ray optics picture, were made to oscillate due to the evanescent propagation of WGMs at the outer b oundary. The estimated cavity quality factors were higher than of resonance mo des in a thin dye-dop ed dielectric ring cavity made on the inner wall of a cylindrical capillary were also studied in [166]. A WGM laser with pulsed optical pumping fabricated by surrounding a small section of a glass capillary with a solution of Rho damine 6G and by coupling the pump light into the capillary wall was demonstrated in [167]. The lasing threshold pump energy was 100 nJ/pulse at a pump pulse duration of 6 ns.

IH

6

. The lasing characteristics

2.1.2

Лазерная генерация в допированных резонаторов ШГ

Another way to create a WGR laser is in use of solids dop ed with active elements, e.g. rare earth ions, as a WGR host materials. A WGM laser based on neo dymium-dop ed silica microspheres with 200 nW threshold was realized [168] with microspheres of radius

a

formed by heat-fusing the tip of a length of dop ed silica b er. Neo dymium ions provide a favorable four-level laser system that can b e pump ed on the
4 4

$ PS SH



m,

I9=2 F3=2

4 4

F5=2 I11=2

transition at

in the 1.06 1.09

a lower level that is depleted by strong phonon relaxation so that p opulation inversion is easily achieved. Similar exp eriments with neo dymium-dop ed silica microsphere laser op erating at 2K and absorbing 200 nW pump p ower were rep orted in [169]. Continuous-wave laser oscillation on b oth the and
4 4

!

$ VIH


nm with a dio de laser. The laser transition

m range connects a long lived upp er level to

F3=

2

3

4

I13=

2 transitions of

Nd

3+

F

3=2

3
I

4

I

11=2

ions in uoride glass WGRs was also
4

achieved [170]. Fabrication of Nd-dop ed tellurite glass WGRs and observations of laser oscillation corresp onding to the optical transition 1.06



F

3=2

3

4

11=2 at

m was rep orted in [171].

A WGM laser utilizing a microsphere made of highly dop ed erbium:ytterbium phosphate glass was used to generate light at pump p ower of

single-mo de op eration were obtained. A bisphere laser system consisting of two microspheres attached to a single b er tap er was also demonstrated. Er
3+

TH

W and b er-coupled output p ower as high as

I:S



m [172]. Laser threshold

Q



W with

-dop ed tellurite glass L-band WGR laser was demonstrated and dis-

cussed in Ref. [173]. The microspheres were made by a "spin metho d". Fib er tap ers were utilized to couple 975 nm pump into the sphere and couple generated light (1.56 1.61 the tellurite glass was

!



A green ro om temp erature up-conversion laser was demonstrated in a

I:U ? IH
Er
3+

m) out of the sphere. The erbium ion concentration of
20

ions/cm .

3

absorb ed pump p ower.

SRH

diameter microsphere of nm with a

VHI

dop ed ZBLAN [174, 175]. Lasing o ccurred around W of

nm dio de laser pump. The lasing threshold was

IPH QH



m

Exp erimental results on the realization and sp ectral characterization of Er:ZBLAN

15


microspherical lasers at threshold as low as

obtained with the external

Green lasing having 4 mW threshold was demonstrated in an erbium-iondop ed uoro-zirconate glass WGR [178]. Perio dic narrow p eaks of the emission sp ectra corresp onding to the WGRs were observed. An erbium-dop ed microlaser on silicon, op erating at wavelength of and characterized with pump threshold as low as [179]. The

THH



I:ST I:RV

m were presented in [176, 177]. The lasing was



m pumping. Multimo de op eration and a laser

W were observed.

of erbium ion implantation, photolithography, wet and dry etching, and laser annealing, using a thermally grown SiO2 lm on a Si substrate as a starting material. Single mo de lasing was observed. Another erbium-dop ed high-Q silica toroidal WGR microlaser (25-80

RH



m diameter toroidal laser WGR was made using a combination

R:S



W was demonstrated in

I:S



m,



m in

diameter) was demonstrated in [180]. The WGR was coupled with a tap ered optical b er. Erbium ion concentrations were in the range 0.009-0.09 at. %. Threshold pump p ower was as low as 4.5 Tm
3+



W.

-dop ed tellurite glass WGR laser was discussed in [181]. The laser,

pump ed at 800 nm with a tap ered optical b er, oscillates in b oth the S band and the 1.9 of the
3 3 3

F4

3

H4 H6

3



m band. The p eak at 1.5
3

F4



m (S band) corresp onds to emission

transition, while the p eak at 1.9



m corresp onds to the

transition.

Numerical analysis of a micro disk laser, taking into account full gain saturation eect and the vector character of the eld was presented in [182]. The authors suggested that Nd:YAG micro disk lasers are the excellent candidates for a light source for optical b er communications op erating at 1.064 and 1.3 was rep orted in [183].



m.

Theoretical study of the inuence of deformation of a WGR on lasing prop erties

2.1.3

Лазерная генерация в резонаторах ШГ с активным покрытием

Instead of use of dop ed materials a passive WGR can b e coated with gain medium. For example, erbium-dop ed solgel lms were applied to the surface of silica microspheres to create low-threshold WGR lasers [184]. Lasing action in an ultra-high-Q spherical WGR coated with gain medium was rep orted in [185]. Lasing in a square cavity with round corners coated with p oly methyl methacrylate and with Rho damine 6G molecules was studied in [186]. A thin gain layer was coated only on the outer b oundary of cavity. The thickness of the gain layer varied from single



m to several



m.

Ultraviolet micro disk lasers on silicon substrate with a layer of zinc oxide gain medium grown on top of the silica micro disks were demonstrated in [187]. Lasing o ccurs in the WGRs at ro om temp erature. The hybrid ZnO/SiO2 WGR was optically pump ed by the third harmonics (355 nm) of a mo de-lo cked Nd:YAG laser with at lens b oth fo cused pump on the resonator and collected the ultraviolet emission

$ IH

Hz rep etition rate and 20 ps pulse width. A microscop e ob jective

$

390 nm.

16


WGM-enhanced inelastic emission from a monolayer of A488 uorophores on the surface of a 9.8



m WGR (p olystyrene b ead trapp ed in an optical trap)

was observed and rep orted [188]. It was p ointed out that it was likely that the WGM-enhanced emission is due to A488 lasing, with a lasing threshold b etween 0.29 W cm



2

and 0.87 W cm



2

.

2.1.4

Резонаторы ШГ с полупроводниковой усиливающей средой

WGM-based lasers can b e created with semiconductor quantum dots coupled to the WGMs. One of the most imp ortant problems here is fabrication of a single quantum dot microlaser. Such a microlaser made by capturing the light emitted from a single InAs/GaAs quantum dot in the WGM of a glass microsphere was prop osed theoretically in [189]. Master equation mo del of a single quantum dot microsphere laser was describ ed in [190]. The op eration of a single quantumdot-microsphere laser and a semiconductor microsphere bistable element was theoretically studied in [191]. A quantum dot-micro cavity system consisting of CdTe nano crystals attached to a melamine formaldehyde latex microsphere was realized exp erimentally [192]. The high optical transparency, thermal and mechanical stability of melamine formaldehyde, make it interesting as a p otential candidate in optical applications. The refractive index of melamine formaldehyde in the visible region ( (

n n

shell of CdTe nano crystals were studied in order to examine the emission intensity as a function of excitation p ower. Ultralow-threshold (the pump was less than

% I:S).

a I:TV

) is greater than that of silica (n

Photoluminescence sp ectra of the microspheres covered by a thin

a I:RU P

) or other glass materials

W) continuous-wave lasing

was achieved at ro om temp erature in a fused-silica microsphere that was coated with HgTe quantum dots (colloidal nanoparticles) [193]. WGRs can signicantly improve op eration of semiconductor quantum well lasers. A microlaser design based on the high-reectivity WGMs around the edge of a thin semiconductor micro disk was describ ed and initial exp erimental results were presented in [194]. It was shown that optically pump ed InGaAs quantum wells provide sucient gain when co oled with liquid nitrogen to obtain singlemo de lasing at

IHH



W.

I:Q

and

I:S



m wavelengths with threshold pump p owers b elow

A realization of an InGaAs/InGaAsP ro om temp erature quantum well disk

laser using [196].

coupling of light output from and to WGR micro disk lasers were describ ed in Optically pump ed, pulsed GaN micro disk laser op erating at ro om temp erature was created [197]. WGMs of the disk had linewidth as narrow as 0.1 nm. WGRs with diameters covering the range 25 750 ing was p erformed p erp endicular to the disk plane by the third harmonic (355 nm) or the fourth harmonic (266 nm) of a Q-switched Nd:YAG laser. The output light emission from these structures was collected by a reecting ob jective

I:T H:VS

m in diameter and



m optical pumping, was rep orted in [195]. Metho ds for directional

H:IV



m in thickness, op erating at

I:SRP



m and

!



m were tested. Optical pump-

17


lo cated 80

o from the surface normal,

Quantum-cascade WGM disk lasers emitting at 9.5 (



m and 11.5



m wave-

length were rep orted in [198]. Taking advantage of the high-quality resonator

Q

reduced b elow the value of the corresp onding ridge waveguide geometry. A "microgear"laser comp osed of a micro disk and a rotationally symmetric Bragg grating was describ ed in [199]. A GaInAsP-InP device with micron size was fabricated and the ro om-temp erature continuous-wave op eration was obtained by Optically pump ed micro disc GaN-based laser was demonstrated in [200]. The optically pump ed WGRs had distinct mo des at excitation p owers ranging from ab out 8 to 16 W cm

$ PHH),

the threshold current density of disk lasers emitting at 9.5



m was

IU



W pumping.



2

. Quality factors for the micro disks were of the order

of 4600. The observed lasing threshold was 12.1 W cm



2

.

2.2

Рассеяние, модифицированное резонатором

There are at least three scattering pro cesses playing signicant role in WGRs. Those are Brillouin, Raileigh, and Raman scattering.

2.2.1

Brillouin scattering

Stimulated Brillouin Scattering (SBS) was demonstrated in liquid droplets [201, 202, 203, 204, 205, 206, 207, 208, 209, 210], though no SBS in high-Q solid WGRs was registered b ecause of selection rules [207].

2.2.2

Rayleigh scattering

Rayleigh scattering leads to the limitation of the Q factor of WGMs as well as to the inter-mo de coupling. The scattering is largely suppressed in high-Q WGRs b ecause of restrictions imp osed on scattering angles by cavity connement, so very high Q WGMs are feasible [211]. The scattering, on the other hand, couples initially degenerate counterpropagating mo des in the WGRs and creates the intracavity feedback mechanism instrumental for the laser frequency lo cking application [66]. Rayleigh scattering mediated intracavity backscattering reaches 100%, as was shown theoretically [211] and demonstrated exp erimentally [212]. In the frequency domain, intracavity backscattering is observed as the splitting of initially degenerate WGM resonances and the o ccurrence of characteristic mo de doublets [213, 214]. Inuence of Rayleigh scattering on Q factors of high refractive index contrast WGRs fabricated from silicon-on-insulator wafers was studied using an external silica b er tap er waveguide [215, 216].

2.2.3

Рамановское рассеяние

Substantial optical p ower enhancement within a high-nesse optical cavity has recently yielded continuous wave Raman lasers with low threshold and large tunability (see, e.g., [217, 218]). Such prop erties make cavity-enhanced cw Raman lasers attractive for high resolution sp ectroscopy, remote sensing, atomic

18


physics, and telecommunications. Reducing the cavity size may further improve the p erformance of the lasers. Op en dielectric spherical micro cavities are promising for those purp oses. An enhancement of stimulated Raman scattering (SRS) is one of the eects demonstrated in spherical micro cavities. Low threshold SRS was observed with pulsed [205, 219, 220, 221, 222, 223, 224, 225] and continuous wave [226, 227] optical pumping in micrometer-size liquid droplets. Theoretical description of the pro cess was presented in [228, 229, 230, 231]. SRS was investigated in a liquid parahydrogen droplet characterized with WGM having Q-factor exceeding vibrational transition but also for rotational transition as in the gas-phase range. SRS in ultrahigh-Q surface-tension-induced spherical and chip-based toroid micro cavities is considered b oth theoretically and exp erimentally in [233]. These fused silica WGRs exhibit small mo de volume (typically whisp ering-gallery typ e mo des with long photon storage times (in the range of 100 ns), signicantly reducing the threshold for stimulated nonlinear optical phenomena. The studies of Raman gain in isolated high-Q WGRs are imp ortant to understand cavity QED prop erties of Raman lasing. Previously, micro cavity QED enhancement of Raman gain has b een inferred as the result of measurements of a dep endence of the SRS threshold on the size and material of the microdroplets, and its comparison with the values of SRS threshold rep orted for liquid core b ers having equivalent interaction length and core comp osition [226, 227]. This enhancement has b een linked to the cavity mo dication of the prop erties of a usual laser. A theory of the Raman gain mo dication that explains the exp erimental results was develop ed [234, 235]. Recent exp eriments with silica microspheres have not shown any signicant change in SRS gain which might b e attributed to quantum eects [233, 236]. This issue was addressed in [237], where it was shown that no cavity QED asso ciated Raman gain enhancement exists, unlike to the cavity enhancement of the sp ontaneous emission.

IH

9

[232]. The SRS was registered not only for

H2

system, leading to multiorder SRS sidebands covering the whole visible sp ectral

IH

3



m ) and p ossess

3

2.3

Переключатели и модуляторы

2.3.1

Переключатели на резонаторах ШГ

WGRs can b e used as ecient and compact optical switches and mo dulators. Nonlinear optical switches based on WGMs are primarily considered in relation with their applications to all-optical computing. A p ossibility of such switching and applications of WGR to create a quantum-mechanical computer was rst recognized in [238]. The ma jority of studies of optical switches that utilize WGMs are theoretical. It was shown theoretically that WGR micro disk lasers are stable and switch reliably [239], and hence, are suitable as switching elements in all-optical networks.

19


An integrated all-optical switch based on a high-Q nonlinear cylindrical micro cavity resonator was prop osed [240]. The switch consists of two planar waveguides coupled to a WGR. It was argued that due to the high Q factor and the small dimensions, fast switching at low p ower is feasible for the devices based on presently available nonlinear p olymers as the active material. A general electro dynamical theory of a high-Q optical microsphere resonator in an external alternating magnetic eld was rep orted in [241]. It was shown that that such a system can change a p olarization state of the WGM photons conned in the sphere due to the Faraday eect. This prop erty was prop osed to b e used in all-optical switches and logical devices. Numerical evaluation of an optical resp onse of a prism-coupled nonlinear microsphere was discussed in [242]. The numerical results have shown that the control and/or the signal lights can induce the optical switching-like variation in the light reectance. This eect was interpreted by the variation in the dielectric constant of the sphere due to its Kerr nonlinearity. Coupled WGRs p ossess dierent, frequently more advanced, prop erties compared with a single WGR. Sequences of optical microresonators can b e used to construct integrated structures that display slow group velo city of light, ultrahigh or low disp ersion of controllable sign, enhanced self-phase mo dulation, and nonlinear optical switching [243]. It was p ointed out that there should b e a reduction in switching threshold for nonlinear optical devices incorp orating b er ring resonators [244, 245]. The circulating p ower in such WGRs is much larger than the incident p ower as well as the phase of the transmitted light varies rapidly with the single-pass phase shift. It was shown that the combined action of these eects leads to a nesse-squared reduction in the switching threshold [244], allowing for photonic switching devices that op erate at milliwatt p ower levels in ordinary optical b ers. A set of coupled dierential equations that describ e Kerr nonlinear optical pulse propagation and optical switching in systems coupled by a few microresonators was derived in [246]. Gap-soliton switching in a system comp osed of two channel waveguides coupled by microresonators was studied in [247]. A numerical demonstration of the feasibility of constructing an all-optical "AND"gate by using a microresonator structure with Kerr nonlinearity was presented in [248]. It was shown that the gate can b e much smaller than similar "AND"gates based on Bragg gratings and has lower p ower requirements. There are a few exp erimental studies of all-optical switches that utilize WGMs. For instance, laser-induced mo dication of cavity Q's was achieved in a micro droplet containing a saturable absorb er [249]. The elastic-scattering sp ectra from such droplets for higher incident intensities shown that cavity Q's are increased when the absorption is bleached. The lasing sp ectra from a droplet containing a saturable absorb er and laser dye were mo died when an intense bleaching eld was injected into the droplet cavity after the pump eld has initiated the lasing. All-optical nonlinear switching in compact GaAs-AlGaAs microring resonators at the 1.55-mum wavelength was demonstrated in [250]. Switching was accomplished in the pump/prob e conguration in which the pump/prob e signals were

20


tuned to dierent resonance wavelengths of the microring. Refractive index change in the microring due to free carriers generated by two photon absorption was used to switch the prob e b eam in and out of resonance. An all-optical switching technique utilizing a silica microsphere optical resonator coated by a conjugated p olymer was develop ed in [43]. A 250-mum diameter silica microsphere was coated by dipping into a toluene solution of the p olymer. WGM resonant frequency shifts as large as 3.2 GHz were observed when 405 nm pump light with a p ower density of the order of 10 W/cm
2

was

incident on the microsphere. The time constant of the observed frequency shifts was approximately 0.165 s, leading us to attribute the frequency shift to thermooptic eects. Such a system is capable of switching the WGM resonant frequency having 2 MHz linewidth at sp eeds on the order of 100 ms. Finally, optical memory elements were develop ed using WGM devices. A memory element constructed by interconnecting WGM microscopic lasers was demonstrated in [251]. The device switches within 20 ps with 5.5 fJ optical switching energy. On the other hand, it was shown theoretically and demonstrated exp erimentally that a random distribution of spherical microparticles may b e used as a sp ectral hole burning memory [114, 115].

2.3.2

Модуляторы на резонаторах ШГ

Microwave cellular phone systems and p ersonal data assistant networks require devices capable of receiving, transforming and pro cessing signals in millimeter wavelength domain [252]. Electro-optic mo dulators based on electromagnetic wave interaction in nonlinear optical cavities with high-Q WGMs will play an enabling role for these and similar applications. An approach to create coupling b etween light and microwave eld in a WGR was recently prop osed [72, 73]. In that study, an ecient resonant interaction of several optical WGMs and a microwave mo de was achieved by engineering the shap e of a microwave resonator coupled to a micro-toroidal optical cavity. Based on this interaction a new kind of electro-optic mo dulator as well as photonic microwave receiver was suggested and realized [253, 254, 255, 256, 257, 258, 259, 260].

2.4

Опто-электронные генераторы

Besides the sources of coherent optical radiation, i.e. lasers, optical WGRs can b e used in sources of coherent microwave radiation. An opto-electronic oscillator (OEO) is an example of such a source. An OEO pro duces microwave signals using photonic techniques [54, 56, 57, 58, 59, 60, 61]. The mo dulator is one of the main sources of p ower consumption in the OEO b ecause of the large p ower required to drive the conventional mo dulators. Both broadband Mach-Zender mo dulators and free space microwave cavity-assisted narrow-band mo dulators typically require one to a few Watts of microwave p ower to achieve a signicant mo dulation. This means that either the photo current in OEO the system should

21


b e amplied signicantly, or p owerful laser should b e used as the source of the drive p ower for the OEO. OEO based on WGM resonant mo dulator was recently prop osed and fabricated [261]. The device is characterized by low threshold and low p ower consumption. The disadvantages of the device are low saturation and low output p ower, and a p ossibility of transforming the noise of the light eld into the microwave signal. In general, resonant and conventional OEO's have nonoverlapping characteristics and are b oth useful dep ending on the application.

2.5

Распространение и генерация короткиз импульсов

It is convenient to distinguish b etween two regimes of optical pulse propagation in a WGR: i) the pulse duration exceeds the inverse of the free sp ectral range (FSR) of the cavity, and ii) the pulse duration is shorter than the inverse cavity FSR. Studies presented in [262, 263, 264, 265] are primarily fo cused on the rst regime. Sp ecically, the transient b ehavior of light intensity inside a dielectric sphere excited by a light pulse was discussed in [262, 263]. Long optical pulses were used for pumping of p olymer microlasers [264]. Linear and nonlinear optical prop erties of a waveguide coupled WGRs has also b een studied theoretically [265]. The second case, propagation of short pulses in WGRs, was also examined [266, 267, 268, 269], and a general theoretical analysis of the propagation was presented in [266]. Time resolved measurements of picosecond optical pulses propagating in dielectric spheres [267] and subpicosecond terahertz pulse propagation in a dielectric cylinder [268, 270] were recently rep orted, and micro cavity internal elds created by picosecond pulses was discussed theoretically [269]. The b ehavior of ultra short light pulses coupled into the resonant mo des of spherical micro cavities was explored in [271]. A noninvasive pulse-tracking technique was exploited to observe the time-resolved motion of an ultrashort light pulse within an integrated optical microresonator [272]. The minimum pulse width as well as the p erio d of the optical pulse train generated by a system that involves a high-Q cavity is determined by the resonator disp ersion. Dep ending on the dielectric host material and the geometric size, a WGR may p ossess either a p ositive, or a negative, or a zero group velo city disp ersion (GVD) [273]. This disp ersion is imp ortant when the pulse duration is shorter than the inverse cavity FSR. Resonators p ossessing a p ositive group velo city disp ersion may b e used for GVD comp ensation in optical b er links. Negative GVD cavities with Kerr nonlinearity (e.g. fused silica cavities) sustain nonlinear Schro dinger soliton propagation and may b e used for pulse shaping and soliton shortening in conventional mo de-lo cked lasers (see, e.g. [274, 275, 276]). Zero GVD cavities may b e used as high-nesse etalons to stabilize actively mo de-lo cked lasers (as in [277]). Integrated optical WGM allpass lters can also b e used for tunable disp ersion comp ensation in the optical transmission line if the pulse duration exceeds the inverse of the FSR of the resonator [243, 278]. Small resonators, like WGRs, are imp ortant for the stable generation of optical pulses with high rep etition rates. This is conrmed by the exp eriments

22


with planar, not WGM, small resonators. For example, rep etition rate were obtained for a laser [279];

laser cavity of total length approximately

RPH

GHz subharmonic synchronous mo de lo cking was realized in a

P:S

mm-long actively mo de-lo cked monolithic

P

ps pulses at a

IT:Q

GHz

noise suppression was demonstrated by inserting a small high-nesse FabryPerot resonator to the cavity of an actively mo de-lo cked laser [277, 281]. It was prop osed to use WGRs to generate short optical pulses [273, 282]. The idea of this laser is based on two recently realized WGM devices: the electro-optic mo dulator and the erbium-dop ed microsphere glass laser [72, 172, 175, 178, 184]. It is also known that an electro-optic mo dulator placed in an optical resonator can generate a frequency comb [283, 284, 285, 286], and that the output of such a device is similar to that of a mo de lo cked laser. However, unlike the mo de-lo cked laser the pulse duration is not limited by the bandwidth of the laser gain b ecause the system is passive. The pulse width decreases with the mo dulation index increase and the overall cavity disp ersion decrease. The mo dulation index may b e very large in a WGM mo dulator, which may signicantly improve the p erformance of the system [273].

IUR



m [280]. A signicant sup ermo de

2.6

Смешение частот и генерация

WGRs were used in optical parametric as well as hyp er-parametric wave mixing pro cesses.

2.6.1

Гиперпараметрический генератор

Hyp er-parametric optical oscillation [287], also known in b er optics as mo dulation instability [288], is based on four-wave mixing (FWM) among two pump, signal, and idler photons, and results in the growth of the signal and idler optical sidebands from vacuum uctuations at the exp ense of the pumping wave. The hyp er-parametric oscillations are dierent from the parametric ones. The parametric oscillations i) are based on

(2)

nonlinearity coupling three photons, and

ii) have phase matching conditions involving far separated optical frequencies, that can only b e satised in birefringent materials in the forward direction. On the contrast, the hyp er-parametric oscillations i) are based on

(3)

nonlinearity

coupling four photons, and ii) have phase matching conditions involving nearlydegenerate optical frequencies that can b e satised in most of the materials b oth in the forward and backward directions. Recently, the study of hyp er-parametric oscillations had a new stage connected with the development of WGM as well as photonic crystal micro-resonator technology [289, 11]. The oscillations o ccurring in cavities or cavity-like systems lled with transparent solids were analyzed theoretically, e.g., in isotropic photonic crystals [290], and were observed exp erimentally in crystalline WGM resonators [291, 292]. It was suggested, in particular, that the narrow-band b eat-note signal b etween the optical pump and the generated sidebands emerging from a high-Q WGM resonator could b e used as a secondary frequency reference [292, 293].

23


The phase stability of the frequency reference signal increases with increase of the Q-factor of the resonator mo des for the same given value of the pump p ower. There exists a maximum of the phase stability (minimum of the phase diusion) of the b eat-note signal that do es not dep end either on the pump p ower or Q-factor of the mo des. Keeping in mind that WGM's Q-factor can exceed

IH

10

(a few tens of kilohertz resonance linewidth) [53], it was found that the

Allan deviation factor of the oscillations to b e smaller than for reasonable exp erimental parameters.

milliWatt optical pumping. The pump threshold could reach microWatt levels

IH

12

s

1

=2 for sub-

2.6.2

Праметрические процессы

Optical parametric oscillators (OPO) have b een extensively studied since the discovery of lasers [294, 295, 296]. Prop erties of OPO are well understo o d by now [297, 287, 51]. The cw-OPO is considered as an ideal device that can generate a broad range of wavelengths. Ecient frequency doubling at



[49] using the same WGR made of p erio dically p oled LiNbO3 (PPLN) [298]. The WGR was doubly resonant, b oth at fundamental and second harmonic frequencies. The follow-up studies of the parametric pro cesses in PPLN WGRs are imp ortant b ecause it has b een predicted that an optical parametric oscillator based on the resonator might have p ower threshold b elow a microWatt [299] orders of magnitude less than that of the state-of-the-art OPOs, typically at

a I:SS



m and



a I:QIW



m was realized

H:S

mW level [300]. It was shown theoretically [301] that a nondegenerate multi-frequency para-

metric oscillator has dierent prop erties compared with the usual three-wave parametric oscillator. A scheme for a resonant cw monolithic microwave-optical parametric oscillator based on high-Q WGMs excited in a nonlinear dielectric cavity. Such an oscillator may have an extremely low threshold and stable operation, and may b e used in sp ectroscopy and metrology. The oscillator mimics devices based on resonant

(3)

nonlinearity (hyp er-parametric pro cess) and can

b e utilized for ecient four-wave mixing and optical comb generation.

2.7

Фундаментальная физика с активными резонаторами ШГ

WGRs can b e used for generation of nonclassical states of light. For instance, it was shown a p ossibility for the generation of heralded single photons and of subPoissonian laser light in the electrically pump ed single-quantum-dot microsphere laser [190]. The reduced density matrix metho d was used to calculate the quantumstatistical prop erties of the radiation of a quantum-dot laser op erating on the WGM of a dielectric microsphere [302]. It was shown that under the conditions of strong coupling b etween the quantum dot and an electromagnetic eld the radiation of such a laser can b e in a nonclassical (sub-Poissonian) state. The laser scheme was characterized by an extremely low lasing threshold and a small

24


numb er of saturation photons, as a result of which lasing is p ossible with close to zero p opulation inversion of the working levels.

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