Photonic atoms and molecules

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  • Laser & Photon. Rev. 4, No. 2, 179191 (2010) / DOI 10.1002/lpor.200910001 179

    Abstract Recent advances in the investigation of optical modesin single three-dimensional whispering-gallery microcavitiesand coupled microresonators are reviewed. Due to the smallsize and high quality factor these resonators have the potentialfor development of single-photon emitters, nanojets, sensorsand low-threshold lasers and for fundamental studies of opticalmatter interactions.

    Photoluminescence confocal image of a spherical whispering-gallery microcavity with monolayer of CdTe quantum dots.

    2010 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

    Photonic atoms and moleculesYury P. Rakovich* and John F. Donegan

    School of Physics and Centre for Research on Adaptive Nanostructures and Nanodevices, Trinity College Dublin, Dublin 2, Ireland

    Received: 2 January 2009, Revised: 28 February 2009, Accepted: 25 March 2009Published online: 2 June 2009

    Key words: Photonic atoms, photonic molecules, spherical microcavities, whispering-gallery modes, semiconductor quantum dots.

    PACS: 42.60.Da, 61.46.Df, 83.85.Ei, 87.85.fk

    1. Introduction

    In the heart of the Temple of Heaven in Beijing (constructedbetween 1406 and 1420) there is a famous circular stonewall, which surrounds the Imperial Vault. It is named theEcho Wall for its very curious and interesting acousticalproperties a whisper spoken at one end can be heardclearly from the other. In other words, it makes possiblewireless communication between two people separatedby more than 100 m.

    A comprehensive explanation of this effect has beenprovided by Lord Rayleigh who investigated the propa-gation of acoustic waves over an arch wall surface in theWhispering Gallery under the dome of St. Pauls Cathe-dral [1]. Lord Rayleigh gave an explanation of this phe-

    nomenon as being due to the curvilinear propagation ofsound, the waves that proceed from a source placed closeto the wall of the gallery clinging to its surface and creep-ing tangentially along it. The phenomenon observed andexplained by Lord Rayleigh has now been observed in circu-lar and spherical optical microresonators. This is the originof the term whispering-gallery modes (WGMs) that iscommonly used to denote electromagnetic modes in theseoptical resonators. Although they did not using this term forthe modes, at the beginning of the 20th century, Debye [2]and Mie [3] derived equations for the eigenfrequencies andscattering resonances of free dielectric and metallic spheres,which naturally take WGMs into account.

    In the case of dielectric microspheres the closed tra-jectories of light are supported by total internal reflectionwhen boundaries of the microparticles and the air have high

    * Corresponding author: e-mail: Yury.Rakovich@tcd.ie

    2010 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

  • 180 Y. P. Rakovich and J. F. Donegan: Photonic atoms and molecules

    Figure 1 (online color at: www.lpr-journal.org) Light confine-ment inside a microsphere in a geometric-optics picture a) and ina wave-optics description b).

    refractive index contrast and the radius of curvature of themicrosphere exceeds several wavelengths.

    In the geometric-optics picture, WGM can be viewedas light rays that, once within the spherical particle, inter-sect repeatedly with the interface above the critical angle,undergoing total internal reflection (Fig. 1a). In a wave-mechanics description, WGMs are internal standing waves,with an integer number of wavelengths circulating near themicrosphere or droplet interface (Fig. 1b).

    The radiative losses from these small optical micro-cavities can be very small, and the quality factor Q of themicrosphere modes becomes limited only by material atten-uation and scattering caused by geometrical imperfections(e.g., surface roughness). An extremely small linewidth ofthe WGMs (and therefore high values of the Q-factor) werereported in the first experimental observations of WGMsin elastic light scattering from spherical dielectric parti-cles [4, 5]. The strong influence of WGMs on the lumi-nescence and Raman scattering was first studied in [6, 7]and [810], respectively.

    Microsphere resonators have found applications as ul-trasensitive mechanical [11,12], refractometric [13,14] andbiological [15, 16] sensors, optical waveguides [17, 18],narrowband passive photonic devices such as bandpass fil-ters [19] or add-drop devices [20, 21]. They were also usedto study cavity quantum electrodynamics effects [22,23]. Ifan active material is coupled to whispering-gallery modes,ultralow threshold lasing is achieved [24, 25]. The rangeof WGM microcavity applications is so wide that in thisreview we will restrict ourselves to the case of small (210 m size) spherical microcavities integrated with highlyemitting semiconductor quantum dots.

    For a much more complete description of the physicsand applications of WGM resonators we refer to otherrecent reviews [26, 27].

    2. Spherical microcavities as photonic atoms

    Extending the ideas of the linear combination of atomicorbitals method to the classical wave case, it was recentlysuggested that Mie resonances (WGMs) of a single spheri-cal microcavity play the same role as the atomic orbitals in

    Figure 2 Schematic ofa spherical microcav-ity, showing distributionand the spatial orienta-tion of modes.

    the electronic case and the spatial distributions of WGMscan be described by analogy with the orbitals in a hydrogenatom [28]. In the absence of gain, the WGM resonances canbe characterized by a mode number n (angular quantumnumber), a mode order l (radial quantum number), and anazimuthal mode numberm (azimuthal quantum number).The value of n is proportional to the circumference dividedby the wavelength of the light propagating within the micro-sphere, the mode order l indicates the number of maximain the radial distribution of the internal electric field, andthe azimuthal mode numberm gives the orientation of theWGMs orbital plane. The modes offering the highest spatialphoton confinement correspond to high values of angularmomentum n, smaller values of l and to azimuthal quantumnumbersm close to n.

    The electromagnetic fields confined in a microsphereare given by [29]

    E

    TE

    = j

    n

    (nkr)X

    nm

    (; ') ; (1)

    for modes having no radial components of the electric field(transverse electric or TE modes) and

    E

    TM

    = r j

    n

    (nkr)X

    nm

    (; ') ; (2)

    for the transverse magnetic (TM) modes (no radial compo-nent of the magnetic field).

    Here, Xnm

    (; ') are the vector spherical harmonicsgiven in spherical polar coordinates, jn

    (n

    r

    kr) and repre-sent the spherical Bessel functions, where k = 2= is thewave number, and is the free-space wavelength.

    On the other hand, the wavefunctions slm

    for the elec-tron confined in the hydrogen atom are given by [30]

    slm

    = R

    sl

    (r)

    lm

    (; ') ; (3)

    where Rsl

    (r) are known as the Laguerre polynomials. Theeigenfunctions (1) and (3) are very similar and their spatialdistributions are characterized by the three integers s, l andm (for the hydrogen atom) and n, l andm (for the micro-sphere), which correspond to total angular, radial and theazimuthal quantum numbers, respectively. Based on all theabove similarities, this approach has enabled small dielec-tric spheres to be considered as photonic atoms [28, 31].However, unlike energy states of electrons in the atom,photonic states in spherical microcavities are not local-ized, due to the finite storage time of photons in the reso-nant mode. This photon lifetime is controlled by qualityfactor Q of the WGMs and therefore can be limited by

    2010 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.lpr-journal.org

  • Laser & Photon. Rev. 4, No. 2 (2010) 181

    Figure 3 (online color at: www.lpr-journal.org) PL spectra ofcolloidal CdTe QDs (NCs) in water (black) and micro-PL spectrafrom a single melamine formaldehyde (MF) latex microsphereof 3 m diameter covered by one monolayer shell of CdTe NCs(red). PL spectra of QDs were recorded using a Spex Fluorologspectrometer. The micro-PL spectra from the single microspherewere recorded using a RENISHAW micro-Raman system. An Ar+

    laser ( = 514.5 nm) was used in the micro-PL experiments.

    diffractive losses, absorption, gain, shape deformation orrefractive index inhomogeneities. As a result, the resonantinternal field of a spherical cavity is not completely con-fined to the interior of the microparticle. Decaying exponen-tially, the evanescent field extends a couple of micrometersinto the surroundings and this field distribution permitsefficient coupling of the emission of semiconductor quan-tum dots (QDs) with WGM by placing the emitter (shellof QDs) on the surface of the high refractive index mi-crosphere. These quantum dots represent the ultimate insemiconductor-based quantum-confined system with atom-like energy levels, large optical transition dipole momentand high photoluminescence quantum efficiency [32]. Dueto their unique optical properties, semiconductor QDs havefound application in biology and photonics [3335].

    Theory suggests that by placing an excited emitter(such as an atom or molecule or QDs) near the surfaceof a spherical dielectric, the interaction of the emitterwith its self-scattered field can produce enormous enhance-ments in its emission rate at frequencies associated withWGM [36]. This result can be clearly seen in Fig. 3. Thishybrid QDs/microsphere structure was fabricated as de-scribed elsewhere [37].

    In contrast to the broad, featureless PL band in thespectra of the colloidal nanocrystals (NCs) or quantumdots, the emission spectra of a single melamine formalde-hyde (MF)=CdTe microsphere exhibit a very sharp periodicstructure. The observed WGM peak structure is a result ofcoupling of electronic states in the QDs and photon states

    of the microsphere. The placement and spacing betweenWGM peaks presented in Fig. 3 are determined by the sizeand refractive index of the microsphere, while the spectralintensity distribution depends on the parameters of the QDsand can be easily modified by using QDs of different size.

    The narrow optical resonances presented in Fig. 3 canbe identified as optical modes with TE and TM polariza-tions. For spherically symmetric microparticles the extinc-tion cross section is derivable from the LorenzMie the-ory [38]

    C

    ext

    =

    2

    k

    2

    Re

    1

    X

    n=1

    (2n+ 1) (b

    n

    (x; n

    r

    ) + a

    n

    (x; n

    r

    )) ;

    (4)where x = 2R= is the size parameter, R is the radiusof the microsphere and the Mie scattering partial waveamplitudes an

    (x,nr

    ) and bn

    (x,nr

    ) can be expressed inthe form [39]

    a

    n

    (x; n

    r

    ) =

    A

    n

    (x; n

    r

    )

    A

    n

    (x; n

    r

    ) + i C

    n

    (x; n

    r

    )

    and

    b

    n

    (x; n

    r

    ) =

    B

    n

    (x; n

    r

    )

    B

    n

    (x; n

    r

    ) + iD

    n

    (x; n

    r

    )

    :

    (5)

    The pairs of functions An

    , Cn

    and Bn

    , Dn

    are deter-mined by the Mie scattering problem [40] and resonancestructure in scattering spectra can be expected when thereal part of an

    or bn

    reaches its maximum value of 1 andthe imaginary part passes through 0 from the positive tothe negative side. In other words, the resonances in the Miescattering characteristics occur when Cn

    = 0 or Dn

    = 0,which gives the following mathematical condition for aresonance

    n

    r

    n

    (x)

    0

    n

    (n

    r

    x)

    n

    (n

    r

    x)

    0

    n

    (x) = 0 (6)

    or

    n

    r

    n

    (x)

    0

    n

    (n

    r

    x)

    n

    (n

    r

    x)

    0

    n

    (x) = 0 : (7)

    Note, that for given n and m, these equations have in-finitely many solutions at discrete values of x for TM andTE WGMs respectively. Also, Eqs. (6) and (7) are inde-pendent of the azimuthal mode numberm because of thespherical symmetry implying that them-modes are wave-length degenerate. The RiccatiBessel functions of the firstand second kind can be introduced as:

    n

    (z) = zj

    n

    (z) =

    r

    z

    2

    J

    n+

    1

    2

    (z) ;

    and

    n

    (z) = zy

    n

    (z) =

    r

    z

    2

    Y

    n+

    1

    2

    (z) ;

    (8)

    where J(z) and Y (z) are the cylindrical Bessel functionsof the first and second kind, respectively. The use of Besselfunctions for systems with cylindrical symmetry together

    www.lpr-journal.org 2010 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

  • 182 Y. P. Rakovich and J. F. Donegan: Photonic atoms and molecules

    with the reccurence relation 0n

    (z) =

    n

    z

    n

    (z)+

    n1

    (z)

    enables us to reduce Eqs. (6) and (7) to a form convenientfor practical calculation of the position of the WGMs. Thus,if the refractive index is real (no absorption) the TM andTE resonances can be taken in the form:

    1

    x

    n

    n

    r

    n

    r

    n

    J

    n+1=2

    (x)J

    n+1=2

    (n

    r

    x)

    +n

    r

    J

    n+1=2

    (n

    r

    x)J

    n1=2

    (x)

    J

    n+1=2

    (x)J

    n1=2

    (n

    r

    x) = 0 ; (9)

    n

    r

    Y

    n+1=2

    (x)J

    n1=2

    (n

    r

    x)

    J

    n+1=2

    (n

    r

    x)Y

    n1=2

    (x) = 0 : (10)

    These conditions are transcendental equations, which canbe solved for the size parameter x (position of a resonance)for given values of refractive index and for given angularquantum number n. Thus, comparing the calculated resultswith the spectral positions of the WGM in the experimentalphotoluminescence (PL) or emission spectra of the micro-sphere we can identify the indexes n and l for each modeand estimate the size of the sphere. Additionally, absorptivelosses can be accounted for by taking into consideration animaginary part of the refraction index.

    The algorithm for the mode assignment can be de-scribed as follows. 1) The resonant wavelengths corre-sponding to the WGM resonances expi

    (i = 1; 2: : :; N )are determined from a PL spectrum of a single microsphere.2) We assume an approximate value of the microsphereradius based on the technical specification within the dis-tribution of sizes that are specified by the manufacturer.3) Theoretical resonance positions theori

    are then calcu-lated using Eqs. (8) and (9). 4) The two lists are comparedand for each value of expi

    , the closest value theori

    is sug-gested and the difference between them i

    is calculated.5) Taking into account the spectral resolution , the cor-

    relation S = 1N

    N

    P

    1

    (1 +

    i

    =)

    1 is then maximized by

    adjusting only one fitting parameter, namely the size ofthe microsphere. Fig. 4 shows a result of WGM identifica-tion in the PL spectra of QDs attached to two spheres ofdifferent sizes.

    Among all the factors that cause the modification in thePL emission pattern due to the spatial photon confinementin spherical microcavities, the nature of the polarization ofthe WGM plays an important role because of the potentialfor their use in all-optical switches and logical devices [41].Direct experimental identification of the polarization stateof WGMs can be provided by inserting a polarizer into theoptical beam path in front of the detection system. In thiscase the polarizer selects only signals from WGMs thatemit components of the electromagnetic field parallel to theorientation of the polarizer axis.

    For a focal spot on the top rim of a microsphere (Fig. 5,inset), the emission spectrum detected with the polarizerorientation parallel to the polarization plane of the laseris shown (Fig. 5a). Due to the higher excitation efficiency

    Figure 4 Micro-PL spectra of two single microspheres witha monolayer of CdTe QDs, with diameters of 1.98 m a) and3.06 m b) with modes identified from Eqs. (8) and (9).

    Figure 5 Micro-PL spectra of a single microsphere with a mono-layer of CdTe NCs with the polarizer orientation parallel...