* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Whispering gallery modes of microspheres in the presence of a
Fourier optics wikipedia , lookup
Silicon photonics wikipedia , lookup
Magnetic circular dichroism wikipedia , lookup
Ultrafast laser spectroscopy wikipedia , lookup
Fiber-optic communication wikipedia , lookup
Thomas Young (scientist) wikipedia , lookup
Nonimaging optics wikipedia , lookup
Atmospheric optics wikipedia , lookup
Fiber Bragg grating wikipedia , lookup
Interferometry wikipedia , lookup
Astronomical spectroscopy wikipedia , lookup
Photon scanning microscopy wikipedia , lookup
Optical aberration wikipedia , lookup
Dispersion staining wikipedia , lookup
Ellipsometry wikipedia , lookup
Harold Hopkins (physicist) wikipedia , lookup
Ultraviolet–visible spectroscopy wikipedia , lookup
Birefringence wikipedia , lookup
Surface plasmon resonance microscopy wikipedia , lookup
Opto-isolator wikipedia , lookup
Nonlinear optics wikipedia , lookup
Refractive index wikipedia , lookup
JOURNAL OF APPLIED PHYSICS 107, 103105 共2010兲 Whispering gallery modes of microspheres in the presence of a changing surrounding medium: A new ray-tracing analysis and sensor experiment Tindaro Ioppolo,1,a兲 Nirod Das,2 and M. Volkan Ötügen1 1 Department of Mechanical Engineering, Southern Methodist University, Dallas, Texas 75205, USA Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Brooklyn, New York 11201, USA 2 共Received 9 November 2009; accepted 7 April 2010; published online 20 May 2010兲 A simple plane wave, ray-tracing approach was used to derive approximate equations for the dielectric microsphere whispering gallery mode 共WGM兲 resonant wavenumber and quality factor, as dependent on the surrounding medium’s refractive index. These equations are then used to determine the feasibility of a micro-optical sensor for species concentration. Results indicate that the WGMs are not sensitive enough to refractive index changes in the case of gas media. However, they can be sufficiently sensitive for measurements in liquids. Experiments were carried out to validate the analysis and to provide an assessment of this sensor concept. © 2010 American Institute of Physics. 关doi:10.1063/1.3425790兴 2Rn1 = l, I. INTRODUCTION Micro-optical sensor concepts based on the so-called whispering gallery mode 共WGM兲 shifts in spherical dielectric resonators were recently demonstrated for temperature,1 force,2,3 pressure,4 and electric field.5 In these studies, a single mode optical fiber is used to couple tunable laser light into the sphere. The WGMs are observed as sharp dips in the transmission spectrum through the fiber. The observed linewidth, ␦, of these dips is related to the quality factor, Q = / ␦. A small perturbation in the shape, size, or the index of refraction of the microsphere causes a shift in the resonance positions allowing for the precise measurement of the external condition causing the change in the morphology of the sphere. In addition to the morphology of the sphere, the WGMs are affected also by changes in the refractive index of the surrounding medium. Thus, a change in the composition of the surrounding medium, if sufficiently large, will induce a detectable shift in the WGMs. This effect can be used, in principle, to monitor species concentration changes in the medium surrounding a dielectric microcavity. Krioukov et al.,6 recently proposed a species concentration sensor using this approach. They monitored refractive index changes around an integrated microdisk-waveguide system by exciting the microdisk WGM and observing them through the scattered light from the disk. In the present approach, we use a dielectric sphere along with a single mode optical fiber as a light input-output device. Figure 1共a兲 depicts the geometric optics view of the WGM resonance condition. In this view, the laser light circles the interior of the sphere through multiples of total internal reflections and returns in phase. The approximate condition for optical resonance 共or WGM兲 is a兲 Electronic mail: [email protected]. 0021-8979/2010/107共10兲/103105/8/$30.00 共1兲 where, R and n1 are the sphere radius and refractive index, is the vacuum wavelength of light, and l is an integer. An incremental change in the refractive index of the sphere will result in a shift in the resonant wavelength; ⌬n1 / n1 ⬃ ⌬ / . Similarly, a change in the radius of sphere will also result in a change in the resonant wavelength; ⌬R / R ⬃ ⌬ / . Therefore, any change in the physical condition of the surrounding that induces a ⌬n1 or ⌬R of the microsphere can be sensed by monitoring the WGM shifts. Equation 共1兲 is an approximation that helps us evaluate the first order effects on WGMs of the sphere 共those due to a a) n1>n2 i 1 R n1 2 … N 2 1 2 1 … 1 Evanescent Wave 1 b) i i FIG. 1. 共a兲 Ray optics model of light traveling inside sphere 共N is the number of reflections for one round trip兲; 共b兲 reflection of a plane wave from a dielectric interface. 107, 103105-1 © 2010 American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 103105-2 J. Appl. Phys. 107, 103105 共2010兲 Ioppolo, Das, and Ötügen change in R or n1兲. However, each time the light is reflected off the inner surface of the sphere, the reflected wave experiences a certain “phase delay.” This phase delay is a function of the light wavelength, , incidence angle, i as well as the sphere-to-surrounding refraction index ratio, n1 / n2. Therefore, for a given laser wavelength and coupling angle, the total optical path length for the round trip of the light is somewhat different from 2Rn1 and is a function of the refractive index of the surrounding medium, n2. By tracking the WGM shifts induced by perturbations in n2, the surrounding medium can be monitored provided that the induced shifts are in the order of ␦ or larger. However, n2 changes also affect the quality factor, Q and hence ␦. In the following the effect of n2 on both WGM resonance and the quality factor is analyzed using a simple geometric optics and basic electromagnetic equations. ZTE1 = Z0 , n1 cos i ZTE2 = ZTM1 = Z0 cos i , n1 ZTM2 = i ⬎ c = sin−1 冑 2 n2 = sin−1 , 1 n1 共2兲 where n and are the refractive indices and the dielectric constants of the resonator 共1兲 and surrounding media 共2兲. The reflection from such a planar interface should accurately model each individual reflection in the ray-optics description shown in Fig. 1共a兲. The reflection coefficients ⌫TE and ⌫TM for the TE and TM polarizations are expressed using equivalent impedances of the two media as10 ⌫TE = where, ZTE2 − ZTE1 , ZTE2 + ZTE1 ⌫TM = ZTM2 − ZTM1 , ZTM2 + ZTM1 共3兲 sin2 i − n22 共4兲 , − jZ0冑n21 sin2 i − n22 n22 , 共5兲 with Z0 = 377 ⍀ as the wave impedance of free space. Our interest is the phase associated with each reflection coefficient for large N. Combining Eqs. 共3兲 and 共4兲 for the TE polarization, ⌫TE = n1 cos i + j冑n21 sin 2i − n22 n1 cos i − j冑n21 sin 2i − n22 II. ANALYSIS Starting with a quantum analogy-based theory developed by Johnson7 describing the optical resonances of a sphere, Teraoka et al.,8 carried out a perturbation approach to study WGM shifts due to changes in the refractive index of the medium surrounding the sphere. More recently, Schweiger and Horn9 used a wave theory approach to obtain similar results. We present here a simple approach using geometric optics, along with basic electromagnetic wave reflection expressions, to obtain equations that accurately describe the resonant wavenumber and quality factor Q of the WGM. That is, using simple physical modeling of reflections at an interface, we derive a modified version of Eq. 共1兲, together with an expression for the associated Q, in which the effect of n2 is included. The analysis is valid for R Ⰷ and the sphere is treated as a cylinder for conceptual simplicity with the light traveling on a plane parallel to the end surfaces of the cylinder. When the order number l of resonance is large, the cylindrical resonance condition would accurately model a spherical resonance as well. The analysis yields two distinct equations for the transverse electric 共TE兲 and transverse magnetic 共TM兲 modes. Consider the reflection of a plane wave from a dielectric interface, as shown in Fig. 1共b兲, when the angle of incidence i is larger than the critical angle. That is − j冑 and arg共⌫TE兲 = − 2 tan−1 A. WGM shift Z0 n21 冑 共6兲 , cos i 冉冊 n2 sin i − n1 2 2 共7兲 . Referring to Fig. 1共a兲, we note that, for N Ⰷ 1, the incidence angle i is i = − ⬵ . 2 N 2 共8兲 Therefore, arg共⌫TE兲 can be approximated as follows: arg共⌫TE兲 ⬵ − 2/N 冑 冉 冊 n2 1− n1 2 共9兲 . Similarly, combining Eqs. 共3兲 and 共5兲 for the TM polarization, ⌫TM = − n22 cos i − jn1冑n21 sin 2i − n22 + n22 cos i − jn1冑n21 sin 2i − n22 共10兲 , yielding the argument arg共⌫TM兲 = − 2 tan−1 cos i 冉 冊冑 n1 n2 2 冉冊 n2 sin i − n1 2 2 . 共11兲 Again, Eq. 共11兲 can be approximated using Eq. 共8兲 as, arg共⌫TM兲 ⬵ − 2/N 冉 冊冑 冉 冊 n1 n2 2 n2 1− n1 2 . 共12兲 Equation 共9兲 shows that the reflection phase is close to for the TE mode. We also know that the magnitude of reflection is unity 共total internal reflection兲. Therefore, we may consider the case as a perturbation of an ideal conducting wall for which 兩⌫兩 = 1 and arg共⌫兲 = . Accordingly, the TE resonant condition for the dielectric sphere of Fig. 1共a兲 may also be considered as a perturbation of that for a sphere with an ideal conducting wall. Using multiple ray-reflection model shown in Fig. 1共a兲, we consider the resonant condition for the conducting sphere first. The resonance radius is analytically known from the first zero of the lth order Bessel function10,11 that is k1R ⬵ l + 1.856l1/3. Here, k1 = 2 / 1 is the Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 103105-3 J. Appl. Phys. 107, 103105 共2010兲 Ioppolo, Das, and Ötügen wavenumber and 1 is the wavelength in the sphere. Thus, from Fig. 1共a兲 and using Eq. 共8兲, the resonant condition can be expressed as N关2k1R cos i − arg共⌫TM兲兴 = 2共l + 1.856l1/3兲, and N2k1R cos i ⬵ N2k1R = 2k1R = 2共l + 1.856l1/3兲. N k 1R ⬵ 共13兲 The factor 1.856l1/3 in Eq. 共13兲 can be interpreted as a first order correction that accounts for the spherical nature of the wave front and the curvature of the resonator boundary, which is different from the simple plane wave treatment of Fig. 1共b兲. For large mode numbers 共l Ⰷ 1兲 this term would be small compared to l, resulting in the approximate resonance condition of Eq. 共1兲. The resonant condition for the dielectric cavity of Fig. 1共a兲 may now be determined by including in Eq. 共13兲 the phase contribution of Eq. 共9兲 2共l + 1.856l1/3兲 + arg共⌫TM兲N 2N N ⬵ l + 1.856l1/3 − ⌬共k1R兲 ⬵ − 共14兲 2共l + 1.856l1/3兲 + 关arg共⌫TE兲 − 兴N 2N N 冑 1 n2 1− n1 冉冊 2. =− 共15兲 Note that the final resonant condition does not depend on N, which is the number of reflections the light ray experiences for one full round trip in sphere as shown in Fig. 1共a兲. 共Also, in principle, N need not be an integer.兲 The change in the resonant condition due to an incremental change in n2 can be evaluated by differentiating Eq. 共15兲. ⌬共k1R兲 ⬵ − n2⌬n2 n21 冋 冉 冊册 n2 1− n1 2 3/2 =− n1n2⌬n2 , 关n21 − n22兴3/2 共16兲 or, 1 n2⌬n2 ⌬共k1R兲 ⌬共k0R兲 = =− 2 , 2 3/2 k 1R k 0R 关n1 − n2兴 k0R n2 n1 2 1 n2 1− n1 冉冊 共19兲 2. 冦 冉冊 冋 冉 冊册 冋 冉 冊册 冋 冉 冊册 2n2⌬n2 n2⌬n2 2 n1 n21 + n2 2 3/2 n2 2 1− 1− n1 n1 n2 n1 2 n1n2⌬n2 2 − 冉冊 ⬵ l + 1.856l1/3 − 冉冊 冉 冊冑 The above equation is the TM counterpart of Eq. 共15兲. Again, the resonance shifts due to an incremental change in the refractive index of the surrounding medium can be obtained by differentiating Eq. 共19兲 N兵2k1R cos i − 关arg共⌫TE兲 − 兴其 = 2共l + 1.856l1/3兲, k 1R ⬵ 共18兲 n2 n1 ⌬共k1R兲 ⌬共k0R兲 = =− k 1R k 0R , 冋 冉 冊册 n2⌬n2 2 − 关n21 冧 2 关n21 − n22兴3/2 or 1/2 − n2 n1 n22兴3/2 共20兲 2 1 . k 0R 共21兲 Equations 共17兲 and 共21兲 describe the dependence of the resonant frequency 共WGM兲 shift on the refractive index of the medium surrounding the sphere for TE and TM polarizations, respectively. Note that l is kept constant in obtaining the above derivatives. In an earlier study, Teraoka et al.8 carried out a perturbation analysis based on an equivalent quantum theory model of the optical modes of spheres7 to obtain expressions describing WGM dependence on ⌬n2. For k0R Ⰷ 1, Eq. 共16兲 and the implicit formula obtained by Ref. 8 differs by the ratio of 冑兵关共l − 0.5兲 / 共k0R兲兴2 − n22其 / 共n21 − n22兲. However, when the order number is approximated by l = n1k0R their expression for WGM shift becomes identical to Eq. 共16兲. 共17兲 where, k1 = k0n1. The treatment for the TE polarization presented above can be repeated for the TM polarization, noting only the key differences. From Eq. 共12兲 we see that reflection phase for TM polarization is close to zero for large N. Therefore, the resonant dielectric sphere of Fig. 1共a兲 may be considered as a perturbation of a sphere with an ideal magnetic wall for which 兩⌫兩 = 1 and arg共⌫兲 = 0. As in the TE resonance case, the TM resonant radius for the magnetic-walled resonator would also correspond to the first zero of the lth order Bessel function,10,11 with k1R ⬵ l + 1.856l1/3. So we have, B. Quality factor The resonance analysis presented above models the interface between the microsphere and the external medium 关Fig. 1共a兲兴 as a planar surface 关Fig. 1共b兲兴. This results in total internal reflection, with magnitudes of the reflection coefficients equal to unity. However, each reflection generates some radiation, attributed to the curvature of the interface. We model the radiation loss with an equivalent radiation resistance Rre and Rrm, for TE and TM polarizations, respectively. The reflection coefficients for the TE and TM modes, taking into account the equivalent radiation resistance are given as Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 103105-4 J. Appl. Phys. 107, 103105 共2010兲 Ioppolo, Das, and Ötügen ⌫TE = ZTE2 + Rre − ZTE1 , ZTE2 + Rre + ZTE1 ⌫TM = ZTM2 + Rrm − ZTM1 . ZTM2 + Rrm + ZTM1 共22兲 The impedances ZTE1, ZTE2, ZTM1, and ZTM2 are given by Eqs. 共4兲 and 共5兲. For N Ⰷ 1 and i = / 2 − / N, the magnitudes of the reflection coefficients can be approximated as 兩⌫TE兩 ⬇ 1 − 2Rre Z 1N , 兩⌫TM兩 ⬇ 1 − and 2RrmZ1 , 共23兲 N兩ZTM2兩2 where Z1 = 冑0 / 1 is the intrinsic impedance of the microsphere material. Equation 共23兲 indicates reflection coefficients smaller than unity, which is physically more realistic. The complex resonance condition for the TE mode may now be accurately established, using the reflection coefficient magnitude of Eq. 共23兲 exp关− jN2k1R cos i + jN arg共− ⌫TE兲兴兩⌫TE兩N = f共r + ji兲 = exp共− jl2兲 = 1, with 冉 f共r兲 = 兩⌫TE兩N = 1 − 2Rre Z 1N 冊 冉 f共r + ji兲 = 1 ⬇ f共r兲 + ji 冉 ⫻ 1− 冊 共24兲 N ⬇ 1− 冊 2Rre , Z1 冉 f共r兲 2Rk1i = 1+ r Prad = Pd2r2 = 共26兲 = 共27兲 The above equation shows that in order to evaluate the quality factor, the equivalent radiation resistance Rre has to be calculated. With this objective, we consider a TE spherical mode for r ⬎ R, with electric field distribution for = / 2 expresses as10 E = jl Ah共2兲 l 共k2r兲e , 共28兲 here A is a constant and h共2兲 l is the outgoing spherical Hankel function of order l. When r → ⬁ the above equation may be simplified to E共r → ⬁兲 ⬇ Ajl+1 −jk2r e . k 2r 2 兩A兩2兩h共2兲 l 共k2R兲兩 Rre2R2 . 兩ZTE2兩2 共32兲 Combining Eqs. 共31兲 and 共32兲, the equivalent radiation resistance can be expressed as Rre = 兩ZTE2兩2 2 2 兩h共2兲 l 共k2R兲兩 Z2共k2R兲 共33兲 . Using the above expression for Rre in Eq. 共27兲 the quality factor Q can be derived as follows: 2 2 r Z 1Rk1 兩h共2兲 l 共k2R兲兩 Z2Z1共k2R兲 共Rk1兲 = = i Rre 兩ZTE2兩2 ⬇ y 2l 共k2R兲共k2R兲3 冋冉 冊 册 r1 −1 , r2 共34兲 where y l共 . 兲 is the spherical Bessel function of the second kind. Note that we approximated 兩ZTE2兩 in the above derivation for i ⬇ / 2 to obtain 兩ZTE2兩 = where, r and i are real and imaginary angular frequencies. Then, the quality factor, Q can be expressed as r Z 1Rk1 = . Q= i Rre 兩E共R兲兩2 Rre2R2 兩ZTE2兩2 Prad = 兩H共R兲兩2Rre2R2 = 共25兲 2Rre 2Rk1i 2Rre ⬇1+ − , Z1 r Z1 共31兲 The above Prad can also be written in terms of the electric or magnetic field components at r = R, and the equivalent radiation resistance Rre. Q= 冊 兩A兩22 . Z2共k2兲2 0 冑k21 sin2 i − k22 ⬇ Z2 冑冉 冊 k1 k2 = 2 −1 Z2 冑冉 冊 r1 −1 r2 共35兲 Using a similar procedure as for the TE mode, an expression for the quality factor Q for TM mode can also be obtained. exp关− jN2k1R cos i + jN arg共− ⌫TM兲兴兩⌫TM兩N = f共r + ji兲 = exp共− jl2兲 = 1, 冉 f共r兲 = 兩⌫TM兩N = 1 − 2Z1Rrm 兩ZTM2兩2N 冊 冉 N ⬇ 1− 共36兲 冊 2Z1Rrm , 兩ZTM2兩2 共37兲 f共r + ji兲 = 1 ⬇ f共r兲 + ji 冉 共29兲 ⫻ 1− 冉 f共r兲 2Rk1i = 1+ r 冊 冊 2Rrm 2Rk1i 2Z1Rrm ⬇1+ − , 兩ZTM2兩2 r 兩ZTM2兩2 The power density associated with the field is given by 兩E兩2 兩A兩2 = , Pd = Z2 Z2共k2r兲2 Z2 = 冑0 / 2 . 共38兲 共30兲 is the intrinsic wave impedance of the where medium surrounding the sphere. The power radiated from the sphere, per unit polar angle ⌬ = 1, can now be expressed in terms of the power density. Q= r 兩ZTM2兩2Rk1 = . i RrmZ1 共39兲 As in the TE mode case, we consider here a TM spherical mode for r ⬎ R, with its magnetic field expressed for =/2 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 103105-5 J. Appl. Phys. 107, 103105 共2010兲 Ioppolo, Das, and Ötügen jl H共r ⬎ R, = /2兲 = Ah共2兲 l 共k2r兲e . 共40兲 The above equation may be simplified for r Ⰷ R, and the corresponding power density Pd and radiated power Prad per unit elevation angle can be expressed as Pd = Z2兩H兩2 = 兩A兩2Z2 , 共k2r兲2 Prad = Pd2r2 . l=1000 l=500 共41兲 Prad may also be expressed using the fields at r = R and the equivalent resistance Rrm. l=100 2 2 Prad = 兩H共R兲兩2Rrm2R2 = 兩A兩2兩h共2兲 l 共k2R兲兩 Rrm2R . 共42兲 Combining Eqs. 共41兲 and 共42兲, yields the equivalent resistance, Rrm, which can then be used in Eq. 共38兲 to find the expression for the quality factor Q for the TM mode. Rrm = Q= Z2 , 共2兲 兩hl 共k2R兲兩2共k2R兲2 共43兲 2 2 2 Rk1兩ZTM2兩2 兩h共2兲 l 共k2R兲兩 兩ZTM2兩 共k2R兲 共Rk1兲 = RrmZ1 Z 1Z 2 ⬇ y 2l 共k2R兲共k2R兲3 冉 r1 −1 r2 冊冉 冊 r1 . r2 共44兲 Again, we approximated 兩ZTM2兩 in the above derivation for i ⬇ / 2 to obtain: 兩ZTM2兩 = 冑k21 sin2 i − k22 = Z2 2 冑冉 冊 r1 − 1. r2 ⬇ Z2 冑冉 冊 k1 k2 2 −1 共45兲 Fig. 2 shows the TE resonance quality factor as a function of microsphere radius for a silica sphere surrounded by air 共n2 / n1 = 0.69兲. In the Q calculation, the order number, l, is determined using 共a兲 the zeroth order approximation that includes only the reflection at the interface 关Eq. 共1兲兴; 共b兲 first order approximation that includes reflection and surface curvature at the interface 关Eq. 共13兲兴; and 共c兲 second order approximation that includes reflection, surface curvature as well as phase 关Eq. 共15兲兴. The resonance condition of Eq. 共1兲 significantly overestimates the quality factor. The conditions of Eqs. 共13兲 and 共15兲 provide comparable values of the quality factor. However, the phase contribution in Eq. 共15兲 in- FIG. 2. TE quality factor as a function of sphere radius. Order number l is calculated using: 共a兲 Eq. 共1兲; 共b兲 Eq. 共13兲; and 共c兲 Eq. 共15兲. FIG. 3. TE mode quality factor as a function of n2 / n1. cludes the additional effect of the external medium, which is valuable in the study of resonance perturbation due to variation in the external medium. The results using Eq. 共15兲 are comparable to those from a much involved theory12 using a volume-current analysis of the spherical mode. This illustrates the accuracy of the present method, obtained with analytical and conceptual simplicity. Figure 3 shows the effect of refractive index ratio 共n2 / n1兲 and order number 共l兲 on TE resonance quality factor. The quality factor, Q, increases with increasing l and decreasing n2 / n1. For large l and small n2 / n1 the Q value due to radiation is much larger than that due to material loss. Accordingly, in this range the overall Q is primarily determined by the material loss, and is essentially insensitive to the variations in the surrounding medium. On the other hand, for n2 / n1 ratios close to unity, the contribution of radiation to Q value would be stronger making the overall quality factors smaller and more sensitive to n2. This may adversely affect the accuracy of n2 measurements using WGM shifts. Yet, Q changes may be significant enough to use them as a sensing parameter. III. EXPERIMENTAL INVESTIGATION A. Experimental setup The output of the tunable distributed feedback laser diode was coupled into a single mode optical fiber with core diameter of ⬃9 m and cladding of 125 m. The laser beam is split in two by a beam splitter 共90%/10%兲 so that the 10% of the beam intensity goes to a photodiode to monitor the laser beam intensity and use it as a reference signal. The fiber with the 90% intensity was sent to the “sensor” side of the fiber with the microsphere in contact with it. At its output end, this fiber is connected to a fast photodiode to monitor the transmission spectrum. This signal through the sensor fiber was normalized using the reference signal. The microsphere was brought very close to the tapered fiber at a distance of the order of several nanometers to facilitate good light coupling between fiber and resonator. The laser is tuned by ramping the current into it using a laser controller. The controller, in turn, was driven by a function generator that provided a saw-tooth input to the controller. The output of both photodiode was monitored by 16 bit digitizer 共data acquisition card兲 that was part of a host PC. The microspheres Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 103105-6 J. Appl. Phys. 107, 103105 共2010兲 Ioppolo, Das, and Ötügen Solution a) Microsphere 0.004 To temperature controller n2 From laser Wolf et al (1965) 0.0035 0.003 = 1312.59 nm 0.0025 = 1312.46 nm 0.002 0.0015 0.001 0.0005 To photodiode 0 0 Distilled water 0.05 0.1 0.15 Molar concentration, g-mols/liter FIG. 4. 共Color online兲 Details of the test section. b) 0.004 were manufactured by melting a small section of an optical fiber using a microtorch as was done in our previous studies.2–5 The sphere diameters ranged between 350 and 500 m in the present experiments. 0.0035 0.0025 n2 The details of the test section are shown in Fig. 4. In these experiments, the effect of potassium carbonate concentration 共in water兲 on the WGM shifts is studied. The microsphere is placed in deionized water in a container. A potassium carbonate solution was gradually added to the container and allowed to diffuse for uniform concentration before the WGM shifts are observed. The temperature of the solution was monitored by a thermocouple and kept a constant value of 22 ° C. The transmission spectrum was digitized and stored on the host PC to determine the WGM shift. The measurements were repeated with increasing concentrations of the solution. Using k = 2 / , Eqs. 共18兲 and 共22兲 can be rewritten in terms of wavelength, as; ⌬ n2⌬n2 = 2 , 2 3/2 共n1 − n2兲 2R ⌬ = 冋 冉 冊册 n2⌬n2 2 − 共n21 − n2 n1 n22兲3/2 = 1312.59 nm = 1312.46 nm 0.002 0.0015 B. Experimental results and Wolf et al (1965) 0.003 共46兲 2 , 2R 共47兲 for the TE and TM modes, respectively. Again, R and n1 are the radius and index of refraction of the sphere, respectively. 共Note that the relationship between the WGM shifts and index of refraction change is not a function of the coupling 共launch兲 angle of the light into the sphere.兲 Figures 5共a兲 and 5共b兲 show the experimentally determined refractive index of potassium carbonate solution. From the measured WGM shifts 共⌬兲, the changes in the refractive index 共⌬n2兲 are determined using both the TE mode using Eq. 共46兲 and shown in Fig. 5共a兲 and the TM mode 关using Eq. 共47兲 and shown in Fig. 5共b兲兴. At this point, we cannot determine if a mode observed in the transmission spectrum is TE or TM. As one would expect, the trends given by the two modes are essentially the same. In Fig. 5, refractive index dependence on potassium carbonate concentration 0.001 0.0005 0 0 0.05 0.1 0.15 Molar concentration g-mols/liter FIG. 5. 共Color online兲 共a兲 Dependence of n2 on potassium carbonate concentration 共TE mode used for experimental results兲; 共b兲 dependence of n2 on potassium carbonate concentration 共TM mode used for experimental results兲. from the chemical literature13 is also presented for comparison. 共Note that the data in Ref. 13 are for ⬃ 590 nm.兲 The agreement between the experiments and the literature data is stronger in Fig. 5共a兲 than that in Fig. 5共b兲. This seems to indicate that in all the experiments, the observed WGMs happen to be TE modes. However, in the results summarized by this figure, we cannot independently identify the observed WGMs as either TE or TM modes. In order to clarify this point, an additional experiment was carried out where ⌬ shifts were measured while simultaneously identifying the observed WGMs as TE or TM. To accomplish this, in addition to the transmission spectrum through the fiber, scattered light from the microsphere was also monitored. This made it possible to choose the appropriate equation 关Eq. 共46兲 or 共47兲兴 to determine ⌬n2. The experimental arrangement is shown in Fig. 6. At resonance, the constructive build-up of the electromagnetic field in the sphere leads to increased scattering 共radiation兲 through the sphere surface 共corresponding to the WGM dips in the transmission spectrum兲. The scattered light from the sphere is filtered spectrally 共using a laser line filter兲 and spatially 共using a pinhole兲 and then imaged on a photomultiplier tube using a pair of lenses. A linear polarizer that is placed between the sphere and the front collecting lens allows for the identification of the observed mode as TE or TM. Note that the polarization of a mode in the sphere is predominantly in the direction normal to the plane in which the light circulates 关as defined in Fig. 1共a兲兴. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 103105-7 J. Appl. Phys. 107, 103105 共2010兲 Ioppolo, Das, and Ötügen ½ wave plate solution Solution From laser Photo diode Linear polarizer DAQ Collecting lenses PC PMT FIG. 6. 共Color online兲 Setup for index of refraction measurements with simultaneous TE and TM detection. The container in Fig. 6 is initially filled with an aqueous solution of methanol 共CH3OH兲. Pure water is then gradually added and from the observed WGM shifts 共⌬兲 the refractive index, n2, is determined using either Eq. 共46兲 or 共47兲, as appropriate. Figures 7共a兲 and 7共b兲 show typical scattering spectra for two mixture-concentrations. Figure 7共a兲 is for the case when the polarizer is adjusted such that only the TM modes are observed. In Fig. 7共b兲, the polarizer is adjusted such that the TE spectra is observed. In both figures, the simultaneously observed transmission spectra are also plot- ted. In these figures, C1 and C2 refer to two different concentrations of methanol solutions 共C2 representing a higher concentration of water兲. Figure 8 shows the experimentally determined refractive index of different concentrations of aqueous solution of methanol along with those values from the literature. Experimental data is obtained by the sensor using both the TE and TM modes. The figure shows that when the appropriate mode equation is used 关i.e., Eq. 共46兲 for the TE modes and Eq. 共47兲 for the TM modes兴, the agreement between the published literature and the present results is strong. This result shows the potential of the proposed microsphere concentration sensor. However, additional work is needed to develop robust methods either to identify TE and TM modes in the transmission spectra or to selectively excite the sphere with TE or TM modes. IV. CONCLUSIONS The simple ray tracing, plane wave approach used in the present analysis results in equations for ⌬ dependence on ⌬n2 that are identical to those approximate explicit expressions previously derived from a perturbation analysis based on a quantum-analogy model. However, we derive the explicit Eqs. 共15兲 and 共19兲 for the resonant wavelength using simple reflection principles, from which the ⌬ dependence 0.8 a) 0.7 3.5 3 0.5 Transmitted C1 Transmitted C2 Scattered C2 Scattered C1 0.4 2.5 0.3 Scattered light (au) Transmitted light (au) Trasmitted light (au) 0.6 0.2 2 0.1 1.5 1.31278 0 1.31279 1.3128 1.31281 1.31282 1.31283 1.31284 1.31285 l, mm m 1 b) 0.9 9 0.8 Trasmitted light (au) Transmitted light (au) 8 0.7 Transmitted C1 Transmitted C2 7 0.6 Scattered C1 6 Scattered C1 0.4 5 0.3 4 0.2 3 2 1.31286 0.5 Scattered light (au) 10 0.1 1.31286 1.31287 1.31287 1.31288 1.31288 1.31289 1.31289 1.31290 1.31290 0 1.31291 l, mm FIG. 7. 共Color online兲 Typical scattering spectrum along with the corresponding transmission spectrum; 共a兲 TM mode and 共b兲 TE mode. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp 103105-8 1.344 1.343 n2 J. Appl. Phys. 107, 103105 共2010兲 Ioppolo, Das, and Ötügen Wolf et al (1965) 1.342 Experiment, TM mode 1.341 Experiment, TE mode 1.34 1.339 1.338 1.337 low atmospheric pressures. However, this also means that in other sensor applications 共such as temperature, pressure, or force兲 gaseous contaminants in air would not have a significant effect on signal. For a typical liquid application Eqs. 共46兲 and 共47兲 are distinct enough that the nature 共TE or TM兲 of the WGM tracked needs to be known so that the appropriate equation is used. 1.336 1.335 15.000 17.000 19.000 21.000 Molar concentration of AM g-mols/liter 23.000 FIG. 8. 共Color online兲 Dependence of refractive index on molar concentration of aqueous methanol solution. ACKNOWLEDGMENTS This research effort was supported by the National Science Foundation through Grant No. CBET-0502421. G. Guan, S. Arnold, and M. V. Ötügen, AIAA J. 44, 2385 共2006兲. T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Ötügen, and V. Sheverev, Appl. Opt. 47, 3009 共2008兲. 3 T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, J. Appl. Phys. 105, 013535 共2009兲. 4 T. Ioppolo and M. V. Ötügen, J. Opt. Soc. Am. B 24, 2721 共2007兲. 5 T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, Opt. Express 17, 16465 共2009兲. 6 E. Krioukov, D. J. W. Klunder, A. Driessen, J. Greve, and C. Otto, Opt. Lett. 27, 512 共2002兲. 7 B. R. Johnson, J. Opt. Soc. Am. A 10, 343 共1993兲. 8 I. Teraoka, S. Arnold, and F. Vollmer, J. Opt. Soc. Am. B 20, 1937 共2003兲. 9 G. Schweiger and M. Horn, J. Opt. Soc. Am. B 23, 212 共2006兲. 10 R. F. Harrington, Time-Harmonic Electromagnetic Fields 共Wiley Interscience-IEEE, New York, 2001兲. 11 Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun 共Dover, New York, 1970兲. 12 B. E. Little, J.-P. Laine, and H. A. Haus, J. Lightwave Technol. 17, 704 共1999兲. 13 A. V. Wolf, G. Morden, and M. G. Brown, Handbook of Chemistry and Physics, 46th ed. edited by R. C. Weast and S. M. Selby 共CRC Press, Boca Raton, FL, 1965兲, p. D-129. 1 on ⌬n2 can be obtained by differentiation. Equations 共15兲 and 共19兲 are essentially more accurate form of Eq. 共1兲, that include additional terms. The second term is identical for both the TE and TM modes, and represents the effect of the curvature of the sphere surface. For large l, this term becomes negligible compared to the first term. The last term is the effect of refractive index of the surrounding medium on the resonance condition. For large l, this term also is dominated by the first term on the right hand side. The quality factor analysis shows that the energy losses due to surface radiation are very small 共the Q-factors are generally large even for small order numbers兲. Equations 共46兲 and 共47兲 describe the dependence of WGM shifts on the refractive index change in the medium surrounding the sphere. The equations suggest that the proposed concentration sensor would not be sensitive enough for applications in a gas medium at or be- 2 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp