Fundamental limit of the microresonator field uniformity and slow light enabled angstrom-precise straight-line translation
FFundamental limit of the microresonator field uniformity and slow light enabled angstrom-precise straight-line translation M. S UMETSKY
Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, UK [email protected]
We determine the fundamental limit of the microresonator field uniformity. It can be achieved in a specially designed microresonator, called a bat microresonator, fabricated at the optical fiber surface. We show that the relative nonuniformity of an eigenmode amplitude along the axial length π³π³ of an ideal bat microresonator cannot be smaller than π π ππ ππππ ππ βππ πΈπΈ βππ π³π³ , where ππ ππ , ππ and πΈπΈ are its refractive index, the eigenmode wavelength and Q-factor. In the absence of losses ( πΈπΈ = β ), this eigenmode has the amplitude independent of axial coordinate and zero axial speed (i.e., is stopped) within the length π³π³ . For a silica microresonator with πΈπΈ = this eigenmode has the axial speed ~ 10 -4 c , where c is the speed of light in vacuum, and its nonuniformity along the length π³π³ = ππππ at wavelength ππ = . Β΅ππ is ~ 10 -7 . For a realistic fiber with diameter 100 Β΅m and surface roughness 0.2 nm, the smallest eigenmode nonuniformity is ~ 0.0003. As an application, we consider a bat microresonator evanescently coupled to high Q-factor silica microspheres which serves as a reference supporting the angstrom-precise straight-line translation over the distance π³π³ exceeding a hundred microns. A monochromatic optical field in an ideal unconfined uniform medium with refractive index ππ ππ can have the form of a plane wave πΈπΈ = π΄π΄ exp( ππππππ ) with propagation constant ππ = 2 ππππ ππ / ππ and wavelength ππ . The amplitude of this field, | πΈπΈ | = π΄π΄ , is independent of coordinates . The question if the amplitude of a confined optical field β e.g., an eigenmode of an optical resonator β can be uniform in a finite spatial area is less intuitive: this field is localized due to reflections which cause interference and spatial variation of its amplitude. For example, light confined in one dimension along axis ππ oscillates, roughly, as Acos( ππππ ) and the condition of its constant amplitude requires ππ = 2 ππππ ππ / ππ = 0 . Thus, for finite ππ ππ , the amplitude of a confined field cannot be spatially uniform in 1D unless its frequency ππ = ππ / ππ tends to zero, i.e., unless the field is a stationary electric field. The situation in 2D and 3D is different since then the field can rotate along a closed path with a finite propagation constant without reflections and, simultaneously, can have zero propagation constant along the direction transverse to this path. As an example, consider a SNAP bottle microresonator (BMR) introduced along the surface of an optical fiber [1]. Whispering gallery mode (WGM) eigenstates of such resonator with azimuthal, radial, and axial quantum numbers ππ , ππ , and ππ are determined in cylindrical coordinates ( π§π§ , ππ , ππ ) , as πΈπΈ ππππππ ( π΅π΅π΅π΅π΅π΅ ) ( π§π§ , ππ , ππ ) = ππ ππππππ πΉπΉ ππππ ( ππ ) Ξ¨ ππππππ ( π§π§ ) . The eigenwavelengths ππ ππππππ of this BMR are the eigenvalues of the one-dimensional wave equation for Ξ¨ ππππππ ( π§π§ ) [1]: ( ) .( ) ( ) ( ) 0,( ) 2 ( ( ) ) mpq mpq m mpqmpq mpq mpcr p mp d z z zdzz n z i Ξ²Ξ² Ο Ξ» λλ Ξ³ β Ξ¨ + Ξ¨ == β + (1)
Here πΎπΎ ππππ determines the WGM propagation loss, ππ ππππ ( ππ ) ( π§π§ ) is the fiber cutoff wavelength depending on the axial coordinate π§π§ , and ππ ππ is the effective refractive index of the fiber. For a fiber having refractive index distribution ππ ππ ( ππ ) , function πΉπΉ ππ , ππ ( ππ ) satisfies the equation [2]: mp mp r mp d F dF mk n Fd d Ο Ξ²Ο Ο Ο Ο ο£Ά+ + β β = ο£·ο£ ο£Έ . (2) In Ref. [3] we presented an analytical example of a BMR having an eigenmode with ππ = 0 (fundamental axial eigenmode) which amplitude is uniform along a fraction of the resonator length π§π§ < π§π§ < π§π§ . It was shown that the BMR should be uniform along this length and have βearsβ at edges (Fig. 1(a)). Since the profile of this BMR resembles the profile of a bat, this BMR was called the bat microresonator (BatMR). It is straightforward to generalize results of Ref. [3] and design a BatMR having an eigenmode with arbitrary ππ which amplitude is uniform along the fiber segment π§π§ < π§π§ < π§π§ . Below, we call this eigenmode the bat mode . For larger quantum numbers ππ , the ears of BatMRs should be appropriately larger. As an example, Figs. 1(a)-(c) show the nanoscale profile, variation of cutoff wavelength ππ ππππ ( ππ ) ( π§π§ ) , and the bat mode profile determined from Eq. (1) for ππ = 2 . We constructed the profile of ππ ππππ ( ππ ) ( π§π§ ) shown in Fig. 1(b) numerically by varying the size of ears at the edges of the segment π§π§ < π§π§ < π§π§ until the eigenwavelength ππ ππππ2 becomes equal to the value of ππ ππππ ( ππ ) ( π§π§ ) at this segment. Experimentally, a BatMR can be created using the SNAP technology [1, 4, 5, 6, 7] as follows. First, a BMR with sufficiently large slopes at the edges and uniform center part (e.g., a rectangular BMR [6]) is formed. Next, the ears at the BMR edges can be introduced by iterations with CO laser beam shots [5], so that ππ ππππππ rises and, finally, coincides with ππ ππππ ( ππ ) ( π§π§ πΆπΆ ) . In the ideal case of no losses, πΎπΎ ππππ = 0 , a BatMR can have a series of bat modes corresponding to different azimuthal quantum numbers ππ , which amplitude is independent of axial coordinate and has zero axial speed (i.e., is stopped) along the segment π§π§ < π§π§ < π§π§ . For a lossy BatMR with an ideally uniform section along the segment π§π§ < π§π§ < π§π§ , he best possible WGM uniformity is determined by the value of losses πΎπΎ ππππ = ππ ππππ ππβ , where ππ is the BatMR Q-factor. From Eq. (1), the smallest possible variation of an eigenmode Ξ¨ ππππππ ( π§π§ ) is achieved for smallest possible propagation constant π½π½ ππππππ ( π§π§ ) β‘ π½π½ ππππππ ( ) = 2 / ( ππ +1) ππππ ππ ( ππ ππππ ( ππ ) ) β3 / ( πΎπΎ ππππ ) / , i.e., when the cutoff wavelength ππ ππππ ( ππ ) ( π§π§ ) is constant and equal to the eigenwavelength of this eigenmode, ππ ππππ ( ππ ) ( π§π§ ) β‘ ππ ππππππ (Figs. 1(b) and (c)). The axial speed of this bat mode, at the segment π§π§ < π§π§ < π§π§ is ππ ππππππ = (2 ππ ) β1 ππππ ππππ Re οΏ½π½π½ ππππππ ( ) οΏ½ =(2 ππ ) β1 / ππ ππ ππ , where ππ is the speed of light. For a silica microresonator with ππ ππ = 1.44 and ππ = 10 this speed is ~ ππ /10000 . Using Eq. (1) and the approximate expression for the bat mode radial dependence (see, e.g., [8]), we determine the evanescent WGM inside this segment assuming that it is symmetric with respect to the segment center π§π§ πΆπΆ = ( π§π§ + π§π§ )/2 as ( ) ( )
0( ) 3/2 1/2 2 1/2( ) (0)( )0 ( 1) ~ exp( )exp ( ) cos ) ,( ) (1 ), 2 ( 1) ( ) .( , , ) (2
BMRmpq mpq mpqmpq Cc cr mp rmp mp mp
E im r zz n zi n Ο ΞΆ ΟΞ» Ξ³ ΞΆΟΟ Ο Ξ²Ξ² Ο Ξ» β β β β= β= β+ (3)
From Eq. (3), in a close vicinity of the BatMR surface, when ππ ππππππ ( ππ β ππ ) << 1 , and | π½π½ ππππππ ( ) ( π§π§ β π§π§ πΆπΆ )| << 1 , we find the relative variation of the amplitude of the evanescent field along the segment π§π§ < π§π§ < π§π§ as ( 4 4 42 ( ) 4) ( )0 0 ( ) | ( , , ) / ( 16 ( )3 (, ) | ), r CBMR BMRmpq mpq cmp z E z r E z zr n zQ Ξ΅ Ο Ξ»Ο Ο β= β = , (4) while the field amplitude is constant at the surface r C cmpr n z zz z r Q n ΟΟ Ο Ξ»ββ = β = β . (5) From Eq. (4), we find the smallest possible field uniformity along the length πΏπΏ of the BatMR surface as ππ οΏ½ πΏπΏ2 οΏ½ = ππ ππ ππ4 ππ β4 ππ β2 πΏπΏ . Fig. 1(g) shows the profiles of ππ ( π§π§ ) and βππ ( π§π§ ) for a silica BMR with r n = at wavelength ππ ππππππ = 1.55 Β΅m for different Q-factors, ππ = 10 , 10 , 10 , and . Generally, it is possible to design a BMR having an eigenmode which amplitude is uniform within a fraction of its surface, cross-section, and volume . We assume, as previously, that at the cutoff wavelength ππ = 2 ππ ππ β = ππ ππππ ( ππ ) the propagation constant of this mode π½π½ = 0 and this WGM is uniform along the BMR segment π§π§ < π§π§ < π§π§ . Provided that π½π½ = 0 , it follows from Eq. (2) that oscillations along a radial segment ππ < ππ < ππ can be suppressed if the squared effective refractive index, ππ ππππππ2 ( ππ ) = ππ ππ2 ( ππ ) β ππ ( ππ ππ ) β , is zero at this segment (see Supplementary Material in [9] where π½π½ ππππππ for radially nonuniform ππ ππ ( ππ ) was determined). Analogous to the design shown in Fig. 1(a)-(c), we introduce two βearsβ at the edges of this segment (Fig. 1(d) and (e)) and adjust their sizes to ensure | πΈπΈ ππππ ( π§π§ , ππ , ππ )| =| πΉπΉ ππππ ( ππ )| = ππππππππππ for ππ < ππ < ππ (Fig. 1(f)). To minimize the required modification of the refractive index and ensure the condition of total internal reflection, this radial segment should be sufficiently short and close to the fiber surface. Besides the fundamental limit of the microresonator field uniformity caused by optical losses, the uniformity of BMR field πΈπΈ ππππππ ( π΅π΅π΅π΅π΅π΅ ) ( π§π§ , ππ , ππ ) is determined by the uniformity of the fiber. Here we are interested in the nanoscale effective radius variation (ERV) of an optical fiber Ξππ ππππππ ( π§π§ ) which is determined through the profile of its cutoff wavelength variation Ξππ ππππ ( ππ ) ( π§π§ ) as Ξππ ππππππ ( π§π§ ) = ππ Ξππ ππππ ( ππ ) ( π§π§ )/ ππ ππππ ( ππ ) , where ππ is fiber radius. The predetermined Ξππ ππππππ ( π§π§ ) and Ξππ ππππ ( ππ ) ( π§π§ ) can be introduced with subangstrom precision using fabrication methods of the SNAP technology (see e.g., [1, 3 - 7]). The possible measurement precision of Ξππ ππππ ( ππ ) ( π§π§ ) is determined by the fiber and microresonator Q-factors. Assuming ππ = 10 [10], ππ = 50 Β΅m and ππ ππππ ( ππ ) = 1.5 Β΅m, we find that Ξππ ππππ ( ππ ) ( π§π§ ) can be measured with the precision better than ππ ππππ ( ππ ) ( π§π§ )/ ππ = 0.01 pm and the ERV Ξππ ππππππ ( π§π§ ) can be measured with the precision better than 0.3 pm. The ERV experimentally measured in [11] was less than 0.2 nm over the axial lengths of hundreds of microns. Fig. 1. (a), (d) Illustration of an optical fiber with (b) cutoff wavelength distribution (solid black curve) and (e) original (dashed black curve) and modified (solid black curve) cross-sectional effective refractive index squared distribution designed to have a portion with (c) uniform axial distribution and (f) uniform radial distribution of the WGM amplitude. Notice asymmetry of βearsβ of ππ ππππππ2 ( ππ ) in (e) in contrast to symmetric ears of ππ ππ ( π§π§ ) in (b). Inset in (a) β ERV profile of a BatMR. (g) Relative nonuniformity ππ ( π§π§ ) of an eigenmode and constant eigenmode amplitude profile βππ ( π§π§ ) at the BMR surface for different Q-factors of a silica BatMR with ππ ππ = ππ ππππππ = The original ERV of optical fibers is primarily caused by the frozen-in capillary waves having the order of an angstrom [12-14]. It can be found from Eq. (1) and rescaling equation
Ξππ ππππππ ( π§π§ ) = ππ Ξππ ππππ ( ππ ) ( π§π§ )/ ππ ππππ ( ππ ) that the perturbation of the field along the uniform BatMR segment by the ERV spatial spectral component Ξππ ππ ( π§π§ ) = βππ exp ( πππππ§π§ ) results in relative variation of the field magnitude
02 2 ( ) 2 1( ) ( )0 00 0 0 ( ) ( , , ) / ( , exp( ),8 ( ) . , ) kBMR BMRk m kk mpq pqmpcr z E z r E z ikzn k r r r Ξ΅ Ο Ξ» Ο Ο Ξ΅Ξ΅ β β = β= β = (6)
For a silica fiber with characteristic
Ξππ = 0.2 nm, ππ = 50 Β΅m, ππ ππππ ( ππ ) = 1.5 Β΅m, and frozen-in wave spatial frequency ππ = 1 Β΅m -1 , we find the relative field nonuniformity ππ β . uppressing this ERV is currently challenging for lengthy optical fibers [14]. However, we suggest that it can be reduced to less than 1 β« by surface postprocessing developed in SNAP technology [1, 4- 7]. Fig. 2. (a) Translation of a BatMR coupled to five SMRs, 1,2,3,4, and 5. Inset β ERV profile of the BatMR. (b) SMR and a BatMR in coordinate systems. (c) Dependence of eigenwavelength splitting βππ and splitting variation πΏπΏππ caused by β« change of BatMR-SMR separation ππ = 200 nm. (d) A BatMR with asymmetric cross-section coupled to three SMR before and after small rotation and displacement. The unique uniformity of the bat mode amplitude suggests the application of a BatMR as an angstrom-precise translation reference . Besides fundamental interest, reaching the angstrom and eventually picometer precision of translation is critical for several applications in nanotechnology and nanoscience. In particular, solution of this problem is important in semiconductor manufacturing [15], atomic-scale electronic engineering [16, 17] as well as for manufacturing of metamaterial, plasmonic and nanophotonic devices [18]. Conventional approaches developed for ultraprecise linear translation are based on capacitive and piezo sensors and optical interferometers (see [19-24] and references therein). The stages with 10 pm resolution enabling translation over distances of 10 Β΅m are available on the market [25, 26]. However, to our knowledge, the problem of translation with the subnanometer precise straightness and flatness has not been satisfactory explored. The reason is presumably in the absence of the reference which allows to follow the straight direction along the required length and with the required precision. The best optical flats, including those fabricated of silica, have the flatness of around 1 nm over the sub-millimeter areas [27], and their application to support the subnanometer-precise straight-line translation at microscale is problematic [28]. In contrast, we show below that a BatMR can be used as a reference for angstrom-precise straight and flat translation along its axial segment with length πΏπΏ ~100 Β΅m. The device proposed here consists of a BatMR and a set of spherical microresonators (SMRs), which are positioned in a submicron distance from the BatMR as illustrated in Fig. 2(a). Light is coupled into SMRs through prisms (shown in this figure) or fiber tapers with micron diameter waist [29]. The output light is detected by an optical spectrum analyzer not shown in Fig. 2(a). Each of SMR controls one degree of the BMR freedom by monitoring the splitting of resonance wavelengths [30, 31]. Therefore, in the absence of rotational BatMR symmetry (see below) we need five SMRs illustrated in Fig. 2(a) to fix all five BMR transverse degrees of freedom and enable its precise translation along a straight line. We assume that each of the SMRs have a WGM πΈπΈ ππ π π ππ π π ππ π π ( πππ΅π΅π΅π΅ ) ( ππβ² , ππβ² , ππβ²οΏ½ with the wavelength eigenvalue ππ ππ π π ππ π π ππ π π ( πππ΅π΅π΅π΅ ) which has or tuned to have very small separation ΞΞ from bat mode eigenvalues ππ ππππππ chosen different for different SMRs (it should have the same axial quantum number ππ but different azimuthal quantum numbers ππ ). A better accuracy of simultaneous matching of the eigenwavelengths of these bat modes and corresponding cutoff wavelengths of the fiber can be achieved for shallower and wider BatMR ears. In close proximity of a SMR to the BatMR, the bat mode eigenvalue splits into two eigenvalues, which separation is determined as ( Ξππ + Ξπ¬π¬ ) where [32]: ( ) ( )( ) * ( ) *2 20 s s s s s s SMR BMRm p q mpqBMR SMRmpq m p qr S dE dEdxdz E En k dy dy Ξ»  ο£Άβ = β  ο£·ο£ ο£Έ β«β« . (7) Here the wavenumber ππ = 2 ππ ππ ππππππ β , the BatRM and SMR modes are normalized, and the integral is taken along the plane π¦π¦ = ππ π π + ππ /2 where ππ π π is the SMR radius and ππ is the BatMR-SMR separation (Fig. 2(b)). The refractive index ππ ππ of all microresonators is assumed to be the same. To enable the full control of the BatMR translation, below we explore a BatMR with an asymmetric cross-section. However, to estimate the SMR-BatMR coupling sensitivity, it is sufficient to use the axially symmetric model. We assume that the length πΏπΏ of the BatMR segment with axially uniform πΈπΈ ππππππ ( πππ΅π΅π΅π΅ ) ( ππ , ππ , πποΏ½ is much greater than the length of bumps enabling this uniformity (see insert in Fig. 2(a)). Under the assumptions made, the integral in Eq. (7) is found analytically. As the result, the wavelength splitting for the SMR radial and axial quantum numbers ππ π π , ππ π π = 0,1,2,3,4 and BMR radial quantum numbers ππ , ππ =0,1,2,3,4 is ( ) s s p q p rr r s s B C B k d nn n k r r r r L ΟΞ» β ββ = β + (8) where π΅π΅ = (2.018, 1.762, 1.634, 1.552, 1.494) and πΆπΆ =(3.733, 0, 3.257, 0, 3.088) . Due to the antisymmetric behavior of SMR modes on π§π§ β² for odd ππ π π , the splitting is zero for ππ π π = 1 and 3. The plots of Ξππ as a function of length πΏπΏ for SMR radius ππ π π = 50 Β΅m and BatMR radii ππ π π = 50 , 100 and 200 Β΅m are shown in Fig. 2(c). The values of Ξππ are in reasonable agreement with the splitting between microspheres experimentally measured in Ref. [33]. In addition, Fig. 2(c) shows dependencies of the splitting variation πΏπΏππ corresponding to the change of 1 β« in SMR-BatMR separation ππ = 200 nm. These variations can be measured for microresonators with ππ β³ and SMRs with precisely tuned eigenwavelengths to ensure ΞΞ β² Ξππ . Indeed, then for ππ ππππ = 1.55 Β΅m the FWHM of the resonance is πΎπΎ ππππ β² pm and variations πΏπΏππ can be measured with a power meter having smaller than 10% relative error. For precise measurement of Ξ΄ππ , the free spectral range along the axial quantum number ππ of the WGM considered should be greater than or comparable with the resonance width πΎπΎ ππππ . This condition leads to the characteristic maximum length of translation πΏπΏ ππππππ = ππ ππππ ππ / ππ ππβ1 . For example, for the same parameters as in Fig. 2, πΏπΏ ππππππ β ππ m for ππ = 10 . The splitting in Eq. (8) depends on the separation between a SMR and BatMR. Keeping it constant will force the BatMR to follow its surface profile. The fundamental quantum limit of the measurement precision of the wavelength splitting Ξππ determined by Eq. (8) is [30, 31] /2 1 1/2 1/2 ( ) 3/2( ) 2 1/2 1/2 1/2 cmnquant cmn r c cQ W d n Q W Ξ» λδλ Ο Ξ» Ο Ξ»Ο β  ο£Ά ββ ο£Ά= = ο£·  ο£·β β βο£ ο£Έο£ ο£Έ ο¨ ο¨ . (9) Here ππ is the speed of light β is the Plank constant, ππ is the power of light in the optical spectrum analyzer used, Ξππ is determined from Eq. (8), and ππ is the measurement time. Setting ππ = 10 , Ξππ = 0.1 nm, ππ = 10 mW, and ππ = 1 ms, we find πΏπΏππ ππππππππππ ~10 β4 pm, which is significantly smaller than the precision πΏπΏππ ~10 β2 pm required for our measurements (Fig. 2(c)). The major deviation from the straight-line translation is caused by the variation of BatMR local surface height and variation of SMRs and BatMR dimensions in time. While the angstrom-scale surface height variation of an actual BatMR can be recorded and taken into account in the process of translation, the temperature variation affects the measurement precision randomly. For the microresonator radii ππ , ππ π π ~
100 Β΅m and the temperature change β² β , we find their radius variation β² β« , which insignificantly affects the angstrom-precise straight-line translation. Finally, we determine the cross-sectional asymmetry of the BMR required to suppress its axial rotation in the process of translation. We assume that the BMR cross-section has the elliptic shape with semi-major and semi-minor axes ππ and ππ shown in Fig. 2(d). We position SMR 1 at the vertex and co-vertex of the ellipse and determine the position of SMR 3 to arrive at the maximum possible separation βππ between it and BMR after the BMR is rotated by small angle βππ . It is assumed that the separations BMR and SMR 1 and between BMR and SMR 2 is kept constant during this rotation. Cumbersome calculations yield the following simple result. The maximum separation βππ = ( ππ βππ ) βππ is achieved for the SMR 3 located at angle πΌπΌ = atan[( ππ / ππ ) / ] with respect to major axis of the elliptic cross-section (Fig. 2(d)). For the BMR with small asymmetry, πΌπΌ β and 1 β« rotational displacement at the BMR surface corresponds to Ξππ β (1 β« )/ ππ . For example, assuming ππ β ππ = 0.1 ππ we find βππ β pm. It follows from the above calculations that this small variation can be measured with microresonators having ππ β₯ and relative measurement precision error smaller than 1%. For ππ ~10 or for BMR with a greater rotational asymmetry, the required measurement precision can be relaxed to 10%. Thus, maintaining constant values of splitting βππ for all SMRs during translation allows us to use the constant-amplitude bat mode field as a translation reference enabling angstrom-precise straightness and flatness. Funding.
Engineering and Physical Sciences Research Council, (EP/P006183/1), Wolfson Foundation (22069).
Disclosures . The author declares no conflicts of interest.
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