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We fabricated nanometer- and micrometer-order diameter optical fibers (NMOFs) by drawing them in a microfurnace comprising a sapphire tube heated with a CO2 laser. Using very short - a few mm long - fiber biconical tapers having a submicron waist, which can be bent locally in a free space by translation of the taper ends, we studied the effect of bending and looping on the transmission characteristics of a free NMOF. In particular, we have demonstrated an optical interferometer built of a coiled self-coupling NMOF.
©2004 Optical Society of America
1. Introduction
Silica nano- and micrometer-order-diameter optical fibers (NMOFs) fabricated by drawing [1–7] have much better transmission characteristics than similar optical waveguides fabricated lithographically [8]. NMOFs have found applications as efficient couplers to high-Q microcavities [1,3], sensors [9], and enhancers of light intensity and non-linear effects, in particular for supercontinuum generation [3,6]. In the future, NMOFs may be considered as building blocks for microphotonics applications such as waveguides and microdevices enabling filtering and delay of light [5,10].
In this paper, we investigate the transmission spectrum of a free NMOF, which is bent or coiled. We define a free NMOF as a microfiber, which is not supported and/or touched by other elements at any point along its length except for the input and output connections. An example of a free NMOF is a micrometer-order diameter part of a biconical fiber taper. Investigation of transmission characteristics of the free NMOFs is important for basic understanding and development of the NMOF-based photonic devices.
Our technique for fabrication of NMOFs, which employs a sapphire tube (microfurnace) heated by a CO2 laser, is described in Section 2. In Section 3, we present the geometric and transmission characteristics of two short fiber tapers containing single-mode NMOFs, which are used in our experiments. Because the tapers are short, having lengths of a few mm, they are convenient for bending and looping of the NMOFs in free space. In Section 4, we experimentally determine the bend loss of the NMOF and compare it with theoretical calculations. Our results show that the transmission spectrum allows to independently calculate the diameter of the NMOF and its bend radius. In Section 5 we study the propagation of light along the coiled self-touching NMOF. It is shown that self-coupling of the propagating mode near the contact point can give rise to strong interference oscillations in the transmission spectrum. The experimental results demonstrating a self-coupling microloop interferometer are in good agreement with theoretical predictions. Finally, the results of the paper are summarized in Section 6.
2. Drawing NMOF with indirect CO2 laser heating
The NMOFs investigated in this paper are drawn with laser heating, which is an alternative to the fabrication method of NMOF employing flame heating [1–7]. In the fabrication of the high-quality microfibers with flame heating, the flame should be clean and low-flow so that the burning gas and air convection do not break the fiber [2,7]. Careful control of the gas flow is critical for the reproducible drawing process [7]. In order to reduce the effect of flame, and, in particular, the air convection, authors of Ref. [5] developed the technique of two-step NMOF fabrication. At the first step, the conventional optical fiber was drawn down to a NMOF with a micron-order-diameter using direct flame heating. At the second step, the obtained microfiber was wound on a sapphire tip, which was heated by a flame at a distance from the fiber, and drawn further to a submicron diameter. Using this technique one can fabricate very thin NMOFs. Being a part of a conical taper, these NMOFs need additional coupler to be connected to the light source and drain from both ends [5].
Laser heating solves the problem of cleanness, makes it easier to achieve reproducibility, fully eliminates the flow of the burning gas, and reduces the air convection. However, it is known that drawing of very thin microfibers using direct heating by a laser beam is impossible: the minimum diameter of the microfiber is limited by the power of the beam [11,12]. Actually, for the NMOF heated with a CO2 laser, the power acquired from the laser beam is proportional to the volume of the fiber i.e. it drops as the fiber radius squared. At the same time, the power dissipated by the fiber is proportional to the surface area of the fiber, i.e. it drops linearly with the fiber radius. At certain radius the dissipated power equals the acquired power and melting the fiber is no longer possible [11,12].
Nevertheless, there exists a way of fabricating the NMOF by indirect melting of the fiber using a laser. Our setup for NMOF fabrication employing a CO2 laser is illustrated in Fig. 1. Building on the idea of Ref. [5], we place the regular optical fiber into the sapphire capillary tube (microfurnace), which is heated by a CO2 laser beam. The beam size along the tube is controlled by its focusing or defocusing with a lens. The setup is operated by displacement of four translation stages, which allow fabrication of the biconical tapers and program their diameter variation (Fig. 1). The fiber ends are fixed at stages 1 and 2. The fiber can be translated as a whole with respect to the laser beam using stage 3. Stage 4 governs the displacement of the sapphire tube with respect to the fiber and the laser beam. This stage allows the removal of the tube from the tapered fiber segment after drawing is completed. In the experiments of this paper, the outer and inner diameters of the tube are 0.9 mm and 0.6 mm, respectively. For these tube dimensions, the mass of the melting fiber is much smaller than the mass of the corresponding heated section of the sapphire capillary. Therefore, the temperature inside our microfurnace is not affected by the fiber radius variation in the process of microfiber drawing. Notice that heating the sapphire tube from only one side introduces a temperature gradient along the cross-section of the tube. The latter effect is small for a thin microfiber and can be further reduced by the axially symmetric heating technique [13,14], which was not employed in the present paper. As a simple illustration of the result of the drawing process, Fig. 2 shows a scanning electron microscope (SEM) image of a submicron-diameter section of a NMOF, which was drawn from a regular single mode fiber by simple pulling with stage 1 only. The NMOF diameter decreases from 700 nm to less than 100 nm.
3. Short NMOFs
In order to simplify local bending of a free NMOF and achieve a small bend radius, we employ short biconical tapers containing NMOF. We bend NMOF by translation of the taper ends with respect to each other using a manually operated translation stages as illustrated in Fig. 3. The result of bending of the NMOF is controlled visually by an optical microscope. The spectrum of transmitted light is detected by an optical signal analyzer (OSA). Fig. 4(a) shows the diameter variation for two short fiber tapers, having lengths of 6 mm and 8.5 mm, fabricated from a standard single mode fiber, which were used in the experiments described below. The tapers were drawn by symmetrically pulling the ends of the fiber in opposite directions using translation stages 1 and 2 in Fig. 1. The laser power was about 20 W. We used a cylindrical lens to focus light along the fiber length (see Fig. 1). The characteristic width of the laser beam near the sapphire tube was about 0.5 mm. In the fabrication of taper 1, the laser beam was more focused than in the fabrication of taper 2. For this reason, taper 1 is more rapidly contracting than taper 2. The diameter variation was determined by a SEM with an axial measurement interval of 200 µm. The minimum diameters of the tapers 1 and 2 are approximately 700 nm. The transmission spectrum of the fibers in the bandwidth from 1520 nm to 1620 nm is shown in Fig. 4(b). The low transmission values of tapers 1 and 2, which are close to -13 dB and -9 dB, respectively, are primarily caused by transitions from the fundamental mode to cladding modes and radiation of light in the short (non-adiabatic) conical regions. In this study we are interested only in the change of the transmission due to bending and looping of the NMOF. All the measurements displayed below report the difference between the bent/looped NMOF and the originally straight NMOF.
Fig. 3. Illustration of the setup for NMOF bending. The ends of biconical fiber taper are glued to the stages. The waist of taper can be bent by translation of one of the stages in three dimensions.
Fig. 4. — variation of diameter of fiber tapers 1 and 2 used in our experiments as a function of length; insert shows variation of diameter for the NMOF parts of these tapers. b — transmission spectrum for the same fibers. Solid curves — fiber 1; dashed curves — fiber 2.
4. Transmission of a bent NMOF
In this section we experimentally study a NMOF, which is bent locally with a characteristic bend radius of a few tens of microns, and compare the experimentally observed transmission spectrum with theoretical predictions. Bent dielectric waveguides and, in particular, bent fibers experience loss, which decreases exponentially with the curvature radius R [15]. The loss can be viewed as tunneling of light out of the fiber due to existence of an “effective potential” which is linear in the coordinate along the normal of the curved fiber axis. The power attenuation of the fundamental mode after traveling a distance L of a microfiber, which is bent with a constant radius R, is determined as a function of radiation wavelength, λ 0, by the equation:
where the power attenuation coefficient, γ, can be calculated by the asymptotic formula [15]:
Here r is the fiber radius, n is the fiber refractive index, K 1(x) is the modified Bessel function of the second kind, and the parameters V(λ 0), U(λ 0) and W(λ0) are expressed in terms of the radiation wavelength, λ 0, as follows:
In Eq. (3), function β(λ 0) is the propagation constant of fundamental mode of microfiber. Eq. (2) is derived using the weak guiding approximation when the distribution of electromagnetic field is assumed to be symmetric with respect to the fiber axis. For the strongly guiding HE11 mode considered in this paper, the pre-exponential factor in Eq. (2), which is a weak function of NMOF parameters, is accurate within a factor of ~ 1. However, the most important strong exponential dependence in Eq. (2) is asymptotically accurate if the propagation constant β(λ 0) is determined by solution of the transcendental equation, which is exact for the HE11 mode [15]:
where J 1(U) is the Bessel function of the first kind and ΄ denotes a derivative. The asymptotic Eq. (2) is valid only if the argument of the exponent is large. Then the dependence on the parameters of the NMOF is primarily determined by the exponential term in Eq. (2) and this equation can be used for estimation of the bend radius R as well as the NMOF radius r from the transmission spectrum of a microfiber.
Fig. 5. Transmission spectrum and digitally sharpened optical microscope images for different configurations of NMOF contained in taper 1. a — bent NMOF; b and c — coiled NMOF; d — NMOF with straight central part.
We used taper 1 from Fig. 4 for the experimental investigation of bending losses in a free NMOF. The spectrum of straight taper 1 containing a NMOF is used as a reference. The NMOF was bent by translation of one of the taper ends towards the other end (Fig. 3). The shape of the NMOF was determined with an optical microscope and its spectrum was measured with OSA. Moving one of the taper ends we have bent the NMOF locally as shown in Fig. 5(a). From this figure we could suggest that the bent segment of the NMOF is close to a semicircle with radius of curvature around 10 µm. In the actual optical microscope image, which was digitally sharpened for clarity in Fig. 5(a), one of the arms of the bent NMOF was blurred indicating that the NMOF was bent out of the plane of the image. For this reason, the actual radius of curvature can be estimated to be around 10 µm or greater. It is reasonable to suspect that the waist of the taper was bent in its thinnest part. Next, translating the taper end further towards the other end, we displaced the NMOF further out of the image plane so that the sharp bend was transformed into a less bent loop shown in Fig. 5(b). The enhancement of transmission is caused by an increase of the bend radius. Moving the taper end further along the same direction we increased the bend radius of the loop and its transmission as shown in Fig. 5(c). Continuing the displacement of the taper end we arrived at the configuration of Fig. 5(d) when the thinnest part of the NMOF became straight. In this case, the transmission spectrum was practically restored to the original one of the straight taper demonstrating no visible losses. Fig. 5(d) confirms that the bend loss away from the thinnest center part of NMOF is small.
We compared the experimental results for case a in Fig. 5 with the bend losses calculated from Eqs. (1)–(4) for the silica refractive index n=1.44. We assumed that the NMOF was approximately uniform along the bent segment where its diameter is close to 700 nm. For the NMOF with diameter 2r=700 nm determined in Fig. 4, the bend radius R=10 µm, and the length of curved segment L=3R=30 µm we found that the theoretically predicted loss exceeds the observed loss by orders of magnitude. This confirms that the NMOF is bent out of the plane of the image and has a larger bend radius. Using the bend radius, R, and the microfiber diameter, 2r, as fitting parameters on Eqs. (1)–(4), and assuming that the bent segment is close to a semicircle so that L=3R, we found that the values that best fit the experimental transmission measurements are R=50.1 µm and 2r=690 nm (curve 1 in Fig. 6). The latter NMOF diameter is in good agreement with the result of the SEM measurement shown in Fig. 4. If we assume a shorter length of the bent segment, L=2R, and use R as a fitting parameter, curve 2 in Fig. 6 shows the best fit, which is somewhat worse than curve 1 and gives a radius of curvature close to the one obtained for L=3R : R=44.8 µm. Curves 3 and 4 are the best fits of the experimental data using R and L=3R as fitting parameters for two diameters of the microfiber: 2r=600 nm and 2r=800 nm, respectively. It is seen that these curves fit the experimental result worse than curve 1, which is another confirmation that the average NMOF diameter of the bend segment is close to 700 nm. Curve 5 in Fig.6 shows the best fit of the experimental data for R=10 µm, 2r=690 nm, using L as fitting parameter. Obviously, this fit, which corresponds to L=0.2R, is not acceptable because of both the small value of L and the behavior of transmission as a function of wavelength. In case c, when the bend radius is around 100 µm, the theoretical values of transmission are several orders of magnitudes less than the experimental data. Presumably, the reason for the disagreement is that the thinnest part of NMOF is bent along the plane normal to the image plane that is invisible in Fig. 5(c). In case d, Eqs. (1)–(3) predict negligible losses in agreement with the experimental data.
Fig. 6. Comparison of the experimental transmission spectrum a, shown in Fig. 5 with theoretical fits obtained using Eqs. (1)–(3) for transmission loss. The bend radius, length of bent NMOF segment, and NMOF diameter for curves 1–5 are specified in the insert.
The results of this section show that that using short fiber tapers it is possible to create tightly bent NMOFs and independently calculate the correct diameter of a microfiber and its bend radius from the experimental transmission spectrum. A more detailed characterization of the shape of the NMOF, e.g. by obtaining additional images taken in orthogonal planes, can provide better understanding of relations between the geometrical and optical transmission properties of NMOFs. For accurate analysis, one should apply a more precise numerical description of bend losses rather than simple estimate we performed using Eq. (2) and take into account the variation of the NMOF diameter along the length of the bent segment.
5. Interference effects in transmission of a coiled self-coupling NMOF
The interference phenomenon can be observed when a coiled NMOF touches itself and causes self-coupling of propagating light. Under certain conditions, strong self-coupling of a microfiber loop with small losses can generate high-Q resonances [16,10]. For a loop having relatively large losses and/or small self-coupling amplitude, a simplified description of propagating light can be applied. Then, as illustrated in Fig. 7, the power spectrum of light transmitted through the loop is a result of the interference of two coherent beams, which can be written in the form:
Beam 1 propagates along the loop with amplitude a 1(λ 0) exp(iβ(λ 0)S), where S is the loop length, and β(λ 0) is the propagation constant defined by Eq. (4). The magnitude of the pre-exponential factor a 1(λ 0) is determined by the bending loss along the loop length and can rapidly decrease with wavelength if the loop radius is small as discussed in the previous section. Beam 2 propagates directly through the self-coupling region with an amplitude a 2 (λ 0), by-passing the loop. Even if the coupling amplitude a 2 (λ 0) is small it can be compared to the amplitude a 1(λ 0) for a loop with a small radius. In particular, if the amplitudes of beams 1 and 2 are close and their phases are opposite the beams will cancel each other. In this case, which is known as the condition of critical coupling in the theory of microresonators (see, e.g., Ref. [4]), the oscillations of the power spectrum determined by Eq. (5) will display a large number of maxima and minima and the microloop will behave as an optical interferometer. In this section, the microloop optical interferometer of this kind is demonstrated experimentally.
Fig. 7. Illustration of two interfering waves (dashed) contributing to the transmission spectrum of the self-coupling microloop. 1 — a wave propagating through the loop; 2 — a wave propagating through the self-coupling region.
By manipulating NMOFs in free space as described in Section 3 one can create a loop like the ones shown in Fig. 5 and move one end of the taper so that the coiled microfiber touches itself. For the fiber 1 considered in section 3, the diameter of the NMOF near the touching point is well above 1 µm and the magnitude of the fundamental mode near the surface of the microfiber is very small [2,17]. For this reason, self-touching does not introduce any visible change in transmission. However, for taper 2 from Fig. 4, which contains a NMOF with smaller diameter, the interference effect described by Eq. (5) can be observed. Fig. 8 shows the results of bending for this fiber. Moving the taper ends towards each other we first create a bent NMOF (shown in Fig. 8(a)) having strong bend losses growing from 1.5 to 7.5 dB in the bandwidth considered. Further translation creates a less bent loop having smaller bend losses shown in Fig. 8(b). Then we moved the taper end in the direction normal to the fiber axis until the NMOF touched itself and the oscillations in transmission spectrum were observed. In case c, the maximum magnitude of oscillations in the bandwidth considered was 5 dB achieved for λ0=1620 nm (Fig. 8(c)). Using Eq. (5) we find that for this magnitude the ratio a 2 (1620 nm)/a 1 (1620 nm) is 0.28. Next, moving the fiber ends away from each other we constricted the loop and decreased the amplitude a 1 (λ 0). In case d, the maximum magnitude of oscillations achieved for λ 0=1612 nm is 18 dB, which corresponds to closer values of amplitudes a 1 (λ 0) and a 2 (λ 0) having a ratio of 0.78. Notice that the reduction of the magnitude of oscillations for λ 0>1612 nm, corresponding to a power of less than 25 dB, is an artifact caused by reduction of source power of the LED, which was used in our experiments, in this region of spectrum. Finally, in case e, we made a more detailed comparison of theoretical predictions obtained from Eq. (5) and the experimental data. The experimental transmission spectrum was fitted assuming that the amplitude of propagation along the loop, a 1(λ 0), decreases exponentially with wavelength while the coupling amplitude, a 2 (λ 0), is constant. It follows from Fig. 8(e) that the loop length is around 200 µm or greater. We estimated the average NMOF diameter along the loop from the insert in Fig. 4 and assumed that it is equal to 800 nm. In very good agreement with the experimental spectrum, the best fit for the transmission spectrum determined by Eq. (5) was achieved for a 1(λ 0)=0.12 exp[-0.015(λ 0-1520 nm)], a 2 (λ 0)=0.028, and the loop length S=215 µm. The latter corresponds to the average bend radius Rav=215/(2π) µm=34.2 µm. Similar to case d, we observed a measurement artifact when the spectrum power became less than 25 dB. In order to estimate the accuracy of using the average NMOF diameter instead of the actual diameter, which is changing along the loop length, we performed a fit with a NMOF diameter equal to 700 nm and found the same parameters for a 1(λ 0), a 2 (λ 0), while the best fit for the length of the loop became S=262 µm.
Fig. 8. Transmission spectrum and sharpen optical microscope images for different configurations of NMOF contained in taper 2. a, b — bent and coiled NMOF without self coupling; c,d,e — self-coupling microcoils with successively decreasing bend radius. Fig. 8e shows comparison of the experimental data with theoretical dependence defined in the text.
Thus, in this section we experimentally observed strong oscillations in the transmission spectrum caused by self-coupling of the propagating mode in a free coiled NMOF and found good agreement between the experimental data and the theoretical model. A more accurate description of the transmission should take into account the variation of NMOF diameter along the loop and the contribution of the resonance interference between higher-order turns of propagating light [10].
6. Summary
In this paper we investigated the effects of bending and looping of free NMOFs on their transmission spectrum. The NMOFs were fabricated by drawing in a microfurnace comprising a sapphire tube heated with a CO2 laser. It is reasonable to assume that after appropriate optimization and development this novel technique of microfiber fabrication can become advantageous as compared to the conventional use of flame heating in terms of achievable cleanness, reproducibility and low air convection.
With the suggested method we fabricated very short biconical tapers a few mm in length and containing NMOF. Using short tapers, we bent NMOF locally in free space by translation of the taper ends. The latter suggests a simple method for investigation of the effects of bending and looping on transmission characteristics of a free NMOF. For a tightly bent NMOF having a bend radius of a few tens of microns, the transmission power spectrum strongly depends on wavelength and can be used to independently calculate the correct NMOF diameter and bend radius.
Manipulating the taper ends we formed a coiled self-touching NMOF, which displays strong oscillations in its transmission spectrum. These oscillations are caused by interference between the light which propagates through the contact point via self-coupling (by-passing the microloop), and the light propagating along the microloop. The observed interference oscillations are in good agreement with the theoretical predictions. As a result, we have demonstrated an optical interferometer built of a coiled self-coupling NMOF.
Further investigation of more complex microfiber-based elements and devices, in particular, the study of NMOF mutual coupling and interaction with other optical elements (e.g., a NMOF situated at or close to flat and corrugated surfaces, NMOF wound on a rod [10], etc.) could be important steps in the development of future NMOF-based microphotonics.
Acknowledgments
We thank D. DiGiovanni, M. Fishteyn, S. Ghalmi, S. Ramachandran, D. Trevor, and A.Yablon for useful discussions and consultations.
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