Design and optimization of broadband tapered optical fibers with a nanofiber waist

The control over the transmission properties of tapered optical fibers (TOFs) is an important requirement for a whole range of applications. Using a carefully designed flame pulling process that allows us to realize preset fiber radius profiles, we fabricate TOFs with a nanofiber waist. We study the spectral transmission properties of these TOFs as a function of the taper profile and the waist length and show how the transmission band of the TOF can be tuned via different fiber profile parameters. Based on these results, we have designed a nanofiber-waist TOF with broadband transmission for surface spectroscopy of organic molecules. Moreover, our method allows us to analyze the loss mechanisms of optical nanofibers. © 2010 Optical Society of America OCIS codes: (060.2270) Fiber characterization; (060.2280) Fiber design and fabrication; (060.2310) Fiber optics References and links 1. G. Brambilla, “Optical fibre nanowires and microwires: a review,” J. Opt. A 12, 043001 (2010). 2. F. 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Introduction
Optical nanofibers with diameters smaller than the wavelength of the guided light attract considerable interest due to their unique properties and wide range of applications [1].The pronounced evanescent field surrounding the fiber and the strong radial confinement of the light [2] make nanofibers well suited for an efficient and controlled interaction of the guided light with matter.Applications include, among others, optical sensing [3][4][5], nonlinear optics [6][7][8][9][10][11], nanofiberbased dye lasers [12,13], nanofiber-based evanescent wave spectroscopy [14,15] and cold atom physics [16][17][18][19][20].These applications demand high-quality nanofibers with a high diameter uniformity and a low surface roughness, thus rendering the technical realization a demanding process.
The standard fabrication method for optical nanofibers is flame pulling of a commercial single mode optical fiber resulting in a tapered optical fiber (TOF) with a nanofiber waist [8,[21][22][23][24][25][26].In order to efficiently couple light into and out of the nanofiber waist, the transformation of the fiber mode within the taper sections has to be adiabatic, thereby ensuring a high transmission of the overall structure [27].At the same time, it is desirable for technical reasons to keep the TOF as short as possible.The combination of the two requirements makes it essential to carefully design the taper profile [28].Moreover, some of the applications cited above require low losses not only for a single wavelength but for several distinct wavelengths or broad spectral bands which makes the TOF fabrication process even more challenging.
We present a systematic study of the influence of the fiber profile parameters on the transmission properties of the TOF.We demonstrate how the transmission band can be shifted, extended and optimized.The obtained information enables us to tailor the radius profile of a TOF according to the desired transmission properties and to study the loss mechanisms connected to the nanofiber waist of the TOF.

Transmission properties of TOFs of predetermined shape
We fabricate TOFs by stretching a commercial single mode optical fiber while heating it with a travelling hydrogen/oxygen flame [28].We use an analytic algorithm to calculate the trajectory of the heat source for a predetermined fiber shape.This algorithm and the technical realization of our fiber pulling rig will be described elsewhere.
Figure 1 shows the measured radius profile of a TOF with the following preset parameters: a 320 nm-diameter waist of 1 mm length, a three-fold linear taper transition with local taper angles Ω 1 = 3 mrad, Ω 2 = 1 mrad and Ω 3 = 5 mrad, and a change of the local taper angles at the fiber radii r 1 ≈ 42 μm and r 2 ≈ 20 μm.The solid line indicates the fiber radius profile as predicted by our algorithm.Throughout this paper, r denotes the radius of the fiber cladding, and Ω(r) denotes the local taper angle of the fiber cladding.We found empirically from our simulations that the profile exhibits local deformations if Ω(r)/r becomes too large.We therefore limited Ω(r)/r to less than 0.3 mrad/ μm.For the local taper angle Ω 3 = 5 mrad used for our TOFs, the profile thus has an exponential shape for radii smaller than r 3 ≈ 17 μm.The inset of Fig. 1 shows a magnification of the submicron-radius region of the profile.
The TOF characterized in Fig. 1 was fabricated from a Nufern 460-HP fiber, deposited on a gold coated substrate, and its radius was then measured with a scanning electron microscope.
The van-der-Waals-force between the fiber and the substrate fixes the fiber during the acquisition of electron micrographs.The measurements were performed with an electron energy of 2 keV and a current of 53 pA.For these values, distortions of the electron micrograph due to charges in the insulating fiber turned out to be negligible.To determine the local fiber radius, we successively took electron micrographs along the TOF with a step size varying between 0.25 and 1 mm.The measured radius profile closely follows the predicted profile with deviations smaller than ±10 %.For the two linear sections with local taper angles Ω 1 and Ω 2 , they even fall below ±5 %.From this excellent agreement and the reproducibility of the spectral transmission characteristics (see below), we conclude that our production procedure for TOFs indeed realizes the predicted profiles.We therefore only consider the predicted profiles of the discussed TOFs throughout this paper.Note that the radius and homogeneity of the waist of our TOFs have been measured by U. Wiedemann et.al. using a more precise method [11].
The transmission properties of the fiber radius profile presented in Fig. 1 are shown in Fig. 2. We measure the transmission spectra using a tungsten halogen light source (Ando AQ-4303B) and a spectrometer (Avantes AvaSpec-2048-2) and normalize them to the transmission spectra of the untapered fiber.Figure 2 displays the normalized transmission of three different samples with the same radius profile.The transmission exhibits a very high degree of reproducibility with variations of only ±(2 ± 1) %.Moreover, the TOFs have a peak transmission of over 90 % around 560 nm and a transmission of over 70 % in a broad spectral band between 470 and 690 nm.
The spectral shape of the transmission can be understood qualitatively by considering the characteristics of light propagation through the TOF.In the untapered part of the fiber, the light is guided by the fiber core in the fundamental mode.As the core size decreases along the taper sections of the TOF, the mode extends further into the cladding until it is effectively guided by the cladding-air interface.Since the cladding radius is much larger than the single mode cutoff radius, taper-induced coupling to higher transverse modes can occur at this point.When the TOF then reaches the single mode cutoff radius, it acts as a mode filter and all the light guided in the higher order modes is reflected or scattered out of the TOF and therefore lost.The coupling between modes depends linearly on the taper angle and increases as the difference between the propagation constants of the modes becomes smaller [29].Following this reasoning, a simple adiabaticity criterion for TOFs has been deduced [27].It compares the local taper length scale z t (r) with the beat length z b (r, λ ) of the two coupled modes leading to an approximate delineation between adiabatic and lossy tapers for z t = z b , where λ is the vacuum wavelength of the propagating mode.The local taper length scale is defined through the local taper angle Ω(r) and the local fiber radius r according to z t (r) = r/Ω(r).Since the beat length of two modes with propagation constants β 1 (r, λ ) and β 2 (r, λ ) is given by z b (r, λ ) = 2π/(β 1 (r, λ ) − β 2 (r, λ )), the delineation condition is fixed by If the left hand side of Eq. ( 1) is much smaller than unity, the mode of the untapered region of the fiber is adiabatically transformed into the strongly guided mode of the nanofiber with negligible coupling losses.However, when it approaches or even exceeds unity, the transformation will deviate from this ideal behaviour.
In the wavelength range between 470 and 690 nm, the fiber radius profile presented in Fig. 1 only exhibits small deviations from adiabaticity.The residual losses are determined by the beat length z b (r, λ ) and the length over which coupling to higher order modes is possible.Since both parameters depend on the wavelength, we obtain a periodic modulation of the transmission within this wavelength range in agreement with [30].
We note that the delineation condition given by Eq. ( 1) depends on the characteristics of the single mode fiber used for fabricating the TOF: the smaller the cutoff wavelength of this fiber, i. e., the smaller the core diameter and/or the refractive index difference between core and cladding, the stronger the coupling to higher order modes.The Nufern 460-HP fiber used for our experiments provides a short-wavelength cutoff of about 430 nm which leads to strong constraints for the taper angle.For fibers with a different cutoff wavelength, on the other hand, adiabaticity can be achieved for different taper angles.

Influence of taper radius profile
According to Eq. ( 1), the only free parameter to control the adiabaticity of a fiber radius profile is the local taper angle Ω(r).For a three-fold linear taper shape, tuning of the local taper angle can either be achieved by changing the slope of the individual linear sections given by Ω i or by shifting the boundaries between the linear sections determined by r 1 and r 2 .
Figure 3 illustrates the effect of tuning the slope of the central linear section for the taper profile presented in Fig. 1.The predicted profiles for three different values of the local taper angle Ω 2 are shown in Fig. 3a).The corresponding measured transmission spectra are displayed in Fig. 3b).Compared to the value of Ω 2 = 1 mrad, the transmission in the wavelength range  between 470 and 690 nm rises for a smaller taper angle of 0.8 mrad, whereas it drops significantly for a higher taper angle of 2 mrad.Moreover, the modulation period increases for smaller angles and decreases for higher angles, as expected from [29,30].
The transmission for wavelengths smaller than 470 nm or larger than 690 nm drops to less than 40 % and is almost unaffected by the changes presented in Fig. 3a).This means that the adiabaticity criterion is not satisfied by the shown radius profiles outside the wavelength range between 470 and 690 nm.We hence conclude that, for wavelengths lower than 470 nm or higher than 690 nm, the difference in propagation constants reaches a minimum at radii below 23 μm or above 43 μm fiber radius, where the taper angle is 5 and 3 mrad, respectively, causing the left hand side of Eq. ( 1) to exceed unity.
In order to modify the spectral band of high transmission, we therefore have to shift the boundaries of the central linear section.In Fig. 4a) and c), we show radius profiles for which either r 1 (Fig. 4a)) or r 2 (Fig. 4c)) are varied.The corresponding changes on the measured transmission band are shown in Fig. 4b) and d), respectively.On the short wavelength side, this tuning is limited by the single mode cutoff of the untapered fiber.The mode filter character of the nanofiber waist of the TOF [30,31] therefore allows us to determine the single mode cutoff wavelength of the standard fiber used for tapering to be (430±7) nm.The value agrees very well with the cutoff wavelength for the Nufern 460-HP fiber of (430±20) nm specified by the manufacturer.Fig. 5. Normalized transmission spectra for different waist diameters of 120 nm (solid line), 150 nm (dashed line), 180 nm (dotted line), and 320 nm (dash-dotted line) (a).Threshold behaviour for coupling to radiative modes for the same waist diameters (b).

Influence of waist diameter
In subwavelength diameter fibers, the mode extrudes considerably into the surrounding air, causing the effective refractive index n eff = β 1 λ /2π to decrease.The propagation constant β 1 of the guided mode will thus approach the propagation constant of the radiative modes which is given by β rad = 2π/λ .In this regime, coupling to radiative modes can become a significant loss mechanism of the TOF [24,32].In analogy to Eq. ( 1), a delineation condition can be determined by comparison of the beat length z b (r, λ ) = 2π/(β 1 (r, λ ) − β rad (λ )) between the two modes and the local taper length scale z t (r) [32].For small radii below r 3 ≈ 17 μm, our TOFs have an exponential shape and the local taper length scale is therefore constant with a value of z t (r) = r/Ω exp (r) ≈ 3.3 mm.The beat length between the guided mode and the radiative modes however increases monotonically with decreasing fiber radius, causing the strongest coupling between the modes to occur when the fiber diameter reaches the waist diameter d.This leads to a delineation condition between adiabatic and lossy tapers which only depends on the waist diameter of the TOF and the wavelength of the guided light: Figure 5 illustrates this dependence.In Fig. 5a) the normalized transmission spectra of three TOFs with the fiber radius profile presented in Fig. 1 and waist diameters of 120 nm, 150 nm and 180 nm are shown.For comparison, the transmission spectrum of a TOF with 320 nm waist diameter from Fig. 2 is depicted, where losses due to coupling to radiative modes can be assumed to be negligible.It is clearly apparent how adiabaticity breaks down at a different characteristic threshold wavelength for each waist diameter of the TOFs.The resulting drop in transmission as a function of the ratio d/λ is shown in Fig. 5b).Taking the 50 % transmission level for reference, the threshold values for d/λ range from 0.23 to 0.25, which agrees well with the values predicted in [32].Due to dispersion of the fiber material, the beat length between the guided and the radiative modes is longer for longer wavelengths, even if the ratio d/λ is fixed, causing a shift of the threshold to higher values for d/λ for higher diameters.

Influence of waist length
In addition to the taper transitions, also the nanofiber waist itself can influence the transmission properties of the TOF [21].Figure 6b) shows the normalized transmission spectra of fibers with the radius profile presented in Fig. 1   radius profiles are depicted in Fig. 6a).Within the reproducibility determined from Fig. 2, an increase in the nanofiber length does not change the transmission properties.The variations of ±(2 ± 1) % match those typically obtained in the fabrication of fibers with identical profiles.Therefore, we conclude that, for the waist lengths up to 1 cm considered here, the transmission losses are mainly due to deviations from adiabaticity in the taper transitions.Losses due to scattering of light by surface contaminants which increase with waist length as suggested by [21] turn out to be smaller than about 9 dB/m in this waistlength regime.This implies that the surface of our nanofiber waist is very smooth and clean.

Conclusions
Summarizing, we have tailored the overall transmission properties of a TOF with nanofiber waist via its radius profile.Our predetermined fabrication process allows us to tune the transmission for different wavelengths individually by modifying the local taper angle.As an example, we have designed a broadband TOF with a transmission exceeding 70 % over a wavelength range from 470 nm to 690 nm.This fiber has been optimized for highly sensitive surface spectroscopy of organic molecules [15] by matching the transmission band with the absorption and emission band of the molecular species under study.The waist diameter of 320 nm was chosen to maximize the evanescent field at the fiber surface and thereby the interaction between the molecules and the light guided by the nanofiber.In order to achieve the highest mechanical stability, the overall length of the TOF has to be small and could be reduced to 81.2 mm.Further, we have shown how to shift and extend the transmission band of the TOF, thus paving the way for the use of this method on other molecular species.Beyond that, numerous other applications of optical nanofibers as, for example, optical sensing or nonlinear optics could benefit from such a deterministic design of the TOF transmission properties.Losses due to scattering of light from the surface of the nanofiber waist have been shown to be on the order of 9 dB/m or less for the waist lengths considered here.Moreover, we could systematically study the limit of light guidance in optical nanofibers due to coupling to radiative modes [24,32] for a wide range of diameters and wavelengths.Exploring this limit, we have realized a TOF with a nanofiber waist of 120 nm diameter and a transmission of 70 % at 485 nm.For this wavelength, more than 99 % of the optical power of the guided light travels outside of the nanofiber.This remarkable value illustrates how well the mode transformation in TOFs can be controlled through careful design of the fiber shape.

#Fig. 1 .
Fig. 1.Comparison of predicted and measured radius profile.See text for details.

Fig. 2 .
Fig. 2. Normalized transmission spectra for three different TOFs with the same radius profile as shown in Fig. 1.

Fig. 3 .
Fig. 3. Effect of the central linear slope of the taper profile (a) on the measured transmission spectra (b).The taper angle Ω 2 has been tuned from 2 mrad (solid line) over 1 mrad (dashed line) to 0.8 mrad (dotted line).

Fig. 6 .
Fig. 6.Transmission properties for different waist lengths varying from 1 mm (solid line) over 5 mm (dashed line) to 10 mm (dotted line): fiber taper profiles (a) and corresponding normalized transmission spectra (b)