## Abstract

The feasibility of a broadband, miniature, and low loss optical delay line composed of a coiled microfiber with several-micron radius is demonstrated theoretically. Under the introduced low-loss condition, the fundamental mode of the microfiber is shifted away and does not scatter from the interfaces with the central rod and adjacent turns. Dimensions of the designed 100 ns feedforward microfiber delay line containing 20 m of microfiber are 5 mm × 5 mm × 20 mm at the radiation wavelength 1.5 μm and can be much smaller for the recirculating loop delay line. These dimensions can be further optimized by varying the radii of the microfiber and coil. The predicted insertion loss of this device is ~ 0.004 dB/ns, which is two orders of magnitude smaller than the loss achieved presently for the miniature delay lines. A curved microfiber taper is proposed as a compact, low-loss, and broadband connection to this optical delay line. The taper adiabatically converts the input fundamental mode of a straight microfiber into the shifted mode of the coiled microfiber.

© 2009 Optical Society of America

## 1. Introduction

An optical buffer, i.e., a device enabling delay of optical packets, is a key element of the future photonic circuits for optical signal processing [1,2]. Optical buffering has received much attention because of its potential applications in telecommunications and optical computing. Recently, the problem of optical buffering was readdressed based on the concept of slow light. Various novel approaches for fabrication of optical buffers were suggested and demonstrated experimentally. They include electromagnetically induced transparency, stimulated Brillouin and Raman scattering, photonic crystal structures, optical microresonator delay lines, and other methods [3]. In spite of significant progress in this rapidly eveloping field, the demonstrated optical buffers are still significantly limited in the achievable bandwidth, delay time, and insertion loss.

Among the important examples of low-loss devices enabling delay of light are the microsphere resonator [4–7] and the microtoroid resonator [8–13] shown in Fig. 1(a) and (b), respectively. An optical fiber taper with a micron-diameter waist is often used for coupling to these resonators. The microsphere resonator is usually fabricated of a fused silica and has the diameter from several tens of microns to several millimeters. The microtoroid resonator consists of a closed silica microfiber with the diameter of several microns. The microfiber is created at the edge of a silica disk mounted on a chip. The largest achieved Q-factor of the microtoroid (microsphere) resonator is ~ 5·10^{8} [10] (~5~10^{9} [4,5]), which corresponds to the propagation loss ~ 0.05 dB/m (0.005 dB/m). The delay time of these resonators can be up to ~ 100 ns, however, the delay time-bandwidth product limitation [14] restricts the corresponding bandwidth of pulses only by ~ 1 MHz.

On the other hand, the conventional optical buffer based on a single-mode optical fiber [1,2,15,16] is a broadband and, simultaneously, a low-loss device. Yet, large dimensions of this device make it unfeasible for application in compact photonic circuits. Therefore, its miniaturization while maintaining the low-loss and broadband properties would be very important. One way of reducing the dimensions of an optical delay line is to fabricate it of planar optical waveguides [2]. The small loss ≤ 1 dB/m can be achieved for waveguides with very small index contrast. Application of such waveguides for long and miniature delay lines is problematic because their insertion loss dramatically grows when the bend radius becomes less than a centimeter [17]. The smaller bending radius of a few mm is achievable by the planar waveguides with larger index contrast, which have larger propagation loss >1 dB/m due to the increased effect of the surface roughness. Recently, the first on-chip silica waveguide optical buffer with 4 dB/m propagation loss has been demonstrated [18]. An alternative way of miniaturization of the optical fiber spool can be achieved with the use of an optical microfiber coil delay line (MCDL) illustrated in Fig 1(c). In theory, the smallest MCDL can be composed of a very thin coiled single-mode microfiber with ~ 1 μm diameter [19,20]. However, scattering from the interface between the single-mode microfiber and the central rod causes significant loss, which makes this device impractical, at least at the current stage of development of the microfiber photonics [20]. In addition, in order to avoid the resonant interturn coupling effects [19], several micron spacing between turns is required. Furthermore, similarly to the planar optical waveguides, the smallest transmission loss demonstrated for a free single-mode microfiber, 1.4 dB/m [21], is still ~ 30 and 300 times greater than the loss of a multi-mode toroidal microfiber and a microsphere, respectively.

The low loss of the microsphere and toroidal resonators can be explained by the characteristic distribution of the fundamental mode illustrated by the surface plot in Fig. 1(a) and (b). For the toroidal microfiber of several micron diameter, this mode is radially shifted clear of the supporting central disk and, therefore, does not experience scattering from its nonuniformities [11,12]. In addition, this mode has a larger size and, hence, a smaller relative field intensity at the microfiber (microsphere) surface, compared to the fundamental mode of a single-mode microfiber [22]. For this reason, this mode experiences smaller losses causes by scattering from the surface nonuniformities. Similarly, the fundamental mode of an MCDL, which is made of a several micron diameter microfiber (Fig. 1(c)), is clear of the interfaces with the central rod and adjacent turns. Therefore, such MCDL can exhibit the propagation loss similar to that of the toroidal microresonator and a microsphere. Advantageously, unlike the microtoroid and microsphere resonators, the MCDL is a broadband device.

Section 2 of this paper presents theoretical investigation of the MCDL composed of a several microns diameter microfiber. It further shows that the dimensions of the 100 ns delay time MCDL, which scale with radiation wavelength *λ*, can be as small as 5 mm × 5 mm × 20 mm for *λ* =1.5 μm. The loss of this delay line can be as small as ~ 0.4 dB. Section 3 introduces a coiled microfiber taper as a means for low loss, broadband, and compact connection to the fundamental mode of an MCDL. This taper adiabatically converts the input fundamental mode of a straight microfiber into the radially shifted mode of the MCDL. Section 4 shows that the MCDL dimensions can be optimized by variation of the microfiber and coil radii and discusses methods of MCDL fabrication. Section 5 summarizes the obtained results.

## 2. Optical microfiber coil delay line (MCDL)

The fundamental mode and nearby higher-order modes of a curved microfiber can be viewed as modes that propagate in the vicinity of a geodesic at the external part of the microfiber surface. Fig. 2 introduces the local orthogonal coordinates near this geodesic: the longitudinal coordinate s and transversal coordinates *x* and *y*. Using the short wavelength scalar diffraction theory [23], a simple asymptotic solution can be derived for the propagation mode with transverse quantum numbers *m* and *n*:

$$X={\left(\frac{2{\beta}^{2}}{R}\right)}^{\frac{1}{3}}\left(x-\frac{{y}^{2}}{2r}\right),\phantom{\rule{.2em}{0ex}}Y={\left(\frac{{\beta}^{2}}{\mathrm{Rr}}\right)}^{\frac{1}{4}}y,\phantom{\rule{.2em}{0ex}}\beta =\frac{2\pi {n}_{f}}{\lambda}.$$

In Eq. (1), *r* is the microfiber radius, *R* is the coil radius, *λ* is the radiation wavelength, *H _{m}*(

*x*) is the Hermite polynomial,

*Ai*(

*x*) is the Airy function,

*t*is the root of Airy function (

_{n}*t*

_{0}= 2.338,

*t*

_{1}= 4.088,

*t*

_{2}= 5.52,…),

*n*is the microfiber refractive index [24], and mn

_{f}*β*is the propagation constant of the mode (

_{mn}*m,n*) given as:

For simplicity, it is assumed that the MCDL is uniform so that *r* and *R* are independent of *s*. However, the theory [23] provides solutions for the case of a nonuniformly coiled microfiber with varying radius. Eqs. (1) and (2) are valid for the modes with *m,n* ~ 1 under the condition of strong localization near the geodesic *s*,

In this approximation, the boundary condition is replaced by zeroing the mode values at the microfiber surface, *X* = 0 , which is valid for the large index contrast, *n _{f}*

^{2}≫. For

*r*=

*R*, Eqs. (1)–(3) correspond to the eigenmodes of a microsphere. Then Eq. (3) coincides with the condition of the short-wave approximation,

*Rβ*≫ 1 and Eq. (2) coincides with the known result for the spherical microresonator [25] with neglected constant phase shift ~

*n*

_{f}^{-2}. In a more general case, Eq. (2) is similar to that obtained in [26] for the toroidal microresonator. In the spherical and toroidal microresonators, the considered modes are not coupling due to the the spherical and toroidal microresonators, the considered modes are not coupling due to the rotational symmetry and/or adiabaticity of bending. In the MCDL, these modes are not coupling due to the helical (rotational plus translational) symmetry and/or adiabaticity of bending.

A mode that satisfies Eq. (3) is exponentially small at the interfaces of the microfiber with the central rod and adjacent turns, i.e., it satisfies the low loss condition mentioned above. In fact, from Eq. (1), the characteristic dimensions of the mode along the *x* and *y* coordinates are(*R* / *β ^{2}*)

^{1/3}and (

*Rr*/

*β*)

^{2}^{1/4}, respectively. The mode is clear of the mentioned interfaces if these dimensions are small compared to the microfiber radius,

*r*. The latter conditions are equivalent to the inequality (3).

Eqs. (1)–(3) are useful for modeling of MCDLs and are quite accurate. Let us choose the coil radius *R* = 2.5 mm. For radiation wavelength *λ* =1.5 μm and refractive index of the microfiber 1.5 *n _{f}* = , this radius corresponds to the right hand side part of Eq. (3) (

*R*/

*β*)

^{2}^{1/3}≈ 4 μm. In order to satisfy the low-loss condition expressed by Eq. (3), let us choose the microfiber radius

*r*=15 μm. Fig. 3(a) shows the surface plot of the fundamental quasi-TE mode

*E*

_{00}numerically calculated for this microfiber using the vector beam propagation model (BPM). Fig. 3(b) shows the distribution of the same mode calculated with Eq. (1). The cross-sections of these two distributions along the axes

*x*and

*y*

_{1}, which cross the mode maximum point (Fig. 3(a) and (b)), are compared in Fig. 3(c) and are remarkably similar.

More importantly, Fig. 3(d) shows the log-scaled profiles of the surface plot of Fig. 3(a) along the axes *x* and *y*
_{0} crossing the microfiber center. It is seen that the values of the field at the interface with the central rod and at the points touching the adjacent turns drastically decrease by more than 55 dB and 30 dB, respectively. These values can be further decreased
for a microfiber with a larger radius *r*, which better fits Eq. (3).

Dimensions of the MCDL with time delay *T* are estimated as 2*R* × 2*R* × (*crT* / *πn _{f}R*), where

*c*is the speed of light. The transmission loss of the microfiber can be estimated from the transmission losses of the microtoroid and microsphere resonators [4,5,10]. Following [5], the record loss of the several micron diameter toroidal microfiber (~0.05 dB/m [10]) is an order of magnitude larger than the loss of a several hundred micron diameter microsphere (~0.005 dB/m [4,5]) because of the larger fraction of the propagating mode scattered from the silica surface nonuniformities. From [5], the transmission loss of the coiled microfiber with radius 15 μm is estimated as ~ 0.02 dB/m, which corresponds to the insertion loss of MCDL ~ 0.004 dB/ns. Consider, for example, the 100 ns time delay line fabricated of an optical fiber of ~ 20 m length. The dimensions and insertion loss of such MCDL are 5 mm × 5 mm × 20 mm and 0.4 dB, respectively. With the same insertion loss, the dimensions of the MCDL can be made proportionally smaller for smaller wavelength.

## 3. Optical coiled microfiber taper

As opposed to the microsphere and microtoroid resonator, the low-loss excitation of the fundamental mode of the MCDL by side-coupling (e.g., using a micron diameter microfiber illustrated in Fig. 1(a) and (b) [6–9] or a prism [4,5]) is problematic. In fact, the MCDL is not an optical resonator and its transmission mode cannot be resonantly excited. Furthermore, for the highly multimode microfiber, the separation between the mode propagation constants defined by Eq. (2) is very small. For this reason, the low loss phase matched coupling to the designated mode is hardly possible.

To solve this problem, a simple, low-loss, and compact method of connection to the fundamental mode of the MCDL is suggested. Let us consider a coiled microfiber taper shown in Fig. 4(a). This taper adiabatically transforms the fundamental mode of a straight microfiber into the shifted fundamental mode of a coiled microfiber described in Section 2. One end of this taper can be a single-mode microfiber with ~ 1 μm diameter. The single-mode microfiber is convenient for connection of microfiber photonic devices because of its small propagation and bend losses [21,27]. It can be side coupled to a planar waveguide or be a waist of an adiabatic taper fabricated from a conventional single-mode fiber (Fig. 4(b)). The lower three diagrams in Fig. 4(b) illustrate the evolution of the transverse eigenvalues along the bend and tapered microfiber. When the taper becomes larger in diameter, new modes appear. However, for the adiabatic taper, the fundamental mode of an initially thin microfiber evolves into the fundamental mode of the thicker microfiber. As another possibility, a coiled microfiber taper can be made of a microfiber coil with constant microfiber radius and variable coil pitch or radius. Then the fundamental mode of a straight multimode microfiber can be transformed into the shifted mode of the MCDL by the adiabatic variation of the microfiber bend radius from a very large value to the value of coil radius, R. An example of such connection between a conventional single mode fiber and an MCDL is illustrated in Fig. 4(c). Recently, a special case of this taper made of a microfiber with the uniform radius was studied theoretically [28] and experimentally [29]. The authors of [29] considered a single mode core-guided fiber with the outer diameter 35 μm. In order to ensure the adiabatic transition, it was critical to adiabatically pass the point of pseudo-intersection of propagation constants, which appeared at a certain radius of curvature. The demonstrated adiabatic fiber coil taper was fairly long because it had the constant rate of curvature variation, which was determined by the adiabatic condition at the point of pseudo-intersection. However, as will be shown below, for a microfiber with a relatively small radius, the core is not needed and the fundamental mode can be adiabatically transformed from the straight section to the curved section of the microfiber.

Adiabatic propagation along the fundamental mode of the coiled microfiber taper is performed with exponentially small losses if the characteristic taper length, *L _{t}*, satisfies the adiabatic condition

*L*≫ 1/(

_{t}*β*

_{1}-

*β*

_{0}). Here

*β*

_{1}-

*β*

_{0}is the smallest separation between the propagation constant of the fundamental mode,

*β*

_{0}, and the closest propagation constant,

*β*

_{1}, of the mode, which can be excited by the introduced deformation [30,31,28,29]. For a coiled microfiber taper with monotonically varying separation of propagation constants, the adiabatic condition can be derived with Eq. (2):

As an example of the microfiber taper illustrated in Fig. 4(b), consider a taper with radius variation *r*(*s*) = *r* + 2(*r*
_{0} - *r*) / [1+ exp(*s*
^{2} / *L*
^{2})] where *r*(0) = *r*
_{0} and *r*(∞) = *r* . Here *L* is the characteristic length of the taper. Assume that this taper is circularly bent with the bend radius
*R*. The initial taper radius at *s* = 0 is set to 0 *r* = 0.5 μm. Consider the transformation of the fundamental mode along this taper into the shifted fundamental mode of the MCDL described in Section 2 with the parameters *R* = 2.5 mm, *r* =15 μm, *λ* =1.5 μm, and 1.5 *n _{f}* = 1.5. The
condition of adiabaticity for this taper, Eq. (4), yields

*L*≫ 0.2 mm. The numerical simulation shown in Fig. 5 confirms that this transformation becomes low loss at

_{t}*L*~ 1 mm. Modeling of the bent microfiber taper was performed with the vector BPM and conformal transformation, where a bend taper with refractive index

_{t}*n*was replaced by a straight taper with the effective refractive index

_{f}*n*(1 +

_{f}**x**/

*R*) [32] as illustrated in Fig. 5(a). Fig. 5(b) shows the propagation of the fundamental mode launched at

*s*= 0 along the taper with

*L*= 0.3 mm. It is seen that this taper is not adiabatic and that several interfering modes are excited. Fig. 5(c) shows the propagation of the same mode along the longer taper with

*L*= 1 mm. This taper adiabatically transforms the initial mode into the shifted fundamental mode of the coiled microfiber. No transmission loss was detected within the accuracy of calculations. The size of the adiabatic taper shown in Fig. 5(c) is negligible compared to the dimensions of the MCDL: its total length is only ~ 2.5 mm, which is close to the MCDL diameter.

As an example of the bent uniform-radius microfiber illustrated in Fig. 4(c), consider a microfiber, which bend radius *R*(*z*) is changing from a very large *R*
_{0} at *s* = 0 (corresponding to the straight microfiber) to the coil radius *R* according to the law *R*(*s*) = *R* + 2(*R*
_{0} – *R*) / [exp(*s*
^{2}/*L*
^{2}) +1] . Here *L* is the characteristic length of the transition region. Similarly to the previous example, bending is modeled with the vector BPM using a local effective index that corresponds to the local microfiber curvature (Fig. 6(a)). The microfiber radius is set to *r* = 15 μm and the final coil radius is set to *R* = 2.5 mm. Fig. 6(b) shows the evolution of the fundamental mode for the transition length *L* = 3 mm. In this case, the fundamental mode of a straight microfiber launched at *s* = 0 experiences transformation, which causes excitation of two modes, *E*
_{00} and *E*
_{01}. The relative power of the *E*
_{01} mode is fairly small, ~ 2%. The interference of *E*
_{00} and *E*
_{01} modes shows up in oscillations of the field amplitude in the region of the coiled microfiber. The period of these oscillations can be calculated from Eq. (2): ∆*s* = 2^{1/3}(*t*
_{1}-*t*
_{0})^{-1}(2*πR*)^{2/3}(*λ*/*n _{f}*)

^{1/3}=451.6 μm. This value of ∆

*s*is in excellent agreement with the value ∆

*s*= 450 μm found from the surface plot of Fig. 6(b). For

*L*= 6 mm shown in Fig. 6(c), the evolution of the fundamental mode is much more adiabatic and the power fraction of the excited

*E*

_{01}mode is only ~ 0.04%. Very small oscillations are still visible in the region of the coiled microfiber and their period ∆

*s*= 450 μm is, again, in excellent agreement with that given by Eq. (2).

The examples of this section show that the length of the bent microfiber, which transforms the axially symmetric fundamental mode into the shifted fundamental mode with negligible losses, can be less than the length of a single turn of the microfiber coil.

## 4. Discussion

The MCDL dimensions considered in Section 2 can be further optimized by varying the
microfiber radius, *r*, and coil radius, *R*. In particular, the MCDL volume significantly decreases with *R*. The MCDL volume is expressed through the delay time *T* as *V* = *cRrT*/*n _{f}*. The low loss condition, Eq. (3), can be written as

*r*= (

*R*/

*β*

^{2})

^{1/3}, where

*C*is a large dimensionless parameter. In the example considered in Section 2,

*C*≈ 4 . Using the same

*C*, we get

*V*= 4

*cR*

^{4/3}

*T*/(

*n*

_{f}*μ*

^{2/3}). This expression shows that, for the same delay time, the volume of the delay line decreases with the coil radius

*R*. At 1.5 μm radiation wavelength,

*R*= 2500 μm, and

*r*=15 μm (parameters considered in Section 2), the dimensions and volume of a 100 ns MCDL are 5 mm × 5 mm × 20 mm and 0.5 cm

^{3}, respectively. At the same wavelength of 1.5 μm and a smaller

*R*= 100 μm, corresponding to

*r*= 4(

*R*/

*β*

^{2})

^{1/3}= 5.6μm, the volume of the 100 ns MCDL is 7.5 mm

^{3}only (more than 60 times smaller than the volume of the MCDL considered in Section 2) and its size is 0.2 mm × 0.2 mm × 190 mm. For this delay line, the values of the field at the interface with the central rod and at the points touching the adjacent turns are calculated to be less than 70 dB and 40 dB, respectively, i.e., even smaller than those of the MCDL considered in Section 2. However, the length of this delay line is fairly long (note that it is still 100 times smaller that the length of the microfiber comprising the coil). The dimensions of this delay line can be further reduced by postprocessing, which includes softening followed by bending or coiling. In addition, the circulated loop MCDL with the same delay time can have much smaller dimensions. Looping of the MCDL can be achieved with tapers described in Section 3. Another way to reduce the delay line length is to assemble it of shorter parallel delay lines. In addition, a set of relatively short MCDLs with different delay times can be connected to a photonic circuit, which performs discrete tuning of the delay time by switching between these MCDLs, in analogy with the methods developed for the optical fiber delay lines [1,2,15,16].

This paper does not consider the polarization effects in the coiled microfiber, which can be an interesting topic for further investigation. These effects are important for the coiled microfiber taper shown in Fig. 4(a) as well as for the MCDL. In the numerical examples considered in Section 3 (Figs. 5 and 6), the microfiber axis is planar, so that the polarization states are separated and can be considered independently.

The MCDL can be created directly from a conventional single mode optical fiber by local
softening (using, e.g., a CO_{2} laser), tapering down to the requested diameter of the MCDL and, simultaneously, coiling on a central rod. Most likely, the MCDL can be created from a prefabricated optical microfiber that is locally softened and coiled on a central rod. Similarly to the fabrication of high Q-factor microsphere and microtoroid resonators, the low loss MCDL should be created in a clean environment by drawing from an extra pure silica preform followed by fire/laser assisted reflow of the device [4–10].

## 5. Summary

The earlier experimental demonstration of extra high Q-factor microsphere and microtoroid resonators supports the idea of a miniature, low-loss, and broadband optical delay line fabricated of a coiled microfiber put forward in this paper. In designing the suggested delay line, the relationship between the microfiber radius and the coil radius is chosen so that the propagating fundamental mode is shifted clear of the interfaces with the central rod and adjacent turns. At this condition, scattering from the interfaces and microfiber surface nonuniformities is minimized and the local geometry of the fundamental mode of the microfiber coil delay line is similar to that of the fundamental mode of the microtoroid and microsphere resonators. Based on this, the achievable propagation loss of the microfiber coil is estimated as 0.02 dB/m which corresponds to the 0.004 dB/ns insertion loss of the delay line. In particular, the designed 100 ns time delay MCDL made of a 20 m microfiber has the dimensions 5 mm × 5 mm × 20 mm and volume of 0.5 cm^{3} at radiation wavelength 1.5 μm. These dimensions will be proportionally smaller for smaller wavelengths. Further optimization and miniaturization of the MCDL is possible. The low loss connection to the shifted fundamental mode that propagates along the MCDL can be achieved with an adiabatic coiled microfiber taper. The length of this taper is fairly short and comparable with the coil diameter. The progress in the high Q factor microresonator fabrication and in the fiber and microfiber drawing technology will hopefully bring the MCDL-based optical buffers to reality in the near future.

## Acknowledgments

The author is grateful to David DiGiovanni and Siddharth Ramachandran for numerous fruitful discussions.

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