This paper presents a model of a subwavelength diameter adiabatic microfiber taper (nanotaper), which allows an asymptotically accurate solution of the wave equation. The evanescent field near the nanotaper is expressed through a Gaussian beam having a singularity at the nanotaper axis. For certain values of parameters of the nanotaper, when it has a swell in the middle and narrows down to zero at the infinity, the nanotaper is lossless. For other values, when the nanotaper has a biconical shape, it exhibits an exponentially small radiation loss, which is determined as a tunneling rate through an effective parabolic potential barrier. The latter case represents an exceptional example of the radiation loss being distributed along the length of an adiabatic nanotaper rather than being localized near focal circumferences in the evanescent field region.
© 2007 Optical Society of America
This paper is concerned with the problem of propagation of radiation along an adiabatically tapered microfiber having the diameter much less than the radiation wavelength. There has been growing interest in investigation and applications of such optical microfibers, which are usually fabricated by drawing conventional silica optical fibers to diameters on the order of a micron [1–8]. Deformation of a dielectric optical microfiber usually causes radiation loss. Radiation loss grows dramatically and is very sensitive to deformations when the microfiber diameter is noticeably smaller than the radiation wavelength. The simplest types of deformation are bending and tapering. Pure adiabatic bending assumes that the radius of the curvature of the fiber is a slow function of the coordinate along the fiber length. It conserves the local cylindrical symmetry and the radius of the fiber r. Pure adiabatic tapering assumes slow dependence r(z) on the coordinate along the microfiber axis, z, and conserves the cylindrical symmetry. Radiation loss in bent uniform microfibers is understood quite well. Adiabatically bent microfibers allow us to find the propagating mode in a curvilinear coordinate system localized along the fiber axis [9,10]. As a result, an effective potential barrier can be introduced. The barrier has a finite width and its external edge determines a caustic surface separating the tunneling or classically forbidden region adjacent to the microfiber from the external classically allowed region. Tunneling through the barrier determines the radiation loss. Usually, the radiation loss is distributed along a bent microfiber and can be described by the attenuation constant, which depends on the coordinate along the microfiber, z .
The problem of radiation loss in an adiabatically tapered microfiber, which has a diameter much less than the radiation wavelength, called a nanotaper (NT), is more complex and has been solved only recently [11–13]. It was found that, for a NT, one cannot introduce an adjacent potential barrier of finite width that determines the local attenuation constant. Instead, the radiation loss takes place locally in the neighborhood of focal circumferences surrounding the NT. Away from these locations, a NT is lossless.
The NT considered below represents an interesting exceptional situation when the tunneling dynamics of light is similar to that of a bent microfiber: The coordinates in the wave equation can be asymptotically separated and the effective potential barrier can be introduced. The evanescent part of the fundamental mode of this NT is determined in the form of a singular Gaussian beam. The existence of an effective potential barrier adjacent to the NT implies the uniform transmittance of this barrier along the taper, i.e., the distributed radiation loss and possibility of introduction of the local attenuation constant.
Section 2 describes the basic parameters and discuses the general properties of nanotapers. Section 3 introduces a singular Gaussian beam and compares it with a regular Gaussian beam. Section 4 considers the behavior of a singular Gaussian beam supported by a NT and compares it with the electromagnetic field supported by a bent microfiber. Section 4 also considers a singular Gaussian beam supported by a nanoswell and compares it with a regular Gaussian beam. Section 5 discusses and summarizes the obtained results. To simplify understanding of the paper and to avoid cumbersome equations in the main text, the details of calculations are moved into Appendices 1, 2, and 3.
2. General properties of adiabatic nanotapers
The fundamental mode of a NT propagates primarily outside the NT. In an adiabatic axially symmetric NT, the propagation constant of the fundamental mode, β(z), is a slow function of the coordinate z along the axis of NT. The propagation constant β(z) is close to the propagation constant of the ambient medium β 0:
Here γ(z) is the transversal component of the propagation constant, which is relatively small, γ(z)≪β 0. For a given β 0, the function γ(z) is uniquely determined through the NT radius r(z) and can be asymptotically expressed as :
where n 1 is the refractive index of the NT and n 2 is the refractive index of the ambient medium.
Generally, the radiation loss in a NT happens locally near focal circumferences surrounding the NT . After splitting off the NT, the radiation wave propagates under a small angle to the NT axis z and slowly expands. Also, the radiation wave exponentially vanishes along the radial directions from the NT. Therefore, for a NT of a common shape, the effective transversal potential barrier, which determines the tunneling dynamics of light, is infinitely wide. For this reason, radiation from the adiabatic NT happens primarily along the longitudial direction.
3. Regular and singular Gaussian beams
This paper shows that there exists an exceptional example of a NT for which one can introduce an effective potential barrier of finite width that is adjacent to the NT and is responsible for distributed radiation loss. As opposed to the NT of a common shape, this NT exhibits radiation, which is transversal (i.e. taking place through the adjacent barrier) rather than longitudial to the NT. The shape of this NT is found in the following way. First, an adiabatic NT of a general shape is considered, and the corresponding adiabatic solution of the wave equation is written out in the form of Eq. (A1.5) of Appendix 1. Then, the anzatz for the solution of the wave equation away from the NT is chosen in the form similar to the form of the Gaussian beam. The details are given in the Appendix 2. This anzatz allows factorization of the solution of the wave equation into the product of a known function and a function Λ(ν) that obeys the etalon ordinary differential equation with respect to the effective transversal coordinate ν:
Here ν is expressed through the cylindrical coordinates (ρ, z) along the NT as follows:
where a, b, and c are free parameters. Function P(ρ,z) = λ2(ν)/ρ 2 determines the distribution of the electromagnetic field power outside of the NT. In order to determine the regular Gaussian beam, one demands that P(ρ, z) is finite for ρ → 0 and P(ρ, z) → 0 for ρ → ∞. The latter conditions lead to the equations b = a 2/4 and λ(ν) = exp(-aν 2/4) which yield the familiar form of a Gaussian beam. As opposed to this regular Gaussian beam, here we look for a solution of the wave equation that does have finite asymptotic near the center of the beam. Instead, our solution matches the asymptotics of the adiabatic solution near the NT which behaves as a function of ρ as K 0(γ(z)ρ) (see Appendix 1). Therefore, it is singular for ρ → 0 . It is shown in Appendix 2, that matching the adiabatic solution with the solution defined with Eqs.(3)-(5) is only possible if the shape of NT corresponds to the transversal propagation constant
where the characteristic nonuniformity of the NT, L, and the value of the transversal propagation constant in the middle of NT, γ 0, are expressed through the parameters of the surrounding electromagnetic field as follows:
Function γ -(z) as well as the corresponding shape of the NT, r -(z), which can be determined from Eq. (2), have the shape shown in Fig. 1(a). This shape shows up as the shape of a thread drawn out of a liquid material reservoir (see Ref.  for an example of drawing nanofibers from bulk glass). Function γ +(z) and the corresponding shape of the nanoswell, r +(z) have the shape shown in Fig. 1(b).
4. Transmission properties of a NT and nanoswell supporting singular Gaussian beams
This paper investigates the transmission properties of a NT and a nanoswell illustrated in Fig. 1 and having the shapes r -(z) and r +(z), respectively. The considered problem can be better understood with illustrations shown in Fig. 2 and 3. Fig. 2 compares the radiation loss in a bent microfiber (a) and in a NT of shape r -(z) (b). Fig. 3 compares the lossless propagation of a regular Gaussian beam (a) and of a beam supported by a nanoswell r +(z)(b).
Consider, first, Fig. 2. In the case of a bent microfiber [9,10], illustrated in Fig. 2(a), the electromagnetic field decays exponentially both into the internal direction (towards the center of microfiber curvature) and into the external direction. In the external direction, there exists a finite effective barrier. Inside the barrier the field deceases exponentially, while outside the barrier the field oscillates and has the form of outgoing wave. Figure 2(a1) shows variation of the effective transverse dielectric constant along the radial direction. Figures 2(a) and 2(a2) illustrate the distribution of the electromagnetic field near the bent microfiber. The negative sign of the effective dielectric constant corresponds to the classically forbidden region and to the exponential decrease of the electromagnetic field. The positive dielectric constant corresponds to the classically allowed region. The radiation loss in this case is determined by tunneling through a potential barrier. Figures 2(b) and (b2) illustrate a similar spatial behavior of the field distribution near the NT r -(z) shown in Fig. 1(a). Figure 2(b1) shows the corresponding behavior of the effective transverse dielectric constant, which is determined by Eq. (3) and is proportional to a - bν 2. In this case, a < 0 and b < 0 and the NT is surrounded by the barrier (classically forbidden) region ν < (a/b)1/2. In this region, the field exponentially decreases along the radial direction. The radiation loss is determined by the outgoing wave in the classically allowed region ν > (a/b)1/2. Calculation of the flux density along the radial direction (see Appendix 3) yields the following expression for the attenuation constant of the NT:
This result has a clear physical meaning. The attenuation constant grows with the pre-exponential factor in Eq. (8), which is proportional to the transversal component of the propagation constant in the evanescent region (see Eq.(1)). This factor has a minimum in the center of the NT, where the NT has the smallest radius, and tends to infinity near the edges of the NT, for z → ±L, where the NT radius becomes large and Eq.(8) fails. The exponent in Eq.(8) is independent of z, which indicates the uniform transparency of the effective potential barrier surrounding the NT. The latter follows from separation of variables in the wave equation, possibility of introduction of the effective transversal coordinate ν, and the etalon Eq. (3).
Consider now Fig. 3, which compares a regular Gaussian beam (a) and a beam supported by a nanoswell r +(z) (b). As opposed to the NT shown in Fig. 2(b), Fig. 3 illustrates the cases when the parameter b in Eq. (3) is positive, b > 0 . Then at ν, which is large enough, the dielectric constant is negative in both Fig. 3(a1) and Fig. 3(b1) and the electromagnetic field density exponentially decreases with growth of the distance from the nanoswell axis (Figs. 3(a), (a2) and Figs. 3(b) and (b1)). For this reason, propagation of the evanescent field along the nanoswell is similar to the propagation of a regular Gaussian beam. In particular, it is lossless in the approximation considered. However, the field in the immediate neighborhood of a nanoswell behaves more singular than the field of a regular Gaussian beam (compare Figs. 3(a2) and (b2)). It is known that tapers can be lossless in special cases (see  and references therein). The nanoswell r +(z) presents an interesting example of a lossless adiabatic taper which has a swell in the middle and, as follows from Eq. (2), for large z, decreases logarithmically as 1/√ln(∣z∣/L) to zero. At first sight, the absence of losses in this taper contradicts the existence of a threshold radius of a NT, below which the propagation of the fundamental mode is impossible . However, Ref.  considers a biconical taper, which possesses a minimum radius, while the minimum radius of the nanoswell does not exist.
5. Discussion and summary
At first sight, the presented examples of a NT and a nanoswell, which are, respectively, lossy and lossless, are very special and can hardly be directly applied to the calculation of losses in realistic ultra-thin nonuniform dielectric waveguides. Nevertheless, the tunneling dynamics of light in the considered waveguides significantly complements and refines our understanding of the structure of the evanescent field and mechanism of radiation losses in adiabatic nanotapers. In fact, generally, tapering of a photonic nanowire gives rise to focal circumferences, in the neighborhood of which the radiation waves split off . For this reason, the radiation of a common NT happens locally. The focal circumferences represent an intersection of complex caustic surfaces with real space. In the example of a NT considered in this paper, the caustic becomes real and its intersection with real space is a surface of rotation defined by ν = ρ/σ(z) = (a/b)1/2 rather than a circumference (see Eqs. (3), (4), and (5)). It is natural to suppose that for a NT, whose shape is slightly different from r -(z), the caustic, though becomes complex, will be still situated in proximity to real space. The radiation loss in the latter case will be still smoothed out along the length of a NT. In particular, there may exist a situation when the local attenuation constant of NT is independent on the coordinate z, similar to the attenuation constant of a microfiber with constant bending radius. For the NT r -(z), the attenuation constant is not uniform along the NT due to the z-dependence of the pre-exponent factor in Eq. (8). However, the transparency of the introduced effective parabolic potential barrier is independent of z.
The nanoswell r +(z) is lossless in the considered approximation. However, the radiation loss of this nanoswell of a higher-order of smallness may be determined with the higher-order terms which were ignored in our derivation based on the first-order approximation of the parabolic equation method . The latter is basically the paraxial approximation. Similarly, a regular Gaussian beam, which is the solution of the wave equation in the paraxial approximation, is lossless along the transversal directions. However, taking into the account the higher-order terms beyond the paraxial approximation it can be shown that a Gaussian beam may exhibit extremely small transversal radiation losses, which grow with simultaneous decreasing of parameters a and b in Eq. (3) . The latter effects are beyond the scope of this paper.
In summary, a model of a NT, which allows asymptotically accurate separation of variables in the wave equation, has been investigated. The evanescent part of the fundamental mode of this NT has the form of a singular Gaussian beam. For certain values of parameters, when the NT has a swell in the middle and narrows down to zero at the infinity, it is lossless. For other values, when the NT has a biconical shape, it exhibits the exponentially small radiation loss, which is determined as a tunneling rate through an effective parabolic potential barrier. The latter represents an exceptional example when the radiation loss is distributed along the length of an adiabatic NT rather than happens locally near focal circumferences in the evanescent field region.
The fundamental mode in the vicinity of an adiabatic NT
Consider the propagation of light along a smoothly deformed weakly guiding axially symmetric dielectric NT. The refractive indices of the NT and of the ambient medium are n 1 and n 2, respectively. The radius of the NT, r(z), is assumed to be an adiabatically slow function of the coordinate z along the fiber axis. Also, it is assumed that r(z) is significantly less than the characteristic radiation wavelength, β 0 r(z)≪1, where β 0 = 2πn 2/λ is the propagation constant and λ is the radiation wavelength in free space. In this situation, at the distances from the NT ρ ≫ r(z) the fundamental mode is linearly polarized and can be described by a scalar wave equation:
Here and below the cylindrical coordinates (ρ,φ,z) are used. The condition β 0 r(z)≪1 implies smallness of the local radial component of the propagation constant outside the NT, γ(z), compared to the longitudinal component, β(z), i.e.
In the adiabatic approximation, function γ(z) is determined through the NT radius r(z) by Eq. (2). In particular, for a glass NF in air, n 1 = 1.45 and n 2 = 1 and
The shape of the NT is determined through γ(z) by the asymptotic inversion of Eq. (2):
More accurate inversion of Eq. (2) should be performed numerically.
In a relatively close vicinity of the NT, the normalized evanescent component of the fundamental mode can be found in the form:
where K 0(x) is the modified Bessel function of the second kind. For γ(z)ρ ≫ 1 the asymptotics of Eq. (A1.5) is:
The adiabatic solution defined by Eq.(A1.5) is valid only in the vicinity of the NT ρ ≪ ρL - Lγ/β 0, where L -∣γ/(dγ/dz)∣ is the characteristic length of the NT nonuniformity . Due to γ/β 0 ≪ 1 this vicinity is relatively small. Furthermore, the radiation loss of a NT is predominantly determined by the behavior of solution in the region ρ ≳ ρL where Eq. (A1.5) fails . The global asymptotic solution of the wave equation can be found using Eq.(A1.5) as a boundary condition at ρ ≪ ρL. The semiclassical approach developed in Ref.  allows to determine this solution. Generally, the latter cannot be found by separation of variables in the wave equation. However, the shape of a NT considered in this paper does allow the asymptotic separation of variables (see Appendix 2).
Singular Gaussian beam model
and function Λ(ν) satisfies the equation
where the coordinate ν is the effective transverse coordinate which has been asymptotically separated from the coordinate z. The constants Aij in Eq. (A2.2) are related as follows:
The free parameters of the solution defined by Eqs. (8)-(11) are A 11 , A 12, a, and b. In the main text, Eq. (5) is equivalent to Eq. (A2.2). However, in Eq. (5) the coefficient A 12 has been eliminated by translation along the axis z.
Eq. (A2.1) represents the Laguerre–Gaussian beam under the condition that this solution is finite for ρ → 0 and tend to zero for ρ → ∞. In this case, solution of Eq. (A2.3) should be finite for ν → 0 and tends to zero for ν → ∞. Then Λ(ν) can be expressed through the generalized Laguerre polynomials. Alternatively, in our situation, solution (A2.1) should be matched with solution (A1.6), which is singular at ρ → 0 . Due to the cylindrical symmetry of Eq. (A1.6) we have to set m = 0 in Eq. (A2.1). Then function Ψ(ρ, φ, z) is independent of φ and, for simplicity, will be written as Ψ(ρ,z). Solutions of Eq. (A2.3) can be exactly expressed through the confluent hypergeometric functions. Matching the solution (A2.1) with the evanescent wave (A1.6) can be performed only if a < 0. The corresponding asymptotic evanescent solution of Eq. (A2.3), which is valid in the classically forbidden (barrier) region, bν 2 - a > 0 , and for -1/a ≪ ν , is
In the region ν > νturn =(a/b)1/2, Eq. (A2.1) should contain only the waves that are outgoing along the axis ρ. In order to satisfy this condition, one has to consider a linear combination of the solution defined by Eq. (A2.6) and a solution which is similar to Eq. (A2.6) but exponentially grows along the axis ρ (i.e. has a positive sign in front of ∫ν 0…). The latter solution is exponentially small in the region close to the NT taper, and does not affect satisfaction of the boundary condition. Direct application of the matching rules  near the turning point ν = ν turn yields the following asymptotic form of the solution in the classically allowed region ν > ν turn:
Radiation loss of a biconical taper r-(z)
The expression for the fundamental mode (A2.10) allows us to find the radiation loss of the NT taper. Provided that the transmission loss is small it can be defined as follows. The total power of radiation at the cross-section of the NT taper having the coordinate z is
The attenuation constant α(z) is determined from the equation:
which is equivalent to
Calculation of I(z) in Eq. (A3.3) can be performed by substitution Ψ(ρ,z) → Ψ(0)(ρ,z) and using Eq. (A1.1), which gives I(z) = 1 with exponential accuracy. Calculation of dI(z)/dz in Eq. (A3.3) can be performed using Eq. (A3.4). As the result the expression for α(z) takes the form:
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