## Abstract

Radiation losses of optical nanofibers are investigated in assumption of Gaussian statistics of distorted glass/air interface. Nonlinear relationship between the radiated power and roughness power spectrum is established. The losses in the single mode silica nanofibers are estimated for the case of inverse-square law of the roughness power spectrum.

© 2008 Optical Society of America

## 1. Introduction

Progress in fiber optics technologies caused attention to the problem of radiation losses induced with a roughness of a waveguide surface. The open surface of the waveguide core has a structure of frozen capillary wave [1–3]. As formation of such structure results from equilibrium thermodynamics, it can not be eliminated by means of technology. Therefore, the roughness induced radiation losses are intrinsic in the waveguides with glass/air interface. These losses are considered to be critical when essential part of the light power is carried with an evanescent wave, particularly in photonic crystal waveguides [1,2], and optical nanofibers [4,5].

In recently published paper [6], the radiation losses in nanofibers are estimated in assumption of sinusoidally perturbed surface. It is demonstrated that for realistic perturbation amplitude, the typical values of the loss coefficient are about 1–10^{-3} dB/mm and substantially depend on the perturbation period and refractive index contrast. Besides, the paper contains the conclusion that the model of sinusoidal perturbation can be generalized for all kinds of the surface deformation by means of Fourier analysis. This statement is controversial because direct generalization of the represented model assumes linear relationship between the amplitude of perturbation and the power of light scattered by this perturbation. Meanwhile, the radiated power is expressed as the square of linear functional of perturbation amplitude and thus can not be represented as a sum of independent contributions of perturbation’s Fourier components. This problem is solved with a help of statistical approach, when the perturbed surface is treated as a random field and the radiated power is determined with averaging over perturbation ensemble.

The statistical approach to the problem of light scattering in randomly perturbed waveguides, first developed in early studies of Marcuse [7,8], was used in [1,2] in order to estimate the lower bound of intrinsic losses in photonic crystal fibers. The mean radiated power is expressed through the power spectral density of the random roughness and this relation has the linear character. Marcuse’s approximation is quite good for weakly guiding fibers but it sets aside the following.

It is well known [9] that the solution of Maxwell equations for inhomogeneous medium in the first approximation of the small perturbation method has a form of linear relation between the mean power of the scattered wave and the power spectral density of refractive index fluctuations. It is demonstrated [8] that the perturbation of the waveguide surface can be represented with equivalent perturbation of the refractive index. The refractive index perturbation is expressed as discontinuous function of the amplitude of the surface roughness. As this relation can not be linearized, the radiated power must have nonlinear form of dependence on the roughness power spectrum even in the first approximation of the small perturbation method. This means that a part of the scattered power, proportional to low frequency components of the roughness spectrum, is lost in the linear solutions. As the power spectrum has inverse-square law of dependence on the spatial frequency [3], the scattering on harmonics of the small frequencies can be considerable.

It was demonstrated earlier [10] that the power spectrum of refractive index fluctuations in perturbed planar waveguide can be expressed through the roughness correlation with a help of Price’s theorem about nonlinear operations over Gaussian random field [11,12]. In the present paper, we intend to establish the relation between the radiation losses and the power spectrum of the rough surface of nanofiber, and to estimate the loss value.

## 2. Radiation losses in approximation of weakly perturbed fiber surface

Following [6, 13], the rough surface of a fiber waveguide is represented as a variation of its radius ρ

where ρ_{0} is the radius of unperturbed waveguide, the function ξ(*z*, φ) describes the perturbation (Fig. 1).

We take that ξ is the uniform Gaussian random field [12] with zero mean and statistical correlation function, which depends on the residual coordinates Δ_{z}=*z*
_{2}-*z*
_{1}, Δ_{φ}=φ_{2}-φ_{1}

where σ^{2}
_{ξ} is the roughness variance, γ_{ξ} denotes normalized correlation. We suppose the perturbation magnitude ξ being small. The small order of ξ means that the amplitude of electric field of both propagating and radiation modes can be considered constant within the perturbed region. As in nanofibers the perturbation is less in scale than 1 nm [4], the small order condition holds with high accuracy.

We confine the consideration to the most important case of single-mode operation of the waveguide. So, the propagating light wave has the appearance of the fundamental HE_{11} mode with electric component **E**
_{11}=**e**
_{1}(*r*, φ) exp(*i*β_{1}
*z*). The scattered wave is expressed as a superposition of radiation ITE and ITM modes **E**
^{r}_{ν}=**e**
^{r}
_{ν}(*Q*;*r*, φ)exp(±*i*β(*Q*)*z*), where *Q*
^{2}=ρ^{2}
_{0}(*k*
^{2}
*n*
^{2}
_{c1}-β^{2}), the signs +/- correspond to the forward and backward propagating modes, respectively. We choose the modal fields in the form with exponential dependence on the azimuthal coordinate [8], (Appendix)

$${\mathbf{e}}_{v}^{r}(Q;r,\phi )={\mathbf{\psi}}_{v}^{r}(Q;r)\mathrm{exp}\left(\mathrm{iv}\phi \right),v=0,\pm 1,\pm 2\dots $$

The mode normalization is chosen as

$$\underset{0}{\overset{2\pi}{\int}}d\phi \underset{0}{\overset{+\infty}{\int}}rdr{\mathbf{e}}_{v}\left(Q\right)\times {\mathbf{h}}_{\mu}^{*}\left(Q\prime \right)\xb7\mathbf{z}=\delta \left(Q-Q\prime \right){\delta}_{v\mu}$$

where **h** denotes the magnetic component of the modal field.

An amplitude of the radiation mode, exited by the fiber of length *L*, is given by [6, 13]

where *ā* is the amplitude of the propagating HE_{11} mode and the following designations are introduced

$${u}_{\mathrm{cl}}=C\left({\mathbf{\Psi}}_{v}^{r*}(Q;{\rho}_{0}+0)\xb7{\mathbf{\Psi}}_{1}\left({\rho}_{0}+0\right)\right),$$

$$C=\frac{ik{\rho}_{0}}{4}{\left(\frac{{\epsilon}_{0}}{{\mu}_{0}}\right)}^{\frac{1}{2}}\left({n}_{\mathrm{co}}^{2}-{n}_{\mathrm{cl}}^{2}\right)$$

We notice that the function *f*
_{ν} (*Q*;ξ) appears as a transform of the random perturbation ξ. Therefore, the amplitude of radiation mode is a random variable with statistical properties dependent on the perturbation statistics. Losses for radiation of the definite mode are given by the mean square of the proper modal coefficient

The total mean power, radiated with the random roughness, is expressed with integral over the whole of system of radiation modes [13]

$$+\sum _{v}\underset{0}{\overset{{k\rho n}_{\mathrm{cl}}}{\int}}{p}_{v}^{(+\mathrm{ITM})}\left(Q\right)dQ+\sum _{v}\underset{0}{\overset{{k\rho n}_{\mathrm{cl}}}{\int}}{p}_{v}^{\left(-\mathrm{ITM}\right)}\left(Q\right)dQ$$

The loss coefficient is obtained as

where *P̅*=*ā*
^{2} is the power of the fundamental mode.

Substituting (5) into (8), carrying out transformations similar to [8], we express the mean power of the radiation mode through the correlation of *f*
_{ν} (*Q*;ξ)

where

Deriving (11) takes into account periodic behavior of *G _{f}* in the azimuthal coordinate

*G*(Δ

_{f}_{φ}, Δ

_{z})=

*G*(Δ

_{f}_{φ}±2π, Δ

_{z}). Also, it is assumed that

*G*takes significant values only when Δ

_{f}_{z}≪

*L*. If the last condition is violated, the integral for dΔ

_{z}should be taken in limits -

*L*≤Δ

_{z}≤

*L*with the window

*w*(Δ

_{z})=1-|Δ

_{z}|

*L*

^{-1}.

The expression (11) has a quite expected structure. As the amplitude of exited mode is the Fourier component of the uniform random field *f*
_{ν} (6), the mean square of the amplitude is given by Fourier transform of the correlation *G _{f}*. In addition, it follows from (11) that radiation losses are linear in

*L*.

## 3. Dependence of radiated power on roughness correlation

The mean power of the radiation mode depends on the perturbation ξ through the random field *f*
_{ν} (6). The multipliers *u*
_{co},*u*
_{cl} (7) are the scalar products of electric fields of fundamental and radiation modes in core and cladding. As the radial component of the electric field has discontinuity in core/cladding interface, the expression (6) defines sectionally linear mapping ξ→*f*
_{ν}. The functional dependence of correlation *G _{f}* on the roughness correlation is established with Price’s theorem as follows

In accordance with (6), we have $\frac{{d}^{2}{f}_{v}}{d{\xi}^{2}}=\left({u}_{\mathrm{co}}-{u}_{\mathrm{cl}}\right)\delta \left(\xi \right)$ . Therefore, averaging in (13) is elementary and gives

Solving this equation under the boundary conditions

$${G}_{f}\left({\gamma}_{\xi}=1\right)={\u3008\mid f\mid \u3009}^{2}=\frac{{\sigma}_{\xi}^{2}}{2}\left({\mid {u}_{\mathrm{co}}\mid}^{2}+{\mid {u}_{\mathrm{cl}}\mid}^{2}\right)$$

results in

Relations (11), (16) establish the desired dependence of the mean power of the radiation mode on the roughness correlation.

It is convenient to represent the expression (16) in the following form

where

Note that in (11) β_{1}∓β(*Q*)>0, because |β(*Q*)|<*kn*
_{cl}<β_{1} [13]. Therefore, the constant term from (17) does not contribute to the power *p*
_{ν}(*Q*) and can be eliminated. The function $\tilde{\gamma}$
(Δ_{z},Δ_{φ}) is well approximated with a square law $\tilde{\gamma}$
≈γ^{2}
_{ξ} (Fig. 2). Assuming such approximation, the power of radiation mode is expressed through the roughness power spectrum *S*
_{µ} (β) by

where

According to (20), the radiated power is a sum of two terms, the first term being linear and the second one being quadratic in correlation γ_{ξ}. Note that both terms are of the same order of smallness in roughness variance σ^{2}
_{ξ}, but are different in dependence on the refractive index contrast because Γ_{L}~(*n*
^{2}
_{co}-*n*
^{2}
_{cl})^{2}, Γ_{N}~(*n*
^{2}
_{co}-*n*
^{2}
_{cl})^{4}. Therefore, the quadratic term in (20) is negligibly small for weakly guiding fibers, but it becomes significant in the particular case of nanofibers, in which *n*
_{co} and *n*
_{cl} are considerably different in value.

To explain the nature of the second term in (20), let us represent the perturbation ξ with equivalent current sources with density [13]:

In the last expression, the electric field **e** equals to unperturbed field of the fundamental mode in the core if ξ<0, and it is equal to the field in the cladding otherwise:

Consider the current induced with a separate Fourier component of the perturbation, which has the spatial frequency Ω:ξ(*z*;Ω)=ξ_{Ω}sinΩ*z*. Since *r* - component of the electric field of the fundamental mode is discontinuous in core/cladding transition, the correspondent component of the field **e** is of the form of raised meander function with the period Ω^{-1}, and can be expanded in Fourier series: *e _{r}*=

*c*

_{0}+

*c*

_{1}sin(Ω

*z*)+

*c*

_{2}sin(2Ω

*z*)+…. Therefore, the induced current (23) is expressed as superposition of harmonic currents: $\mathbf{J}=\sum _{m}{\mathbf{J}}_{m}\mathrm{exp}\left(i\left({\beta}_{1}-m\Omega \right)z\right)$ . Thus, the harmonic perturbation is represented with the equivalent anharmonic induced current, and the linear and the quadratic terms in (20) correspond to the currents

**J**

_{1}and

**J**

_{2}respectively.

In context of this paper, the losses, which are caused with low frequency perturbations, are of the special importance. The current **J**
_{m} exp(*i*(β_{1}-*m*Ω)*z*) radiates in resonance the plain waves in directions, which subtend an angle θ with *Oz* axis. This angle satisfies the relation: *kn*
_{cl} cos θ=(β_{1}-*m*Ω). As β_{1}>*kn*
_{cl}, the losses are caused with currents **J**
_{m} with *m*≥1. It is easy to show, that **J**
_{1}~(*n*
^{2}
_{co}-*n*
^{2}
_{cl}), **J**
_{m}~(*n*
^{2}
_{co}-*n*
^{2}
_{cl})^{2}, *m*=2,3…. Therefore, in weakly guiding fibers, the induced currents **J**
_{2}, **J**
_{3}… could be neglected. Then, the minimal frequency of perturbation, which still satisfies the resonance condition, equals Ω_{min}=β_{1}-*kn*
_{cl}. The lower frequencies of the perturbation cause no radiation losses. When refractive index contrast is high, the currents **J**
_{1} and **J**
_{2} have the same order of smallness, and produce two meaningful components in the expression for total radiation losses. It should be noticed that the minimal frequency of the resonant perturbation is halved, when the current **J**
_{2} is taken into account: Ω_{min}=(β_{1}-*kn*
_{cl})/2.

## 4. Losses from perturbation with inverse square-law power spectrum

In accordance with [3], unbounded glass surface has a random roughness with spectral power density *S*(**β**)

where *T* is the glass transition temperature, α is the surface tension in transition, **β** is bidirectional vector of spatial frequencies, bounded in the absolute value as β_{low}<|**β**|<β_{high}. The lower cutoff frequency is conditioned with gravity
${\beta}_{\mathrm{low}}=\sqrt{\frac{\mathrm{gd}}{\alpha}}$
, where *d* is the glass density. Its value is small in comparison with the typical frequencies of the optical range. Thus, for α=0.1 J·m^{-2} [3], *d*=2500kg·m^{-3}, the estimation gives β_{low}≈0.5 mm^{-1}. The high-frequency cutoff is specified with atomic structure of the matter.

In a round fiber the spectrum must be quantized in the azimuthal coordinate because of periodicity condition, and can be approximated as

In the fiber, the cutoff frequency β_{cut} is not equivalent to β_{low} of the unbounded plane surface, and is introduced into (26) phenomenologically in order to avoid singularity of *S*
_{0} (β). With that, a numerical value of β_{cut} remains uncertain. This uncertainty, nevertheless, presents no obstacle to estimation of the radiated power. Let us assume that the lower cutoff β_{cut} is small enough against the inverse radius of the fiber ρ^{-1}
_{0}. In such approximation, the roughness variance σ^{2}
_{ξ} and the convolved spectrum *S̃*
_{µ}(β) (22) are given by

where *M*=2, if µ=0, and *M*=1, otherwise. The power *p*
_{ν}(*Q*) (20) depends on the parts of the spectra (26), (28), which correspond to the spatial frequencies β=β_{1}∓β(*Q*)>0. Therefore, the points *S*
_{µ}(0), *S̃*
_{µ}(0) lie aside the integration boundaries (9), which gives an opportunity to calculate the radiation losses in the limit β_{cut}→0.

In the single-mode operation of the waveguide, the spectral components *S*
_{0}(β_{1}∓β(*Q*)), *S̃*
_{0}(β_{1}∓β(*Q*)) are considerably larger than the rest of *S*
_{µ}, *S̃*
_{µ}. These components “are responsible” for excitation of the radiation modes, which have the same azimuthal number with the fundamental mode. So, the terms with ν=1 are the most significant in the expression for the total radiated power (9).

## 5. Numerical results, discussion and conclusion

We applied the proposed theoretical model to estimate the radiation losses caused with the rough glass/air interface of the nanofibers. The physical parameters of the model were chosen so as to make possible comparison with the reported experiments [4, 5]. For the lack of data, we confined the calculation to the case of the silica fiber, for which we accepted *n*
_{co}=1.46, *T*=1500K, α=0.3 J·m^{-2} [1]. The calculated dependence of the loss coefficient on the fiber diameter at the wavelength λ=633nm is represented at Fig. 3. The chosen diameter range corresponds to the single-mode operation. Figure 3(a) shows separately the losses calculated in linear in *S*
_{0} (β) approximation. As follows from these calculations, the total loss estimation increases essentially when the term, proportional to the convolved spectrum *S̃*
_{0}(β), is taken into account. So, the surface induced radiation losses are considerably nonlinear in the perturbation spectrum.

The calculations, carried out for various values of the lower cutoff β_{cut}, demonstrate that the choice of this parameter makes effect only when the fiber diameter is especially small, in our case being less than 180 nm (Fig. 3(b)). The result, which fits qualitatively to the experiment [4, 5], is obtained in the limit β_{cut}→0.

The plot of losses at Fig. 3(b) can be separated into three parts which demonstrate substantial difference in behavior. The losses go up with decreasing waveguide diameter from 450 nm to approximately 180 nm. They form pseudo-plateau in the range 180 nm<*D*<300 nm. For very small diameters *D*<180 nm, the plot of losses becomes very sensitive to the choice of the lower cutoff frequency of the perturbation. To explain the plot’s behavior, we shall notice at first that the radiated power (20) is proportional to the power density *w*
_{1} of electric component of the fundamental mode, taken in the perturbed area. Dependence of *w*
_{1} on the waveguide diameter is nonmonotonic, and has the maximum at *D*≈300 nm (Fig. 4(a)). We notice also that the radiation arises from the spectral components of the perturbation which have the spatial frequencies β exceeding Ω_{min} : β>Ω_{min}=(β_{1}-*kn*
_{cl})/2. As the gap (β_{1}-*kn*
_{cl}) decreases with the decrease of the waveguide diameter, the low frequency perturbation components add their contribution into the total radiation losses. The perturbation power spectrum is expected to have inverse-square low, therefore, the contribution of the low frequencies is very critical, and could be explained in terms of the effective depth of the perturbed layer Δ_{ξ}.

Let us define Δ_{ξ} as:

Dependence of Δ_{ξ} on the waveguide diameter is demonstrated at Fig. 4(b). One can see that Δ_{ξ} increases with decreasing diameter, and essentially depends on the choice of β_{cut} when *D* becomes less than 180 nm. Thus, the growth of losses with decreasing *D* from 450 nm to 300 nm is explained with simultaneous growth of the power density of the fundamental mode in the perturbed layer and of the depth of this layer. In the range 180 nm<*D*<300 nm, the trends of Δ_{ξ} and *w*
_{1} have the opposite character and compensate each other. For *D*<180 nm, the loss estimation depends on the phenomenological parameter β_{cut}. Passage to the limit β_{cut}→0 gives the reasonable estimation of losses, but leads to the growth of the effective depth of the perturbation to the values which exceed the waveguide diameter. To our opinion, the extremely low frequency perturbations with large amplitudes should be understood as random waveguide bends. Naturally, there are no reasons to describe such perturbations with the inverse-square law spectrum. So, the estimation of radiation losses in the silica nanofibers with *D*<180 nm has rather qualitative character.

The calculated losses are somewhat less in value than those observed experimentally. Apparently, the suggested model is approximate, and sets aside some important particularities of the experiment. With it, the model must be useful in understanding of losses in optical nanofibers, caused with intrinsic roughness of glass/air interface.

## Appendix A

A modal field of a round fiber waveguide is expressed in transversal cross-section with a product of two functions. The first function depends on *r*, the second one depends on φ. A choice of the modal system is ambiguous because the modes are degenerate in propagation constant. In [6, 14], they used the system of even and odd modes, which is described in detail in [13]. The azimuthal dependence of these modes is expressed with sine and cosine functions. In the problems of light scattering and mode coupling, it is convenient to use an alternative set of orthonormal modes with the exponential type of the azimuthal dependence. The relation between these two equivalent modal sets is established with the following expressions:

for the pair of the fundamental modes

for radiation modes

where **e**
_{11, even}, **e**
_{11, odd} are the fields of the even and the odd HE_{11} mode, **e**
^{r}
_{ν, even}, **e**
^{r}
_{ν, odd} are ITE or ITM modes, *N*, *N*
_{ν}(*Q*) are the normalization factors [6, 13].

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