## Abstract

We present a simple variational method for the analysis of UV-side-illuminated single-mode fibers. We use a shiftable, two-dimensional elliptical Gaussian function as a field trial for the fundamental-mode. By this method, the actual UV-induced peak refractive-index increase can be quantitatively related to the measured effective-index increase. The asymmetry of the UV-induced refractive-index profile due to the absorption of the writing UV light causes both form birefringence and transition losses between UV-illuminated and non-illuminated fiber sections. These characteristics are easily calculated from the results of the variational method. We show that UV-illumination of the fiber from two opposite sides reduces both the form birefringence and the transition losses.

©2001 Optical Society of America

## 1 Introduction

The UV-writing process is widely used to fabricate Bragg gratings or long-period gratings in single-mode fibers [1]–[4]. UV-illumiation of a photosensitive fiber (Fig. 1) results in an increase of the refractive index in the photosensitive fiber core. Thus, the effective index of the fundamental mode in the fiber is also raised, either by the zero-order Fourier (or average) component of the refractive-index modulation along the grating [5] or by the longitudinally constant index increase in homogeneously UV-illuminated fiber sections [6]. The quantitative relations between the UV-induced refractive-index change and the propagation characteristics of the fundamental mode are investigated theoretically in this paper.

First, the UV-induced increase of the effective index of the fundamental mode is considered. Usually it is the quantity actually assessible by the various measurement methods [7, 8]. However, it does not give direct information on the actual UV-induced refractive-index change of the fiber material, the latter being of interest for optimizing the photosensitivity of fibers. Much more, the actual UV-induced peak refractive-index change in the fiber can be perceptibly larger than the increase of the *effective* index of the fundamental fiber mode. Thus, a knowledge of the quantitative relation between the two index increases is desirable in order to realistically interprete experimental results. (Throughout the paper, we will strictly distinguish between the UV-induced change of the effective index as a characteristic of the fundamental mode and the UV-induced change of the *refractive* index as a characteristic of the fiber material.)

Further, the UV-induced refractive-index profile is non-symmetric across the fiber core (Fig. 1a) due to the absorption of the writing UV-laser beam. As has been shown both by atomic-force microscopy [9] and by refracted-near-field measurements [8], the UV-induced refractive-index profiles have an exponential-like form with a 1/*e* width of only a few microns. Such asymmetries cause form birefringence [10], which can lead to an unwanted spectral splitting of the transmission dip of narrow-band Bragg gratings or to inacceptable polarisation-mode dispersion in chirped dispersion compensating Bragg gratings.

Finally, the changed refractive-index profile in a UV-illuminated fiber section can also produce perceptable transition losses at the transition to a non-illuminated fiber section. Besides the contribution from the enhanced field confinement [12, 11], mainly the field shift out of the fiber axis due to the profile asymmetry leads to a field mismatch and may cause transition losses of up to the order of 1 dB [12, 13].

In order to evaluate the effective-index increase, the form birefringence and the transition losses for given UV-induced index profiles, various numerical models could be chosen. In a rough but simple approximation, the fiber could be modelled by an equivalent planar waveguide with a UV-induced exponential profile function, for which analytical characteristic equations in terms of Bessel functions are available for TE and TM modes [14]. A more complicated Fourier-Bessel expansion has been applied for calculating the effective-index increase and the transition losses in a UV-illuminated fiber modelled by a step-delta function profile [12], which, however, does not yield any information on the form birefringence. A vectorial finite-element method as one of the most general techniques could also be used [15, 16], but it is rather complicated and gives less insight than analytical methods.

In this paper, we apply a simple variational method with a Gaussian fundamental-mode field function to analyse UV-side-illuminated single-mode fibers (Section 3). The field function chosen here is able to stretch into an elliptical form and to shift out of the fiber axis, both to follow the asymmetry of the UV-induced index profile. This method allows us to study easily the propagation characteristics of the fundamental mode in UV-side-illuminated fibers: First, we calculate quantitatively the relation between the UV-induced refractive-index profile and the effective index of the fundamental mode (Section 4). Further, the form birefringence in UV-side-illuminated fibers is easily evaluated from the two field widths of the non-circular scalar mode field (Section 5). Third, the transition losses at the transition between UV-illuminated and non-illuminated fiber sections are calculated from the two field widths and the field shift in the UV-illuminated fiber section and the field radius in the non-illuminated fiber (Section 6). We start with specifying the refractive-index profiles of our UV-illuminated fibers.

## 2 UV-induced refractive-index profiles

Before UV illumination, the optical fibers considered here have circularly symmetric step-index profiles

where *r*=(*x*
^{2}+*y*
^{2})^{1/2}, and (*x*, *y*) are cartesian coordinates in the plane of the fiber cross section. Such a profile can be realized by homogeneous germania doping of the core. Accordingly, we also assume a homogeneous photosensitivity profile across the core, and no photosensitivity in the cladding, and a similar step-profile distribution of the absorption coefficient for the UV-laser light. Thus, the UV-induced refractive-index profile written from one side will have an exponential-like shape (Fig. 1a) according to the absorption of the UV-laser light during illumination [8, 10]. The refractive-index profile of the UV-illuminated fiber can be modelled as

where *δn*
^{2}+(*x*, *y*) and ${\delta n}_{-}^{2}$(*x*, *y*) are the profile contributions induced by UV-illumination from the left (UV light propagating along the +*x* direction, Fig. 1a) and from the right (UV light propagating along the -*x* direction, Fig. 1b), respectively. These UV-induced profile contributions can be written

Here, 2*α* is the power attenuation constant for the UV laser light, and ${\delta \widehat{n}}_{+}^{2}$ and ${\delta \widehat{n}}_{-}^{2}$ denote the peak profile changes at (*x*=-*ρ*, *y*=0) and (*x*=+*ρ*, *y*=0) for UV-illumination from the left and the right, respectively. If the fiber is UV illuminated from the left side only, for example, we have *δn̂*^{2}-=0. Symmetrical illumination means *δn̂*^{2}-=${\delta \widehat{n}}_{+}^{2}$. Because the profile differences ${\delta \widehat{n}}_{\pm}^{2}$, ${\delta n}_{\pm}^{2}$(*x*, *y*) and Δ*n̄*^{2} are small, they can simply be related to corresponding refractive-index differences as ${\delta \widehat{n}}_{\pm}^{2}$⋍2*n*
_{2}
*δn̂*_{±}, ${\delta n}_{\pm}^{2}$(*x*, *y*)⋍2*n*
_{2}
*δn*±(*x*, *y*) and Δ*n̄*_{2}⋍2*n*
_{2}Δ*n̄*. In the following, we analyse the properties of the fundamental mode in UV-side-illuminated fibers.

## 3 Variational analysis using a Gaussian field function

Because of the small refractive-index differences, we assume our fiber to be weakly guiding. The fiber axis coincides with the *z* axis of a cartesian coordinate system. In order to obtain the characteristics of the fundamental mode propagating as Ψ(*x*, *y*) exp(-*iβz*) along the fiber, one has to solve the scalar wave equation [17]

Here, Ψ(*x*, *y*) is the scalar mode field, *β* is the propagation constant, *k*=2*π*/λ is the free-space wave number and λ is the free-space wavelength. The refractive-index profile *n*
^{2}(*x*, *y*) may have an arbitrary distribution over the transverse coordinates *x* and *y*. According to the variational principle [18], the solution of the wave equation (4) is equivalent with maximizing the expression

$$=\frac{1}{N}\underset{-\infty}{\overset{\infty}{\int}}\underset{-\infty}{\overset{\infty}{\int}}\left[-{\left({\nabla}_{t}\Psi \right)}^{2}+{k}^{2}\Delta {n}^{2}(x,y){\Psi}^{2}\right]dxdy$$

over all parameters describing the field distribution Ψ(*x*, *y*). Here,

serves as a normalization, ∇t is the transverse part of the Nabla operator and (∇
_{t}
Ψ)^{2}=(*∂*Ψ/*∂x*)^{2}+(*∂*Ψ/*∂y*)^{2}. Further,

denotes the profile elevation above the cladding level ${n}_{2}^{2}$. The exact value for *β* or *γ* is obtained only, if the exact field distribution is inserted into the right-hand side of Eq. (5). For all other field approximations, the values obtained for *β* or *γ* are smaller than the exact ones. In order to find accurate results at a simultaneously small number of parameters describing the field, one has to construct a field trial function for Φ(*x*, *y*), which is similar and flexible enough to fit closely to the exact field distribution. In a round step-index fiber, the fundamental-mode field has a circular symmetry and a nearly Gaussian radial dependence [17]. The analysis in Ref. [12] showed that the field in a UV-illuminated fiber is shifted out of the fiber axis and is no longer circular, but approximately elliptical. In order to combine simplicity and the described field behavior, we choose an elliptical and shiftable Gaussian field approximation as the trial function in our variational analysis,

Here, *w*_{x}
is the field width along the *x*-direction, *w*_{y}
is the field width along the *y*-direction, and *s* is the field shift along *x*-direction out of the fiber axis. With this field function, we obtain for general refractive-index profiles

$$+\frac{{k}^{2}}{\pi {w}_{x}{w}_{y}}\underset{-\infty}{\overset{\infty}{\int}}\underset{-\infty}{\overset{\infty}{\int}}\Delta {n}^{2}(x,y)\mathrm{exp}\{-\left[\frac{{\left(x-s\right)}^{2}}{{w}_{x}^{2}}+\frac{{y}^{2}}{{w}_{y}^{2}}\right]\}dxdy$$

and *N*=*πw*_{x}*w*_{y}
${\mathrm{\Psi}}_{0}^{2}$. The variational principle for the present field trial requires the expression (9) to become stationary over the three parameters *w*_{x}
, *w*_{y}
and *s*, i.e.

and thus yields a system of non-linear equations for *w*_{x}
, *w*_{y}
and *s*. We obtain the two field-width equations

$$\times \underset{-\infty}{\overset{\infty}{\int}}\underset{-\infty}{\overset{\infty}{\int}}\Delta {n}^{2}(x,y)\left[{w}_{x}^{2}-2{\left(x-s\right)}^{2}\right]\mathrm{exp}\{-\left[\frac{{\left(x-s\right)}^{2}}{{w}_{x}^{2}}+\frac{{y}^{2}}{{w}_{y}^{2}}\right]\}dxdy,$$

$$\times \underset{-\infty}{\overset{\infty}{\int}}\underset{-\infty}{\overset{\infty}{\int}}\Delta {n}^{2}(x,y)\left[{w}_{y}^{2}-2{y}^{2}\right]\mathrm{exp}\{-\left[\frac{{\left(x-s\right)}^{2}}{{w}_{x}^{2}}+\frac{{y}^{2}}{{w}_{y}^{2}}\right]\}dxdy,$$

and one symmetry equation

It is easy to see, that the latter symmetry equation is solved by *s*=0 for all profiles symmetric with respect to the *y*-axis. In order to find a solution of the variational problem, the non-linear system (11)–(13) can be solved and the results for *w*_{x}
, *w*_{y}
and *s* be inserted into Eq. (9), or Eq. (9) can be directly maximized. In either case, Eqs. (11) and (12) will be used in Section 5 for deriving a simple expression for the form birefringence.

For the UV-induced refractive-index profiles considered here, Eq. (9) becomes

$$+\frac{{k}^{2}\Delta {\overline{n}}^{2}}{\sqrt{\pi}{w}_{y}}\underset{0}{\overset{\rho}{\int}}\mathrm{exp}(-\frac{{y}^{2}}{{w}_{y}^{2}})\left[\mathrm{erf}\left(\frac{s+\sqrt{{\rho}^{2}-{y}^{2}}}{{w}_{x}}\right)-\mathrm{erf}\left(\frac{s-\sqrt{{\rho}^{2}-{y}^{2}}}{{w}_{x}}\right)\right]dy$$

$$+\frac{{k}^{2}\delta {\hat{n}}_{+}^{2}}{\sqrt{\pi}{w}_{y}}\mathrm{exp}\left({\alpha}^{2}{w}_{x}^{2}-2\alpha s\right)\underset{0}{\overset{\rho}{\int}}\mathrm{exp}\left(-2\alpha \sqrt{{\rho}^{2}-{y}^{2}}-\frac{{y}^{2}}{{w}_{y}^{2}}\right)\times $$

$$\times \left[\mathrm{erf}\left(\frac{s-\alpha {w}_{x}^{2}+\sqrt{{\rho}^{2}-{y}^{2}}}{{w}_{x}}\right)-\mathrm{erf}\left(\frac{s-\alpha {w}_{x}^{2}-\sqrt{{\rho}^{2}-{y}^{2}}}{{w}_{x}}\right)\right]dy$$

$$+\frac{{k}^{2}\delta {\hat{n}}_{+}^{2}}{\sqrt{\pi}{w}_{y}}\mathrm{exp}\left({\alpha}^{2}{w}_{x}^{2}+2\alpha s\right)\underset{0}{\overset{\rho}{\int}}\mathrm{exp}\left(-2\alpha \sqrt{{\rho}^{2}-{y}^{2}}-\frac{{y}^{2}}{{w}_{y}^{2}}\right)\times $$

$$\times \left[\mathrm{erf}\left(\frac{s+\alpha {w}_{x}^{2}+\sqrt{{\rho}^{2}-{y}^{2}}}{{w}_{x}}\right)-\mathrm{erf}\left(\frac{s+\alpha {w}_{x}^{2}-\sqrt{{\rho}^{2}-{y}^{2}}}{{w}_{x}}\right)\right]dy.$$

In case of symmetrical UV-illumination from the right and the left side of the fiber, ${\delta \widehat{n}}_{+}^{2}$=${\delta \widehat{n}}_{-}^{2}$=*δn̂*^{2} and *s*=0, and Eq. (14) simplifies to

$$+\frac{2{k}^{2}\delta {\hat{n}}^{2}}{\sqrt{\pi}{w}_{y}}\mathrm{exp}\left({\alpha}^{2}{w}_{x}^{2}\right)\underset{y=0}{\overset{\rho}{\int}}\mathrm{exp}\left(-2\alpha \sqrt{{\rho}^{2}-{y}^{2}}-\frac{{y}^{2}}{{w}_{y}^{2}}\right)\times $$

$$\times \left[\mathrm{erf}\left(\frac{\sqrt{{\rho}^{2}-{y}^{2}}+\alpha {w}_{x}^{2}}{{w}_{x}}\right)+\mathrm{erf}\left(\frac{\sqrt{{\rho}^{2}-{y}^{2}}-\alpha {w}_{x}^{2}}{{w}_{x}}\right)\right]dy.$$

Although Eqs. (14) and (15) look somewhat lengthy they are not really complicated. They contain merely one-dimensional integrals over simple exponential functions and error functions erf(*ζ*)=2*π*
^{-1/2}${\int}_{0}^{\mathrm{\zeta}}$ exp(-*ξ*
^{2})dξ [20], which can easily be calculated by numerical integration. We solve the variational problem for given fiber profiles by numerically maximizing Eq. (14) over the set of parameters (*w*_{x}
, *w*_{y}
, *s*).

In our analysis, we consider a step-index fiber [19] the exposure characteristic of which was measured by a Fabry-Perot method in Ref. [21]. The fiber parameters are *ρ*=3.06*µ*m, Δ*n*=0.0077, i.e. Δ*n*
^{2}=0.0225, and the wavelength is λ=1550 nm. For this fiber an effective-index increase Δ*n*_{e}
of 0.012 was achieved by UV illumination after H2-loading. In order to relate our analysis to that practical situation, we will plot the numerical results for the various modal characteristics as functions of the fundamental-mode effective-index increase up to this value. This way, we can illustrate how the peak refractive-index change and the fundamental-mode field deformation (Section 4), the growth of birefringence (Section 5) and the increase of the transition losses (Section 6) accompany the effective-index increase induced by the UV-illumination of the fiber.

## 4 Effective-index increase and field shape

The effective index of the fundamental mode follows from the results of the variational method Eq. (14) as *n*_{e}
=*β*/*k*=(${n}_{2}^{2}$+*γ*
^{2}/*k*
^{2})^{1/2}. In Fig. 2, the UV-induced peak refractive-index change *δn̂*_{+}=${\delta \widehat{n}}_{+}^{2}$/2*n*
_{2} in the fiber profile is plotted as a function of the UV-induced effective-index increase *δn*_{e}
of the fundamental mode for various values of the UV-light absorption constant *α*. The fiber is illuminated from the left side. Obviously, the peak refractive-index change *δn̂*_{+} can be significantly larger than the effective-index change *δn*_{e}
. For *α*=1/*ρ*, i.e. for a UV-intensity attenuation by a factor of 1/*e*
^{2} over a distance of one core radius, *δn̂*_{+} is larger than *δn*_{e}
at least by a factor of 3.

The field shift *s* and the two field widths *w*_{x}
and *w*_{y}
are plotted in Fig. 3 as functions of the UV-induced effective-index increase with the UV-absorption constant as a parameter. The fiber is UV-illuminated from the left side in Fig. 3a, and equally illuminated from the two opposite horizontal sides in Fig. 3b. In case of the one-sided illumination in Fig. 3a, the field shifts to the left out of the fiber axis more and more for increasing asymmentry both due to an increase of the UV-absorption constant α and due to the growth of the UV-induced profile with increasing ${\delta \widehat{n}}_{+}^{2}$. The field widths decrease and the field confines more and more as the peak refractive-index difference increases. The field width *w*_{y}
along the *y* direction is scarcely affected by the strength of the UV attenuation. For larger UV-attenuation values *α*, the field confinement along the *x*-axis becomes stronger, i.e. *w*_{x}
decreases, since the refractive-index profile becomes more and more peaked (at the UV-illuminated side of the core-cladding interface) and thus becomes effectively higher, but also narrower along direction of UV-light propagation. If the fiber is equally illuminated from the left and the right (Fig. 3b), the fiber profile is symmetric and there is no field shift (*s*=0). The field confines more and more with growing effective-index increase, but remains nearly circularly symmetric (*w*_{x}
⋍*w*_{y}
). The ring-shaped refractive-index profile (Fig. 1b) associated with high UV-absorption constants tends to spread the field in contrast to the field-confining effect of the total refractive-index increase. Accordingly, the field confinement with increasing profile height is slowed down for higher UV-absorption values.

## 5 Form Birefringence

Now we consider the form birefringence in UV-side-illuminated fibers. In Section 5.1, we review some effects of birefringence in fiber gratings in order to illustrate the importance of this propagation charcteristic. In Section 5.2, we calculate the UV-induced form birefringence by a perturbation technique, which uses the scalar fields obtained by the variational method in Section 4.

## 5.1 Effects of birefringence in fiber gratings

We summarize some effects of birefringence in fiber gratings. Birefringence, i.e. a non-vanishing difference of the propagation constants of the two fundamental-mode polarizations, may result in different spectral responses for the two polarizations both in Bragg gratings and in long-period gratings [22, 23]. Birefringence in chirped Bragg gratings can result in dramatically different group delays for the two polarizations [24], which are unwanted in gratings used as compensators for the chromatic dispersion [25]. On the other hand, the UV-induced birefringence can be exploited to affect the polarization behavior of fibers and gratings, e.g. for a control of the output polarization of fiber lasers [26].

We define the birefringence as

where *β*_{x}
and *β*_{y}
are the propagation constants and *n*_{x}
=*β*_{x}
/*k* and *n*_{y}
=*β*_{y}
/*k* are the effective indices of the *x* and *y* polarized fundamental modes, respectively.

The Bragg wavelength λ_{B} at which a longitudinally homogeneous (non-chirped) Bragg grating has maximum reflectivity, is given by the resonance condition

where Λ is the grating period. For birefringent fibers, this condition generalizes to

and is fulfilled at two different resonance wavelengths λ_{Bx} and λ_{By}. In terms of effective indices this condition implies

In order to evaluate the birefringence-induced spectral splitting Δλ_{B}=λ_{Bx}-λ_{By} of the Bragg grating spectra, we apply a Taylor expansion around the mid wavelength λ_{m}=(λ_{Bx}+λ_{By})/2,

where ${n}_{x}^{\prime}$
=*dn*_{x}
/*d*λ and ${n}_{y}^{\prime}$
=*dn*_{y}
/*d*λ are the derivatives of the effective indices *n*_{x}
and *n*_{y}
with respect to the wavelength at λ=λ_{m}. Inserting these expansions into Eq. (19) yields the splitting of the Bragg wavelengths

where *n*_{gx}
=*n*_{x}
-λ${n}_{x}^{\prime}$
and *n*_{gy}
=*n*_{y}
-λ${n}_{y}^{\prime}$
are the group indices [17, 27] of the two polarization modes. For practical weakly guiding fibers, the group indices are close to each other and to the cladding refractive index [27]. Thus, we can set *n*_{gx}
+*n*_{gy}
=2*n*_{g}
⋍2*n*
_{2}, and Eq. (23) simplifies to

For typical experimental values of the birefringence in UV-side-illuminated fibers in the range of |*B*|=1…5×10^{-5} the splitting is |Δλ_{B}|⋍10…50 pm [22].

A chirped Bragg grating with a *z*-dependent grating period Λ(*z*)=Λ_{0}+Λ_{1z} can be used to compensate for chromatic dispersion of a dispersive single-mode fiber [7]. For a grating length *L*, roughly an accumulated delay of Δ*τ*=2*Ln*
_{2}/*c* over a spectral width Δλ=2*n*
_{2}
*L*Λ_{1} can be compensated [7], where *c* is the vacuum speed of light. In order to compensate the delay accumulated in a 100 km long standard single-mode fiber with a chromatic dispersion of 16 ps/(km nm) at λ⋍1550 nm over a channel width of 1 nm, the grating must have a chirp rate of Λ_{1}=2.1×10^{-9} and a length of *L*=165 mm. In such a chirped grating the effect of birefringence is dramatic, because it separates the reflection points for the two polarizations within the grating. This may lead to an unacceptably large differential group delay Δ*τ*
_{PMD} of pulses propagating in the two polarizations, known as polarization mode dispersion: At the wavelength λ, the resonance condition (17), generalized to

is fulfilled at different positions *z*_{x}
and *z*_{y}
along the grating. The separation of these two reflection points Δ*z*=*z*_{x}
-*z*_{y}
=-λ*B*/(2Λ_{1}
${n}_{2}^{2}$) causes an approximate differential delay of the pulses reflected in the two polarizations of about Δ*τ*
_{PMD}⋍2*n*
_{g}Δ*z*/*c*⋍-λ*B*/(*c*Λ_{1}
*n*
_{2}). For the parameters of the chirped grating assumed above, this corresponds to Δ*τ*
_{PMD}=*B*×1.7*µ*s. A birefringence of only 10^{-5} produces a differential group delay of 17 ps which is inacceptable for multi-Gb/s transmission [28].

For long-period gratings, the resonance condition for maximum mode coupling at λ=λ_{L} is [4]

where *β*
_{0} and *β*
_{h} are the propagation constants of the fundamental mode and the coupled higher mode, such as a cladding mode [4] or the core-guided LP_{11} mode [29]. A mutual wavelength shift of the transmission spectra of the two orthogonal polarizations by a few nanometers has been measured for birefringent long-period gratings [23, 30]. Similar mathematical manipulations as done above for Bragg gratings yield the birefringence-induced splitting Δλ_{L}=λ_{L,x}-λ_{L,y} of the resonance wavelengths λ_{L,x} and λ_{L,y} for *x* and *y* polarized light, respectively,

where *B*
_{0}(λ_{m})=*n*
_{0,x}(λ_{m}) - *n*
_{0,y}(λ_{m}) and *B*
_{h}(λ_{m})=*n*
_{h,x}(λ_{m}) - *n*
_{h,y}(λ_{m}) denote the birefringence of the fundamental mode and the higher mode, respectively, and *n*
_{g0x}(λ_{m}), *n*
_{g0y}(λ_{m}), *n*
_{ghx}(λ_{m}) and n_{ghy}(λ_{m}) are the corresponding group indices of the two orthogonal polarizations of each mode. Here, the splitting is highly sensitive to the difference of the mean group indices of the fundamental mode *n*
_{g0}=(*n*
_{g0x}+*n*
_{g0y})/2 and of the higher mode *n*
_{gh}=(*n*
_{ghx}+*n*
_{ghy})/2 appearing in the denominator. It is interesting, that the splitting of the resonance wavelength vanishes if the two modes have the same birefringence, *B*
_{0}(λ_{m})=*B*
_{h}(λ_{m}), but different mean group indices, *n*
_{g0}(λ_{m})≠*n*
_{gh}(λ_{m}). For a non-birefringent cladding mode as the higher mode with the group index close to the cladding refractive index *n*
_{2} and a weakly birefringent fundamental mode, the relation (27) simplifies to

The mean group index *n*
_{g0} of the fundamental mode can be evaluated by numerical differentiation of the effective index obtained by the present variational method with respect to the wavelength with a wavelength-dependent refractive-index profile.

After this short overview on some birefringence effects in fiber gratings, we calculate the form birefringence by a perturbation method using the scalar fields of the variational analysis.

## 5.2 Perturbation method for the form birefringence

In order to calculate the form birefringence of the fundamental mode in the UV-illuminated fiber, we apply the usual perturbation method [31] but use the scalar field obtained by the present variational method as the unperturbed field. To this aim, we start from the exact vectorial wave equation for a fiber mode [17]

Here, **E**
_{t} is the transverse part of the electric field and *β*_{ξ}
is the propagation constant of the vector mode. Further, *ξ*=*x* and *ξ*=*y* denote the predominantly *x*- and *y*-polarized fundamental mode, respectively. The corresponding unperturbed linearly polarized fields are

where *x* and *y* denote the unit vectors along the *x* and *y* coordinates, respectively, and Ψ(*x*, *y*) is the scalar field of the wave equation (4). Since our fiber geometries are symmetric with respect to the *x* axis, these two directions describe the two possible polarizations of the fundamental mode. For our practical situation, that the right-hand side of Eq. (29) is small, the perturbation method [17] yields a small correction *δβ*_{ξ}
for the calculation of the vector-mode propagation constant *β*_{ξ}
=*β*+*δβ*_{ξ}
[31], where

and *β* is the propagation constant of the scalar fundamental mode obtained from the scalar wave equation (4). Partial integration in the right-hand side of Eq. (31) yields

Inserting the Gaussian field ansatz of Eq. (8) with

and comparing with the field-width equations (11) and (12), we obtain particularly simple polarization corrections in terms of the field width parameters *w*_{x}
and *w*_{y}
,

Formally, these results are identical with those already obtained in the analysis [32] of fibers which are symmetrical with respect to both the *x* axis and the *y* axis and thus have a well-defined centre of symmetry. In that former analysis it was clear that the centre of the scalar mode field Ψ(*x*, *y*) always coincides with the center of the refractive-index profile *n*
^{2}(*x*, *y*) at (*x*, *y*)=(0, 0) and no field-shift parameter appeared in the field description. In the present more general analysis, the field can experience a lateral shift out of the fiber axis for increasing asymmetry of the UV-induced refractive-index profile, and the field shift s occurs as an additional free parameter in the description of the scalar field trial in Eq. (8). The formal coincidence of the results (35) of the present analysis with those derived in [32] can be understood by considering the derivations in more detail: The analysis described in [32] does not actually rely on any assumption about the fiber symmetry, but is best suited only to fiber profiles with two orthogonal symmetry axes such as the elliptical or homogeneous-function refractive-index profiles serving as examples. The field ansatz was chosen with its maximum fixed at the origin of the coordinates coinciding with the well-defined fiber centre in the examples. However, in general one does not *a priori* know the position of the field maximum, for instance in case of the UV-induced profiles described in Section 2. In order to keep the field position flexible, we have introduced here the shift parameter *s*. By carrying out the present more general variational method, we find optimum values for the field shift s and for the two field widths *w*_{x}
and *w*_{y}
, and the field is approximated with high accuracy. We could afterwards take the shift s fixed at this optimum value, and start the variational analysis again but with only the two field widths *w*_{x}
and *w*_{y}
as free parameters. In this case, this reduced variational method would be formally similar to that in [32], but with a position of the underlying refractive-index profile more suitably chosen with respect to the field maximum. From this point on, the rest of the calculation would be identical with that presented in [32]. Therefore the introduction of an additional field-shift parameter *s* in the field description cannot change the principal form of the polarization corrections Eq. (35) in comparison with those given in [32]. Hence the field-shift parameter *s* does not explicitely enter the formula. The field shift only indirectly affects the polarization correction via the field widths *w*_{x}
and *w*_{y}
, which are now obtained such that the resulting field approximates the exact field with higher accuracy.

With Eq. (35) the form birefringence can be given simply in terms of the field widths *w*_{x}
and *w*_{y}
,

In Fig. 4, the form birefringence is plotted as a function of the UV-induced effective-index increase. In Fig. 4a, the fiber is UV-illuminated from the left side only as illustrated in Fig. 1a. Depending on the UV-attenuation constant, the birefringence can assume values up to 8×10^{-5}. For *α*=0, there is no asymmetry and thus no birefringence. The birefringence grows with increasing asymmetry both due to an increasing UV-attenuation constant and to an increasing content of the asymmetric profile. If the fiber is equally UV-illuminated from two opposite sides (Fig. 1b), the birefringence is drastically reduced (Fig. 4b), e.g. by a factor of 10 for *α*=1/2*ρ* and *δn*_{e}
=0.01, or even reversed and 50 times smaller in magnitude for *α*=1/*ρ* and *δn*_{e}
=0.01.

Photoinduced birefringence in fiber gratings has already been observed in 1985 [33] and 1990 [29]. It was either attributed to an asymmetric, but isotropic change of the fiber geometry [10], or to a creation of dipole moments depending on the polarization of the illumating UV laser light [34, 35].

Our results agree with the observation [10] that the UV-illumination of the fiber from two opposite sides reduces the birefringence. The agreement is also good with respect to the magnitudes for one-sided and two-sided illumination of 0…8×10^{-5} and 0…3×10^{-6}, respectively. The model cannot explain the observation made in [35], that the birefringence induced by an *s*-polarised UV-laser beam can be erased by a *p*-polarized post-illumination. It is noteworthy, that the magnitude of the birefringence obtained here agrees with the values measured both in [10] and [22] as well as with those measured in Ref. [35] for the fiber after s-polarised UV-illumination. The present model supports the theory that birefringence can be reduced by UV-illumination from two opposite sides [10] at least with equal polarisation. The theory proposed in [35] assumes that an anisotropy caused by UV-induced dipole moments [34] predominates the total birefringence, and form birefringence is only a minor effect. Consequently, this theory should also predict a further creation of such dipole moments and hence a further increase of the birefringence for UV-illumination of the fiber from the opposite side, in contrast to the birefringence reduction observed experimentally in [10] and calculated by the present theoretical approach. Apparently, a more comprehensive physical model is required to explain the two seemingly contradicting observations of the birefringence behavior.

## 6 Transition losses

At a transition between a UV-illuminated and a non-illuminated fiber section, the incoming fundamental mode looses power because of the field mismatch to the fundamental mode in the fiber after the transition. The transition loss can be calculated as

where

is the excitation coefficient between the mode Ψ(*x*, *y*) of the UV-illuminated fiber with the index profile of Eqs. (2) and (3) and the mode Ψ(*x*, *y*) of the non-illuminated fiber with the index profile of Eq. (1). The field of the latter is circularly symmetric and can be approximated as [17]

where the field radius *w̄* can be obtained from the variational method as *w̄*=*ρ*/(2 ln V)^{1/2} for the step-index fiber, and *V*=*kρ*(${n}_{1}^{2}$-${n}_{2}^{2}$)^{1/2} is the normalized frequency of the fiber. This approximation is the limit of the present field analysis for vanishing UV-induced refractive-index changes. With the fundamental-mode field of the UV-illuminated fiber in Eq. (8), the excitation coefficient becomes

and the transition losses are given by Eq. (37).

Fig. 5 shows the transition losses for various UV-attenuation values. In Fig. 5a, the transition losses for one-sided UV-illumination of the fiber increase with growing effective-index change. The higher the UV-attenuation the stronger is the growth of the losses because of the stronger asymmetry of the field and the corresponding larger field shift s out of the fiber axis. Even in the absence of any profile asymmetry (*s*=0, *w*_{x}
=*w*_{y}
=*w*_{r}
≠*w*) the transition losses do not vanish [12, 11], in which case Eq. (40) simplifies to *C*=2*w̄**w*_{r}
/(*w̄*^{2}+${w}_{r}^{2}$)≤1, since the mode in the UV-illuminated fiber has a stronger field confinement than that in the non-illuminated fiber. However, the shift of the field out of the fiber axis causes a much stronger contribution to the field mismatch and thus to the loss at a transistion between UV-illuminated and non-illuminated fiber sections [12]. This behavior is mirrored by the exponential dependence of the excitation coefficient *C* on the field shift *s* in Eq. (40). UV-illumination of the fiber from two opposite sides cancels the lateral field shift and thus reduces the transition losses. In this case a large UV-attenuation constant even prevents high transition losses, since it goes along with a ring-shaped refractive-index increase mainly near the core-cladding interface (Fig. 1b), which tends to spread the field in competition to the field-confining effect of the raised overall index difference. At the same effective-index increase, the transition losses induced by highly absorbed UV light are larger for one-sided illumination and smaller for two-sided illumination as compared to the situations with low UV-light absorption (Figs. 5 a and b).

## 7 Conclusions

We have presented a variational method for the analysis of UV-side-illuminated fibers. The chosen Gaussian field function is able to shift out of the fiber axis and to stretch into an elliptical form to follow the asymmetry of the UV-induced refractive-index profile caused by the UV-laser light absorption. The present model provides a quantitative relation between the actual refractive-index increase and the effective-index increase of the fundamental mode, the latter being usually the experimentally assessible quantity. The UV-induced peak refractive-index increase can be significantly larger than the effective-index increase of the fundamental mode, in particular for strong UV-attenuation. Simple formulas for the calculation of the form birefringence and the transition losses between UV-illuminated and non-illuminated fiber scetions are given in terms of the two orthogonal field widths and the transverse field shift. A high absorption value for the UV-laser light goes along with a strong profile asymmetry and thus with a large form birefringence. UV-illumination of the fiber from two opposite sides strongly reduces the form birefringence due to the reduced profile asymmentry. The magnitude of the calculated birefringence of several 10^{-5} agrees with experimental results reported by other authors. Transition losses increase both with increasing UV absorption and with growing height of the UV-induced refractive-index profile and can reach values up to the order of 1 dB. Besides the confinement of the modal field, mainly the lateral field shift out of the fiber axis causes transition losses due to the mismatch to the field of the non-illuminated fiber. UV-illumination from two opposite sides reduces the transition losses. In this case, a larger UV-attenuation even leads to smaller transition losses, since the more pronounced ring-shaped UV-induced refractive-index profile tends to spread the field and thus counteracts the mode-confining effect of the increasing overall index content.

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