## Abstract

We report on the inscription of chirped fiber Bragg Gratings (FBGs) with a phase mask and a deformed wavefront using a femtosecond laser. A qualitative model is developed to predict the behavior of the resulting grating period for a deformed wavefront. In addition the quantitative change of the period was simulated based on a ray optical solution of the diffraction behind the phase mask. For deforming the wavefront experimentally a cylindrical tuning lens was used. Tilting of the lens increased the higher order aberrations like coma and spherical aberration, which leads to chirped FBGs. A chirped FBG with a FWHM bandwidth of 2.5 nm could be realized. The change of the resulting fiber Bragg grating period was measured using a side diffraction setup yielding good agreement with the measured spectra.

© 2011 OSA

## 1. Introduction

Ultrashort lasers provide extraordinary means for micro- and nano structuring of transparent materials [1, 2]. Over the last decade, ultrashort pulse direct writing has enabled various tightly integrated waveguide geometries in glasses and non-linear crystals. The flexibility of the direct write approach to realize fully three-dimensional structures within the volume set it apart from conventional planar fabrication techniques [2]. For example, ultrashort pulse written waveguide arrays have fueled fundamental research of discrete optical systems [3].

Of special interest is the ability to inscribe structures in non-photosensitive, rare-earth doped glasses [4]. Here, potential applications are monolithic fiber laser cavities, realized by inscribing the reflecting fiber Bragg gratings (FBGs) directly into the active fiber core [5]. Continuous wave (cw) lasers based on that concept proved to be of extraordinary stability even at high temperatures [6], allowing for power-scaling to > 100 W [7, 8] output power.

There are two fundamental approaches to realize FBGs with ultra-short pulses: the direct write method [9] and the side-illumination with a phase mask [10]. The direct write approach offers extraordinary flexibility for tailoring the FBG properties as well as access to higher order modes [11]. Phase mask techniques on the other hand provide reproducibility for mass production of FBGs. They allow for efficient inscription of FBGs that homogeneously extend far into the cladding. Such large cross-section FBGs (LCFBG) represent efficient reflectors even in large mode area (LMA) fibers. Furthermore, they can be employed for stable operation even in few- and multimode fibers, since mode-conversion can be suppressed [12, 13].

While the narrow spectral response of the FBGs is perfect for cw fiber lasers, much broader reflection spectra are required for ultrashort pulse fiber lasers. This can be achieved with chirped FBGs, which are widely used in all-fiber chirped pulse amplifier (CPA) systems [14]. In order to realize such FBGs with ultrashort lasers, an inscription technique is required, that provides all the benefits of the phase mask approach combined with more flexibility to tune the period of the FBG.

During the telecommunication boom, phase mask inscription methods with conventional UV lasers were extended to allow for chirped FBGs [15]. In recent years, these methods were adapted for ultrashort laser inscription: Inscription with a chirped phase mask yielded the best results on the expense of being non-flexible [16]. More flexible, but less robust approaches like bending the fiber [17] or tuning the phase mask were also successfully demonstrated [18]. Prohaska et al. [19] showed, that the period of the grating can be tuned by placing a lens in front of a phase mask with constant period. In this paper, we demonstrate a chirped FBG by manipulating the wavefront in front of the phase mask.

## 2. Theory

The phase mask is the key element for this inscription technique generating an interference pattern, which is responsible for producing the refractive index modulation of the FBG. The interference pattern is generated by overlapping the different diffraction orders [20]. Here we assume a pure two beam interference from the −1st and +1st diffraction orders. Thus, the Talbot effect can be neglected. This is valid as for the inscription of FBGs with ultrashort laser pulses the order walk-off between the zeroth and the first orders leads to a segregation, due to the short coherence length [21, 22].

Usually the phase mask is illuminated by a plane wave generating an interference pattern with half of the phase mask period. However, a non-plane wavefront modifies this pattern. Here, we assume a one dimensional wavefront *W* (*x*) with the origin on the surface of the phase mask propagating in *z*-direction (see Fig. (1)(a)). The resulting Bragg grating period at a distance *z*
_{0} is given by the magnification factor

*f*[19].

Wavefronts of higher orders can be approximated by spherical waves with different curvature radii. We assume, that the slope of the wavefront at two different points *x*
_{1}, *x*
_{2} (see Fig. (1)(b)) can be expressed by

*γ*and the focal distance

*f*. Solving this to

*x*

_{2}−

*x*

_{1}, Eq. (3) merges to Thus, the reciprocal of the focal length can be approximated by the second derivative of the wavefront. For quadratic wavefronts of the form

*x*

^{2}, the second derivative is constant yielding a different, but constant grating period. To achieve a chirped period wavefronts with higher order (n > 2) are necessary.

However, this simple model is based on the paraxial Huygens-Fresnel approximation, where the wavefronts are assumed by parabolic waves. This limits the model for an accurate calculation of the variation of the period for arbitrary wavefronts. Therefore, the diffraction of the wavefront at the phase mask has to be numerically analyzed.

For this the incident wave is described by the complex field amplitude

with the propagation direction*z*and wave vector

*k*

_{0}= 2

*π*/

*λ*(see Fig. (2)). The factor

*k*

_{0}

*W*(

*x*) expresses the phase difference of the wavefront on the surface of the phase mask. After propagating through the phase mask the field is diffracted into the ±1st orders

The intensity distribution is given by the square of the sum of both fields [23]. For an arbitrary wavefront there is no analytical solution of the interference pattern. Hence for simplification the wavefront is investigated using ray optics (see Fig. (2)). At every point along the x-axis of the phase mask the wavefront is expressed by a beam which is diffracted by the mask fulfilling the grating equation. Now we assume, that in every point (*z*
_{0}, *x*
_{0}) behind the phase mask two beams of the order −1 and +1 with the angle *θ*
_{−1}(*x*
_{−1}) and *θ _{x}*

_{+1}(

*x*

_{+1}) overlap. The position

*x*

_{z0}follows the equation

*x*

_{z}_{0}. This yields the wave vectors k and depending on the phase difference between both beams they interfere constructively or destructively to

*z*

_{0}from the phase mask. This numerical approach can be used to compute the interference pattern and thus the grating period for arbitrary wavefronts.

Here we describe the wavefront by Zernike polynomials [24]. Since the wavefronts are only dependent on *x* the Zernike polynomials can be reduced to *Z _{j}*(

*ρ*, ϕ) =

*Z*(

_{j}*x*) with

*x*=

*ρ*cos ϕ and ϕ = 0. In Table (1),

*Z*(

_{j}*x*) is given for three well-known aberrations of the order n = 2, 3, 4.

Figure (3) shows the wavefronts and the simulated periods for the listed Zernike polynomials including a plane wavefront for comparison. The maximum amplitude of the wavefront was set to 5 *μ*m. As predicted, the grating period for the defocus term is changed but constant, while coma (Fig. (3(b)) red line) produces a linear chirp along the grating. A spherical aberration yields a quadratic chirp. Therefore, a Zernike polynomial *x ^{n}* with

*n*≥ 3 is necessary for generating a chirped FBG.

## 3. Experimental Realization

For the realization of the FBGs the phase mask scanning technique is used [22]. in general, a collimated femtosecond laser beam is focused by a cylindrical lens (here f = 20 mm) through the phase mask into the fiber core (Fig. (4)). The distance between the phase mask and the fiber was set to 1.5 mm, to ensure a pure two-beam interference, due to the order walk-off [21, 22]. The cylindrical lens focuses the beam perpendicular to the fiber axis ensuring a plane wavefront along the x-axis at the diffraction grating. As the refractive index change is generated only in the focus of the cylindrical lens, the width of the modification is usually smaller than the diameter of the fiber core, leading to a small overlap between the fundamental mode and the refractive index modulation. This reduces the coupling strength of the grating on the one hand and leads to cladding mode resonances on the other hand [11]. This can be overcome by moving the laser beam across the fiber core and extend the FBG over the complete mode field diameter [12]. By fixing phase mask and fiber with respect to each other both can be moved underneath the laser beam to extend the grating size without losing the phase stability. Here we scanned the focus 30 *μ*m across the fiber core in y-direction.

The laser system used for the inscription of the FBGs is a commercial regenerative amplified Ti:Sapphire system (CPA 2110, Clark-MXR) with a pulse energy of 1 mJ and a repetition rate of 1 kHz. The pulse duration is 150 fs. For positioning the focus with respect to the fiber a three-axis air bearing translation stage (AEROTECH ABL90300) is used. The spectra of the inscribed FBGs are characterized by a broadband SLD-source and an optical spectrum analyzer (Yokogawa AQ6375).

However, just from the bandwidth of the measured spectrum it is not possible to obtain information about the change of the period along the inscribed FBGs. Therefore, a side diffraction setup was used to evaluate the local change of the period of the inscribed FBGs [25]. The schematic setup is shown in Fig. (5). The period change over the grating length is so small, that it is not possible to measure the change in the diffraction angle directly. Thus the interference between a diffracted and a reference beam was investigated. The incident beam hits the FBG under the diffraction angle *θ* and is diffracted under the same angle. It is overlapped with the reference beam transmitted under the zeroth order without any diffraction. From the lateral period of the interference pattern the period of the grating Λ* _{FBG}* can be calculated. The grating period is evaluated by Fourier transforming the modulation of the interference pattern and calculating the spatial frequency of it. As the laser beam of the HeNe-laser used, is much smaller than the grating length, we translated the FBG in 250

*μ*m steps and calculated from every point the change of the period along the grating.

In order to deform the wavefront usually spatial light modulators or deformable mirrors can be used [26]. Instead, here we simply used an additional cylindrical lens to induce the required wavefront aberrations for tailoring the FBGs. This additional lens, called tuning lens in the following, is added between the phase mask and the inscription lens as shown in Fig. (6). It is orientated perpendicular to the inscription lens. The cylindrical tuning lens has the advantage, that it does not affect the focusing condition of the inscription lens and the previous theory based on wavefront deviations only in the x-z-plane is still valid. The lens was placed directly on top of the phase mask.

Figure (7) shows the transmission and reflection spectra of two FBGs inscribed with a tuning lens with a focal length of f = −50 mm. The lens causes a defocusing of the wavefront, which shifts the Bragg peak from the original wavelength of 1555 nm to 1596 nm for the given distance of 1.5 mm between phase mask and fiber (cf. Eq. (1)). The reflection spectrum (Fig. (7(b))) is normalized with the strength of the transmission dip. This is valid, because the written gratings do not exhibit any measureable out of band scattering or absorption losses (<0.1 dB) and the grating strength is not stronger than 99 %. Both gratings were written through the center of the tuning lens (Δ*x* = 0). The variation of the inscribed period measured with the side diffraction method as described above is shown in Figure (8). The blue line corresponds to the grating, which was written with the non-tilted tuning lens. However, weak higher order aberrations are already present leading to variations of the period of about 0.3 nm. The change in the grating period along the fiber is in good agreement with numerical simulations based on the theory above (dashed line in Fig. (8)). For this simulation the required Zernike polynomials for the wavefront have been calculated from the parameters of the tuning lens with the help of the commercial program ZEMAX.

Higher order aberrations can be increased by tilting the tuning lens. The effect is shown by the red line in Fig. (7) and (8), where the lens was tilted by *α* = 5.7°. The spectrum shows an increase in the bandwidth of the FBG, while the measured period of the FBG shows a significantly steeper change of the period along the fiber compared to *α* = 0° and thus a significantly stronger chirp. This is caused by stronger higher order aberrations like spherical aberration and coma.

The small difference between the measured and the numerical simulated grating period in the red line in Fig. (8) can be explained by the strong chirp of the interference pattern in z-direction behind the phase mask. A change of only 50 *μ*m in height shifts the period by 0.9 nm. However, the tilted tuning lens also causes a tilt of the focus position with respect to the x-axis leading to a FBG extending further into the cladding. For the measurement of the period with the side diffraction setup the beam of the HeNe-laser is focused by a cylindrical lens into the fiber core (see Fig. (5)). Here, small differences in the height of the probe beam can lead to changes of the measured period of a few hundred of pm.

Since the beam diameter of the inscription laser (5.6 mm) is smaller than the width of the tuning lens (20 mm), the bandwidth of the FBG can be increased by scanning also along the x-direction. A spectrum for such a (Δ*x* = 2 mm) scanned chirped FBG can be seen in Fig. (9). The 3-dB bandwidth is increased to 2.5 nm, while the 10-dB bandwidth is already 4.7 nm. However, due to the varying thickness of the tuning lens along the x-direction the focus moves out of the core. This causes a weaker grating strength at the edge of the lens compared to the center.

## 4. Conclusion

In conclusion we have inscribed chirped FBGs in non-photosensitive fibers using a deformed wavefront and the phase mask scanning technique. A quantitative model predicts the necessity of deviations of the wavefront with curvatures higher than the second order. Numerical simulations based on a ray optical diffraction model were used to calculate the period behind the phase mask for arbitrary wavefronts. An experimental realization was demonstrated by placing an additional cylindrical tuning lens in front of the phase mask. Higher order aberrations were generated by tilting the tuning lens. Therewith, a chirped grating with a bandwidth of 2.5 nm could be realized. Additionally the period of the grating was measured with a side diffraction technique. Numerical simulations based on the calculated Zernike polynomials of the tuning lens yield good agreement with the measured variation of the FBG period. The approach for using a tuning lens is a simple way to generate aberrations. However, even more flexibility in tuning the grating properties can be obtained by using a deformable mirror or a spatial light modulator for deforming the wavefront.

## Acknowledgments

We acknowledge financial support by the German Federal Ministry of Education and Research (BMBF) under contract No. 13N9687 and the German Research Foundation (DFG).

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