1. Introduction

Long-period fiber gratings (LPGs) arise from periodic variations of the refractive index [1]. If the grating period equals the beating length of two transverse modes, a conversion can be obtained, e.g. the fundamental mode can be converted to a higher-order either guided or leaky mode. The grating periods of LPGs typically are in the range between few hundred μm and a few mm. Various applications for LPGs have been shown, ranging from absorption filters, temperature, strain and refractive index sensors [2] to optical switches [3]. Mode conversion in few-mode fibers offers a high potential for many applications [4] like signal power management or dispersion compensation [5]. Different generation techniques have been demonstrated: written in photosensitive fibers by UV pulses [6], written by femtosecond pulses [7], or induced by stress [8], acoustic waves [9] or electro-optical effects [10].

In this letter, we propose and demonstrate a LPG induced by the optical Kerr effect. A high power beam, which will be called writing beam, temporally induces a grating into the fiber making use of the beating between the LP₀₁ and the LP₁₁ mode. A counter-propagating low power beam, denoted as probe beam, is diffracted from this grating, i.e. it is converted to another mode. Compared to other techniques this optically induced LPG (OLPG) is only induced for transient periods, as a pulsed laser beam is used to achieve the necessary peak power of the writing beam. The necessity of a high power writing beam is limiting for some applications, but may be reduced by a high-nonlinearity fiber as for example used in [3]. As the proposed OLPG is not side-written but induced by a guided beam, it potentially allows an easy generation of very long LPGs, which offer a very small bandwidth for LPG-based absorption filters. A tunability of LPGs has for example been shown with mechanical, acoustical and tilted gratings [8, 9, 10]. The proposed OLPG is optically tunable via the properties of the writing beam.

2. Basic setup

Figure 1 illustrates the experimental setup. A pulsed laser beam, which serves as the writing beam, is coupled to a few-mode fiber in the way that two guided modes (LP₀₁ and LP₁₁) are excited. Due to different propagation constants the relative phase between the modes changes during the propagation, what causes a grating-like intensity distribution with the period equal to the beating length. A counter-propagating continuous-wave (cw) probe beam is carefully aligned to excite only the fundamental mode (LP₀₁). After the transmission through the few-mode fiber the probe beam is coupled to a single-mode fiber (SMF) serving as a spatial filter, so that for the case of a mode conversion a drop of the power, which is detected behind the SMF, is expected. We used the same wavelength (λ = 1.064μm) for the writing and the probe beam, which is not a mandatory choice, but ensures that the grating period naturally matches the beating between the LP₀₁ and the LP₁₁ of the probe beam.

 

Fig. 1. Experimental setup: The pulsed laser induces a grating that diffracts the counter-propagating continuous-wave (cw) laser beam. NPBS/PBS: non-/polarizing beam splitter, SMF: single-mode fiber, PD: photodiode.

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The combination of a half-wave plate and a polarizing beam splitter is used to control the power of the writing beam. As the fiber itself as well as the induced index change are in general birefringent a half-wave plate and a quarter-wave plate are used to optimize the mode coupling. The light is coupled into the SMF with a non-polarizing beam splitter in order to prevent the unintended detection of polarization changes.

3. Numerical simulation

The mode conversion with an OLPG has been modeled using a beam propagation method [11] and assuming a Kerr-induced index change of Δn = n₂I with the nonlinear refractive index $n_2=2.6\bullet {10}^{-26}\frac{{\mathrm{m}}^2}{\mathrm{W}}$ and I being the intensity. Within the model the fiber had a core diameter of d = 25μm and a numerical aperture of NA = 0.06 in accordance with the experimentally used fiber (see below). The mode propagation has been simulated firstly only in two dimensions, as it was recently shown that there is only a negligible difference to three dimensional simulations if only two modes are excited [12]. First, the writing beam with a power of 200kW is excited (as a quasi-cw beam) with 50% of the power in the LP₀₁ as well as in the LP₁₁ mode. The different propagation constants of both modes cause a mode beating with a beating length of about 3mm. The beating results in a grating-like intensity distribution that induces a spatially alternating refractive index change shown in Fig. 2(a). Secondly, the probe beam is excited with all power completely in the fundamental mode and propagated through the fiber with the modified refractive index (Fig. 2(b)). The grating causes a modal power transfer from the LP₀₁ to the LP₁₁ mode (Fig. 2(c)). Considering a pulse instead of the quasi-cw writing beam, the grating length is determined by the pulse duration. The conversion depends on the propagation distance, which corresponds to a certain pulse energy for a constant pulse power. A complete conversion is achieved with a pulse energy of 80μJ. Actually, a pulse with lower peak power but longer duration, causes the same conversion if the pulse energy is the same. We also tested the dependence on the modal power distribution of the writing beam. For 25% of the power in the LP₁₁ mode the conversion efficiency decreases to a value of about 90%. Notice that the writing beam barely diffracts itself as phase matching is not fulfilled.

 

Fig. 2. Numerical simulation of the OLPG: (a) Refractive index change induced in the fiber by the mode beating of a 200kW quasi-cw beam, (b) intensity of the probe beam that is propagated through the fiber with modified refractive index, and (c) normalized modal power of cw beam, the fundamental mode (LP₀₁) is converted to the LP₁₁ mode.

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4. Experimental details and results

In the experiments we used a Q-switched Nd:YAG laser in order to induce the LPG. The repetition rate was f_rep = 10Hz, the maximum used pulse energy E_p = 70μJ and the pulse duration T = 14ns, which corresponds to a pulse length of 2.9m in the fiber. Compared to the simulations such a long pulse is necessary as the maximum peak power of the Gaussian-shaped pulse was only about 5kW. The beam profile of the available laser was not diffraction-limited (M² ≈ 2), so that the coupling efficiency into the fiber was only about 35%. Due to this beam profile always both modes, the fundamental as well as the higher-order mode were excited. The few-mode fiber had a step index profile, a core diameter of d = 25μm, a numerical aperture of NA = 0.06 and a length of L = 8m. The fiber had a V-number $V=\frac{\pi }{\lambda }\bullet d\bullet \mathrm{NA}$ of 4.4 and therefore nominally supports the propagation of LP₀₁, LP₁₁, LP₂₁ and LP₀₂. The fiber was coiled to a diameter of 15cm, what accomplishes suppression of LP₂₁ and LP₀₂. In former experiments [13] we measured only a very small excitation of LP₂₁ and LP₀₂ in the same type of fiber. The fiber had a high index acrylate coating, so that light in the cladding was not guided. In order to reduce the reflections of the pulse into the detection branch the fiber ends were angle-polished and the optical components were slightly inclined. The beam of a cw non-planar ring oscillator with diffraction-limited beam quality and a power of P = 40mW served as the probe beam. The probe beam alignment was carefully adjusted, while the writing beam was blocked, to excite only the fundamental mode. The coupling efficiency was about 70%. In order to measure only the power in the fundamental mode a part of the transmitted probe beam was reflected at a non-polarizing beam splitter, coupled into the SMF (actually a strongly coiled part of the few-mode fiber) and detected with a photodiode. In this way the SMF serves as a spatial mode filter, whose efficiency can be described by the suppression of the LP₁₁, which depends on the precision of the alignment. With a perfect alignment LP₁₁ is completely suppressed. Assuming a maximum misalignment of 3μm the transmission of the LP₁₁ has been calculated for the used fiber to be still smaller than 10%.

Turning on the cw and the pulsed beam at pulse energies greater than 3μJ strong back-reflections were observed for some pulses due to seeded stimulated Brillouin scattering (SBS) [14]. SBS occurs if a forward propagating beam is scattered back by an acoustic wave, which itself is formed by the interference of the forward and backward propagating wave. The threshold of this nonlinear process can be lowered by an additional backward propagating wave with similar frequency. However, as the writing and the probe laser oscillated independently from each other, the seeding of SBS occurred randomly, so that we chose pulses without appearance of SBS for the measurement. An explicit avoidance of SBS was possible by detuning the frequencies of the writing and the probe laser.

In Fig. 3(a-h) the power of the probe beam, which was detected behind the SMF, is shown for different pulse energies. Despite angle-polishing of the fiber ends parts of the pulse were reflected from both fiber ends and reached the detector. This amount of signal corresponding to these reflections was subtracted as an offset from all measurements. Moreover, each trace is averaged over ten pulses in order to reduce effects of pulse-to-pulse fluctuations. Then, a characteristic minimum in the probe beam power was observed, as expected for a conversion into a higher-order mode. The minimum of the dip decreased with higher pulse energies down to about 40% of the unconverted power at a pulse energy of about 60μJ (Fig. 3(i)) and slightly increased again at energies of about 70μJ. Higher pulse energies damaged the fiber front facet, so that a clear back conversion could not be observed so far. Fiber end caps and a better beam profile of the writing beam should accomplish to increase the destruction threshold. As the modal distribution and modal polarization can change due to variations in the refractive index, the induced grating can vary during propagation and thereby over time, what is considered to be the reason for the temporal variation of the dip.

 

Fig. 3. (a-h) Power of the probe beam behind the SMF for different pulse energies, (i) dip minimum over pulse energy with a third-order fit, and the calculated decrease of the fundamental mode in dependence on the pulse energy.

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Comparing the measured data and the calculated decrease of the fundamental mode a similar shape and a relative good accordance concerning the pulse energies are found. Nevertheless, two main differences are observed: First, the measured minimum is reached at a smaller pulse energy than in the simulation. This deviation can result from the uncertainty of the measured pulse energy (of about 10%) and the uncertainty of the actual core diameter (±2μm), which affects the intensity in the core and thereby the strength of the grating. Moreover, there can be remaining deviations as the model has been calculated in only two dimensions. Secondly, the measured data does not reach zero, which means that a full conversion is not observed in the experiments. The probable reason for this reduced efficiency are imperfections of the induced grating, which can result from different modal polarizations (that occur as the modes experience different birefringences) or from the excitation of additional modes (LP₂₁ or LP₀₂). Additional modes, which can be excited at the incoupling or within the fiber due to variations of the index profile, generate an additional beating that causes coupling of the probe beam into further modes. In this more complicated case of a coupling of three or more modes, the power of the fundamental mode does not necessarily reach zero.

In order to confirm that the decrease of detected probe power is caused by a guided mode conversion the spatial dependence of the converted and unconverted probe beam was measured. As we had no camera available, which was fast enough to resolve the conversion, the SMF was scanned over the beam. Therefore, it was considered that a superposition of the LP₀₁ and the LP₁₁ modes causes a shift of the beam center depending on the relative phase. Fig. 4 shows the calculated intensities of (a) the fundamental mode, (b) an example of the superposition of LP₀₁ and LP₁₁ with equal powers, and (c) the calculated transmission through a SMF scanned over these two beam profiles. Experimentally, the incoupling end of the SMF was laterally shifted and the transmitted power of the converted and unconverted beam was measured. The results (each data point is a result of averaging over six pulses) are shown in Fig. 4(d) and (e). A third-order polynomial was fitted only to guide the eye. While the scan across the unconverted beam resulted in a symmetric decrease of the transmission, the scan over the converted beam is shifted to the left (Fig. 4(d)). After horizontally displacing the writing beam a shift to the right was observed (Fig. 4(e)). Depending on the incoupling of the writing beam it was also possible to shift the probe beam center up or down. The shift of the beam center is an evidence for the occurrence of the first higher-order mode [15]. From these scanning experiments we estimate that about 50% of the fundamental mode power have been converted to the LP₁₁ mode. The transmission measurement suggests a slightly higher efficiency of about 60%, but this value includes possible conversions to further higher-order modes.

 

Fig. 4. Spatial dependence of the converted beam: Calculated intensity profiles of (a) the fundamental mode and (b) a superposition of LP₀₁ and LP₁₁ with equal powers; (c) calculated transmission scanning a SMF across (i) the fundamental mode and across (ii) the mode superposition; (d) measured transmission of the converted and unconverted probe beam across the position over the SMF (lines are third-order polynomial fits to guide the eye); (e) analogous to (d) after a horizontally shifted incoupling of the writing beam.

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5. Conclusion and outlook

In conclusion, we have demonstrated for the first time, to the best of our knowledge, an optically induced long-period fiber grating. The concept was modeled with a beam propagation method that showed that a full conversion from the fundamental to the LP₁₁ mode should be feasible. Related to the constraints of the writing beam quality and low damage threshold of the fiber facet a mode conversion efficiency of about 50% has been experimentally achieved. Further improvements are expected for experiments with a better beam quality of the writing beam. Moreover, a high-nonlinearity fiber and a smaller core diameter (but still a few-mode fiber) can significantly reduce the necessary power of the writing beam. The study of the spectral properties of an OLPG will be a further interesting subject. The grating is instantaneously and only temporally induced while the writing pulse is in the fiber. This can also be regarded as a possibility to switch the OLPG on and off. The strength and the length of the OLPG can be controlled by the power and the duration of the writing pulse offering a high flexibility, which for example would allow to tune the bandwidth of a LPG-based absorption filter.