1 Introduction

Optical bandpass filters are used in a wide range of fields, extending from wavelength-division multiplexing in telecommunication to spectroscopy and sensing applications. In fiber optics, the most common way of wavelength filtering is to use fiber Bragg gratings (FBGs). However, besides being intricate in production, FBGs usually require quite some effort (e.g., an optical circulator) if they are to be used in a transmission line. Furthermore, they exhibit a rather narrow bandwidth, which becomes a problem as soon as ultrashort, i.e., broadband, pulses are to be filtered. On the other hand, it has been shown that implementing a spectral filter in a fiber laser cavity can significantly improve the performance of pulsed systems, e.g., for wavelength tuning, or can greatly facilitate pulse generation as, e.g., in all-normal dispersion (ANDi) [1] or soliton-similariton [2] lasers. However, bulk filters are mostly used in these cases, realized as Lyot-filters, free-space interference filters, or by placing a slit in the spectral plane behind a prism. Fiber-based approaches are not too widely spread, but there have recently been some advances with fiber-based Lyot-filters [3] and Sagnac loop mirrors [4]. A promising alternative are filters based on multimode interference (MMI), which results in periodic imaging of the input mode field in a multimode waveguide [5] due to the different propagation constants of the transverse modes. As the distances at which these so-called self-images appear depend on the wavelength, it is possible to design spectral filters based on MMI. Several approaches have been presented exploiting a wavelength-dependent lens in front of a mirror [6] or, in an all-fiber manner, based on a single-mode/multi-mode/single-mode (SMS) fiber combination [7–10], where a piece of multimode fiber (MMF) was spliced between two pigtails made of single-mode fiber (SMF). Wavelength tuning in such filters can be achieved by changing the length of the MMF segment, which has in the past been accomplished by mechanical stretching [7]. However, the mechanical limits of the fiber elongation restricted the tuning range to less than 1.3 nm. Another approach to change the MMF lengths was to use a ferrule filled with index-matching liquid for a virtual elongation of a no-core fiber segment [9, 10], but this does not allow the use of a standard MMF, and no-core fibers are very sensitive to dirt. Temperature control can also be used for tuning [11], but this requires an oven and additional control and power supply electronics, increasing size and complexity of the setup. The fiber diameter is another parameter that can be used for designing the transmission wavelength, and by choosing the diameter and the length of the MMF correctly, the bandwidth of such an MMI filter (MMIF) can also be designed [7]. However, the fiber diameter is fixed after production and can thus not be used for wavelength tuning.

In order to circumvent the limits of stretching a fiber and the necessity to use index-matching liquid and no-core fibers, we employ a different method for the tuning of SMS fiber bandpass filters. Our approach is based on the change of the propagation constant of each fiber mode when the fiber is bent. Due to wavelength-dependence of the differences of the propagation constants of the transverse modes (which cause MMI), this effect can be used to tune the central transmission wavelength of an MMIF, because a shift of the wavelength can cancel out the bending-induced changes. We have recently introduced an all-fiber mode-locked laser that could be spectrally tuned employing such a filter [15]. In this paper, we analyze the properties of the filter in more detail and deepen the understanding of the underlying physics. Toward this aim, we present for the first time, to the best of our knowledge, an analysis of the transmission of a bent MMI structure. Based on the mode fields of a bent fiber derived by Garth [12], we employ a mode expansion approach for numerical simulations. The results show excellent agreement with experimental measurements. We also provide an analysis of the governing equations to deduce design parameters for MMI fiber filters. Finally, we investigate experimentally whether the MMIF can be used for filtering of femtosecond pulses, and in particular inside a fiber laser cavity. For this, we analyze the polarization changes imposed by the fiber bending, which is relevant because mode-locked lasers generating ultrashort pulses often use nonlinear polarization rotation (NPR) for mode-locking and are thus sensitive to birefringence.

2 Setup

The design of the filter is depicted in Fig. 1(a). It was manufactured by splicing a piece of multimode fiber between two pigtails made of SMF. Standard SMF28 (core diameter 8.2 μm, NA 0.14, germanium-doped core) was used for the pigtails, while we employed AFS50/125Y with a core diameter of 50 μm (NA 0.22, silica core and F-doped cladding) and a length of around 48.5 mm for the MMF section. A total length of L 0=70 mm of fiber, with the MMF segment in the center, was fixed between two fiber clamps (FC). One of these clamps was located on a translation stage. Thus, by moving the translation stage, the fiber could be bulged. Note that the curve of the fiber bending has cosine shape with a geometric wavelength λ geo=L 0−Δs given by the distance of the fiber clamps theoretically, as it is actually given by the fourth Euler case known from mechanics. Since the dependence of the amplitude of a cosine function on the wavelength for a constant curve length cannot be expressed in elementary functions, the exact local curvature for different translation stage positions could not be derived. The curve’s shape can, however, be approximated by four identical circular arc segments, where the outer two have positive curvature and the inner two are negatively curved. This is indicated by the dashed circular arc segments in Fig. 1(a). In that case, the bending radius R can be calculated from the total fiber segment length L 0 and the translation stage position Δs from geometric considerations by first order expansion:

$$ R=\frac{L_0}{4} \biggl[10-\sqrt{120\frac{L_0-\Delta\mathrm{s}}{L_0}-20} \biggr]^{-1/2}. $$
(1)
Fig. 1
figure 1

(a) Design of the MMIF. The upper part shows the case of the unbent fiber with length L 0, held by fiber clamps (FC) on both sides. By moving the translation stage by a distance Δs, the fiber is bent and assumes cosine shape with a geometric wavelength λ geo, depicted in the lower part of the figure. Circular arc segments with radius R (dashed black lines) are used to approximate the curve. (b) Measured fiber bending, depending on the translation stage position. Black squares: measured bending radii, fitted using Eq. (1) (red); blue dashed line: square of the curvature, calculated from the red fit curve; green triangles: amplitude of a cosine fit (λ geo=(70 mm−Δs)) to the bending curves, with power law fit (see text)

Full size image

In order to verify Eq. (1), we measured the bending radius by illuminating the fiber from above and drawing the bending curve from the shadow. The central part of each curve was then individually fitted with a circular arc, giving the results shown in Fig. 1(b). Note that the square of the curvature, κ 2=R −2, increases almost linearly with Δs, as can be seen from the blue dashed line in the figure. Indeed, using Eq. (1) and first order expansion, we get

$$ \kappa^2 \approx\frac{96}{L_0^3}\Delta\mathrm{s} = \tilde{c}_s \Delta\mathrm{s}. $$
(2)

As one can see, a large range of bending radii is continuously accessible in this geometry, which makes it superior to standard coiling on a drum. For comparison, we also fitted the measured curves with a cosine-shaped function and evaluated the peak amplitude. These data are represented by the green triangles. They have been approximated by a power law fit a|x−x 0|p with an exponent of p=0.53, in order to receive additional values (via interpolation) for the numerical simulations presented in Sect. 4.

3 Experimental results

For transmission measurements, we used amplified spontaneous emission (ASE) from an erbium-doped fiber pumped by a 976 nm diode laser and an optical spectrum analyzer at the filter output. A stepper motor was attached to the translation stage and allowed automatic scanning of the bending radius. The results are depicted in Fig. 2. There the transmission of the filter is plotted color-coded in Fig. 2(a). As one can see, self-images of three different orders were observable in the range between 1480 nm and 1600 nm, labeled in Roman numbers and separated by the dashed white lines in the figure.

Fig. 2
figure 2

(a) Transmission spectrum of the MMIF depending on the translation stage position. The transmission is color-coded. Regions separated by white lines and labeled I–III correspond to different filter orders. Green dotted lines indicate linear dependence of the peak transmission wavelength on the translation stage position. (b) Peak wavelength with transmission higher than 0.5. Dashed lines separate the different filter orders. (c) Transmission at the peak wavelength. All three filter orders are evaluated in (b), while only region III is considered in (c) for reasons of clarity. Dotted lines are to guide the eye, indicating regions where the peak transmission in order III exceeded 50 %

Full size image

Standard two-dimensional MMI theory (e.g., [5]) predicts a wavelength difference of approximately 74 nm for self-imaging with the given fiber parameters, which matches the distance between the lowest (I) and the highest (III) observed filter order. In fact, as will be seen later, this is the wavelength difference that originates from the interference between the LP01- and the LP02-mode (see Fig. 5). However, investigations by Zhu et al. [8] have shown that the actual imaging distance in three-dimensional systems with multiple modes can be much shorter than predicted by the theory, and thus the same holds for the wavelength difference between the filter orders. As is indicated by the green dotted lines, the peak wavelength of each filter order changed approximately linearly with the translation stage position. Considering Fig. 1(b) and Eq. (2), this indicates linear dependence on the square of the curvature κ 2. One can also see that the regions of high transmission repeated periodically when the fiber was bent. As the propagation constant increases linearly with κ 2 (see next section), these regions corresponded to different self-image orders for the respective optical wavelength. The appearance of the separated regions of high transmission was due to the number of excited modes, as will be shown by simulations in Sect. 4. When the bending became too strong, the overall losses of the MMIF increased significantly, exceeding 90 % around Δs=3.234 mm. This was related to the fact that a growing fraction of the LP01 mode (which is most strongly excited) of the MMF was guided in the cladding in that case [13], and that the mode fields of the other modes were also shifted due to bending (thus lowering the mode overlap with the fundamental mode of the SMF).

The linear dependence of the transmission wavelength on Δs can also be seen in Fig. 2(b), where the peak transmission wavelength of the filter is plotted for all translation stage positions at which the peak transmission [see Fig. 2(c)] exceeded 50 %. One can observe that a total wavelength range of 9.7 nm was accessible with a peak transmission over 50 % in the highest filter order (III). This value decreased for the subsequent transmission regions (for higher Δs) in the same filter order, as it is also the case for the other filter orders. For the lower filter orders (I & II), the first transmission regions exhibited a two-peak structure [Fig. 2(a)], which corresponds to the discontinuities in the peak transmission wavelengths [Fig. 2(b)], where the peak transmissions dropped below 50 %. Thus, although these filter orders covered ranges of up to 20.8 nm, continuous tuning with high transmission was only possible over 9.3 nm, similar to filter order III. For the subsequent transmission regions, the two-peak structure became less pronounced, resulting in fewer and smaller discontinuities of the peak transmission wavelength. Again, an approximately linear dependence of the transmission wavelength on the translation stage position Δs was visible for all tuning regions. If the limitation to a peak transmission exceeding 50 % is discarded (which can, e.g., be done in the cavity of a continuous-wave fiber laser due to the high gain), a maximum continuous tuning range of 13.6 nm was achieved.

The transmission for the respective peak wavelength of filter order III is plotted in Fig. 2(c). It reached 86 % close to the center of the tuning region. In this filter order, the spectral full width at half maximum (FWHM) of the transmission decreased linearly from about 22.5 nm to 12.5 nm. Higher peak transmissions can be achieved by choosing a different fiber length and thus a different self-image order, which can have a different image quality [8]. This is because the accumulated phase difference between two modes is never exactly 2π for all pairs of modes after a given fiber length, so the interference of the modes is never purely constructive for all modes.

As such a filter with a bandwidth of 10–20 nm could well be employed in a mode-locked fiber laser [2], we also investigated the polarization changes due to bending-induced birefringence, which are relevant in case of polarization mode-locking. For this purpose, we used a polarized continuous-wave laser based on erbium fiber (λ cw=1552.8 nm) and measured the polarization behind the filter as a function of Δs with a fiber polarimeter [14]. Due to the birefringence of the pigtail fibers, no absolute measurements were possible, but the bending-induced changes could nonetheless be determined. The results are shown in Fig. 3 in the Poincaré representation. As one can see, the measured polarization states were confined in a rather small segment of the Poincaré sphere, which indicates that the polarization changed only a little. Since mode-locking by nonlinear polarization rotation (NPR) in fiber ring lasers is possible in larger regions on the Poincaré sphere [14], it should be feasible to integrate an MMIF into such a laser to achieve wavelength-tunable mode-locked operation. So far, an MMIF in an NPR-mode-locked laser has only been implemented in a σ-shaped cavity, where the bending-induced birefringence was cancelled out by using a Faraday rotator mirror (FRM) [15]. Getting rid of the FRM and the required optical circulator by using an MMIF in a ring cavity would greatly simplify the setup.

Fig. 3
figure 3

(a) Polarization changes inside the bent MMIF at 1552.8 nm for a total translation stage range of 4.2 mm. No absolute states of polarization are given because of the unknown birefringence of the transport fibers (see text). (b) Enlarged part from (a), showing closed loops (indicated by red lines) in the polarization evolution. The arrows mark the direction of the polarization change with increasing Δs

Full size image

When taking a closer look at the output polarization during fiber bending, it was observed that the state of polarization (SOP) occasionally evolved in closed loops for a short interval of translation stage positions. These loops are depicted as solid lines in Fig. 3(b), the arrows mark the direction of the polarization change with increasing Δs. Apparent discontinuities are caused by the different change of the SOP per measurement step. In order to elucidate the origin of these loops, we performed the same polarization measurement while bulging a simple piece of SMF28 instead of the MMIF. No loops could be observed then, while the remaining data looked similar. Thus, the loops obviously originated in the MMF. This assumption was further supported by the fact that the appearance of the loops coincided with the raising edges of the peak transmission [compare Fig. 2(c)], which were caused by MMI. We assume that the closed loops were caused by the field distortion due to bending in the multi-mode fiber [13]. As was shown by Hatta et al. [16], the polarization dependent loss of an SMS structure is sensitive to the transverse offset of the single-mode fibers from the center of the core of the MMF. As the mode field in the MMF is shifted towards the outer side of the bending when the fiber is bent, it is thus viable that this transverse offset also results in polarization dependent loss, which causes a change of the SOP.

4 Theoretical results

In general, any normalized mode field Ψ inside an MMF can be described by mode expansion using the orthonormal base of the transverse eigenmodes ϕ k of the MMF:

$$\varPsi= \sum_{k} \langle\varPsi, \phi_k \rangle\phi_k = \sum_{k} \biggl(\int_A\varPsi^* \phi_k\,\mathrm{dA} \biggr)\phi_k. $$

The power fraction in the kth mode is then given by \(c_{k}^{2} = \langle\varPsi,\phi_{k} \rangle^{2}\). If we consider Ψ SMF to be the LP01-mode of the input single mode fiber and denote by β k the propagation constant of the kth eigenmode of the MMF, then the field after propagation through a length L of MMF is given by

$$\varPhi(L) = \sum_k \langle\varPsi_{\mathrm{SMF}}, \phi_k \rangle\exp (i\beta_k L) \phi_k . $$

For an MMI structure, where the input fiber equals the output fiber, the transmission is determined by the overlap of Φ(L) and Ψ SMF. Taking into account mode orthonormality, this results in a total power transmission of

$$ T = \biggl\vert \sum_k c_k^2 \exp(i\beta_k L)\biggr\vert^2 . $$
(3)

For the case of a bent step-index fiber, Garth has shown that the change of the propagation constant can be derived using a perturbation approach [12, 17]. Following [12], the propagation constant of an LP0m -mode in a fiber that is bent at a radius R can be expressed as

(4)

where r is the fiber core radius, β b and β 0 denote the propagation constants in the bent and in the straight fiber, respectively. The second-order correction term β 2 (which must not be confused with the group velocity dispersion) is a non-trivial expression of several Bessel functions, explicitly given in [12]. An image of the input mode field occurs whenever the accumulated phase difference of the modes is a multiple of 2π. In a bent fiber, the difference of the mode propagation constants of LP0m and LP0k can be evaluated from Eq. (4) by first-order expansion:

(5)

For the modeling of the bent MMIF, we took the mode expansion approach by simulating Eq. (3) with the propagation constants received from Eq. (4). Both cosine- and circular-arc-shaped bending curves were used, yielding similar results. In case of the cosine-shaped arc, we measured the amplitude as a function of Δs and used a power-law fit for interpolation (as described above, see Fig. 1) to calculate the local curvature κ along the fiber. The term β(κ)L was then evaluated locally and integrated over the fiber length. It is also important to note that the effective radius R eff=1.27×R geo (and the respective effective curvature κ eff) had to be used instead of the geometric value R geo in these simulations to take into account the bending-induced stress in the material [13]. In the case of an SMF28 input fiber and an AFS50 multi-mode fiber, we found that only the first five radially symmetric modes LP01–LP05 were significantly excited, and thus only those were considered. The propagation constants for the straight fiber were calculated by numerically solving the standard eigenvalue equation for each wavelength, with the wavelength-dependent refractive indices of core and cladding taken from Sellmeier equations for pure fused silica [18] (the core material of AFS50) and the NA of the MMF. The results are shown in Fig. 4(a), with the respective section of the experimental measurement depicted in (b) for comparison. Excellent agreement can be observed in all filter orders up to Δs≈1 mm.

Fig. 4
figure 4

(a) Transmission (color-coded) of the MMIF as calculated by the mode expansion approach, (b) corresponding excerpt from the experimental results for comparison

Full size image

In these simulations, we assumed an MMF length of 48.86 mm and a cosine-shaped arc with a geometric wavelength of λ geo=(76 mm−Δs). In the experiment, the measured geometric wavelength was (70 mm−Δs) and the fiber length was 48.50 mm. The deviation in the fiber length that is required for quantitative agreement of experiment and simulation is easily explained by the error in fiber core diameter, which is given as 50±1 μm by the manufacturer. Since the peak transmission wavelength depends quadratically on the core diameter, while it scales only linearly with the fiber length [9], a small difference in the core diameter requires a more significant change in fiber length to be equalized. The discrepancy in geometric wavelength can be explained by the fact that the fiber section was not fully mechanically isotropic because it was stripped at the splices, but had a coating in the center of the MMF section. As a result, the curvature in the center was reduced in the experiment. This is equivalent to a higher geometric wavelength in the simulations. The quantitative agreement in transmission impairs for lower bending radii (higher Δs), which was to be expected as bending losses and (more important) mode deformation play a role then, and both effects were not included in the numerics.

In order to improve the tuning characteristics of the MMIF, it would be beneficial if only two modes were excited and thus would contribute to the MMI. While this would require beam shaping for a fiber with many modes (like the AFS50) and is therefore hard to achieve in an all-fiber experiment, it could be easily simulated. Figure 5 shows the result for an AFS50 fiber were only the LP01- and the LP02-mode were excited, with 50 % of the power in each mode. Indeed, a continuous tuning over the whole free spectral range of the filter could then theoretically be achieved with only minimal translation stage movement of less than 0.5 mm. Another option to get rid of the excessive modes would be to use an MMF that only supports LP0x -modes for x≤2; this would require a V-number between 4 and 7. Experimental realization of such a filter is subject of our current work.

Fig. 5
figure 5

Simulated transmission (color-coded) of the bent MMIF in the case where only the transverse modes LP01 and LP02 were excited. The crosses mark the peak transmission wavelengths calculated from Eq. (7) for three self-image orders

Full size image

If only two modes are excited inside the MMF, Eq. (3) is essentially reduced to a cosine term. One can then learn more about the parameters of a tunable MMIF by noticing that, for a constant self-image order m, the accumulated phase difference of both modes must remain fixed at a value of 2πm during bending. Thus, the difference of their propagation constants, given by Eq. (5), must be fixed at 2πm/L. For the two excited modes, the dependence of their differences Δβ 0 and \(\Delta\beta_{\mathrm{bc}} = \Delta(\beta_{2}^{2}/(2\beta_{0}))\) (the bending correction) on the wavelength λ can be linearly approximated with slopes c 0 and c bc in the vicinity of λ 0, i.e., Δβ 0(λ)=Δβ 0(λ 0)+c 0(λ−λ 0) and Δβ bc(λ)=Δβ bc(λ 0)+c bc(λ−λ 0). This can be seen by the nearly linear slopes in Fig. 6. Then we can use Eq. (2) and Eq. (5) to find

$$ \Delta\beta(\Delta\mathrm{s},\lambda) = \Delta\beta_0( \lambda_0)+ (c_0+A\Delta\mathrm{s} ) (\lambda- \lambda_0 )+B \Delta\mathrm{s}, $$
(6)

where \(A = c_{s}r^{2}c_{\text{bc}}\), B=c s r 2Δβ bc(λ 0) and \(c_{s}=\tilde{c}_{s}/(1.27)^{2}\) [see Eq. (2)] to account for the effective curvature. Keeping the total phase fixed (and thus preserving the self-imaging condition at the end of the MMF) means that Δβ(Δs,λ)=2πm/L for all allowed combinations of Δs and λ. Note that the sign of m depends on the sign of Δβ 0, and thus on the choice of the modes in Eq. (5), it will be negative for the choice presented in Fig. 6. For the optimum transmission wavelength, we thus get

$$ \lambda_{\mathrm{max}} = \frac{(A \lambda_0-B)}{c_0+A \Delta\mathrm{s}}\Delta\mathrm{s}+ \frac{(c_0\lambda_0-\Delta\beta_0(\lambda_0))L+2\pi m}{(c_0+A \Delta\mathrm{s})L}. $$
(7)
Fig. 6
figure 6

(a) Wavelength dependence of Δβ 0, the difference between the propagation constants of the LP01 and LP02 transverse modes in an AFS50 multimode fiber. The slope of the curve gives c0. (b) Wavelength dependence of the differences between the bending corrections β bc [given by Eq. (5)] of different transverse modes of the AFS50 fiber and that of the respective LP01 mode

Full size image

For the case of a circular bending instead of the cosine shape, Eq. (7) is in very good quantitative agreement with the simulations, as can be seen from the crosses in Fig. 5. It reveals that truly linear dependence of the peak transmission wavelength on Δs can only be achieved if A is zero, which would require c bc to be zero and can thus not be achieved in standard step-index fibers. To calculate the bandwidth, one can consider the period of the cosine that results from inserting Eq. (6) in Eq. (3). This gives

$$ \delta\lambda_{\mathrm{FWHM}} = \frac{\pi}{L(c_0+A \Delta\mathrm{s})}. $$
(8)

It is worth noting that c 0 and A differ in sign in our case. As a result, the bandwidth increases with the curvature, up to the point \(A\mbox{$\Delta$s}=-c_{0}\), where the transmission does theoretically no longer depend on the wavelength. Beyond this point, the bandwidth decreases again. This can also be seen in Fig. 5. The horizontal sections through the figure yield the cosine-shaped transmission curve for the chosen bending radius, starting with a bandwidth of δλ FWHM≈35 nm for the unbent fiber. As the bending gets stronger, the high transmission ridges become more and more parallel to the x-axis, which means that a horizontal section will give a cosine with a longer period and ultimately a spectrally flat transmission, to be seen around Δs≈3.5 mm in the figure. While this limits the use for filtering applications, it could be useful for broadband multimode devices. However, bending losses and mode deformation will likely reduce the overall transmission in that case.

For optimization of the tunability with minimum bending, it is enlightening to consider the derivative of Eq. (7) with respect to Δs. The resulting term also has a singularity at \(A\mbox{$\Delta$s}=-c_{0}\), meaning that the peak transmission wavelength will vary rapidly with Δs in that region. However, as mentioned above, this will be accompanied by a vastly increased bandwidth. For lower values of Δs, i.e., lower curvature, it is useful to evaluate the derivative at the origin:

(9)

This equation reveals that the tunability increases quadratically with the MMF core radius, as expected from Eq. (5), but also scales inversely with c 0, the wavelength dependence of the difference of the propagation constants of both modes in the unbent fiber. These parameters can thus be used to achieve high tunability of an MMIF while keeping bending, and thus bending losses, low. Moreover, if a higher image order is chosen in order to achieve a certain bandwidth [see Eq. (8)] for a given peak wavelength, L also increases with 2πm, and thus the dependence of the tuning slope on the length of the MMF segment effectively vanishes. On the other hand, fine tuning of the fiber length in order to design the unbent filter for a certain peak transmission wavelength only changes the fiber length by a few percent for a high image order. Since both summands in Eq. (9), as well as their difference, are of the same order of magnitude, the tuning slope only experiences minor alterations in that case.

For more than two modes, Eq. (3) leads to a sum of cosine terms, each representing the beating of two particular modes:

$$ T=T_0+2\sum_{i>j}c_i^2c_j^2 \cos \bigl(\bigl(\phi_{0,ij}+f_{ij}(\mbox{$\Delta $s})\lambda+ \phi_{s,ij}\mbox{$\Delta$s}\bigr)L \bigr), $$
(10)

where ϕ 0,ij =(Δβ 0,ij (λ 0)−c 0,ij λ 0), f ij (Δs)=(c 0,ij +A ij Δs) and ϕ s,ij =(B ij −A ij λ 0). Here, indices i and j indicate that the constants are taken with respect to the LP0i and LP0j modes. Since the f ij and ϕ s,ij are different for each pair of modes, the peaks of these cosine curves shift with different slopes when Δs is changed, which can also be extracted from Eq. (9). The maximum transmission can be achieved if all cosine peaks coincide as well as possible, but as soon as the fiber is bent away from that point, the overall peak transmission will be reduced due to the shift of the cosine curves with respect to each other. This explains the transmission behavior seen in Fig. 2(a).

5 Conclusion and outlook

We have presented the tuning of a multimode interference fiber filter by mechanical bending. A tuning range of 13.6 nm was achieved for continuous wavelength shifting. The effect was explained using a perturbational approach for bent fibers and the mode expansion method, leading to good agreement between theory and experiment. In addition, we showed in simulations that excitation of only two modes is sufficient for MMI filtering and qualitatively improves the tuning characteristics of the filter, and we addressed possible parameters for optimization of the tunability. This analysis also predicts a bending radius range where the wavelength dependence of the MMI structure should be greatly reduced, which might be useful for applications apart from filtering, e.g., to include an MMF section in a broadband single mode fiber device.

Although the numerical simulations allow a quick and easy design of tunable MMI filters, several aspects should be considered for their optimization. As it was pointed out in [8], a larger diameter of the pigtail increases the achievable transmission. Thus, employing a pigtail-fiber with increased core size could be useful to improve the transmission of the MMIF. Furthermore, the change of the propagation constant depends on the core radius of the MMF [see Eq. (4)], so a large-core MMF (with an appropriate NA for the desired number of modes) should in general be beneficial to maximize the tunability.