Tunable gratings in a hollow-core photonic bandgap fiber based on acousto-optic interaction

We demonstrate a coherent intermodal coupling in a hollow-core photonic bandgap fibers (PBGF) using an acoustic grating. A theoretical study on the properties of acoustic and optical wave propagating the PBGF is performed indicating that the applied frequency higher than 10 MHz is required to obtain the acousto-optic interaction at the wavelength around 1550 nm. Wavelength-tunable coupling from fundamental mode to higherorder core modes is achieved with more than 90% coupling efficiency at the applied frequency of 11.54 MHz. The low dispersive properties and the mode filed distributions of the higher-order guided modes of the air-core PBGF in the experiment show reasonable agreement with the simulation results. ©2009 Optical Society of America OCIS codes: (060.5295) Photonic crystal fibers; (060.2310) Fiber optics; (050.2770) Gratings; (230.1040) Acousto-optical devices References and links 1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. 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Introduction
Hollow-core photonic bandgap fibers (HC-PBGFs), consisting of central large air-hole defect and air/silica photonic crystal cladding, have attracted considerable interest over last decade because of its novel and unique guiding properties [1][2][3][4].A propagating light whose field intensity is confined mostly in the air-core region by photonic bandgap effect can potentially exhibit low propagating loss down to 1.2 dB/km [5], ultralow material dispersion and high threshold power for nonlinearity compared to the conventional fibers.The suppressed nonlinearity with proper dispersion property of the fiber enables the delivery and the compression of pulses with ultra-high power and ultra-short pulse width [6,7].This type of fiber has also been used as a host material for highly efficient nonlinear interaction of laser light with Raman active gas [8] or for highly sensitive interaction with gas filled in the core, which brought striking impact on recent nonlinear optics and sensing applications [9,10].To provide increased functionality of the devices based on the HC-PBGFs, a tunable optical filter is desirable where the propagating light can be dynamically selected or rejected at specific wavelength.The tunable optical filter can also be useful as a diagnostic tool for probing the modal properties of the air-guiding light in this fiber which has been only partially studied near the bandgap edge by measuring the angular distribution of radiated light [11].In addition to these classical linear filter functions, the tunable optical filter could be used for shaping [12] or controlling [13] the ultra-high power soliton propagating along the HC-PBGF.
One of the attractive tunable filters is an all-fiber acousto-optic tunable filter (AOTF) where the propagating flexural acoustic wave generates traveling long-period fiber grating (LPFG).Here both center wavelength and strength of the grating are tunable by the simple control of the electrical signal applied to the device [14].The periodic micro-bends generated by acoustic wave in the LPFG allows intermodal coupling in various types of fibers including conventional silica fibers [14,15] and photonic crystal fibers (PCFs).In particular, the AOTFs using PCFs is of great interest since the transmission properties of the filter can easily be engineered thanks to the flexibility of PCFs in designing optical parameters.Indeed there have been the demonstrations of several AOTFs using single-mode PCF [16], weakly multi-mode PCF [17], solid-core PBGF [18] and nearly endless two-mode PCF [19].However, an AOTF using the HC-PBGFs has not been reported yet although its possibility was conceptually described in Ref. 17. Recently, an LPFG filter at a fixed wavelength has been reported by collapsing some air holes of HC-PBGFs using side-illumination of CO 2 laser [20].The structural deformation, however, can cause appreciable change in the optical properties of propagating light which is not desirable for probing the optical properties of the HC-PBGFs.In this paper, we demonstrate a tunable optical filter in a HC-PBGF based on acousto-optic (AO) interaction.The optical and acoustic properties of the hollow-core PBGF are theoretically studied and calculated.Based on the calculation, we apply the acoustic wave with a frequency more than 10 MHz and achieve coherent inter-modal coupling in the HC-PBGF.The properties of the fabricated filter including transmission spectrum, near field profile and dispersion properties of the coupled modes are discussed.

Acoustic and optical properties of silica HC-PBGF: theoretical considerations
Optical fibers are considered as good cylindrical waveguides for acoustic waves when the protective polymer jackets are removed.When flexural acoustic wave is launched along the fiber for a low normalized frequency of fR/c t , only the lowest-order acoustic mode can be supported in the fiber.Here, f, R and c t are acoustic frequency, fiber cladding radius, and transverse acoustic wave velocity (3764 m/s in case of silica), respectively.
The acoustic dispersion equation, that relates the acoustic angular frequency ω a and the acoustic wave vector k a , provides the information on the period of an acoustic grating for a given applied frequency of the electrical signal.The exact solution of acoustic dispersion can be solved by using acoustic wave equation where the stress-free boundary conditions are imposed at cladding-air interface [21].Figure 1(a) shows the numerical results of exact solution in the solid fiber.The acoustic dispersion of a silica fiber can be approximately calculated from Euler-Bernoulli (E-B) theory [22] when fR/c t << 1.This approximation is particularly useful for the PCFs or PBGFs possessing relatively complex internal structures.The simplified dispersion form is generally expressed as [17,23] 2 4 where ρ is the density of fiber material, A eff is the effective area of fiber cross section, and I E is the flexural rigidity defined by the following integral equation calculated over the fiber cross section A, 2 ( , )( cos sin ) Here E(x,y) is Young's modulus and θ is the angle between the acoustic oscillation direction and the x-axis.In case of the conventional solid silica fiber with cladding radius of R, the E-B approximation of the acoustic dispersion can be expressed as Here λ a, and f a are acoustic wavelength and frequency, respectively, and c ext (5760 m/s in silica) is the acoustic wave velocity of the fundamental longitudinal mode in low frequency limit, defined as c ext≡ (E silica /ρ silica ) 1/2 .Figure 1(a) compares the E-B approximation of acoustic dispersion with exact solution in the fiber.For example the approximation agrees with the exact solution within 2% at the low frequency (fR/c t < 0.01) but the error becomes more significant as the normalized frequency increase (the error is more than 30% at fR/c t = 0.2).When we consider the acoustic dispersion in the PBGFs, the air holes reduce both the effective area and the flexural rigidity in Eq. (1).Since the portion of the silica in the holey region is estimated to be about 8% in the PBGF (HC 1550-2, Crystal Fiber A/S) used in our experiment, we approximated the PBGF in the Fig. 1(b) as a silica cylindrical shell including a material with a density and E(x,y) of 8% of silica as depicted in the Fig. 1(c) for simplification.Here we assume that the inner radius r and the outer radius R of cylindrical shell are 35 and 60 µm respectively based on the information from Crystal Fiber A/S (diameter of holey region: 70 µm, diameter of silica cladding 120 µm).From this approach we can directly see the effect of inner radius on the acoustic dispersion.The acoustic dispersion based on E-B theory can then be modified as follows.
where R 0 is The equation shows, for a given acoustic frequency, the acoustic wavelength becomes larger in the PBGF than in the solid fiber with same cladding radius, and the difference becomes appreciable as the inner radius increases.For example the acoustic wavelength will increase by 6.8% compared to that in solid fiber at the inner radius r of 35 µm for the same outer cladding radius as shown in the Fig. 1(a).This result is independent of normalized frequency but it should be noted that the E-B theory is accurate at only low frequency limit.Previous report experimentally showed that the acoustic dispersion of the PCF and the PBGF approaches that of the solid fiber at relatively high frequency because the distribution of the acoustic field at central region of the fiber is reduced as the acoustic frequency increases.They experimentally show the difference gradually decreases to 1% at the fR/c t of 0.05 from 3% at low frequency limit for different kinds of PBGFs possessing smaller air-hole region [17].The operating range of our device expected from next section requires much higher value of fR/c t lying between 0.16 and 0.2.Based on this we expect the acoustic wavelength of our PBGF will approach to that of a solid fiber from the difference of 6.8% in maximum at low frequency limit.The optical properties of guided modes in the air-core have been previously studied in many articles, mostly focusing on the guiding properties of fundamental mode such as bandgap diagram, dispersion characteristics and effect of surface modes on the guidance [1,24,25].Recently the detailed analysis of the higher-order mode is reported for the HC-PBGF with a large hollow-core (19-unit-cell air-core) with the focus on the suppression of the higher-order guided modes in the large air-core PBGF [26].Here we numerically investigate the optical properties of the higher-order modes in our HC-PBGF (7-unit-cell-defect core) with the focus on the inter-modal beatlength and dispersion properties between the (C) 2009 OSA fundamental and higher-order guided modes in the fiber.In the simulation using the plane wave expansion method [27], we built the PBGF structure considering the fiber used in our experiment (HC 1550-02) where the period and the air filling ratio of the web-like cladding was assumed to be 3.8 µm and 92% respectively.From the simulation we observed the nearly degenerate higher-order guided modes of which polarizations and field properties resemble those of the typical higher-order modes (TM 01 , HE 21 × 2, TE 01 ) in a step-index two-mode fiber.As an example, the mode field distribution and the polarization direction of the HE 11like and TM 01 -like core mode of the PBGF are shown with the silica structure in the Fig. 2(a) and 2(b).In practice, the PBGF used in our experiment is considered as a single-mode fiber at wavelength around 1550 nm because the higher-order core modes close to the bandgap edge suffers very large bending loss for a long length of the fiber.However, it should be regarded as a few-mode fiber when the AO devices are realized in a straight fiber with a length of several tens-of-cm.We found that the optical beatlength between the fundamental mode and the higher-order modes is about 230 µm at the wavelength around 1550 nm in the PBGF. Figure 2(c) compares the calculated optical beatlength and the acoustic dispersion.From the curve in the Fig. 2(c), we expect the acoustic wavelength meets the optical beatlength at higher frequency more than 10 MHz (fR/c t ~0.16)where the acoustic wavelength of the PBGF will be almost close to that of the solid fiber as expected from previous section.

Experimental analysis and discussion
We fabricated an AOTF using the PBGF (HC 1550-2).The basic configuration of the device is similar to that described in Ref. 14 except that a high-frequency-resonance (10 MHz) piezoelectric washer is used here.We striped off the plastic jacket over 12-cm-long length of the PBGF and then attached it to the acoustic transducer as depicted in Fig. 3(a).We then placed the acoustic damper at the end of bare fiber section to effectively absorb the acoustic wave without a significant reflection.A polarized broadband light source and an optical spectrum analyzer connected with a conventional single mode fiber (SMF) were employed to characterize the performance of the tunable filter.
Figure 3(b) and 3(c) shows the transmission spectra of a notch filter induced by the AO coupling in the PBGF.We applied electric signal of 18 Vpp to the transducer at the frequency of 11.54 MHz.Two main notches at the different spectral position were observed depending on the input polarization states.The maximum notch depths were about −15 dB and −9dB, respectively.Field distributions of the coupled mode were measured using tunable laser source and infrared camera, as shown in the insets of Fig. 3(b) and 3(c).In this case we imaged the light coming out of the end of the PBGF on the camera using a lens.As shown in Fig. 3, it is clearly observed that the mode field distribution changes from fundamental Gaussian-like mode to the next higher-order core mode when we turn on the RF signal.The wavelength separation of the two notches was 52 nm and the corresponding optical birefringence is about 2.5 × 10 −4 .This value is much larger than our expectation from simulation (order of 10 −5 ).We expect the discrepancy mainly come from the strong birefringence in the guided modes induced by the slight structural deformation of the PBGF as in the case of the solid-core PCF [28].In this case the coupled two notches (polarization 1,2) would be close to the LP 11 -like modes rather than the calculated modes (TE 01 , TM 01 , and HE 21 mode) from the simulation of an ideal PBGF without any structural deformation.Thus we expect polarization-dependent two notches are regarded as LP 11 modes for two polarization states provided that the acoustic oscillation direction is properly determined.The notch depth difference, sideband structure and lobe orientations of the two notches may be quantitatively analyzed when considering the acoustic birefringence as in the case of Ref. 28.In future we will investigate more details of the optical and acoustic birefringence by varying the oscillating direction of the acoustic waves.The tuning characteristics of the filter were investigated by varying the applied frequency of the electric signal.In Fig. 4(a), we can see that two main peaks move toward longer wavelength as the applied frequency increases.The sign of the tuning slope is opposite to that in conventional SMF because the higher-order mode is more dispersive than the fundamental mode in the PBGF similar to the normal PCFs.The magnitude of the tuning slope, / f λ ∂ ∂ is 0.040 nm/kHz which is very small value compared to that in conventional SMF (~-0.2 nm/kHz) [14], PCF (0.3 ~0.5 nm/kHz) [16,23] or solid-core PBGF (~0.08 nm/kHz) [18].The value of / f λ ∂ ∂ can be expressed as The slow change of the intermodal beatlength with respect to the wavelength change is considered to be from the less dispersive core-modes which are guided through the air-core region in the PBGF.The tuning range of the AOTF is about 60 nm and 25 nm for the two notches respectively.The tuning range is limited by the efficiency of the acoustic transducer at long wavelength edge and by the finite bandgap width of the PBGF at short wavelength where the higher-order modes suffer significant loss near bandgap edge around 1510nm.
It is interesting to note that our beatlength of ~220 µm at the wavelength of 1595 nm is almost a half of that (395 µm) in the previous hole-collapsed grating for the similar conditions [20].We suspect that this large discrepancy may come from the changed average optical beatlength by the fiber deformation in the previous hole-collapsed grating.Another possibility might be that the harmonic components of the original period (395 µm) in the hole-collapsed PBGF may contribute to the grating spectra in the previous work.
The estimated 3-dB bandwidth of the optical filter obtained from the value of ( / ) 1.06 and 0.74 nm with 12-cm-long interaction length for two polarization states.They are smaller than the measured value of 1.26 and 0.9 nm in the experiment.The discrepancy might be due to the non-uniformity of the fiber along the propagation length.The narrow bandwidth for such a short AO interaction length is from the very small nominal beatlength (~230 µm).

Conclusion and prospects
In summary, we demonstrated tunable filters in HC-PBGF based on the AO interaction for the first time.The coherent AO interaction was obtained at the applied frequency higher than 10 MHz where the corresponding period of the acoustic grating was around 230 µm.The field distributions and dispersions of the coupled modes reasonably agreed with our expectations, indicating that the HC-PBGF supports higher-order guided modes within short length of the fiber.We expect this AO devices can be used for the dynamic spectral devices as well as an efficient tool for probing the properties of the guided modes in the PBGFs.In addition, we expect this periodic perturbation by this linear grating can be potentially used as a dynamic ultrafast pulse shaper for huge power optical soliton propagating the HC-PBGFs.

Fig. 1 .
Fig. 1.(a) Comparison of the exact solution of acoustic dispersion with that using Euler-Bernoulli theory in a conventional silica fiber and a silica cylindrical shell (approximation of the HC-PBGF).(b) Image of fiber cross-section used in the experiment (HC 1550-02, the picture was obtained from www.crystal-fiber.com)(c) Simplified PBGF structure as a silica cylindrical shell model

Fig. 2 .
Fig. 2. Simulation of field distribution and polarization direction of (a) the fundamental (HE11like) core-mode and (b) higher-order (TM01-like) core-mode in the HC-PBGF.(c) Comparison of acoustic dispersion and optical beat length between two coupled modes at the wavelength around 1550 nm.

Fig. 3 .
Fig. 3. (a) Schematics of the AOTF using the PBGF.Optical transmission spectrum of the AO device for different polarization states of (b) polarization 1 and (c) polarization 2. The inset figures show the near field images at the resonance in the absence (or presence) of the acoustic grating.
estimated from the acoustic dispersion shown in the Fig. 2(c) where #110149 -$15.00USDReceived 15 Apr 2009; revised 24 May 2009; accepted 25 May 2009; published 28 May 2009 (C) 2009 OSA we assumed the acoustic dispersion of the PBGF is similar to that in the SMF at the high frequency of more than 10 MHz.The variation of the optical beatlength with the wavelength, ( / ) then obtained resulting in the −0.362 µm/nm and −0.434 µm/nm for each polarization states as shown in the Fig. 4(b).They are smaller than the typical values in the conventional SMF.

Fig. 4 .
Fig. 4. (a) Tuning properties of the AO devices for two polarization states.(b) estimated optical beatlength as a function of the wavelength.