Capacitors go optical: wavelength independent broadband mode cavity

Sébastien Loranger; Mathieu Gagné; Raman Kashyap

doi:10.1364/OE.22.014253

1. Introduction

Resonators, more specifically Fabry-Perot interferometers (FPI), are well known devices in optics and have a range of applications in photonic. The physical properties of the FPI are well known as light accumulators in which specific modes are allowed to exist. These modes are characterized by a bandwidth, which is generally very small (much less than a GHz for FPI lengths of cm to m, when the losses are low), and by a free-spectral-range (FSR), which is the spacing between the modes. We present here a new type of FPI which is in fact a broad-band mode cavity (BBMC), having an oscillating mode which is broadened over a width of several nm (and possibly more), since the wavelength dependence for the mode’s spectral position is eliminated. A very interesting fact about such a macroscopic device –contrary to photonic crystal micro-cavities – is that its accumulation of energy property can be applied to any type of laser source, such as a broadband pump laser or a pulse laser of any given repetition frequency. This could be a major breakthrough for many applications, such as laser cooling of solids [1], non-linear wave-mixing and amplification, etc. The broadband nature of this mode could also open the door to a new class of single-mode laser sources. But mostly, a BBMC has the potential of becoming a capacitive temporal filter and since dispersion can actually be suppressed in such a device, as will be shown, it becomes an attractive tool in optical signal processing, which can be added to other analog optical signal processing tools such as the optical integrator [2, 3] and differentiator [4].

Using chirped structures to attain a broad spectrum for slow light cavities in photonic crystals has been extensively studied in the last decade [5, 6]. In the more macroscopic scale (cm to m scale), this concept is somewhat similar and comes from manipulating a 1D photonic crystal, i.e. a fiber Bragg grating. Modifying FPI parameters with such a device was first demonstrated by Town et al. [7] where the authors used a chirped fiber Bragg grating (CFBG) to modify the FSR of the cavity, which is normally only dependent on cavity length. Thereafter, superimposed CFBG were used to generate interesting filters to control the FSR [8] and a tunable version was recently demonstrated [9, 10]. It has also been proposed to use such a filter in ultrahigh repetition rate pulse generation [11]. A CFBG placed in a Gires Tournois interferometer also brings interesting dispersion properties, which can be used in dispersion compensation in ultra-short pulse applications [12–14]. Other application of FPI cavities modified via CFBG as sensors have been demonstrated [15–17], where the large FSR allows a radical increase in the dynamic range while maintaining the small bandwidth of the mode. A more detailed theoretical study has also been made on field profiles within a FPI cavity with CFBG mirrors by Pereira et al [18] for superimposed CFBG and small cavity lengths (~mm range). Only very recently was a more general theoretical study made on the implementation of a cavity near broad-band broadband operation [19]. However, the broad-band nature of a particular mode of the CFBG FPI filter was not shown or discussed. In this paper, we propose a novel configuration of short CFBGs in long cavities to serve as broad-band light accumulators, for processing broad-band signals, what we refer to as the optical capacitor.

2. Theory

In a simple FPI cavity of fixed length, such as shown in Fig. 1(a), each mode exists at a specific wavelength for a given spacing between mirrors. The concept proposed here is to cancel out this wavelength dependence. This corresponds effectively to change the cavity length for different wavelengths, as conceived in photonic crystals [5, 6], so that a specific mode can exist for the entire wavelength band of interest, such as shown in Fig. 1(b). This can be implemented with a CFBG, in which the chirp must be tuned to the spacing such that the phase accumulation in one round-trip remains constant with wavelength.

Fig. 1 Scheme of concept of broad-band mode cavity (BBMC). In (a) a standard Fabry-Perot cavity is shown, in which the mirror spacing is fixed and different wavelengths oscillate in different modes. In (b), a BBMC is shown, in which different wavelengths oscillate in a same mode, generating a broad-band response.

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In order to demonstrate this principle and find the required chirp, a simple approach is implemented by supposing that the phase accumulation of a monochromatic wave from a reflection in a CFBG can be approximated by a fixed point reflection. This point corresponds to where the period of the CFBG fits the resonance condition of this monochromatic wave. Such an approximation has been validated by Dong et al. [19] who showed that phase accumulation is equivalent to setting an effective length corresponding to the reflection point in the grating. By supposing an ideal case of perfectly apodized gratings, we can neglect reflection and phase ripples normally observed in a chirped grating.

Therefore, a cavity consisting of two CFBG, as shown in Fig. 2(a), would have a phase accumulation per roundtrip of:

φ = β 2 L_{0} + β 2 δ_{C F B G 1} (Δ λ) + β 2 δ_{C F B G 2} (Δ λ)

where, β is the propagation constant of the mode, L₀ is the length of cavity between the center of the two CFBGs and δ_CFBG1(Δλ) and δ_CFBG2(Δλ) are the variations in the reflection point in each CFBG from the center in relation to the wavelength difference from the central wavelength λ₀ (which reflects at the center of the grating). To simplify the calculation and the implementation of the grating, both CFBGs are assumed to be identical (but with opposite chirp):

δ = δ_{C F B G 1} = δ_{C F B G 2}

(as defined by Eq. (1)). By expanding

δ (Δ λ)

into a linear chirp and a non-linear term, we have:

δ (Δ λ) = δ_{1} Δ λ + δ_{2} Δ λ^{2}

where δ₁ is the linear chirp (in mm/nm), while δ₂ is a quadratic correction. By expanding the wavenumber as

β = \frac{2 π n_{e f f}}{λ_{0}} - \frac{2 π n_{g}}{λ_{0}^{2}} Δ λ - \frac{D π c}{λ_{0}} Δ λ^{2}

and by adding the resonance condition of

φ = m 2 π

for one round-trip, where m is the mode number, we get, neglecting the higher order terms:

φ = \frac{2 π n_{e f f} L_{0}}{λ_{0}} + (- \frac{2 π n_{g} L_{0}}{λ_{0}^{2}} + \frac{4 π n_{e f f}}{λ_{0}} δ_{1}) Δ λ + (- \frac{4 π n_{g}}{λ_{0}^{2}} δ_{1} + \frac{4 π n_{e f f}}{λ_{0}} δ_{2} - D \frac{π c L_{0}}{λ_{0}}) Δ λ^{2} = m π

where n_eff is the effective refractive index, n_g the effective group index of the mode and D the dispersion parameter of the fiber (in ps/nm km). What is required in order to make a BBMC, is to suppress the dependence on Δλ from the resonance condition. As can be seen in Eq. (3), a linear chirp would be sufficient to cancel out the linear dependence from Δλ, but a non-linear chirp would be required to go further and suppress second-order dispersion. Therefore, both linear δ₁ and nonlinear δ₂ components can be found by cancelling the second and third term respectively in Eq. (3), thereby giving the following condition:

Fig. 2 Different configurations of a BBMC and experimental characterization system. The different configurations proposed are: (a) two CFBG, thus allowing a guided transmitted output, or (b) a single CFBG used to cancel out the wavelength dependence of the FPI and a fixed reflector (silver-tip mirror). Both configurations have been tested with the experimental setup shown in (c), using a JDSU-OMNI wavelength sweeping laser where the fibered BBMC is spliced to standard SMF-28 fiber.

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δ_{1} = \frac{n_{g} L_{0}}{2 n_{e f f} λ_{0}} δ_{2} = (\frac{n_{g}^{2}}{n_{e f f} λ_{0}^{2}} + D \frac{c}{2}) \frac{L_{0}}{2 n_{e f f}}

To get a feel for the order of magnitude, such a relationship corresponds to a chirp (inverse of δ₁) of 1.50 nm/mm for a cavity length of 1m at 1550 nm, or 15 nm/mm for 10 cm of length. Over 10 nm/mm, such a chirp becomes difficult to achieve for any useful reflection from a CFBG (but may be possible in other material systems), which is why such a system functions mostly for cavities of more than 10 cm in length. However, a problem arises when the length becomes too long, which is seen by the non-linear term in Eq. (3). Indeed, the group delay introduced by the CFBG will limit the width of the BB mode as shown in Fig. 3(b) and introduce dispersion, thus the need for non-linear chirp correction. In this case, δ₂ is in the order of 0.1 to 1 µm/nm², which makes it a very small quadratic correction to add. To put these parameters in a FPI resonator, we need to calculate the output electric field of such a resonator as:

E = \frac{(1 - r_{1}) (1 - r_{2}) e^{i φ}}{1 - r_{1} r_{2} e^{i φ}}

where r₁ and r₂ are the reflection coefficients of each CFBG. By taking the intensity of the field in Eq. (5), the theoretical results shown in Fig. 3 are obtained, where the intensity transmittance as a function of wavelength and cavity length is shown. Three cases are compared: the first one, in Fig. 3(a), shows a standard FPI with a fixed reflector and an FSR of ~1 pm. The second one shows a BBMC with a linearly chirped grating. The BB mode is around 1 nm wide. Of course, the cavity length has to be stabilized to force one particular wide mode to exist. The case in 3c shows a super broad-band dispersion suppressed cavity where the quadratic correction in the chirp enables modes which are ~10 nm wide.

Fig. 3 Theoretical calculation of transmission (in dB) of different types of cavities of 50 cm of length in function of wavelength (Δλ) AND cavity stretching (ΔL): (a) standard Fabry-perot resonator, (b) broad-band cavity with linearly chired gratings (2 gratings at 6.10 nm/mm) and (c) Super-broad-band cavity with a quadratic chirp correction to compensate for group delay (δ₂ = 0.22 µm/nm²). Note the difference in the wavelength scale.

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3. Experimental method

Chirped FBGs were fabricated using an ultra-long FBG fabrication system recently demonstrated by Gagné et al [20] that has been shown to produce ultra-high quality custom made FBGs (short, long, ultra-long, chirped, non-chirped, apodized, distributed feed-back, etc.). Briefly, a phase mask is mounted on a piezoelectric stage driven by a ramp signal to create a moving fringe pattern by mean of a standard Talbot interferometer that can be synchronized to a moving fiber mounted on a linear translation stage. By properly choosing the ramp signal frequency and amplitude, custom grating characteristics can be implemented. This technique lifts the reliance on the phase mask characteristics and allows any desired chirp to be imprinted in the fiber. The chirp here was obtained by sweeping the ramp frequency applied the piezo stage during the writing process, thus creating a variation in the grating period through a Doppler shift. Current system constraints limits the time of frequency sweep to 500 s, thus limiting the writing time, and therefore the strength of the grating.

After grating writing, the length of cavity was adjusted (to a few mm) to match the chirped grating, as shown in Eq. (4), which is not exactly as intended due to fabrication imperfections. This adjustment was made by heat-stretching the fiber under CO₂ laser irradiation, or cutting and resplicing as necessary. The reflectivity spectrums were acquired using a JDSU-OMNI wavelength sweeping laser, such as shown in Fig. 2(c). The laser linewidth is in the order of MHz, while the resolution of the scan is 3 pm. The scanning speed is fast – 20 nm/s – so that thermal fluctuations are not a problem.

4. Results

Different BBMC were fabricated. In all cases, the propagation medium was a single-mode, germanium doped photosensitive fiber and the chirped reflectors to get the broadband FPI mode are CFBGs. The most useful model incorporates two CFBGs with opposite chirp, which is the result shown in Fig. 5. Another possibility, which was tested and showed similar results in Fig. 6 is when a CFBG is replaced with a fixed reflector, i.e. a silver – or gold – tipped fiber end, however the transmitted light is no longer usable, making such device less interesting. We could also imagine a BBMC in the form of a ring cavity, where a circulator could be used to interact with the chirped grating, in a similar scheme to what was shown by Duval et al [21].

To compare with theory and to be able to observe the cavity modes, firstly, the cavity had to be thermally stabilized from environmental fluctuations by placing it in a thermally insulated container. In order to observe the broad band mode, as shown in Fig. 3, the cavity must be stretched to within a micron. Therefore, to simulate stretching, the container was slowly heated electrically with a resistor. This method of testing has the inconvenience of not knowing the exact induced stretch, since the temperature change, although linear with time, is extremely small and not necessarily uniform in the container. A more rigorous method of testing, and eventually to actively stabilize the cavity, would be to use a small piezo transducer glued to the fiber.

In a first case a BBMC consisting of two fabricated CFBG was fabricated (Fig. 2(a)), which spectrum are shown in Fig. 4(a). The total transmission (sum of both gratings) is shown in Fig. 4(b) in resonance and in non-resonance. Those plots were found by considering the maximum (or minimum) intensity at each wavelength during heat stretching of cavity where several mode oscillation periods were observed. In an ideal case, we should expect the resonant transmission (maximum transmission) to be 100% over the entire grating spectrum. This is not the case, as shown in Fig. 4(b), due to inequalities in grating transmission between both gratings, therefore creating a loss at the resonance. Ripples, such as seen in Fig. 4(a), are to be expected from a un-apodized chirped grating. However, in an ideal case, the ripples would be symmetric and repeatable, which is not the case here. Those irregularities can be caused during writing by dust blocking the grating writing UV radiation, vibration, temperature shift, UV laser power fluctuation, etc. This makes the major challenge for BBMC which is to have an exact replica of transmission spectra for both gratings.

Fig. 4 (a) Reflection spectrum of each 2.5 mm long individual grating with a chirp of 3.77 nm/mm. (b) Total transmission of both CFBGs in the experimental BBMC where the resonant and non-resonant cases are compared, thus showing the potential contrast of the modes. This is done by performing a scan at different temperatures (which varies slowly) to observe all the possible mode positions and find the maxima and minima at each wavelength. The resonant case is taken as the maximum while the non-resonant case is taken as the minimum. Ideally a 100% transmission over the whole grating is expected at resonance, which is not the case here because of dissimilarities in grating transmission.

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The grating’s strength was around 10 dB (90% reflectivity), which is sufficient to observe an effect in transmission. A higher strength can be reached by slower writing (our system is currently limited to a frequency sweep of 500 s), higher UV power (currently limited by optical components) and higher photosensitivity of the fiber such as by hydrogen loading, which was not available for these experiments.

Although our gratings were not perfect, we can still observe good agreement in shape with the theoretical reflection and transmission intensities for the non-dispersion suppressed case, as can be seen in Fig. 5. Oscillating modes are shown as a loss of reflection (minimum), or maximum in the transmission in these plots. From these results, this BBMC indicates a broadband mode of 1.5 nm (3x10⁴ times larger than a normal FPI in the same conditions), which is more than sufficient to allow interaction with the entire intensity spectrum of many simple lasers. Therefore, we have proven here the creation of a cavity in which a broad-band resonance mode of >1nm exists, in which energy can accumulate. The good contrast region in the transmission spectrum in Fig. 5 is much narrower than for the reflectivity spectrum, since resonance can still be observed at low reflectivity when far from the central operating wavelength, while it is not the case for the transmission where contrast is lost with a smaller wavelength difference.

Fig. 5 Simulation and measurements of a broad-band cavity of 80 cm in length with two ~10 dB CFBG (~90% reflectivity). The theoretical (a)Reflection and (b)Transmission spectrum is compared with measured (c)Reflection and (d)Transmission spectrum. To simulate cavity stretching, the experimental BBMC is heated in a thermally isolated container. The broadband mode of interest is not perfectly centered on the grating, hence the de-centering in the experimental data. Some regions show lower contrast due to variation in transmission from one grating to another, but the concept of BBMC is still well demonstrated here in the experimental device.

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To demonstrate the generality of the BBMC concept, the CFBG-fixed mirror configuration was also tested (Fig. 2(b)). Results are shown in Fig. 6 in which theoretical and experimental data can be compared. However, because of the important difference in transmission of both reflectors (~95% for the silver-tip and 10% for the CFBG), contrast was rather poor (1 to 3 dB in reflection). Despite this, the same broadband mode phenomena can be observed when the cavity length is tuned properly. The broadband mode here is about 1.5 nm wide, which is larger than for the theoretical BBMC with a linearly chirped CFBG. This broadening is probably due to an unintended, but favorable, non-linear chirp component (δ₂ form Eq. (2)) which cancels out some part of the group dispersion.

Fig. 6 Theoritical calculation (a) and measurements (b) of a broad-band cavity of 80 cm in length with one CFBG of ~1 dB in strength (~10% reflectivity) and a silver-tipped fiber end (fixed mirror) at 90-95% reflectivity. To simulate cavity stretching, the experimental BBMC is heated in a thermally isolated container. The larger bandwidth of the experimental BBMC is probably due to a non-linear chirp which was added by mistake to the grating and which favorably broadens the mode.

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The non-linear chirp correction to get super broadband modes has not been implemented yet, because of the requirements of increased precision on the systems, but is in the process of being dealt with. Indeed, simulation show a radical increase in the matching requirement between the various parameters (operation wavelength, cavity length and chirp) when a favorable non-linearity is added to suppress dispersion.

5. Discussion and applications

A BBMC is foremost a cavity in which power needs to accumulate to achieve interference at the input and output and gain 100% transmission. This makes it a capacitor, analog to its electrical cousin. Therefore, it takes time for this resonance regime to appear, a build-up time corresponding to the life-time of a photon in this cavity and is given by:

τ_{N} = {[v_{g} (α - \frac{1}{2 L} \ln (R_{1} R_{2}))]}^{- 1}

Where α is the intra-cavity losses and v_g the group velocity. This transition time is actually the same as for a FPI, which can also be called a capacitor. However, a normal FPI is limited to a very specific repetition frequencies (dependent on the round-trip time), i.e. the free-spectral range, and mode wavelength position must be matched with the source, a laborious task, limiting its applications. Therefore, the usefulness of a FPI as a capacitor is severely limited. In a BBMC, any laser source could potentially be used, continuous wave or pulsed at any repetition frequency. Therefore the full capacitive nature of a cavity can be exploited.

The accumulation of power itself can be useful for several applications. For example, effects requiring high pump power, such as non-linear effects, would greatly benefit from a BBMC, since the pump intensity circulating in the cavity is much higher than what is injected from the outside (formally known as Q-factor). Also, low absorption interaction, such as in laser cooling of solids, would benefit from a BBMC with the increased number of effectives passes in the medium, as shown in Fig. 7(a). In such schemes, we would want the incoming pump beam to be fully injected within the cavity, i.e. no reflectivity at the input of the device. However, if light interacts with the cavity material (i.e. the cavity is lossy) then the losses of the first grating – reflectivity – must match the round-trip losses in the cavity plus the losses in the second grating. This would ensure a destructive interference capable of countering all reflected input power while in resonance. Therefore, fixing the second grating’s reflectivity at R₂, the reflectivity of the input grating R₁ can be calculated to allow 100% injection of a pump beam for specific intra-cavity absorption per pass, A as:

Fig. 7 Capacitive application of the BBMC. A scheme for local amplification of power of a pump beam in an active material is shown in (a). The total calculated absorption of a BBMC with low single-pass intra-cavity absorption is plotted in (b), showing a considerable improvement in absorption, for any pump laser (no need to match modes). Here, the input reflector is adjusted to cancel out reflection and gain 100% injection. As a different example (without intra-cavity loss), the capacitive nature of the BBMC can be shown in (d), where we see the simulated temporal transient function of the transmission in this optical capacitor for an arbitrarily chosen train of pulse of 750 MHz starting at t = 0, as shown schematically in (c). Sending the same pulse train in such a FPI cavity (no chirped gratings) would result in only 0.25% transmission in the steady state.

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R_{1} = R_{2} {(1 - A)}^{2}

The net transmission of the system is reduced due to multi-pass intra-cavity absorption, as shown in Fig. 7(b). The number of effective passes calculation was performed by comparing the cavity lifetime ( $τ_{N}$ ) with the time for a single-pass ( $τ_{1} = L_{0} / v_{g}$ ).

The most interesting fact about the BBMC is its temporal capacitive nature. As already mentioned, contrary to a FPI, the BBMC can be used at any repetition frequency. This implies that the output transmission is sensitive to phase and amplitude change. If there is a sudden variation in phase or amplitude, this variation will not be transmitted through. This can be shown in simulation results in Fig. 7(d), showing the transient evolution of average power output of a pulse train after it is suddenly turned on from 0 to 100%. We can assume this transient function to be reciprocal if the power drops suddenly. If there is a sudden phase shift, then the previous pulse train will deplete according to this transient function, while the new one with the new phase will grow, giving a gradual phase change in transmission. This will have a stabilizing effect on any spectrum, reflecting noise and transmitting a stable spectrum. However, contrary to a standard FPI, any signal could be filtered out, therefore making it as a true capacitor.

This compatibility to any repetition frequency may seem counter-intuitive, since the round-trip time does not need to match the repetition period of a pulse train. However, what actually happens is that the effective “reflection point” is not at the input mirror, but somewhere in the cavity, which is in the center for a symmetric BBMC. It is this position that serves as phase reference for the destructive interference of the reflected waves, contrary to a standard FPI, where the effective reflection takes place at the first input mirror. The transmitted output pulse, on the other hand, is seen as simply passing through the cavity and transmitted, as it would with a FPI. While the pulses travel in the cavity, their peak power is much higher within due to accumulation of waves, as in a FPI but only at specific wavelengths. This effect is shown in a simulation video (Fig. 8) in which propagating waves inside the cavity were simulated by summing all spectral and reflected components of a pulsed signal.

Fig. 8 Frames of simulation video comparing the growth of a pulsed signal in a BBMC (a) (see Media 1) versus a standard FPI with fixed mirrors (b) (see Media 2). In the last case, the pulse train frequency does not match the FSR of the FPI, therefore there is no transmission after any long period. However, in a BBMC, the pulse train repetition frequency does not have any frequency restrictions, therefore it is transmitted after the build-up time.

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6. Conclusions

We have demonstrated here a new type of FPI cavity, in which chirped reflectors are used to cancel out the wavelength dependence and thus allows a broad resonance mode of at least 1 nm in width. Our experiments have validated the theoretical prediction of such a BBMC with two CFBGs. Although this device still needs improvement, especially in matching of reflectivities of the gratings, the concept of a broadband mode resonance has been demonstrated. Just as a FPI would at specific frequencies, the BBMC acts as an optical capacitor, but at any repetition frequency. Therefore, it can act as a low pass (transmission) or high pass (reflection) filter and it can accumulate power which can be used for many applications such as in non-linear optics and laser cooling of solids. Although dispersion suppression has not yet been demonstrated, we have shown that it can be implemented, opening the possibility of a zero-dispersion optical capacitor.

Acknowledgment

The following sources of funding are acknowledged for their support of this research: Raman Kashyap’s Canada Research Chair on Future Photonics Systems - a program of the Government of Canada, and Sébastien Loranger’s Vanier graduate scholarship and Andre Hamer Prize.

References and links

1. G. Nemova and R. Kashyap, “Laser cooling of solids,” Rep. Prog. Phys. 73(8), 086501 (2010). [CrossRef]

2. N. Quoc Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007). [CrossRef] [PubMed]

3. R. Slavík, Y. Park, N. Ayotte, S. Doucet, T.-J. Ahn, S. LaRochelle, and J. Azaña, “Photonic temporal integrator for all-optical computing,” Opt. Express 16(22), 18202–18214 (2008). [CrossRef] [PubMed]

4. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Express 15(2), 371–381 (2007). [CrossRef] [PubMed]

5. T. Baba, D. Mori, K. Inoshita, and Y. Kuroki, “Light localizations in photonic crystal line defect waveguides,” IEEE J. Sel. Top. Quantum Electron. 10(3), 484–491 (2004). [CrossRef]

6. D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. 85(7), 1101–1103 (2004). [CrossRef]

7. G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7(1), 78–80 (1995). [CrossRef]

8. R. Slavik and S. LaRochelle, “Large-band periodic filters for DWDM using multiple-superimposed fiber Bragg gratings,” IEEE Photon. Technol. Lett. 14(12), 1704–1706 (2002). [CrossRef]

9. Y.-G. Han, X. Dong, C.-S. Kim, M. Y. Jeong, and J. H. Lee, “Flexible all fiber Fabry-Perot filters based on superimposed chirped fiber Bragg gratings with continuous FSR tunability and its application to a multiwavelength fiber laser,” Opt. Express 15(6), 2921–2926 (2007). [CrossRef] [PubMed]

10. X. Dong, P. Shum, C. C. Chan, and X. Yang, “FSR-tunable fabry-Peárot filter with superimposed chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 18(1), 184–186 (2006). [CrossRef]

11. J. Azana, R. Slavik, P. Kockaert, L. R. Chen, and S. LaRochelle, “Generation of customized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg gratings,” Lightwave Technology, Journalism 21, 1490–1498 (2003).

12. X. Shu, K. Sugden, and K. Byron, “Bragg-grating-based all-fiber distributed Gires-Tournois etalons,” Opt. Lett. 28(11), 881–883 (2003). [CrossRef] [PubMed]

13. R. Szipöcs, A. Köházi-Kis, S. Lakó, P. Apai, A. Kovács, G. DeBell, L. Mott, A. Louderback, A. Tikhonravov, and M. Trubetskov, “Negative dispersion mirrors for dispersion control in femtosecond lasers: chirped dielectric mirrors and multi-cavity Gires–Tournois interferometers,” Appl. Phys. B 70(S1), S51–S57 (2000). [CrossRef]

14. R. Szipöcs, K. Ferencz, C. Spielmann, and F. Krausz, “Chirped multilayer coatings for broadband dispersion control in femtosecond lasers,” Opt. Lett. 19(3), 201–203 (1994). [CrossRef] [PubMed]

15. K. P. Koo, M. Leblanc, T. E. Tsai, and S. T. Vohra, “Fiber-chirped grating Fabry-Perot sensor with multiple-wavelength-addressable free-spectral ranges,” IEEE Photon. Technol. Lett. 10(7), 1006–1008 (1998). [CrossRef]

16. R. Silva, M. Ferreira, J. Santos, and O. Frazão, “Nanostrain measurement using chirped Bragg grating Fabry-Perot interferometer,” Photonic Sens. 2(1), 77–80 (2012). [CrossRef]

17. A. Wada, K. Ikuma, M. Syoji, S. Tanaka, and N. Takahashi, “Wide-Dynamic-Range High-Resolution Fiber Fabry?Perot Interferometric Sensor With Chirped Fiber Bragg Gratings,” J. Lightwave Technol. 31(19), 3176–3180 (2013). [CrossRef]

18. S. Pereira and S. Larochelle, “Field profiles and spectral properties of chirped Bragg grating Fabry-Perot interferometers,” Opt. Express 13(6), 1906–1915 (2005). [CrossRef] [PubMed]

19. X. Dong, W. Liu, D. Wang, and M. Wu, “Study on Fabry–Perot cavity consisting of two chirped fiber Bragg gratings,” Opt. Fiber Technol. 18(4), 209–214 (2012). [CrossRef]

20. M. Gagné, S. Loranger, J. Lapointe, and R. Kashyap, “Fabrication of high quality, ultra-long fiber Bragg gratings: up to 2 million periods in phase,” Opt. Express 22(1), 387–398 (2014). [CrossRef] [PubMed]

21. S. Duval, M. Olivier, M. Bernier, R. Vallée, and M. Piché, “Ultrashort pulses from an all-fiber ring laser incorporating a pair of chirped fiber Bragg gratings,” Opt. Lett. 39(4), 989–992 (2014). [CrossRef] [PubMed]

Capacitors go optical: wavelength independent broadband mode cavity

Abstract

Corrections

1. Introduction

2. Theory

3. Experimental method

4. Results

5. Discussion and applications

6. Conclusions

Acknowledgment

References and links

Supplementary Material (2)

Cited By

Figures (8)

Equations (7)

Optics Express