Optical multistability in a silicon-core silicacladding fiber Ivan A. Temnykh,1* Neil F. Baril,2 Zhiwen Liu,3 John V. Badding2 and Venkatraman Gopalan1 1
Department of Material Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA 2 Department of Chemistry, The Pennsylvania State University, University Park, PA 16802, USA 3 Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA *
[email protected]
Abstract: We fabricate a novel silicon-core silica-cladding optical fiber using high pressure chemical fluid deposition and investigate optical transmission characteristics at the telecommunications wavelength of 1550 nm. High thermo-optic and thermal expansion coefficients of silicon give rise to a thermal phase shift of 6.3 rad/K in a 4 mm-long, 6.9 µm diameter fiber acting as a Fabry-Perot resonator. Using both power and wavelength modulation, we observe all-optical bistability at a low threshold power of 15 mW, featuring intensity transitions of 1.4 dB occurring over 1 MW/cm2). Furthermore, if using a cw optical input, the thermo-optic effect would dominate at higher intensities as well, since the additional energy absorbed by TPA would be eventually converted to heat in the indirectband semiconductor core [18]. As we tune the wavelength to approach a resonance, more power builds up in the cavity and is dissipated into heat. The rise in core temperature T increases both Lo and no, thus redshifting the resonance peaks. One can account for this resonance offset effect by adding a thermal phase shift φT to the total phase φtot [Eq. (4)].
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(C) 2010 OSA
Received 11 Jan 2010; revised 18 Feb 2010; accepted 18 Feb 2010; published 26 Feb 2010
1 March 2010 / Vol. 18, No. 5 / OPTICS EXPRESS 5309
1
1 (4) + β Ploss λ λo We take the temperature rise (and thus the thermal phase shift) to be directly proportional to the power lost in the cavity Ploss through the thermal response coefficient β [Eq. (5)]. Using a similar approach as Vienne et al. have with a silica knot resonator [19], we set β to be the product of the heating efficiency of the core ε = ∂T/∂Ploss, multiplied by the phase shift per unit temperature ∂φtot/∂T, which in turn is determined by the material properties and length of the core [Eq. (5)].
ϕtot = [ϕ0 + φλ + ϕT ] = ϕ0 + 2π ( 2 Lo no )
−
∂ϕtot ∂ϕ 2π 2 Lo 1 ∂L ∂n ∂T = × tot = ε × n + (5) ∂Ploss ∂Ploss ∂T λo o L ∂T ∂T Substituting the values for our cavity Lo = 4.0 mm, λo = 1550.000 nm, as well as the standard values for the thermo-optic and thermal expansion effects for silicon: dn/dT = 1.86x10−4 1/K [20], and (1/L)*dL/dT = 2.6x10−6 1/K [21], we get ∂φtot/∂T = 6.33 rad/K. The heating efficiency ε is left as a free parameter, to be fitted to the experimental data. Note the coupled nature of the governing equations, where the transmission Pt depends nonlinearly on φtot [Eq. (1)] and conversely, φtot is linearly proportional to Ploss through Pt and Pref [Eqs. (4), (2a), (2b)]. These equations can be solved through numerical iteration and eventual convergence on a single solution. However, we must be careful not to overlook the remaining solution(s), if they exist. An alternate, more transparent way to solve these equations is by implementing a graphical method, as described by Vaughan [16]. We have pursued the graphical approach, as described in the following section.
β=
5. Bistability: experiments and graphical simulations
5.1 The graphical solution method The graphical solution method is used to solve for the power transmitted Pt as a function of either wavelength λ [Fig. 2(a)–2(c)], or incident power Pin [Fig. 4(a)]. By plotting two independent expressions for the normalized transmitted power Pt/Pin as a function of the total phase shift φl + φT, one finds intersections which indicate possible stable solutions. The first expression is the periodic Fabry-Perot transmission [Eq. (1)]. The second expression is a linear equation for Pt /Pin written explicitly as a function of the thermal phase shift φT [Eq. (6)].
Pt Pt 1 1 = (6) ( β Ploss ) = γ ϕT Pin Ploss β Pin Pin β Equation (6) emphasizes that the slope of the line is inversely proportional to both the incident power Pin, and the thermal response coefficient β. The fraction Pt/Ploss is a constant ratio γ, depending only on the endface reflectivity and material loss parameters [Eq. (7)]. For our silicon-core cavity, substituting R = 0.22 and σ = 0.30 into Eq. (7) yields γ = 0.32. 2 Pt P P (1 − R ) σ = t in = (7) 2 2 2 Ploss Pin Ploss (1 − Rσ ) − (1 − R ) σ − R (1 − σ ) Thus, by plotting the transmission curve for our cavity at a specific wavelength [Eq. (1)], and the linear relation at a specific power [Eq. (6)], we can predict all the stable transmission states that exist at that unique input condition. In the next two subsections, we use the graphical solution method to fit data for both power and wavelength modulation experiments, clearly exhibiting optical bistability. In Section 6, we use our model to predict higher-order multistability (up to the sixth order), and then demonstrate tristability experimentally.
γ =
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Received 11 Jan 2010; revised 18 Feb 2010; accepted 18 Feb 2010; published 26 Feb 2010
1 March 2010 / Vol. 18, No. 5 / OPTICS EXPRESS 5310
5.2 Predicting bistability: power modulation First, we investigated the case where Pin is modulated with the amplifier, while the laser wavelength is fixed at λ = λo = 1550 nm, meaning φλ = 0. Figure 4 shows the incident power scan from 0 mW to 60 mW and back to 0 mW, with delays at each data point to allow the transmitted power to stabilize. After collecting transmission data, the graphical model was fitted, with the heating efficiency ε and the initial phase φo as the free parameters [Fig. 4(b)]. The best fit of the heating efficiency in Eq. (5) was ε = 0.21 K/mW, meaning the core temperature increased by 0.21 K for every milliwatt of power lost in propagation. Also, an initial cavity phase φo = 0.6(2π) was present. Since φo is highly sensitive to the ambient temperature, it was fitted for every individual experiment.
Fig. 4. Transmission through silicon fiber as a function of laser power. (a) Power scan 0 mW to 60 mW, with inset of hysteresis loop around Pin = 42 mW; (b) Using graphical method to model power scan. Inset shows hysteresis loop in four steps: (1) increasing Pin in “low” state; (2) sharp transition to “high” state; (3) decreasing Pin in “high” state; (4) sharp transition back to “low” state.
In Fig. 4 we can see how the graphical solution predicts the transmission through the silicon fiber cavity for any given input power. To generate the predicted curves in Fig. 4(a), we track the intersection of the dotted line [Eq. (6)] with the solid curve [Eq. (1)]. As the power increases beyond 15 mW, the slope of Eq. (6) becomes small enough so that more than one intersection exists. Two intersections lying on a single line [Eq. (6)] signify two stable states, indicated by the red dots in Fig. 4(b), with the third intersection between the “high” and “low” states indicating an unstable state [16]. The agreement between experiment and theory is excellent. 5.3 Predicting bistability: wavelength modulation The procedure for modeling transmission as a function of laser wavelength is similar to that for a power scan. Here we hold the incident power fixed, while scanning the wavelength of the laser from 1549.8 nm to 1550.1 nm and back as described in Section 3. Figure 5(a) shows a wavelength scan at Pin = 19.5 mW, well in the bistable regime. Since the incident power is fixed, the slope of the linear transmission expression [Eq. (6)] in Fig. 5(b) is constant, but the wavelength phase shift φλ is no longer zero. The discontinuous transitions between the high and low states, that define the hysteresis loops as a function of wavelength, are in excellent agreement with the model.
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Received 11 Jan 2010; revised 18 Feb 2010; accepted 18 Feb 2010; published 26 Feb 2010
1 March 2010 / Vol. 18, No. 5 / OPTICS EXPRESS 5311
Fig. 5. Transmission through silicon fiber as a function of laser wavelength. (a) Wavelength scan 1549.8 nm to 1550.1 nm, with hysteresis loop centered at 1549.967 nm; (b) Using graphical method to model wavelength scan. Inset shows hysteresis loop in four steps: (1) increasing λ in “high” state; (2) sharp transition to “low” state; (3) decreasing λ in “low” state; (4) sharp transition back to “high” state.
6. Tristability and higher-order multistability
If we extend the graphical technique outlined in Section 5 to higher input powers, we immediately notice that the model predicts an increasing number of stable states for each Pin. Threshold powers to achieve multistability up to the sixth order are calculated for our siliconcore fiber [Fig. 6(a)].
Fig. 6. (a) Predicting Multistability: Using the graphical solution, we can predict the threshold power required to achieve nth-order multistability. The highest multistability shown here is of the sixth order, meaning six stable states exist for one incident power. (b) Power scan up to Pin = 80 mW, clearly demonstrating regions of tristability. Green lines show theoretical predictions for the middle transmission states.
Extending the power modulation experiment to 80 mW, we indeed observe regions of tristability as Pin increases beyond the predicted threshold of 47.6 mW. Careful scanning of the power through the hysteresis loops demonstrates that three stable states exist at the same Pin in the tristable regions [Fig. 6(b)]. In principle, there is no theoretical limitation to the multistability order. Practical considerations include the optical damage threshold of the cavity, as well as the power stability of the laser and amplifier system.
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Received 11 Jan 2010; revised 18 Feb 2010; accepted 18 Feb 2010; published 26 Feb 2010
1 March 2010 / Vol. 18, No. 5 / OPTICS EXPRESS 5312
7. Conclusions
We have described a novel method to deposit semiconductors and metals in silica templates, as the initial step toward making active in-fiber optoelectronic devices. Here, we have constructed and characterized a 4 mm long, single-core waveguide filled with polycrystalline silicon. Owing to the high reflectivity of the silicon-to-air facets and low material losses of 12.9 dB/cm, the silicon fiber makes an effective Fabry-Perot resonator. Heating of the silicon core by the incident laser beam causes an increase in the effective index mainly by the thermo-optic effect, thus red-shifting the Fabry-Perot resonances and leading to all-optical bistability at input powers as low as 15mW. We model multistable behavior and experimentally confirm multistability up to the third order. Multistability in an optical fiber geometry can have potential applications in multistate optical memory. Acknowledgments
We would like to acknowledge fruitful discussions with Mahesh Krishnamurthi, Rongrui He, Justin Sparks, Pier Sazio, and Anna Peacock. We gratefully acknowledge funding from the National Science Foundation through the grant numbers DMR-0820404 and DMR-0806860.
#122453 - $15.00 USD
(C) 2010 OSA
Received 11 Jan 2010; revised 18 Feb 2010; accepted 18 Feb 2010; published 26 Feb 2010
1 March 2010 / Vol. 18, No. 5 / OPTICS EXPRESS 5313
