## 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 <0.1 pm change in wavelength. Threshold powers for higher-order multistable states are predicted. Tristability is experimentally confirmed.

©2010 Optical Society of America

## 1. Introduction

As the electronics and the optical communications technologies have developed side by side in recent years, the demand for fast optics-electronics-optics (o-e-o) signal conversion has increased. One approach to overcome the conversion bottleneck has been to integrate optical components on a planar silicon chip [1–3]. An alternative approach that we are pursuing is to integrate electronic devices such as amplifiers, modulators, and detectors seamlessly inside an optical fiber waveguide. The basis for in-fiber electronics is to incorporate active optoelectronic materials into the waveguiding core of the fiber.

We have developed a unique high-pressure chemical fluid deposition (HPCFD) technique to incorporate semiconductors (silicon, germanium, zinc selenide) and metals (Au, Pt, W, Ag) into the holes of microstructured optical fibers [4]. The HPCFD technique can be used to deposit multiple concentric layers of semiconductor or metal in a radial geometry, enabling the construction of active optoelectronic junctions—the backbone of o-e-o conversion. Incorporating active semiconductors with extended infrared transparency wavelength range, large nonlinear optical and thermo-optic coefficients as compared with silica, ability to dope and form optoelectronics devices, and incorporating laser gain materials inside optical fibers can provide us an unprecedented opportunity to realize *all-fiber optoelectronics*, where light generation, modulation and detection can possibly be seamlessly integrated within the backbone of the fiber-based optical communications network.

We have recently deposited high-quality, low-loss silicon optical cores inside capillary silica fibers [5]. Here, we propagate 1550 nm light through these silicon waveguides and demonstrate optical multistability. Optical multistability was first observed by McCall et. al. in 1974, and has since been keenly pursued due to its potential application in all-optical logic [6]. A multistable system can arrive at two or more unique output states for the same input state by using a nonlinear feedback mechanism [7]. *Intrinsic* (all-optical) multistable systems rely on a direct interaction of the guiding medium with light to modulate the transmission, while *hybrid* systems require an external electronic feedback loop [8]. Both intrinsic [3,9,10] and hybrid [11] multistable systems have been demonstrated in silicon interferometers and planar silicon geometry in the past. This work is the first demonstration of intrinsic all-optical multistability in a silicon-core optical fiber geometry. These fibers, being circular in cross-section, are free from polarization sensitivity, which is a potential advantage over waveguides in planar geometry.

## 2. Fabrication and waveguiding setup

Solid silicon waveguide cores were formed within 6 μm diameter capillaries of silica fibers using a high pressure supersonic chemical fluid deposition (HPCFD) technique [12]. A reservoir containing a 5% silane:He gas mixture at 35 MPa total pressure was attached to fiber which was placed inside a furnace. The furnace was heated from 450 to 500°C over 48 hours as the high pressure gas mixture flowed through the capillary producing a 2 cm long completely filled silicon wire. The silicon wire was crystallized by ramping the temperature from 500°C to 725°C over 7 days, followed by a rapid recrystallization anneal to 1325°C for 10 minutes. After annealing, Raman and TEM analysis of the wire showed randomly oriented 0.5-1 μm crystalline silicon grains, and no evidence of remaining amorphous material [13]. The measured FWHM Raman values, collected using Dilor XY Raman with 633nm excitation, were 3.0 cm^{−1} for the polycrystalline fiber core, compared to 2.7 cm^{−1} for a single-crystal silicon wafer [5].

After deposition, the fiber was sectioned to locate regions with complete filling of the core. The fully-filled sections, up to 2 cm in length, were encased in a larger glass tube, and the end faces were polished to optical quality grade (colloidal silica 0.05 μm) [Fig. 1(a) ]. The scanning electron microscope image [Fig. 1(b)] displays the 6.9 μm diameter silicon core, with complete filling, and a smooth interface to the silica cladding. Waveguiding loss due to interfacial surface roughness, which is typically a challenge for lithographically patterned Si waveguides, is not an issue since the RMS surface roughness of the silica capillary is less than 0.1 nm [14].

Optical waveguiding at the telecommunications wavelength of 1550 nm was performed in this silicon-core fiber. A cw tunable laser (HP 81680A) provided 1550 nm light at constant 3.0 mW, capable of tuning the wavelength in 0.1 pm increments. Power was modulated from 1 mW to 60 mW with an erbium-doped amplifier (PriTel). Coupling into the silicon core was achieved with a lensed single-mode fiber having a 6 μm spot size and NA = 0.11 (OZ Optics). A 60x microscope objective (Newport) with NA = 0.85 collected a 116° cone of light exiting the fiber, at a transmission efficiency of 78%. The transmitted light was then focused onto either an IR camera (Hamamatsu C2741), or an IR power meter (Thorlabs PM30). Figure 1(c) shows 1550 nm light emerging from the silicon core, with the lensed fiber position optimized to obtain the highest power throughput.

A finite-element method COMSOL simulation of guiding 1550 nm light in a 6.9 μm diameter crystalline silicon-core silica-cladding fiber was performed. The silicon core was assumed to have a refractive index of bulk crystalline silicon: *n _{o}* = 3.48 [15]. The simulated fundamental HE

_{11}mode [Fig. 1(d)] has an effective index of 3.4759, and features a Gaussian intensity profile. Though the output light profile from our silicon fiber is approximately Gaussian, we note that our fiber is highly multimodal, so the output intensity profile is a superposition of many modes. Next, we show wavelength modulation experiments at various input powers that demonstrate Fabry-Perot resonance, with optical bistability occurring for input powers greater than 15 mW.

## 3. Observation of optical bistability

For the following experiments we used a 4 mm long, 6.9 µm diameter silicon fiber that showed complete filling of the silicon core on both polished endfaces. With coupling optimized for maximum throughput at a set input power, we scanned the tunable laser from 1549.8 nm to 1550.1 nm and back in 2 pm wavelength steps [Fig. 2(a) –2(c)]. After a 2 second pause for each step, the transmitted power was recorded.

We note three interesting facts:

We proceed to show that effects (2) and (3) arise from thermo-optic tuning of the cavity by the laser.

## 4. Theory of thermo-optic bistability

#### 4.1 Classical Fabry-Perot resonator

We model our cavity as a classical Fabry-Perot resonator, with the polished facets acting as identical mirrors at each end (Fig. 3 ).

The fraction of power transmitted [Eq. (1)] and reflected [Eq. (2a)] depend explicitly on the facet reflectivity *R*, the single-pass transmission term $\sigma ={e}^{-\alpha *{L}_{o}}$(given material loss *a* and cavity length *L _{o}*), and the phase of the cavity

*φ*[16]. By conservation of energy, the power that is neither transmitted nor reflected, must be either absorbed by the material and converted to heat, or scattered away as light [Eq. (2b)].

_{tot}Maximum transmission (resonance) occurs when the total cavity phase *φ _{tot}* [Eq. (3)], is an integer multiple of 2π, meaning an integer number of wavelengths fit in a round trip of the cavity

*2L*. As the wavelength

_{o}*λ*is scanned, additional phase

*φ*is added or subtracted. We explicitly define $0<{\phi}_{0}<2\pi $ as the initial cavity phase at the reference wavelength

_{λ}*λ*= 1550.000 nm.

_{o}For our silicon-filled fiber, we measured the length to be *L _{o}* = 4.0 mm and used the standard index for crystalline silicon of

*n*= 3.48 [15]. By fitting Eq. (1) to the transmission curves at

_{o}*P*= 2.0 mW [Fig. 2(a)], we found the best-fit values

_{in}*R*= 0.22 and

*σ*= 0.30, corresponding to a material loss of

*a*= 2.97 1/cm, or 12.9 dB/cm. The fitted

*R*value is significantly lower than the ideal Fresnel reflectivity of

*R*= 0.31 due to our waveguide being highly multimodal [17]. The fitted experimental loss of 12.9 dB/cm at 1550 nm is much higher than the simulated HE

_{11}material loss of 0.04 dB/cm for a crystalline silicon core. We attribute the difference to the polycrystalline nature of the fabricated silicon core, with grain sizes between 0.5 μm and 1 μm contributing to light scattering [13].

#### 4.2 The thermal phase shift

However, the asymmetry in transmission at higher powers [Fig. 2(b), 2(c)] cannot be explained by the classical Fabry-Perot theory. To account for the asymmetry, we must realize that fluctuations in the temperature of the silicon core will directly change the core index *n _{o}* by the thermo-optic effect, as well as change the length

*L*by the thermal expansion effect. Index change by the presence of free carriers generated by two-photon absorption (TPA) is not a dominant effect here, since the maximum in-fiber intensity of 200 kW/cm

_{o}^{2}is much less than typical intensities at which TPA becomes significant (>1 MW/cm

^{2}). 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 indirect-band 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 *L _{o}* and

*n*, thus red-shifting the resonance peaks. One can account for this resonance offset effect by adding a thermal phase shift

_{o}*φ*to the total phase

_{T}*φ*[Eq. (4)].

_{tot}We take the temperature rise (and thus the thermal phase shift) to be directly proportional to the power lost in the cavity *P _{loss}* 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/∂P*, multiplied by the phase shift per unit temperature

_{loss}*∂φ*, which in turn is determined by the material properties and length of the core [Eq. (5)].

_{tot}/∂TSubstituting the values for our cavity *L _{o}* = 4.0 mm,

*λ*= 1550.000 nm, as well as the standard values for the thermo-optic and thermal expansion effects for silicon:

_{o}*dn/dT*= 1.86x10

^{−4}1/K [20], and

*(1/L)*dL/dT*= 2.6x10

^{−6}1/K [21], we get

*∂φ*= 6.33 rad/K. The heating efficiency

_{tot}/∂T*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 *P _{t}* depends

*nonlinearly*on

*φ*[Eq. (1)] and conversely,

_{tot}*φ*is

_{tot}*linearly*proportional to

*P*through

_{loss}*P*and

_{t}*P*[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.

_{ref}## 5. Bistability: experiments and graphical simulations

#### 5.1 The graphical solution method

The graphical solution method is used to solve for the power transmitted *P _{t}* as a function of either wavelength

*λ*[Fig. 2(a)–2(c)], or incident power

*P*[Fig. 4(a) ]. By plotting two independent expressions for the normalized transmitted power

_{in}*P*/

_{t}*P*as a function of the total phase shift

_{in}*φ*+

_{l}*φ*, 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

_{T}*P*/

_{t}*P*written explicitly as a function of the thermal phase shift

_{in}*φ*[Eq. (6)].

_{T}Equation (6) emphasizes that the slope of the line is inversely proportional to both the incident power *P _{in}*, and the thermal response coefficient

*β*. The fraction

*P*/

_{t}*P*is a constant ratio γ, depending only on the endface reflectivity and material loss parameters [Eq. (7)]. For our silicon-core cavity, substituting

_{loss}*R*= 0.22 and

*σ*= 0.30 into Eq. (7) yields

*γ*= 0.32.

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.

#### 5.2 Predicting bistability: power modulation

First, we investigated the case where *P _{in}* is modulated with the amplifier, while the laser wavelength is fixed at

*λ = λ*= 1550 nm, meaning

_{o}*φ*= 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

_{λ}*ε*and the initial phase

*φ*as the free parameters [Fig. 4(b)]. The best fit of the heating efficiency in Eq. (5) was

_{o}*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

*φ*= 0.6(2π) was present. Since

_{o}*φ*is highly sensitive to the ambient temperature, it was fitted for every individual experiment.

_{o}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 *P _{in}* = 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 f

_{λ}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.

## 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 *P _{in}*. Threshold powers to achieve multistability up to the sixth order are calculated for our silicon-core fiber [Fig. 6(a)
].

Extending the power modulation experiment to 80 mW, we indeed observe regions of *tristability* as *P _{in}* 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

*P*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.

_{in}## 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.

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