## Abstract

We report a large nonlinear response in a 1.3*mm* long GaInP photonic crystal waveguide. The wide band gap of GaInP (1.9 *eV*) ensures that no two photon absorption takes place for photons at 1.55*μm* improving the nonlinear performance. The nonlinearity is enhanced by a resonance effect due to the waveguide end facet reflectivities as well as by the low group velocity exhibited by the waveguide. A low CW input pump power of ≃2*mW* causes a very large change in the nonlinear refractive index coefficient which manifests itself in a large, ≃*π*/3 phase shift in the Fabry Perot fringes. The extracted effective nonlinear coefficient *γ* varies from 3.4 × 10^{5}*W*^{-1}*m*^{-1} at short wavelengths to 2.2 × 10^{6}*W*^{-1}*m*^{-1} near the band edge. These values are several orders of magnitude larger than those obtained in reported nonlinear experiments which exploit the Kerr effect. We postulate therefore that the observed nonlinearity is due to a hybrid phenomenon which combines the Kerr effect and an index change which is induced by local heating that results from the residual linear absorption. The efficient nonlinear phase shift was also exploited in a fast dynamic experiment where we demonstrated wavelength conversion with 100*ps* wide pulses proving the potential for switching functionalities at multi GHz rates. The index change required for this switching experiment can not be obtained, at the power levels used here, with a *γ* value of a few thousands *W*^{-1}*m*^{-1} which is a typical Kerr coefficient in similar waveguides. Hence, we conclude that the hybrid nonlinearity is sufficiently fast to enable switching with a time scale of at least 100*ps*.

© 2010 Optical Society of America

## 1. Introduction

Photonic crystal (PhC) waveguides based on defect lines in two dimensional periodic hole structures etched into a thin membrane are capable of tightly confining propagating modes [1]. PhC waveguides have small cross sections, typically 0.2*μm*^{2} so that the intensities of propagating fields are high and nonlinearities are vastly enhanced. Further enhancement of nonlinearities results from low group velocities which commonly characterize PhC waveguides [2]. Such PhC devices hold therefore the promise of compact nonlinear functional photonic devices [3, 4, 5] such as ultrafast and ultra small all optical switches and logic devices.

One of the best ways to implement all optical ultrafast switching is to make use of instantaneous nonlinear effects such as refractive index changes in response to an optical pump field [6]. Unfortunately nonlinear index changes are inextricably bound to nonlinear losses. In the semiconductor materials most commonly used for PhC devices operating in the 1.55*μ*m telecommunication regime, Silicon and GaAs, the nonlinear losses stem primarily from two photon absorption (TPA) [7, 5]. Materials having band gaps which are sufficiently wide to avoid TPA of photons at 1.55*μ*m (whose energy is 0.8*eV*) are naturally advantageous for nonlinear PhC devices. One such material is GaInP whose bandgap is 1.9*eV*. PhC waveguides based on GaInP have been recently demonstrated with transmission properties that exhibit nonlinear phase shifts of more than *π* radians with no apparent saturation for pulse peak powers as large as 2.5*W* [8].

This paper reports on the static and dynamic nonlinear response of a GaInP PhC waveguide. The observed nonlinearity is resonantly enhanced due to the reflectivity (≈ 40%) of the cleaved end facets. The Fabry Perot resonances lengthen the effective interaction length thereby enhancing the nonlinear response [9]. Furthermore, the structure enables a highly sensitive measurement of the nonlinear index of refraction by a direct observation of the phase shift experienced by the Fabry Perot fringes. The phase shift was observed here using a static pump probe scheme which exhibits an extremely efficient response where *π*/3 phase shift was obtained in a 1.3*mm* long waveguide with a pump power of ≃2*mW*. The Fabry Perot fringes characterizing the linear transmission spectrum yield directly the group index *n _{g}* and hence the group velocity. The group velocity decreases with increasing wavelength in particular as the band edge is approached [2].

The observed phase shift of the Fabry Perot fringes enables to extract the well known phenomenological nonlinear coefficient *γ*. The extracted values depend quadratically on *n _{g}*, as expected [4, 5]. The

*γ*values are very large at any wavelength across the waveguide transmission band taking on the value of 3.4 × 10

^{5}

*W*

^{-1}

*m*

^{-1}at 1530

*nm*and rising to 2.2 × 10

^{6}

*W*

^{-1}

*m*

^{-1}at 1557

*nm*. The observed nonlinear phase change is significantly more efficient than that reported in experiments which are strictly based on the Kerr effect [4, 6, 8]. We postulate therefore that the large observed nonlinear phase shift is induced by a hybrid phenomenon which combines the Kerr effect with some other, strong nonlinearity that also causes an index increase with power. A likely possibility is a thermal effect due to some local heating resulting from a residual linear absorption. While the nature of this strong nonlinear effect was not fully determined, it allows nevertheless a fast dynamic response where a 100

*ps*wide pulse was wavelength converted. Tuning the wavelength of the converted signal to either a peak or a valley of a single fringe enables the conversion to be in or out of phase with the pump pulse.

## 2. Experimental results

The PhC waveguides we tested were designed for the 1.55*μm* wavelength range. They are airbridge W1 type PhC waveguides comprising GaInP slabs patterned with a triangular lattice of air holes. The waveguide is created by omitting a single row of air holes in the ΓK direction and is 1.3*mm* long. The lattice constant is *a* = 480*nm*, the air holes have a radius of *r* = 0.19*a* and the semiconductor slab is ≃ 170*nm* thick [8]. The device was fabricated using standard processes [10] with the GaInP structure grown by MOCVD, the membrane being etched by a wet process and the holes being defined by E-beam and etched using a dry etch process.

#### 2.1. Static characterization

Linear and nonlinear static characterizations were performed using the static pump probe setup shown in Fig. 1. The pump is a CW tunable laser whose output is amplified and filtered. The pump power is controlled by an in-line variable attenuator. The amplified spontaneous emission (ASE) of an Erbium doped fiber amplifier (EDFA) serves a weak (100*μW*) broad band (1525 – 1565*nm*) probe. Pump and probe are combined by a 50/50 fiber coupler and their polarizations are set to be TE before they are coupled into the PhC waveguide via a lensed fiber. The transmitted light is collected at the waveguide output by a *NA* = 0.86 microscope objective lens and measured by an optical spectrum analyzer (OSA) with a spectral resolution of 0.01*nm*. The fiber to waveguide input coupling efficiency was 10% while the waveguide to lens output collection efficiency was 30%.

### 2.1.1. Linear transmission characterization

Linear transmission spectroscopy of the PhC waveguide was obtained by using the ASE probe with no pump. Figure 2 (a) shows the transmitted spectrum (normalized to the power spectrum of the EDFA) and reveals Fabry-Perot fringes originating from the reflectivity of the waveguide end facets. The spectral distance between fringes defines the group index *n _{g}* =

*λ*

^{2}/2

*L*Δ

*λ*so that the spectral dependence of

*n*across the transmission spectrum is easily extracted. The two inserts in Fig. 2 (a) highlight the dispersion of

_{g}*n*. The measured

_{g}*n*values are marked by circles in Fig. 2 (b) which shows how

_{g}*n*increases with the wavelength in accordance with the classical dispersive band diagram of W1 type PhC waveguides. The value of

_{g}*n*is larger than that of bulk GaInP (

_{g}*n*= 3.37) all across the transmission spectrum. A quadratic fit to

_{o}*n*is represented by a dashed line in Fig. 2 (b). The accuracy of the fit decreases somewhat near the band edge. Close to the band-edge, a slow mode with a group velocity of about

_{g}*V*=

_{g}*c*/

*n*≃

_{g}*c*/13 is obtained. Since nonlinear effects scale with the square of the slowdown factor

*S*(

*S*=

*n*/

_{g}*n*) [2], a large third order nonlinear response is expected for these slow modes. The waveguide linear losses

_{o}*α*were also measured using a tunable single mode laser at specific wavelengths where the experiments were performed (1525

*nm*and 1555

*nm*) and was found to vary between

*α*= 200

*m*

^{-1}to

*α*= 1000

*m*

^{-1}.

### 2.1.2. Nonlinear static characterization

The static nonlinear response of the waveguide was studied by measuring the phase shift of the Fabry Perot fringes induced by a CW pump. Figure 3 (a) displays a typical example of pump power dependent transmission spectra observed at ${\lambda}_{\mathit{probe}}=1530\pm \frac{1}{2}\mathit{nm}$ for a pump wavelength of *λ _{pump}* = 1537

*nm*. The pump powers presented in the legend of Fig. 3 (a), denoted

*P*, are the powers measured at the output facet of the PhC waveguide. The corresponding input powers range from 0 to 2

_{out}*mW*. The pump induces a clear red shift in the phase of the Fabry-Perot fringes, which increases with power. The phase shift

*δφ*is calculated as:

*δφ*= 2

*πϕ*/

*FSR*, where

*ϕ*is the measured shift and

*FSR*is the free spectral range. Figure 3 (b) summarizes the phase shifts as a function of

*P*. The dependence is essentially linear and shows no wiggles or saturation due to the effect of the shifting fringes on the pump. The largest observed phase shift is ≃

_{out}*π*/3 for a power of 600

*μW*. Detailed studies of the induced phase shift revealed two important observations. First, for a fixed pump power and wavelength, the obtained phase shift is the same for all the fringes across the transmission spectrum. Second, a given phase shift, say

*π*/3, is obtained for different pump wavelengths as long as the input pump power is held constant. This is easily understood since both the losses and the group index increase with wavelength so that the nonlinear efficiency remains more or less constant.

We quantify the effective nonlinear coefficient *γ* in a phenomenological manner according to the experimentally observed phase shifts. The nonlinear phase shift *δφ* for one round trip in the cavity [11] induced by the effective nonlinear coefficient *γ*, for a cross phase modulation arrangement within a resonator of length *L* is *γ* = *δφ*/4*P _{circ}*

*L*[12] with

*P*being the power circulating in the cavity which is related to the output power by

_{circ}*P*= 2

_{circ}*F*

*P*/

_{out}*π*[13].

*F*is the finesse defined as

*F*=

*π*√

*R*/(1 −

*R*) with

*R*being the modal power reflectivity which in the present structure is ≃0.4.

*R*≃ 0.4 yields a finesse of about 3, a value which was confirmed in a measurement where the fringes were mapped out at high resolution using a tunable laser.

The obtained value for the mode at *λ _{pump}* = 1537

*nm*, having a group velocity of

*V*≈

_{g}*c*/7, is

*γ*= 3.4 × 10

^{5}

*W*

^{-1}

*m*

^{-1}. The effect of the slowdown factor is demonstrated by calculating 7 values for various pump wavelengths. Figure 4 shows the wavelength dependence of

*γ*, together with a fit (dashed curve) representing the quadratic dependence of the group index on wavelength. The upper axis shows the group index as obtained from Fig. 2 (b). Each point in Fig. 4 was obtained by choosing a pump wavelength which coincides with a fringe peak, measuring the obtained phase shift and calculating

*γ*. Near the band-edge,

*γ*of the slow guided mode (

*V*≈

_{g}*c*/15) reaches the high value of

*γ*= 2.2 × 10

^{6}

*W*

^{-1}

*m*

^{-1}.

#### 2.2. Dynamic behavior

To prove that this large nonlinear phase shift can be exploited for ultrafast active functionalities, we performed a wavelength conversion experiment using 100*ps* wide pulses. The experimental set-up is illustrated in Fig. 5. Here, the pump signal is externally modulated by a Mach Zender modulator driven by a fast pulse generator and the probe is a tunable CW signal. The pulsed pump comprised 100*ps* pulses at a duty cycle of 1 : 16 with a maximum input peak power of 25*mW*. The probe input power was 1 – 2*mW*. The probe signal is filtered at the waveguide output and detected by a preamplified wide band receiver whose output is observed on a fast sampling oscilloscope.

Figure 6 (a) shows, in red, the pump pulse at the input to the waveguide (the pulses are inverted by the receiver). The probe wavelength can be tuned to coincide with either a valley or a peak of a Fabry Perot fringe. In either case, the pump induces a fringe shift, changing the transmission of the probe thereby imprinting on it the modulation of the pump. When the probe coincides with a valley, (Fig. 6 (b), blue curve), the converted signal is in phase with the pump. On the other hand, when the probe is chosen to coincide with a peak of a Fabry Perot fringe, the probe exhibits the complimentary modulation sense as seen in Fig. 6 (b), green curve. The converted signal follows exactly the pump pulse in both cases proving that the nonlinear response is sufficiently fast to enable switching at rates of at least 10 GHz.

## 3. Discussion

The *γ* values observed in the experiments described here are very large compared to corresponding values obtained in experiments which rely solely on the Kerr effect using Si, GaAs, AlGaAs, GaInP and chalcogenide glass nano photonic structures [4, 8, 14, 15, 16, 17]. The hybrid thermal and Kerr nonlinearity which is large to begin with, is enhanced due to several reasons. One is the resonance enhancement [18]; this plays however only a marginal role since the finesse is moderate. The second is the lack of TPA and three photon absorption [5] which we verified experimentally to be totally lacking at the low power levels used here. The third is the well known enhancement due to the slow down factor [2] and the last is related to the waveguide effective cross section. The effective cross section is somewhat ambiguous in membrane PhC structures. It is known from calculations but is nearly impossible to measure accurately. Nevertheless, it is somewhat smaller here than in other reported devices (except in [5, 8] where it is similar).

As for the relative merit and usefulness of the thermal and kerr contributions to the hybrid nonlinearity, we note that the *γ* value due to the Kerr effect was measured in similar waveguides [8] to be one to a few thousands *W*^{-1}*m*^{-1}. It follows that at the power levels used in the dynamic experiment (100*ps* pulses with a peak power of ≈ 25*mW*), such a *γ* value would result in a minuscule phase shift of less than 10^{-2} radians with which the switching experiment would be impossible. The hybrid nonlinear effect must therefore be sufficiently fast to respond on the 100*ps* time scale.

## 4. Conclusions

To conclude, we have described static and dynamic nonlinear properties of an GaInP photonic crystal waveguide. The waveguide exhibits a very large effective third order nonlinearity which is due to a hybrid phenomenon combining local heating and the Kerr effect. The nonlinearity is enhanced resonantly by the reflectivity of the end facets. A low group velocity and the wide band gap of GaInP which prevents losses due to TPA for photons at 1.55*μ*m further enhance the nonlinear response. Static measurements reveal a very large phase shift of ≃*π*/3 in the Fabry Perot fringes for an input power of 2*mW*. The dynamic nonlinear response of the waveguide was demonstrated in a wavelength conversion experiment with 100*ps* wide pump pulses with a ≈ 25*mW* peak power which are imprinted on a CW probe signal. The converted signal can follow the pump pulse or take on its complimentary version depending on wether the probe wavelength coincides with a valley or a peak of a single fringe. The fast response suggests that such nonlinear waveguides hold the promise for compact switching devices operating at multi *GHz* rates.

## Acknowledgment

This research was supported by the project GOSPEL within the seventh framework of the European commission. Dr. Vardit Eckhouse acknowledges the Aly Kaufman fellowship.

## References and links

**1. **Y. Akahane, T. Asano, B. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) **425**, 944–947 (2003)
[CrossRef] [PubMed]

**2. **M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**, 2052–2059 (2002)
[CrossRef]

**3. **C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express **17**, 2944–2953 (2009)
[CrossRef] [PubMed]

**4. **K. Inoue, H. Oda, N. Ikeda, and K. Asakawa, “Enhanced third-order nonlinear effects in slow-light photonic-crystal slab waveguides of line-defect,” Opt. Express **17**, 7206–7216 (2009)
[CrossRef] [PubMed]

**5. **C. Husko, S. Combrié, Q. V. Tran, F. Raineri, C. W. Wong, and A. De Rossi, “Non-trivial scaling of self-phase modulation and three-photon absorption in III-V photonic crystal waveguides,” Opt. Express **17**, 22442–22451 (2009) [CrossRef]

**6. **C. Husko, A. De Rossi, S. Combrié, Q. Vy Tran, F. Raineri, and C. Wei Wong, “Ultrafast all-optical modulation in GaAs photonic crystal cavities,” Appl. Phys. Lett. **94**, 021111-1–021111-3 (2009) [CrossRef]

**7. **P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express **13**, 801–820 (2005) [CrossRef] [PubMed]

**8. **S. Combrié, Q. Vy Tran, C. Husko, P. Colman, and A. De Rossi, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. **95**, 221108-1–211108-3 (2009) [CrossRef]

**9. **E. Inbar and A. Arie, “High-sensitivity measurements of the Kerr constant in gases using a Fabry Perot-based ellipsometer,” Appl. Phys. B **70**, 849–852 (2000)

**10. **S. Combrié, A. De Rossi, Q. N. V. Tran, and H. Benisty, “GaAs photonic crystal cavity with ultrahigh Q: microwatt nonlinearity at 1.55 mm,” Opt. Lett. **33**, 1908–1910 (2008) [CrossRef] [PubMed]

**11. **A. Yariv, *Optical Electronics in Modern Communications* (Oxford University Press, 1997)

**12. **R. W. Boyd, *Nonlinear Optics* (Academic Press, 2008)

**13. **A. E. Siegman, *Lasers* (University Science Books, 1986)

**14. **N. Matsuda, R. Shimizu, Y. Mitsumori, H. Kosaka, A. Sato, H. Yokoyama, K. Yamada, T. Watanabe, T. Tsuchizawa, H. Fukuda, S. Itabashi, and K. Edamatsu, “All-optical phase modulations in a silicon wire waveguide at ultralow light levels,” Appl. Phys. Lett. **95**, 171110-1–171110-3 (2009) [CrossRef]

**15. **P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in GaAs micro-ring resonators,” Opt. Lett. **25**, 554–556 (2000) [CrossRef]

**16. **H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express **13**, 4629–4637 (2005) [CrossRef] [PubMed]

**17. **K. Suzuki, Y. Hamachi, and T. Baba, “Fabrication and characterization of chalcogenide glass photonic crystal waveguides,” Opt. Express **17**, 22393–22400 (2009) [CrossRef]

**18. **A. M. Malvezzi, G. Vecchi, M. Patrini, G. Guizzetti, L. C. Andreani, F. Romanato, L. Businaro, E. Di Fabrizio, A. Passaseo, and M. De Vittorio, “Resonant second-harmonic generation in a GaAs photonic crystal waveguide,” Phys. Rev. B **68**, 161306-1–161306-4 (2003) [CrossRef]