Nonlinear-refractive-index measurement in As2S3 ... - OSA Publishing

Laniel et al.

Vol. 22, No. 2 / February 2005 / J. Opt. Soc. Am. B

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Nonlinear-refractive-index measurement in As2 S3 channel waveguides by asymmetric self-phase modulation Jacques M. Laniel, Nicolas Hoˆ, and Reál Valleé Centre d’Optique Photonique et Laser, Pavillon Vachon, Universite´ Laval, Cite´ Universitaire, Que´bec, Canada G1K 7P4

Alain Villeneuve ITF Optical Technologies, 400 Montpellier, Ville St-Laurent, Ville St-Laurent, Que´bec, Canada H4N 2G7 Received March 17, 2004; revised manuscript received September 7, 2004; accepted September 21, 2004 The nonlinear refractive index (n 2 ) of an As2 S3 channel waveguide at 1.55 !m has been measured. The n 2 was deduced when the spectral broadening and modulation due to self-phase modulation (SPM) were characterized. Owing to the temporal asymmetry of the injected pulse, a specific approach to asymmetric SPM is presented. The measurements have been performed for both photodarkened and nonphotodarkened etched thin films so that the effect of photodarkening on the n 2 could be assessed. © 2005 Optical Society of America OCIS codes: 130.3130, 190.3270, 190.4390.

1. INTRODUCTION The development of ultrafast all-optical switches for a high-bit-rate communication system requires materials with optimal optical properties. These materials must possess strong nonlinearity and low propagation losses (linear and nonlinear). Furthermore, it is required that the materials can be deposited in thin films in order to build photonic integrated circuits. The As2 S3 chalcogenide glass is a good candidate for such applications and can be easily deposited in thin films and etched to create optical waveguides.1 This glass has high linear and nonlinear refractive indices2,3 and a good transmission in the 1–10-!m-wavelength optical window. This amorphous semiconductor is also a photosensitive material that allows the writing of holographic patterns, such as Bragg gratings by a photodarkening process.4,5 The precise measurement of the nonlinear refractive index (n 2 ) is crucial to the proper use of such material. Several measurements of the n 2 were reported for the near-infrared region.3,6–12 These measurements were performed primarily in bulk samples. Because the development of integrated components relies on deposited thin films, which are likely to present altered optical properties with respect to the corresponding bulk material,13 it is of great interest to better characterize the n 2 of As2 S3 thin films. In this paper, we present the results of the experimental measurements of the n 2 of As2 S3 channel waveguides. Furthermore, the measurements were performed on both photodarkened and nonphotodarkened waveguides in order to relate the effects of photodarkening on the n 2 . To our knowledge, this is the first report of the direct n 2 measurements of photodarkened As2 S3 thin films. Our measurements were analyzed when the self-phase 0740-3224/2005/020437-09$15.00

modulation14 (SPM) acquired by the pulses propagating in the channel waveguide was characterized as a function of the injected intensity. It is important to outline the fact that owing to the asymmetrical temporal shape of the input pulses, a new approach has been developed to analyze the modulated spectrum. Section 2 describes the third-order nonlinearity present in As2 S3 and the limiting properties of the photosensitivity at 1.55 !m present in this glass. Section 3 provides a description of the experimental procedure used to measure the n 2 . A complete description of the analysis of an asymmetrical SPM measurement is given in Section 4. The experimental results are summarized in Section 5. These results are discussed in Section 6, and Section 7 concludes the present work.

2. DESCRIPTION OF THE KERR EFFECT IN As2 S3 A. Origin of the Third-Order Nonlinearity The main mechanism responsible for the Kerr optical effect in As2 S3 chalcogenide glass is the two-photon absorption occurring between the valence and conduction band.15 Such a process is resonant when the exciting optical signal energy ("#) is close to half of the bandgap energy (E g /2). 16,17 The bandgap energy of As2 S3 is 2.4 eV. The wavelength associated with the energy at halfbandgap is 1.034 !m. According to this value, the telecommunication wavelengths around 1.55 !m can be considered to be nonresonant. Two models are commonly used to evaluate the magnitude and the dispersion of the nonlinear refractive index of dielectric materials with a bandgap. The first one was derived by Sheik-Bahae et al.17 and is based on the semi© 2005 Optical Society of America

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J. Opt. Soc. Am. B / Vol. 22, No. 2 / February 2005

classical calculation of the two-photon transition between the valence and the conduction bands inside a crystalline semiconductor. By using the nonlinear Kramers–Kronig relations, Sheik-Bahae et al. obtained the dispersion and magnitude of the nonlinear refractive index (n 2 ). The second method is based on a classical calculation involving the electron oscillator model of the atom.18 The nonlinearity is included in the model when the binding force between the electron and the atom are considered as anharmonic.19 In the case where the nonlinearity is nonresonant, this model gives Miller’s rule, which states that the third-order susceptibility tensor (related to the n 2 ) is proportional to the first-order susceptibility tensor (related to the linear refractive index n o ) to the fourth power. These two models can be used as a first approximation to get an evaluation and the dispersion of the n 2 of As2 S3 amorphous semiconductor. It is possible to use these two models to obtain a theoretical evaluation of the n 2 of As2 S3 at 1.55 !m. Such evaluations require two parameters related to the optical properties of the materials: the energy of the bandgap (E g ) and the linear refractive index (n o ) of the material at 1.55 !m. Furthermore, the evaluation of the n 2 for the photodarkened material requires a quantitative evaluation of the modification on both E g and n o . The photodarkening process occurs when the material is exposed to photons with energy close to the bandgap energy.20 In such a case, the photons excite defects inside the material, resulting in the redshift of the bandgap energy. That shift causes an increase of the refractive index in the longer wavelength according to the linear Kramers–Kronig relations. The bandgap energy of As2 S3 is 2.4 eV and the refractive index2 of an As2 S3 annealed thin film is 2.405 at 1.55 !m. When the material’s sensitivity is completely saturated, the bandgap shift20 is !0.045 eV, and the refractive index21 at 1.55 !m is increased by an amount of 0.057. With these parameters, the n 2 of As2 S3 at 1.55 !m obtained with the Sheik-Bahae formula and Miller’s rule are, respectively, 3.12 " 10!14 cm2 /W and 2.41 " 10!14 cm2 /W. The n 2 obtained for the photodarkened material with Sheik-Bahae formula and Miller’s rule are, respectively, 3.26 " 10!14 cm2 /W (5% increase) and 2.88 " 10!14 cm2 /W (20% increase). These evaluations are close to the values reported in the literature.3,6–12 Both models predict that the photodarkening process increases the nonlinear refractive index. However, the quantitative evaluations of the n 2 increase are quite different for both models. B. Photosensitivity at 1.55 !m The nonlinear-refractive-index measurement of a material requires two important experimental conditions: a high intensity in order to get a clear nonlinear effect and low propagation losses that would restrict the effective accumulation of nonlinear phase. The recent observation of photosensitivity22–24 in As2 S3 thin films at 0.80 and 1.55 !m introduces some difficulties to the n 2 measurements. The presence of photosensitivity limits the maximum propagated intensities inside the materials. Hoˆ et al.23 established an intensity threshold for the photosensitive mechanism between 0.8 and 1.3 GW/cm2.

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Above this threshold, microstructures appear inside the thin films and strongly perturb the propagation of light. For precise measurement of the n 2 of As2 S3 channel waveguides, the experiments presented here were all performed below the threshold of the nonlinear photosensitive process. It is important to mention that the photosensitivity of As2 S3 at 1.55 !m is not within the scope of the present paper, we mention the phenomena only to clarify the limits of the SPM experiment presented here.

3. EXPERIMENTAL PROCEDURE A. Waveguide Structure We used a ridge waveguide in order to confine the pulse in both transverse directions. Accordingly, the As2 S3 thin films were first prepared by thermal evaporation of bulk glass samples (from Amorphous Materials Inc., Garland, Texas). The deposition was made at a rate of approximately 20–30 Å/s under a pressure of 10!7 Torr onto an oxidized silicon wafer. The substrate was rotated for better homogeneity and was unheated. The thickness deposited was 0.6 !m. The samples were annealed in vacuum at 120 °C for two hours. To create the ridges, we processed the films by a standard photolithographic process in a clean room environment. The etch depth was 0.15 !m. Two different channel widths were used in the SPM measurements: 4.5 and 5.0 !m. At a wavelength of 1.55 !m, these channel waveguides are multimode. However, the second mode is so close to the cutoff that it is widely spread spatially. Since the excitation of that mode is very weak owing to the waveguide and input beam geometries, the electric field inside the waveguides can be considered to be single mode.

B. Preparation of the Photodarkened Samples An argon ion laser operating at 0.514 !m was used to saturate the photodarkening in the channel waveguides. The experimental setup used to expose the thin films is shown in Fig. 1. The laser beam was spatially filtered and was expanded by the use of a telescope with a pinhole at the common focal point. The intensity impinging on the As2 S3 waveguides is 28 mW/cm2, and the exposition lasted 1 h (total fluence of $100 J/cm2). Such exposition is sufficient to saturate the photodarkening process and obtain a maximum refractive index change.4

Fig. 1. Experimental setup used for photodarkening of the As2 S3 thin films. The intensity impinging on the samples is 28 mW/cm2, and the expositions last 1 h ($100 J/cm2).

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i

&' &z

!

#

Fig. 2. Experimental setup used for measurement of the SPM on pulses propagated inside the As2 S3 channel waveguides. The input beam is initially polarized (POL1). The injected intensity is controlled by a wave plate (%/2) and a polarizer cube (POL2). The input and output power are measured by InGaAs photodetectors with beam splitters (BS1–3). The signal is coupled and decoupled by microscope objectives (OBJ1–2).

C. Experimental Setup The value of the nonlinear refractive index was deduced from a careful analysis of the SPM signature on a pulse propagating inside the Kerr material. The experimental setup used for measuring the n 2 of As2 S3 channel waveguides is shown in Fig. 2. The laser source is an optical parametric oscillator (OPO) pumped by a commercial Ti:sapphire (Model Tsunami from Spectra Physics, Mountain View, California). The OPO is built with a crystal of periodically poled lithium niobate and generates pulses of 800 fs at a wavelength of $1.55 !m with a repetition rate of 81 MHz. The OPO signal was initially polarized. The power injected in the waveguide was controlled by a halfwave plate (%/2) and a polarizing cube. The polarization of the input beam was set to excite only the TE-polarized optical mode inside the waveguide. The laser beam was end-fire coupled with a 40" microscope objective inside the channel waveguide, and the outgoing light was collimated by a 20" microscope objective. The input and output powers were monitored by two InGaAs photodetectors. The outgoing optical mode was observed with an IR camera and characterized by an optical spectrum analyzer. The spectrum and the power measurements were computer controlled. The waveguides were initially cleaved in order to get good input and output facets. The quality of the cleaving is crucial to get good injection and reproducibility between measurements. The typical coupling efficiency achieved is approximately 8%–10%. This evaluation assumes Fresnel losses of 15% at the input facet and a transmission of 70% through the input microscope objective. This experimental efficiency includes the transmission through the input coupling microscope objective, the Fresnel reflection at the input facet, and the overlap integral between the impinging beam and the fundamental optical mode of the guiding structure.

4. NUMERICAL ANALYSIS The propagation of short electromagnetic pulses inside a channel waveguide with Kerr nonlinearity is governed by the nonlinear Schro¨dinger equation (NLS)14:

( 2 & 2' 2 &) 2

!

k on 2 A eff

!

i 6

(3

" '"2 #

439

& 3' &) 3

i*o 2

#

i*2 2A eff

#

" ' " 2 ' $ 0,

(1)

where ' (z, ) ) is the electric field envelope representing the pulse, z is the propagation dimension, ) is the time inside the pulse frame of reference, ( j are the dispersion terms, k o is the wave vector, n 2 is the nonlinear refractive index, A eff is the effective area of the propagating mode, * o is the propagation losses, and * 2 is the two-photon absorption coefficient. The resolution of Eq. (1) is done by using a beam propagation method14 (BPM). Once the input pulse has been propagated, the spectrum is obtained through computation of the Fourier transform of the output solutions. To analyze the SPM experiment, one must have a precise description of the input pulse shape. In the case of pulses with duration less than $10 ps, the temporal profile cannot be measured directly by a photodetector. Alternative methods of measurement need to be used, such as optical autocorrelation.25 The problem with an autocorrelator is that it does not directly give the shape of the pulses. Furthermore, the signal it gives is always symmetrical with respect to time. This becomes of great importance in the present experiment, since the short pulses generated by OPOs are generally asymmetrical. There are few papers26,27 that analytically treat the temporal dynamics of OPO pumped in a synchronous regime by a mode-locked oscillator. This lack of modeling comes from the complexity of the parametric gain relation. Cheung et al.26,27 developed a theoretical approach based on a perturbation derivation. Their model has given many insights on the type of solution describing the output signal of an OPO. The model shows that the cavity can never be synchronized perfectly. This is caused by the fact that the parametric interaction introduces an intensity-dependent phase in the equation describing the signal. This temporal dependence has the effect of time shifting the signal with respect to the pump, causing the signal to see a different gain each time. Such a shift favors one wing of the signal pulse over the other, rendering the signal temporal shape asymmetric. It is important to point out that the signal generated by an OPO is asymmetrical only when the pump pulse durations are less than $10 ps. Since the pulses coming from the Ti:sapphire oscillator shown in Fig. 2 have a duration of 1 ps, the pulses coming from the OPO are asymmetrical, according to the Cheung et al. model. These conclusions on the temporal shape of the signal generated by an OPO are qualitative in nature and therefore do not give an analytical description required for the analysis of the SPM experiments. Furthermore, since the information obtained with an autocorrelator is too indirect, the only reliable experimental fact available on the pulses generated by an OPO is their spectrum. However, the experimental pulse-power spectrum does not provide ˜ ( f ) is any information relative to its phase, i.e., when A defined as the Fourier transform of the pulse centered at the angular frequency # o , the power spectrum is

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˜ + f , $ "A ˜ + f ," 2. G

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(2)

The phase on the spectrum can be included through expression of it with an expansion series in the following way: ˜+f, $ A

$ !- #% .

!G˜ + f , exp

i

j$2

/ jf j

,

(3)

where / j represents the expansion coefficients of the phase profile. The expansion terms of order j $ 0 and 1 have been omitted, since they do not affect the pulse profile. Equation (3) describes completely the pulse. Once the expansion coefficients are known, one can obtain the pulse temporal envelope by taking the inverse Fourier transform. ˜ ( f ) 1 of the pulses produced by The power spectrum 0 G the OPO described in Fig. 2 was obtained experimentally. The spectrum was compared with analytical expression and was found to be close to a hyperbolic secant function in agreement with the fact that the spectrum of the Ti:sapphire oscillator also possesses a hyperbolic secant shape. Equation (3) becomes ˜+f, $ A

!P˜ o sech

! # $ !- #% f

2f o

.

exp i

j$2

/ jf j

,

(4)

˜ is the spectrum peak power and 2f is the specwhere P o o tral width. Further simplification can be made in order to restrict the number of coefficient / j required for the expansion. First, since the spectrum has a finite width, only the first few expansion coefficients are needed. This comes from the relation between the pulse duration and the spectral width within the Fourier theory. However, it is important to keep enough coefficients, because the effect of SPM is to broaden the propagated spectrum. Second, we restricted ourselves to the odd coefficients in order to take the pulses asymmetry into account. The even coefficients are not required to describe the pulses used in this experiment. Their effects in the time domain modify the pulse duration only to the first order. The expression for the experimental spectrum of the OPO pulses can then be simplified to ˜+f, $ A

!P˜ o sech

! # & $f

fo

N

exp i

j$1

/ + 2j#1 , f + 2j#1 ,

%'

poral shape, given by Eq. (5), can be propagated through resolution of the NLS [Eq. (1)] with a BPM algorithm. The numerically propagated pulse can be compared with the experimental spectrum. Numerical analysis of the SPM experiment is done by use of a least-squares algorithm in which the unknown physical parameters are adjusted. From all parameters included in Eq. (1) and Eq. (5), only the following are experimentally undetermined: the n 2 and the phase coefficients / k . These variables are the parameters that will be adjusted during the numerical fit. The ansatz de˜ scribed by Eq. (5) gives the spectrum A BPM( f; n 2 , / ) after being propagated by the NLS. The difference between this analytical spectrum and the experimental ˜ ( f ) is given by spectrum A exp

3 2+ n 2 , / , $

- " A˜ f

BPM+

˜ + f ," 2. f; n 2 , / , ! A exp

(6)

By adjusting the fitting parameters n 2 and / k , one can minimize 3 2 and therefore obtain the nonlinear refractive index of the channel waveguide. At the same time, the fitting procedure yields the coefficients describing the spectrum phase profile. To perform this analysis one must have two spectra: one at high input intensity and one at low input intensity. The low-input-intensity spectrum is used to establish the spectral width given in Eq. (5) by 2f o .

5. EXPERIMENTAL RESULTS The nonlinear refractive indices of both photodarkened and nonphotodarkened As2 S3 channel waveguides have been measured with the SPM characterization method. Three measurements have been performed for each photodarkened and nonphotodarkened film. In all cases, photoinduced microstructures were created as a result of the photosensitivity at 1.55 !m once the intensity thresh-

, (5)

where only the N first odd phase expansion coefficient are kept (beginning at / 3 ). The presence of an asymmetrical spectral phase renders the pulse profile asymmetrical in the time domain. This asymmetry has some consequences on the modulation caused by SPM. In such a case, the modulation becomes asymmetrical, and the broadening of the spectrum is smaller than in the case of symmetrical pulse. It is important to point out that this asymmetry is not caused by higher-order nonlinear effects, such as Raman scattering or self-steepening,14 but rather is a purely SPM-related effect. Furthermore, the spectral width of the pulses is not sufficient to experience Raman shifting along the short propagation distance (which in this work is $2 cm). This fact was also verified numerically. We finally have a complete analytical description of the pulses generated by the OPO. The tem-

Fig. 3. Spectra of the propagated pulses inside a nonphotodarkened As2 S3 channel waveguide for four different average injected powers: (a) to (d) correspond to 0.02, 0.30, 0.99, and 2.02 mW (2-mW average power corresponds to a peak power of 33.0 W). The dashed lines serve as references in order to clearly show the apparent blueshift and the spectral asymmetry.

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old was reached. Typical SPM traces measured at the output of a 2.03-cm-long channel waveguide having an effective area of 2.11 !m2 (corresponding to a ridge width of 4.5 !m) are shown in Fig. 3 for different injected powers. The pulse duration is 800 fs. The output spectrum that is shown for four different injected average powers is characterized by an apparent blueshift and a pronounced asymmetry at high intensity. It is important to point out that the spectra obtained for an average power greater than 2 mW are not shown because they were obtained while the thin film was undergoing structural changes (microchannels) caused by the photosensitivity at 1.55 !m. Accordingly, the numerical fits are performed on the spectrum associated to the highest injected power right before the beginning of the microchannel writing. It was impossible to characterize the propagated pulses with the autocorrelator, because the outgoing power was not sufficient to get an autocorrelation signal. To fit the experimental data, one must define the physical parameters of the experiment. These parameters are the dispersion coefficients, the propagation losses, the effective area of the channel waveguide, and the waveguide length. The repetition rate of the OPO is 81 MHz. The central wavelengths and the spectral widths are determined for each experiment with the spectral data. Asobe et al.28 measured the dispersion coefficient of As2 S3 and reported a value of ( 2 $ 5.3 " 10!3 ps2 /cm. Alternatively, the ( 2 can be derived from the refractive index measurements made by Petkov and Ewen,29 which can be expressed in terms of a Sellmeier relation: n 2o + % , ! 1 $

A% 2 %2 ! B2

,

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to detect the presence of nonlinear absorption. In all cases, no nonlinear absorption was observed, so it was assumed in the numerical fits that * 2 $ 0.0 cm/GW. Two different widths for the channel waveguide were used: 4.5 and 5.0 !m. The effective area for these two waveguide configurations were computed with the mode profile obtained numerically with a commercial mode solver (3D Mode Solver Optiwave Corporation, Ottawa, Canada). The mode profile of the fundamental mode was computed at a wavelength of 1.55 !m by use of refractive indices of 2.405 for the As2 S3 layer and 1.444 for the silica substrate. The effective indices and effective areas for both waveguides are 2.233 and 2.11 !m2 for the 4.5-!m channel waveguide and 2.234 and 2.38 !m2 for the 5.0-!m channel waveguide. With these data, the experimental SPM spectra were all fitted numerically by use of the procedure described in Section 4. We obtained optimum results by keeping only three odd phase coefficients: / 3 , / 5 , and / 7 . The re-

(7)

where % is the wavelength in micrometers, n o is the refractive index, and the fitting parameters are A $ (3.98 % 0.04) !m!2 and B $ (0.268 % 0.005) !m. With Eq. (7), the ( 2 and ( 3 were computed by use of the following relation:

( j+ # , $

dj d#

$ % n+ # ,#

j

c

,

(8)

where # $ 2 4 /% and c is the speed of light. The dispersion coefficient obtained with Petkov and Ewen refractive indices are ( 2 $ 4.76 " 10!3 ps2 /cm and ( 3 $ 4.2 " 10!6 ps3 /cm at 1.55 !m. The value obtained for the ( 2 is close to the one measured by Asobe et al.; therefore the value kept for the group velocity dispersion is an average of these two measurements: ( 2 $ 5.0 " 10!3 ps2 /cm. Starting from the pulse duration, which is 800 fs, the second and third-order dispersion lengths14 can be computed for these two coefficients. They are 2.38 m for ( 2 and 1.2 km for ( 3 . Since the maximum waveguide length used in our experiment is $2 cm, the dispersion effect can be considered to be very small if not negligible. The linear propagation losses ( * o ) inside the channel waveguide have been measured by analysis of the light diffused from the top of the waveguide.1 A value of * o 5 1 dB/cm was obtained. During each SPM experiment, the transmission of the waveguide was monitored in order

Fig. 4. Typical experimental spectra of SPM measurements. (a) and (b) correspond to nonphotodarkened and photodarkened As2 S3 channel waveguides, respectively. In each figure, two plots are shown; the top one corresponds to the low-intensity propagation, and the bottom one corresponds to the highintensity propagation. Dotted curves correspond to experimental data and solid curves correspond to numerical fits.

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Table 1. Fitting Results for the n 2 Measurement in As2 S3 Photodarkened and Nonphotodarkened Channel Waveguides Thin Film # State of the film % o ( ! m) f o (THz) L p (cm) w ( ! m) P moy,max (mW) / 3 (ps3 ) / 5 (ps5 ) / 7 (ps7 ) n 2 " 10!14 (cm2 /W) P peak (W) a b

1

2

3

4

5

6

npa 1.5427 0.35 2.03 5.0 2.06 5.29 !2.39 0.32 4.21 31.8

np 1.5418 0.25 2.03 4.5 1.19 4.76 !1.53 0.00 4.13 15.4

np 1.5419 0.35 2.03 4.5 2.02 4.37 !1.82 0.22 2.94 33.0

pb 1.5418 0.35 1.97 5.0 4.16 5.77 !2.52 0.15 5.22 62.8

p 1.5421 0.35 1.97 5.0 2.59 3.83 !1.23 0.03 6.18 43.7

p 1.5424 0.40 1.97 5.0 3.01 3.97 !1.57 0.15 6.98 56.7

The nonphotodarkened samples are represented by np. The photodarkened samples are represented by p.

Table 2. Results of the Nonlinear Measurements in As2 S3 Channel Waveguides

State of the Film

Nonlinear Refractive Index (n 2 ) ("10!14 cm2 /W)

Intensity Threshold at 1.55 !m (GW/cm2)

3.8 % 0.7 6.1 % 0.9

1.2 % 0.4 2.3 % 0.4

Nonphotodarkened Photodarkened

sults obtained for all six cases are presented in Table 1. The thin films labeled 1 to 3 have been obtained in nonphotodarkened waveguides (np), and the ones labeled 4 to 6 have been obtained in photodarkened waveguides (p). Two typical examples of experimental spectra obtained are shown in Fig. 4. Figures 4(a) and 4(b) correspond to the thin films labeled #1 and #5 in Table 1. The results presented in Table 1 can be used to obtain precise values for the nonlinear refractive index of nonphotodarkened and photodarkened As2 S3 channel waveguides. Furthermore, since all of the high intensity measurements were made directly below the photosensitive threshold, the peak-power measurements give an estimates of the intensity threshold associated with the photosensitivity at 1.55 !m. It also gives an evaluation of the threshold inside a photodarkened thin film. These results are presented in Table 2. The uncertainties on the n 2 are 19% and 15% for the nonphotodarkened and photodarkened waveguides, respectively.

6. DISCUSSION The proposed ansatz Eq. (5) for describing the spectrum of the pulses generated by the OPO allowed the fitting of the propagated pulses with great accuracy. The spectra showed in Figs. 3 and 4 clearly show the asymmetrical broadening. The apparent blueshift of the spectra is not caused by the Raman effect (which usually causes a redshift). The only optical effect present here is SPM, which is responsible for the broadening and the modulation of the spectrum. The apparent blueshift is caused by the SPM interacting with the initially asymmetric spectral phase profile. It is important to point out that asym-

metrical SPM is something that will occur in a particular regime. According to the propagation equation, it is possible to define characteristic lengths14: L NL $ LD $

A eff k on 2P o T 2o " ( 2"

,

,

(9)

(10)

where P o is the peak power and T o is the pulse duration. Equations (9) and (10) are called the nonlinear and dispersion length, respectively. In order for SPM to be observed, the following condition must be respected: L NL & L D and L D ' L p , where L p is the length of the waveguide. If L D is decreased so that it becomes close to L NL and L p , the dispersion will either lead to soliton formation ( ( 2 ( 0) or broaden the pulse dramatically and decrease the peak power along the propagation ( ( 2 ) 0). In the case of a standard optical fiber, the effective area is so large that it is difficult to inject enough power so that nonlinearity will be stronger than the dispersion. One solution would be to use short pulses to get higher peak power, but this will increase the effect of dispersion (or decrease the dispersion length L D ). In short, in standard fibers such an experiment will usually lead to soliton formation, so the asymmetric SPM emphasized in this paper will not be observed. Only special cases such as propagation inside chalcogenide glasses in the near infrared, where n 2 is strong and ( 2 ) 0, will show such asymmetric SPM. The fitting results for the nonlinear spectral phase coefficient / j vary from one experiment to another, as shown in Table 1. These variations are caused by the fact that between each SPM measurement, the length of the OPO cavity needed to be adjusted in order to properly synchronize the cavity with the repetition rate of the pump laser. Such adjustments allowed us to get intense and short pulses from the OPO but were also responsible for giving slightly different pulses shape each time the cavity was tuned. For better understanding of the effect of the nonlinear spectral phase, Fig. 5 shows the interferometric autocorrelation for the experimental pulses as well as for the in-

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verse Fourier transform of the spectral ansatz Eq. (5), in which we used f o $ 0.35 THz, / 3 $ 4.67, / 5 $ !1.84, and / 7 $ 0.15 or / i $ 0 for all i. These values are close to the one obtained in Table 1. The differences between the autocorrelation traces are very small. It is clear that the effect of the nonlinear spectral phase profile is small and almost impossible to observe with an autocorrelator. Thus such small phase effects are difficult to characterize but have a profound effect on pulse propagation in some nonlinear media, such as chalcogenide glasses. The experimental values obtained for the nonlinear refractive index at 1.55 !m of As2 S3 in thin films are close to the values reported in the literature.3,6–12 The n 2 ob-

Fig. 5. Autocorrelation traces. (a) corresponds to the measured autocorrelation of the pulses generated by the OPO. (b) and (c) are numerically evaluated autocorrelations for which the spectral width is f o $ 0.35 THz and the spectral phase coefficients are either / 3 $ 4.67, / 5 $ !1.84, and / 7 $ 0.15 in (a) or / i $ 0 for all i in (b).


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tained in photodarkened thin films clearly shows an increase in comparison with the nonphotodarkened thin films. The only other experimental measurement of the n 2 of As2 S3 channel waveguides is 0.80 " 10!14 cm2 /W, as reported by Cerqua et al.10 Their measurement was also based on a SPM characterization. However, their analysis is not as numerically precise as the one presented here. Their analysis is based on a qualitative determination of the nonlinear phase accumulation, which is less accurate than a direct fitting of the measured spectra. In comparison with the theoretical evaluation of the n 2 that were presented in Subsection 2.A, the following observation can be made: The n 2 obtained in the nonphotodarkened thin films is slightly greater than the one obtained with both theoretical models, whereas the results obtained in the photodarkened is greater by a factor of $2. The experimentally observed increase of the n 2 in a photodarkened thin film is in agreement with the theoretical models. The models showed that the increase of the linear refractive index and the redshift of the bandgap led to an increase of the n 2 . However, the experimental results presented in Table 2 show that the actual increase is 60%, which is much more than the theoretical evaluations of 5% and 20%. Such a discrepancy can be explained by the oversimplifying assumptions behind the theoretical models. These models rely only on the linear refractive index and the energy of the bandgap; they do not include all of the complex electronic structure present inside the chalcogenide glasses. Furthermore, the photodarkening of the material might change the structure of the transition and therefore modify the nonlinear refractive index dispersion and magnitude (via the nonlinear Kramers– Kronig relations). The experimental determination of the n 2 was clearly limited by the photosensitivity at 1.55 !m. The intensity threshold for this phenomenon were measured by Hoˆ et al. and established between 0.8 and 1.3 GW/cm2 for nonphotodarkened As2 S3 thin films. The value reported in Table 2 is of the same order as the values reported by Hoˆ et al. The value obtained for the photodarkened thin film is clearly greater than the nonphotodarkened ones by a factor of $2. One can explain this increase in the intensity threshold by taking a closer look at the photodarkening mechanism. If a thin film is first exposed to photons of energy close to the bandgap energy, the film gets closer to the saturation of its photodarkening process. Under such a condition, it is more difficult for the photons at 1.55 !m to further photodarken the glass; thus the intensity threshold is increased. It is possible to evaluate the nonlinear potential of As2 S3 channel waveguides. One can quantify this potential by looking at the maximal nonlinear phase shift per length unit. This quantity is defined by the following relation:

/ max $ k o n 2 I o .

(11)

This formula is valid in the case where the propagation losses are neglected. With the data of Table 2, the follow-

444


ing values were obtained for / max : 0.64/cm for a nonphotodarkened waveguide and 1.84/cm for a photodarkened waveguide. The results presented in this paper can lead to the conclusion that photodarkening the As2 S3 thin films is the solution to stabilization of the glass matrix and an increase of the nonlinear potential of the channel waveguides, but one must be careful with such a conclusion. The photodarkening process only saturates the creation of defects responsible for the shift in bandgap energy. However, this saturation is not complete, and there is still an intensity threshold that leads to the creation of photoinduced structures inside the material. The mechanisms responsible for the photosensitivity at 1.55 !m are not well understood and require a more thorough study. The real solution resides in the complete elimination of the photosensitivity at 1.55 !m. Previous works3,6,12 in bulk and fiber made of As2 S3 have shown no photosensitivity at 1.55 !m. These facts suggest that the photosensitivity in the As2 S3 thin films comes from the deposition process. It is well known that the structure of thermally evaporated thin films is different from the bulk.13 To avoid the problem caused by the thermal evaporation, one must consider alternative deposition processes. Recently, chalcogenide thin films fabrication using a laser ablation technique have been reported.30–32 This deposition process seems to yield thin films with good optical quality. Furthermore, this process seems to have solved the photosensitivity problem at 1.55 !m, since there has not been any mention of this phenomenon in the deposited films. This can be explained by the fact that this deposition process probably keeps the glass stoichiometry constant so that the thin films are identical to the bulk samples from a chemical perspective.

Laniel et al.

ACKNOWLEDGMENTS The authors thank R. Piche´ for technical support. This work was supported by a strategic grant from the Natural Sciences and Engineering Research Council of Canada (NSERC). J. M. Laniel and N. Hoˆ acknowledge a scholarship from NSERC and the Fond que´bećois de la recherche sur la nature et les technologies (FQRNT). The e-mail address [email protected].

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7. CONCLUSION We have presented the results of the experimental measurements of the nonlinear refractive index (n 2 ) of photodarkened and nonphotodarkened As2 S3 channel waveguides. The photodarkening process was performed through exposure of the thin films to a laser beam at 0.514 !m. The thin films were exposed long enough in order to saturate the photodarkening process inside the material. The results were obtained through analysis of the SPM on the spectrum of pulses propagated inside the chalcogenide channel waveguides. The analysis of the SPM was presented for the case in which the temporal shape of the pulses used was asymmetrical. Such asymmetrical temporal shape is typical for an OPO pumped by short pulses (pulse duration less than 10 ps). Such a condition induces a specific modulation of the propagated spectrum that is asymmetric. The experiments have also provided the intensity threshold associated with the photosensitivity at 1.55 !m for the case of nonphotodarkened and photodarkened thin films. To our knowledge, this is the first report of the n 2 and the intensity threshold inside photodarkened As2 S3 thin films. It has been observed that the photodarkening process increases both the nonlinear refractive index and the intensity threshold of the thin films.

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