Abstract

We study free-carrier nonlinearities in crystalline silicon at 1.064 µm using the Z-scan technique, with special emphasis on the dependence of their nonlinearities on the width of incident pulses. In the Z-scan experiment, the pulse duration was changed from 11.5 ns to 1.6 ns by the pulse compression using stimulated Brillouin scattering in a liquid. At this excitation wavelength, linear absorption is dominant for the creation of electron-hole pairs and the photoexcited carriers can modify the refractive index and absorption coefficient just as a third-order nonlinear effect. The effective nonlinear refractive index n 2eff and nonlinear absorption coefficient β eff are proportional to the pulse duration and optical intensity, i.e. the fluence when the pulse duration is shorter than the carrier recombination lifetime. We can determine the variation of refractive index per unit of photoexcited carrier density σ r and the total carrier absorption cross section σ ab from the dependence of n 2eff and β eff on the pulse width, respectively. In this work we had σ r = -1.0 × 10-21 cm3 and σ ab = 8.4 × 10-18 cm2, which agree well with previous data. We also observed the decrease in the magnitude of n 2eff and β eff at high incident fluence, which is presumably attributed to band filling. This new measurement approach has an advantage of being able to separate an ultrafast Kerr nonlinearity and a cumulative nonlinearity such as the free-carrier nonlinearity treated in this paper and can be utilized to evaluate the optical nonlinearities of other materials.

© 2008 Optical Society of America

1. Introduction

The nonlinear optical properties of semiconductors have been studied extensively because of scientific interests in the materials and their potential applications to optical devices. Especially silicon has attracted attention in recent years and a new branch named silicon photonics is being formed in optics. Silicon photonics could enable a chip-scale platform for highly-integrated, ultracompact optical circuits or microphotonics devices [14]. The third-order nonlinearities available in silicon are the nonlinear refraction due to bound electrons, i.e. Kerr nonlinearity, the plasma effect of free carriers generated by optical absorption, and the stimulated Raman scattering. Although the magnitude of Kerr nonlinearity in silicon is relatively large (about 200 times larger than that of silica for bulk samples at fiber-optic wavelengths) [5,6], its application is still significantly limited. From the standpoint of a low operating power of nonlinear optical devices, the most useful one is free-carrier nonlinearities although they are sometimes harmful to device applications [14,79]. In the free-carrier nonlinearities, refraction (free-carrier dispersion, FCD) and absorption (free-carrier absorption, FCA) are changed in proportion to the density of the charge carriers generated by two-photon absorption (TPA). All-optical switching using FCD on a silicon chip has been successfully demonstrated by using photonic crystal nanocavities [10,11] and microring cavities [12].

About two decades ago, the nonlinearities in bulk silicon were extensively studied at the Nd:YAG laser oscillation wavelength of 1.064 µm for their applications to phase conjugation, pulse shaping, optical limiting, etc. [1316]. Since silicon has a moderate absorption loss of ~10 cm-1 at this wavelength, the free carriers are created by linear absorption (one-photon absorption, OPA) rather than by TPA. The FCD and FCA induced by OPA can be considered as a third-order nonlinearity. The associated material parameters were determined and discussed by using degenerate four-wave mixing and nonlinear transmission. These measurement methods cannot determine the nonlinear refractive index and absorption coefficient including their sign at the same time.

In this paper we propose a new measurement method based on a combination of a pulse compression technique and the Z-scan technique, and investigate the dependence of optical nonlinearities in crystalline silicon on the laser pulse width of the order of nanoseconds using it. The magnitude and sign of effective nonlinear refractive index n 2eff and effective nonlinear absorption coefficient β eff in the bulk sample are measured at a wavelength of 1.064 µm using the Z-scan technique [8,17,18]. In this work stimulated Brillouin scattering (SBS) in liquids is used to compress coherent nanosecond pulses from a Q-switched Nd:YAG laser [1921]. The pulse width is changed from 11.5 ns to 1.6 ns using liquid heavy fluorocarbons as a SBS medium. The dependence of n 2eff and β eff on the incident pulse width is measured at a low incident energy level. Although it is well recognized that the optical nonlinearities of materials generally depend on the pulse duration as well as the frequency and power of an incident pulse, there is no experiment where the pulse duration was changed for the nonlinearity estimation. To change the pulse duration brings in useful information in discussing the optical nonlinearities of materials. It has been found that the measured values of n 2eff and β eff are proportional to the pulse duration when the incident intensity is weak, as predicted by Drude’s model. It has also been found that the magnitude of n 2eff and β eff decreases as the incident pulse energy is increased, which is presumably attributed to band filling.

The proposed measurement method has an important advantage of being able to separate an ultrafast Kerr nonlinearity and a slow (cumulative) nonlinearity such as a thermal nonlinearity and the free-carrier nonlinearity treated in this paper because the magnitude of the cumulative nonlinearity should be proportional to the pulse duration. It can also be utilized to evaluate the optical nonlinearities of potential nonlinear materials such as chalcogenide glasses [22,23]. Since there are always photo-structural changes in chalcogenide glasses and the details of optical nonlinearities depend on fabrication conditions of glasses unlike crystals, discussion of the nonlinearities is more complicated. Another reason why we have chosen crystalline silicon as a substance to be examined is that the mechanisms of the free-carrier nonlinearity are relatively simple.

2. Theoretical background

In order to facilitate understanding of measured nonlinearities in silicon using the Z-scan technique, we briefly explain the origins of third-order nonlinearities and the basic equations describing the related phenomena. Let us consider an optical wave propagating in the z direction within the nonlinear material. The phase change Δϕ and optical intensity I of the optical wave per unit propagation distance are given by

dΔϕdz=k0Δn=k0(γI+σrN(I)),
dIdz=(α+βI+σabN(I))I,

where k 0 is the free-space wave vector, Δn is the change in the refractive index, γ is the nonlinear refractive index due to bound electrons, α is the linear absorption coefficient, β is the TPA coefficient. σ r is the FCD coefficient, i.e. the change in the refractive index per unit of photoexcited charge-carrier density N, and σ ab is the FCA coefficient, i.e. the carrier absorption cross section. The generation of carriers is done by one-photon absorption (OPA) and/or multiphoton absorption such as TPA.

First, we summarize the temporal response of the photoexcited charge-carrier density. We can approximately separate the time and space dependence when the incident pulse width is much longer than the transit time through the sample. Assuming the only OPA as the mechanism for generating carriers and neglecting the lateral diffusion of carriers, we have the following rate equation determining the density N of induced free-carriers.

dN(t)dt=N(t)τ+αI(t)ω,

where ħω = hc/λ is the photon energy, τ is the carrier recombination lifetime, and αI is the optical power density absorbed due to OPA. We consider a Gaussian pulse with the following optical intensity profile:

I(t)=I0exp(t2t02),

where I 0 is the peak intensity and t 0 is the pulse width defined by the half width at e-1 of the maximum. In our experiments, the measured pulse width t FWHM is defined by the full width at half maximum and the relation between t 0 and t FWHM for a Gaussian pulse is given by t 0 = t FWHM/[2(ln2)1/2]. Substituting Eq. (4) into Eq. (3) and then solving Eq. (3), we have the following solution:

N(t)=αI0ω·πt02·exp(t024τ2)[1+erf(tt0t02τ)]·exp(tτ).

The OPA-induced free-carrier density is proportional to the peak intensity I 0 and it behaves as a third-order nonlinearity. In both cases of a long pulse (t 0τ) and a short pulse (t 0τ), Eq. (5) can be approximated as follows:

N(t)={αI0ω·τexp(t2t02)t0>>τ,αI0ω·πt02[1+erf(tt0)]τ>>t0,

In the case of t 0τ, the carrier density N(t) changes with the intensity I(t). However it should be noted that its peak value αI 0/ħω is multiplied by the carrier lifetime τ independently of the pulse width t 0. On the other hand, in the case of τt 0, the carrier density N(t) is temporally-integrated independently of τ and the result is expressed by the error function erf(z), where erf(-∞)=-1, erf(0) = 0, and erf(∞) = 1. It should also be noted that π1/2 t 0 I 0 is the fluence, i.e. integration value of the intensity given by Eq. (4). On the other hand, when the linear absorption of the material is negligible, a main mechanism for generating charge carriers is TPA. Assuming the only TPA and neglecting the lateral diffusion of carriers, the rate equation of the induced free-carrier density N is given by replacing αI(t) and t 0 in Eq. (3) by β I 2(t)/2 and t 0/21/2, respectively. The solution of the resultant equation is given by

N(t)=βI022ω·πt08·exp(t028τ2)[1+erf(2t0tt08τ)]·exp(tτ).

The TPA-induced free-carrier density is proportional to the square of peak intensity I 0. Moreover, for t 0τ or t 0τ, Eq. (7) can be approximated as follows:

N(t)={βI022ω·τexp(2t2t02)t0>>τ,βI022ω·πt08[1+erf(2t0t)]τ>>t0,

As is seen from Eqs. (7) and (8), the plasma effect of TPA-induced free-carriers behaves as a fifth-order nonlinearity.

Next, we definite the effective nonlinear refractive index n 2eff and effective nonlinear absorption coefficient β eff in conjunction with the Z-scan experiment. In the Z-scan technique [17], a sample of thickness L is translated through the focus of a spatial Gaussian beam and the transmission through an aperture behind the sample is measured as a function of the longitudinal coordinate z. In this closed-aperture scan to determine the nonlinear refractive index, the time-averaged nonlinear phase shift of the beam transmitted through the sample is evaluated on the axis at the focus. The on-axis phase shift is defined as

ΔΦ0(t)=k0Δn(t)Leff,

where L eff = [1-exp(-αL)]/α is the effective interaction length. We must average the instantaneous phase shift Δ:0 (t) over the laser pulse shape. For example, in the case of OPA-induced free-carrier nonlinearities with τ≫t 0, i.e. in the case of the short pulse in Eq. (6), we have

ΔΦ0<Φ0(t)>=k0(γI02+σrαI0ωπt02)Leffk0n2effI02Leff,

where γ is the nonlinear refractive index due to bound electrons and I 0 is the peak on-axis intensity. The coefficients 1/21/2 and 1/2 appearing in the third equation are an averaging factor for the instantaneous Kerr nonlinearity and the accumulative (free-carrier) nonlinearity, respectively. The effective nonlinear refractive index n 2eff defined by Eq. (10) is based on the assumption of a fast nonlinearity and a temporally Gaussian pulse. In this way we can define n 2eff for a combination of the Kerr nonlinearity and FCD due to OPA as follows:

n2eff={γ+σrατωt0>>τ,γ+σrαωπt02τ>>t0,

In this closed-aperture Z-scan, the transmission through the aperture reveals a characteristic peak-valley shape, which is well described by

T(z)=14(zz0)ΔΦ0(1+z2z02)(9+z2z02),

where z 0 = kw 2 0/2 is the diffraction length of the beam and w 0 is the 1/e2 beam radius at focus.

On the other hand, in the open-aperture Z-scan to determine the nonlinear absorption coefficient, the time-averaged transmittance through the sample is measured on the axis at the focus. We must here define an effective nonlinear absorption coefficient when the FCA effect is added to the fast TPA. From Eq. (29) of [17], the average nonlinear transmittance can be expressed as

T(z)<T(z,t)>=112<q0(z,t)>+13<q0(z,t)2>14<q0(z,t)3>+....,

and

q0(z,t)=βI(t)Leff1+z2z02,

where β is the TPA coefficient. We can obtain an expression for β eff by approximating Eq. (13) by the first two terms and by adding σ ab N(t) to βI (t). For example, in the case of the slow FCA with τt 0, the average nonlinear transmittance is given by

T(z)=112<βI(t)+σabN(t)>Leff1+z2z02=112(β2+σabαωπt02)I0Leff1+z2z02
1122βeffI0Leff1+z2z02.

In this way we can define β eff for a combination of TPA and FCA due to OPA as follows:

βeff={β+σabατωt0>>τ,β+σabαωπt02τ>>t0.

The values of γ and β are generally small compared to FCD and FCA due to OPA, respectively. The contribution of n 2eff and β eff due to OPA-induced free-carriers is independent of I 0 and is proportional to the pulse width t 0 in the case of τt 0. We can moreover determine the magnitude and sign of parameters σ r and σ ab from the linearity of measured values of n 2eff and β eff with respect to the pulse width t FWHM, respectively.

Finally, we also definite n 2eff and β eff for a combination of the Kerr nonlinearity, TPA, and FCD and FCA induced by both OPA and TPA. The situation is just the case where a third-order nonlinearity and a fifth-order nonlinearity are simultaneously present and the definition of n 2eff and β eff is not easy. Introducing the simplification done in [8], for τt 0, we can get the following equations:

n2eff=γ+σr(αωπt02+Cβωπt0I04),
βeff=β+σab(αωπt02+βωπt0I04),

where C = 0.21/0.406 is a correction coefficient for the difference in Z-scan formula between third-order and fifth-order nonlinearities. The last term in Eqs. (17) and (18) is the contribution of TPA-induced free-carriers, which is proportional to the fluence π1/2 t 0 I 0.

3. Experimental procedure

Figure 1 shows the experimental setup for efficiently compressing coherent nanosecond pulses and for measuring the optical nonlinearities in materials. The pump laser used in this work is an injection-seeded, Q-switched Nd:YAG laser, which provides 400-mJ Gaussian pulses at a 1.064-µm wavelength and a 10-Hz repetition rate. The optical pulses from this laser are compressed by using stimulated Brillouin scattering (SBS) in heavy fluorocarbon liquids [19,20]. To the best of our knowledge, ultrafiltered fluorocarbon is the best medium for SBS temporal compression at a near 1-µm wavelength because of nontoxic material, chemical inertness, high durability, and high compressibility. Pulse compression of 10-ns pump pulses to 0.87-ns Stokes pulses has been successfully carried out using Fluorinert FC-75, which has an acoustic decay time (phonon lifetime) of 0.78 ns near the lasing wavelength of 1.064 µm [20]. Since FC-75 is now not available, we use FC-72 instead, which is inferior to FC-75 in phonon lifetime (1.2 ns). The optical and SBS properties of Fluorinert FC-72 at 25°C and1.06 µm are given in [19]. We utilize the compressed Nd:YAG laser pulses to the Z-scan measurement for determining the nonlinear refraction and nonlinear absorption of silicon.

 

Fig. 1. Schematic diagram of a SBS compressor and a Z-scan experiment.

Download Full Size | PPT Slide | PDF

Although various geometries of SBS pulse compressors have been proposed to date, we use a compact single-cell geometry proposed in [21], where a SBS generator in a two-cell generator-amplifier setup is folded back in an amplifier cell using a concave mirror. In Fig. 1, a combination of a plane mirror (M1) and a spherical convex lens (L1) with a 25-cm focal length is used in place of the concave mirror. After passing through the SBS cell filled with Fluorinert FC-72, the pump wave is focused in the cell by this equivalent concave mirror (M1 and L1). Consequently, a Stokes wave is created from the leading edge of the focused pump pulse near the focus and propagates in the opposite direction. The interaction of pump and Stokes waves limits the input energy into the SBS generator and the Stokes pulse undergoes strong intensity amplification and substantial temporal reshaping in the amplifier cell. In this way, the SBS cell operates as a generator and an amplifier for Stokes waves at the same time. Although the optimum length of the SBS cell is given by L SBS = (c/n) t FWHM and n is the refractive index of the SBS medium and L SBS = 300 cm in the present case, its length was taken at 180 cm from structural constraints. The pump energy can be changed by using a half-wave plate (HWP) and a thin film polarizer (P1). The generated Stokes wave from the SBS cell can be separated from the pump wave using a quarter-wave plate (QWP) and the polarizer (P1) and be attenuated by using a polarizer (P2). The temporal shape of the pump and Stokes beams was measured by using a fast PIN photodiode with a 35-ps rise time and a 500-MHz oscilloscope, or a 50-GHz sampling oscilloscope.

We measure the nonlinear optical properties in silicon by the conventional Z-scan technique [8,17,18]. The sample is mounted on a translation table and is moved along the z-axis through the focal plane of a 20-cm focal length lens (L2). The beam waist w 0 in the focal plane is 40 µm. Closed-aperture Z-san allows determining the sign and magnitude of nonlinear refractive index, while open-aperture Z-scan is used for the measurements of nonlinear absorption. However the closed-aperture Z-san data are divided by the open-aperture Z-scan data to obtain a purely refractive Z-scan. The average on-axis phase change ΔΦ0 is given by the difference in the peak and valley transmittance values: ΔT p-v = 0.405(1-S)0.25ΔΦ0 for Φ0≤π, where S is the aperture linear transmission. We use S=0.40 for our Zscan experiments.

4. Experimental results and discussion

Before presenting the results of the Z-scan experiments of silicon, we describe the electrical and optical properties of a silicon wafer to be examined and the experimental results for the SBS compression using Fluorinert FC-72. The thickness of the wafer is 0.50 mm and the crystal orientation relative to the wafer surface is <100 > . We determined the electrical properties of the sample by the four-point probe and Hall-effect experiments. The resistivity was 7.0 × 103 Ωcm, doping was n-type, and its concentration was 1.5×1013 cm-3. Therefore the sample can be considered to be intrinsic. We also measured the absorption coefficient by use of the Brewster angle [24] and obtained α=11.8 cm-1 at 1.064 µm. The carrier recombination lifetime τ also was estimated to be ~100 µs.

Figure 2(a) shows the dependence of the duration of the generated Stokes pulse on the pump pulse energy. Figure 2(b) shows the temporal intensity profiles of the pump pulse and the Stokes pulses at different pump energies. We can change the pulse duration from 11.5 ns to 1.6 ns using the present SBS pulse compressor and use the compressed pulses as a pump beam for the Z-scan measurements.

 

Fig. 2. (a). Stokes pulse duration t FWHM versus the pump pulse energy. (b) Temporal intensity profiles of the pump and Stokes pulses.

Download Full Size | PPT Slide | PDF

First, we present typical closed- and open-aperture Z-scan traces for the silicon sample in Fig. 3. The incident pulse widths t FWHM are 11.5 ns (Fig. 3(a)) and 1.6 ns (Fig. 3(b)). The closed-aperture data shown in the figures are ones which were already divided by the open-aperture data. An incident energy per pulse (or average power) was fixed at 1 µJ (10 µW) for the two pulse widths. Therefore the peak optical intensity at t FWHM = 1.6 ns is approximately seven times as much as one at t FWHM = 11.5 ns. In the present case, an input pulse energy (outside the sample) of 1 µJ leads to the on-axis fluence (inside the sample) F 0 = 27.3 mJ/cm2 and creates a carrier density N(0) = 8.6 × 1017cm-3. Moreover the incident energy gives the onaxis peak intensity (inside the sample) I 0 = 2.23 MW/cm2 for t FWHM = 11.5 ns and I 0 = 16.0 MW/cm2 for t FWHM = 1.6 ns. The sign of n 2eff is obviously minus from the peak-to-valley transmission change in the closed-aperture Z-scan for two pulse durations. Theoretical fits based on Eqs. (12) and (15) are also shown in Fig. 3, where n 2eff = -5.41 × 10-14 m2/W and β eff = 3.26 × 10-8 m/W are used at t FWHM = 11.5 ns, and n 2eff = -6.80 × 10-15 m2/W and β eff = 6.50 × 10-9 m/W are used at t FWHM = 1.6 ns.

 

Fig. 3. Typical normalized open- and closed-aperture Z-scan data of crystalline silicon at λ = 1.064 µm with an incident energy of 1 µJ. The incident pulse width is (a) t FWHM = 11.5 ns and (b) t FWHM = 1.6 ns. The solid lines are the best fits based on Eqs. (15) and (12).

Download Full Size | PPT Slide | PDF

Next, we examine the dependence of n 2eff and β eff on the incident pulse energy. Figures 4 and 5 show the experimental results at t FWHM = 11.5 ns and 1.6 ns, respectively. The closed-aperture Z-scan should be usually applied for the average on-axis phase change ΔΦ0≤π. However this phase shift increases with increasing incident energy and can easily exceed a critical limit of π. To gain the general tendency, we dare to show the obtained results for ΔΦ0≥π with white circles. On the other hand, there is no critical limitation to the measurement of β eff. Therefore, regarding β eff, reliable results can be expected over the full energy ranges except for ones obtained at low energy levels such as 0.5 or 0.7 µJ. Although the values of n 2eff and β eff are independent of the optical intensity (or pulse energy) under Drude’s model as mentioned in Sec. 2, the experimental results clearly show intensitydependent changes. At t FWHM = 11.5 ns (Fig. 4), the measured values of n 2eff and β eff are almost constant for low energies less than 1 µJ although they are somewhat scattered. These two values then decrease monotonically with increasing pulse energy. The dependence of n 2eff and β eff on the incident pulse energy at t FWHM = 1.6 ns (Fig. 5) is very similar to one at t FWHM = 11.5 ns (Fig. 4) except for the following point: There is no plateau on the measured data of n 2eff and β eff in the low energy regime, which seems to be caused by the measurement errors due to weak energy.

 

Fig. 4. Dependence of n 2eff (<0) and β eff of the incident pulse energy at the pulse width t FWHM = 11.5 ns. The white circles show ΔΦ0≥π.

Download Full Size | PPT Slide | PDF

 

Fig. 5. Dependence of n 2eff (<0) and β eff of the incident pulse energy at the pulse width t FWHM = 1.6 ns. The white circles show ΔΦ0≥π.

Download Full Size | PPT Slide | PDF

We must here discuss the origin of the nonlinear dependence of n 2eff and β eff on the incident energy. Besides the Drude effect of the charge-carriers generated by OPA, other physical effects are responsible for changes in the refractive index and the absorption coefficient at high optical intensities (or high carrier densities): sample heating, band filling, renormalization of the band structure, charge-carriers generated by TPA and so on. These possibilities have been well accepted through long studies of silicon [7,15]. To explain the observed intensity dependence, a mechanism producing a positive change in the refractive index and a negative change in the absorption coefficient with optical intensity is additionally required. In silicon, the band-gap energy decreases with increasing temperature. Therefore the absorption band edge red-shifts with temperature and the thermo-optic coefficient, i.e. a change in the refractive index due to a temperature rise is positive; dn/dT=1.86×10-4 K-1 at 1.5 µm [25]. If thermal contributions are present, both the refractive index and the optical absorption are increased with increasing incident pulse energy. Therefore the increase in the optical absorption is inconsistent with our observation. The band-gap normalization describes a change in the carrier energy due to many-body interactions in the excited carriers. This effect brings the shrinkage of the energy gap, resulting in the increase in the optical absorption [15]. Therefore this also cannot explain the present observation. If TPA took place appreciably, the magnitude of n 2eff and β eff could increase in proportion to the incident pulse energy. It should be pointed out that in Drude’s effect a change in the refractive index Δn and a change in the attenuation constant Δα do not vary as the free-carrier density N at dense density levels [26,27]. However, since the linearity between Δn (or Δa) and N approximately holds up to N=1020 cm-3 at least in silicon [7], we can neglect this effect. Eventually, the observed decrease in the magnitude of n 2eff and β eff with respect to the incident fluence is presumably attributed to band filling.

 

Fig. 6. (a). Dependence of n 2eff (<0) on the incident pulse width t FWHM at an incident energy of 1 µJ. (b)Dependence of β eff on t FWHM. The straight lines are the best fits to extract the coefficients σ r and σ ab.

Download Full Size | PPT Slide | PDF

Finally, we examine the dependence of the nonlinear coefficients n 2eff and β eff on the incident pulse width t FWHM and determine the values of σ r and σ ab. The experiment for deducing these two coefficients must be done at low energy levels where Drude’s model can hold, i.e. the values of n 2eff and β eff can be regarded to be constant. Figure 6 shows the experimental results when an incident pulse energy (or average power) was fixed at a low pump level of 1 µJ (10 µW), which was determined from the experimental date in Figs. 4 and 5. Therefore the incident peak irradiance I 0 increases in inverse proportion to the pulse duration in this measurement. It should be noted that the sign of measured n 2eff values is minus and its magnitude is shown in the figure. It is clearly shown that the measured values of n 2eff change in proportion to the pulse width t FWHM as predicted from Eq. (11). The measured β eff also changes lineally with tFWHM as predicted from Eq. (16), although they are somewhat scattered for t FWHM > 7 ns. The cause of the scattered data is that the transmission change of the sample at focus becomes small with increasing pulse width as shown in Fig. 2 and hence it becomes increasingly difficult to determine the value of β eff with accuracy. It should also be noted that the absolute variance of data increases with pulse width even if the relative error in experimental data is constant. The two straight lines presented in Fig. 6 are the best fit to the experimental data of n 2eff and β eff. Although the Kerr coefficient γ and TPA coefficient β are given by the intersection with the vertical axis of each straight line, we could not determine them because of negligibly small magnitudes. According to [6], their magnitudes of the bulk silicon at 1.06 µm are γ = 4.5 × 10-18 m2/W and β = 1.9 × 10-11 m/W. From the slope of the fitted straight lines, we have the FCD coefficient σ r = -10.0 × 10-22 cm3 and the FCA coefficient σ ab = 8.4 × 10-18 cm2. These values agree well with the known values of |σ r| = 9 × 10-22 cm3 and σ ab = 7 × 10-18 cm2 [16]. The FCD and FCA coefficients obtained at 1.064 µm can be utilized at different wavelengths λ. These two coefficients are proportional to λ2 under Drude’s model [7]. Calculating these coefficients at 1.55 µm, we have σ r = -2.1 × 10-21 cm3 and σ ab = 1.8 × 10-17 cm2, which agree with σ r = -1.35 × 10-21 cm3 and σ ab = 1.45 × 10-17 cm2 used in [12] and [3], respectively.

5. Conclusions

We have proposed a new measurement method based on a combination of the SBS compression technique and the Z-scan technique and have investigated the optical nonlinearities in crystalline silicon at 1.064 µm. At this excitation wavelength, the nonlinearities are attributed to free-carriers created by linear absorption. In this experiment, the duration of incident pulses was changed from 11.5 ns to 1.6 ns by the pulse compression using SBS in liquid Fluorinert. The effective nonlinear refractive index n 2eff and the effective nonlinear absorption coefficient β eff were measured as a function of the incident pulse-width. The measured values of n 2eff and β eff are proportional to the pulse width when it is shorter than the carrier recombination lifetime. From comparison between theory and experiment, we determined the FCD coefficient σ r and the FCA coefficient σ ab, which agree well with the previous values. It has been successfully demonstrated that changing the pulse width brings us useful information in measurements of optical nonlinearities. We also observed that the magnitude of n 2eff and β eff decreased with increasing pump energy, which is presumably attributed to band filling. We are studying the optical nonlinearities in As2Se3 glass using the proposed approach and the obtained results will be reported elsewhere.

Acknowledgment

This work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology in Japan. The authors would like to thank A. Ishida for his help in the measurement of the electrical properties of silicon wafers.

References and links

1. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006). [CrossRef]  

2. M. Lipson, “Guiding, modulating, and emitting light on silicon - Challenges and opportunities,” J. Lightwave Technol. 23, 4222–4238 (2005). [CrossRef]  

3. B. Jalali and S. Fathpour, “Silicon photonics,” J. Lightwave Technol. 24, 4600–4615 (2006). [CrossRef]  

4. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15, 16604–166444 (2007). [CrossRef]   [PubMed]  

5. M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82, 2954–2956 (2003). [CrossRef]  

6. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007). [CrossRef]  

7. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23, 123–129 (1987). [CrossRef]  

8. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9, 405–414 (1992). [CrossRef]  

9. Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All optical switching and continum generation in silicon waveguides,” Opt. Express 12, –4102 (2004). [CrossRef]  

10. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “All-optical switches on a silicon chip realized using photonic crystal nanocavities,” Appl. Phys. Lett. 87, 151112 (2005). [CrossRef]  

11. T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, “Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities,” Appl. Phys. Lett. 90, 031115 (2007). [CrossRef]  

12. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31, 341–343 (2006). [CrossRef]   [PubMed]  

13. R. K. Jain and M. B. Klien, “Degenerate four-wave mixing near the band gap of semiconductors,” Appl. Phys. Lett. 35, 454–456 (1979). [CrossRef]  

14. R. K. Jain, “Degenerate four-wave mixing in semiconductors: application to phase conjugation and to picosecond-resolved studies of transient carrier dynamics,” Opt. Eng. 21, 199–218 (1982).

15. T. F. Boggess, K. Bohnert, D. P. Norwood, C. D. Mire, and A. L. Smirl, “Nonlinear refraction in silicon induced by one-micron picosecond pulses,” Opt. Commun. 64, 387–392 (1987). [CrossRef]  

16. H. J. Eichler, J. Chen, and K. Richter, “Four-wave mixing reflectivity of silicon at 1.06 µm: Influence of free-carrier absorption,” Appl. Phys. B 42, 215–219 (1987). [CrossRef]  

17. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760–769 (1990). [CrossRef]  

18. K.-H. Lee, W.-R. Cho, J.-H. Park, J.-S. Kim, S.-H. Park, and U. Kim, “Measurement of free-carrier nonlinearities in ZnSe based on the Z-scan technique with a nanosecond laser,” Opt. Lett. 19, 1116–1118 (1994). [PubMed]  

19. H. Yoshida, V. Kmertik, H. Fujita, M. Nakatsuka, T. Yamanaka, and K. Yoshida, “Heavy fluorocarbon liquids for a phase-conjugated stimulated Brillouin scattering mirror,” Appl. Opt. 36, 3739–3744 (1997). [CrossRef]   [PubMed]  

20. V. Kmertik, H. Fiedorowicz, A. A. Andreev, K. J. Witte, H. Daido, H. Fujita, M. Nakatsuka, and T. Yamanaka, “Reliable simulated Brillouin scattering compression of Nd:YAG laser pulses with liquid fluorocarbon for long-time operation at 10 Hz,” Appl. Opt. 37, 7085–7090 (1998). [CrossRef]  

21. D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, “SBS pulse compression to 200 ps in a compact single-cell setup,” Appl. Phys. B 68, 671–675 (1999). [CrossRef]  

22. K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, “Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching,” Opt. Lett. 29, 265–267 (2004). [CrossRef]   [PubMed]  

23. V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggelton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15, 9205–9221 (2007). [CrossRef]   [PubMed]  

24. K. Ogusu, K. Suzuki, and H. Nishio, “Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle,” Opt. Lett. 31, 909–911 (2006). [CrossRef]   [PubMed]  

25. G. Cocorullo and I. Rendina, “Thermo-optical modulation at 1.5 µm in silicon etalon,” Electron. Lett. 28, 83–85 (1992). [CrossRef]  

26. C. H. Lee, P. S. Mak, and A. P. DeFonzo, “Optical control of millimeter-wave propagation in dielectric waveguides,” IEEE J. Quantum Electron. 16, 277–288 (1980). [CrossRef]  

27. T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15, 36–38 (2003). [CrossRef]  

References

  • View by:
  • |

  1. R. Soref, "The past, present, and future of silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 1678-1687 (2006).
    [CrossRef]
  2. M. Lipson, "Guiding, modulating, and emitting light on silicon - Challenges and opportunities," J. Lightwave Technol. 23, 4222-4238 (2005).
    [CrossRef]
  3. B. Jalali and S. Fathpour, "Silicon photonics," J. Lightwave Technol. 24, 4600-4615 (2006).
    [CrossRef]
  4. Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon waveguides: Modeling and applications," Opt. Express 15, 16604-166444 (2007).
    [CrossRef] [PubMed]
  5. M. Dinu, F. Quochi, and H. Garcia, "Third-order nonlinearities in silicon at telecom wavelengths," Appl. Phys. Lett. 82, 2954-2956 (2003).
    [CrossRef]
  6. A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (2007).
    [CrossRef]
  7. R. A. Soref and B. R. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. 23, 123-129 (1987).
    [CrossRef]
  8. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, "Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe," J. Opt. Soc. Am. B 9, 405-414 (1992).
    [CrossRef]
  9. Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, "All optical switching and continum generation in silicon waveguides," Opt. Express 12, 4094-4102 (2004).
    [CrossRef]
  10. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
    [CrossRef]
  11. T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
    [CrossRef]
  12. Q. Xu and M. Lipson, "Carrier-induced optical bistability in silicon ring resonators," Opt. Lett. 31, 341-343 (2006).
    [CrossRef] [PubMed]
  13. R. K. Jain and M. B. Klien, "Degenerate four-wave mixing near the band gap of semiconductors," Appl. Phys. Lett. 35, 454-456 (1979).
    [CrossRef]
  14. R. K. Jain, "Degenerate four-wave mixing in semiconductors: application to phase conjugation and to picosecond-resolved studies of transient carrier dynamics," Opt. Eng. 21, 199-218 (1982).
  15. T. F. Boggess, K. Bohnert, D. P. Norwood, C. D. Mire, and A. L. Smirl, "Nonlinear refraction in silicon induced by one-micron picosecond pulses," Opt. Commun. 64, 387-392 (1987).
    [CrossRef]
  16. H. J. Eichler, J. Chen, and K. Richter, "Four-wave mixing reflectivity of silicon at 1.06 ?m: Influence of free-carrier absorption," Appl. Phys. B 42, 215-219 (1987).
    [CrossRef]
  17. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
    [CrossRef]
  18. K.-H. Lee, W.-R. Cho, J.-H. Park, J.-S. Kim, S.-H. Park, U. Kim, "Measurement of free-carrier nonlinearities in ZnSe based on the Z-scan technique with a nanosecond laser," Opt. Lett. 19, 1116-1118 (1994).
    [PubMed]
  19. H. Yoshida, V. Kmertik, H. Fujita, M. Nakatsuka, T. Yamanaka, and K. Yoshida, "Heavy fluorocarbon liquids for a phase-conjugated stimulated Brillouin scattering mirror," Appl. Opt. 36, 3739-3744 (1997).
    [CrossRef] [PubMed]
  20. V. Kmertik, H. Fiedorowicz, A. A. Andreev, K. J. Witte, H. Daido, H. Fujita, M. Nakatsuka, and T. Yamanaka, "Reliable simulated Brillouin scattering compression of Nd:YAG laser pulses with liquid fluorocarbon for long-time operation at 10 Hz," Appl. Opt. 37, 7085-7090 (1998).
    [CrossRef]
  21. D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, "SBS pulse compression to 200 ps in a compact single-cell setup," Appl. Phys. B 68, 671-675 (1999).
    [CrossRef]
  22. K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, "Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching," Opt. Lett. 29, 265-267 (2004).
    [CrossRef] [PubMed]
  23. V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggelton, D. Y. Choi, S. Madden, and B. Luther-Davies, "Ultrafast all-optical chalcogenide glass photonic circuits," Opt. Express 15, 9205-9221 (2007).
    [CrossRef] [PubMed]
  24. K. Ogusu, K. Suzuki, and H. Nishio, "Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle," Opt. Lett. 31, 909-911 (2006).
    [CrossRef] [PubMed]
  25. G. Cocorullo and I. Rendina, "Thermo-optical modulation at 1.5 ?m in silicon etalon," Electron. Lett. 28, 83-85 (1992).
    [CrossRef]
  26. C. H. Lee, P. S. Mak, and A. P. DeFonzo, "Optical control of millimeter-wave propagation in dielectric waveguides," IEEE J. Quantum Electron. 16, 277-288 (1980).
    [CrossRef]
  27. T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, "All-optical switching in a laterally coupled microring resonator by carrier injection," IEEE Photon. Technol. Lett. 15, 36-38 (2003).
    [CrossRef]

Other (27)

R. Soref, "The past, present, and future of silicon photonics," IEEE J. Sel. Top. Quantum Electron. 12, 1678-1687 (2006).
[CrossRef]

M. Lipson, "Guiding, modulating, and emitting light on silicon - Challenges and opportunities," J. Lightwave Technol. 23, 4222-4238 (2005).
[CrossRef]

B. Jalali and S. Fathpour, "Silicon photonics," J. Lightwave Technol. 24, 4600-4615 (2006).
[CrossRef]

Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear optical phenomena in silicon waveguides: Modeling and applications," Opt. Express 15, 16604-166444 (2007).
[CrossRef] [PubMed]

M. Dinu, F. Quochi, and H. Garcia, "Third-order nonlinearities in silicon at telecom wavelengths," Appl. Phys. Lett. 82, 2954-2956 (2003).
[CrossRef]

A. D. Bristow, N. Rotenberg, and H. M. van Driel, "Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm," Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

R. A. Soref and B. R. Bennett, "Electrooptical effects in silicon," IEEE J. Quantum Electron. 23, 123-129 (1987).
[CrossRef]

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, "Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe," J. Opt. Soc. Am. B 9, 405-414 (1992).
[CrossRef]

Ö. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, "All optical switching and continum generation in silicon waveguides," Opt. Express 12, 4094-4102 (2004).
[CrossRef]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic crystal nanocavities," Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

T. Tanabe, K. Nishiguchi, A. Shinya, E. Kuramochi, H. Inokawa, M. Notomi, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Fukuda, H. Shinojima, and S. Itabashi, "Fast all-optical switching using ion-implanted silicon photonic crystal nanocavities," Appl. Phys. Lett. 90, 031115 (2007).
[CrossRef]

Q. Xu and M. Lipson, "Carrier-induced optical bistability in silicon ring resonators," Opt. Lett. 31, 341-343 (2006).
[CrossRef] [PubMed]

R. K. Jain and M. B. Klien, "Degenerate four-wave mixing near the band gap of semiconductors," Appl. Phys. Lett. 35, 454-456 (1979).
[CrossRef]

R. K. Jain, "Degenerate four-wave mixing in semiconductors: application to phase conjugation and to picosecond-resolved studies of transient carrier dynamics," Opt. Eng. 21, 199-218 (1982).

T. F. Boggess, K. Bohnert, D. P. Norwood, C. D. Mire, and A. L. Smirl, "Nonlinear refraction in silicon induced by one-micron picosecond pulses," Opt. Commun. 64, 387-392 (1987).
[CrossRef]

H. J. Eichler, J. Chen, and K. Richter, "Four-wave mixing reflectivity of silicon at 1.06 ?m: Influence of free-carrier absorption," Appl. Phys. B 42, 215-219 (1987).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, "Sensitive measurement of optical nonlinearities using a single beam," IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

K.-H. Lee, W.-R. Cho, J.-H. Park, J.-S. Kim, S.-H. Park, U. Kim, "Measurement of free-carrier nonlinearities in ZnSe based on the Z-scan technique with a nanosecond laser," Opt. Lett. 19, 1116-1118 (1994).
[PubMed]

H. Yoshida, V. Kmertik, H. Fujita, M. Nakatsuka, T. Yamanaka, and K. Yoshida, "Heavy fluorocarbon liquids for a phase-conjugated stimulated Brillouin scattering mirror," Appl. Opt. 36, 3739-3744 (1997).
[CrossRef] [PubMed]

V. Kmertik, H. Fiedorowicz, A. A. Andreev, K. J. Witte, H. Daido, H. Fujita, M. Nakatsuka, and T. Yamanaka, "Reliable simulated Brillouin scattering compression of Nd:YAG laser pulses with liquid fluorocarbon for long-time operation at 10 Hz," Appl. Opt. 37, 7085-7090 (1998).
[CrossRef]

D. Neshev, I. Velchev, W. A. Majewski, W. Hogervorst, and W. Ubachs, "SBS pulse compression to 200 ps in a compact single-cell setup," Appl. Phys. B 68, 671-675 (1999).
[CrossRef]

K. Ogusu, J. Yamasaki, S. Maeda, M. Kitao, and M. Minakata, "Linear and nonlinear optical properties of Ag-As-Se chalcogenide glasses for all-optical switching," Opt. Lett. 29, 265-267 (2004).
[CrossRef] [PubMed]

V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggelton, D. Y. Choi, S. Madden, and B. Luther-Davies, "Ultrafast all-optical chalcogenide glass photonic circuits," Opt. Express 15, 9205-9221 (2007).
[CrossRef] [PubMed]

K. Ogusu, K. Suzuki, and H. Nishio, "Simple and accurate measurement of the absorption coefficient of an absorbing plate by use of the Brewster angle," Opt. Lett. 31, 909-911 (2006).
[CrossRef] [PubMed]

G. Cocorullo and I. Rendina, "Thermo-optical modulation at 1.5 ?m in silicon etalon," Electron. Lett. 28, 83-85 (1992).
[CrossRef]

C. H. Lee, P. S. Mak, and A. P. DeFonzo, "Optical control of millimeter-wave propagation in dielectric waveguides," IEEE J. Quantum Electron. 16, 277-288 (1980).
[CrossRef]

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, "All-optical switching in a laterally coupled microring resonator by carrier injection," IEEE Photon. Technol. Lett. 15, 36-38 (2003).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Schematic diagram of a SBS compressor and a Z-scan experiment.

Fig. 2.
Fig. 2.

(a). Stokes pulse duration t FWHM versus the pump pulse energy. (b) Temporal intensity profiles of the pump and Stokes pulses.

Fig. 3.
Fig. 3.

Typical normalized open- and closed-aperture Z-scan data of crystalline silicon at λ = 1.064 µm with an incident energy of 1 µJ. The incident pulse width is (a) t FWHM = 11.5 ns and (b) t FWHM = 1.6 ns. The solid lines are the best fits based on Eqs. (15) and (12).

Fig. 4.
Fig. 4.

Dependence of n 2eff (<0) and β eff of the incident pulse energy at the pulse width t FWHM = 11.5 ns. The white circles show ΔΦ0≥π.

Fig. 5.
Fig. 5.

Dependence of n 2eff (<0) and β eff of the incident pulse energy at the pulse width t FWHM = 1.6 ns. The white circles show ΔΦ0≥π.

Fig. 6.
Fig. 6.

(a). Dependence of n 2eff (<0) on the incident pulse width t FWHM at an incident energy of 1 µJ. (b)Dependence of β eff on t FWHM. The straight lines are the best fits to extract the coefficients σ r and σ ab.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

d Δ ϕ d z = k 0 Δ n = k 0 ( γ I + σ r N ( I ) ) ,
d I d z = ( α + β I + σ ab N ( I ) ) I ,
d N ( t ) d t = N ( t ) τ + α I ( t ) ω ,
I ( t ) = I 0 e x p ( t 2 t 0 2 ) ,
N ( t ) = α I 0 ω · π t 0 2 · exp ( t 0 2 4 τ 2 ) [ 1 + erf ( t t 0 t 0 2 τ ) ] · exp ( t τ ) .
N ( t ) = { α I 0 ω · τ exp ( t 2 t 0 2 ) t 0 > > τ , α I 0 ω · π t 0 2 [ 1 + erf ( t t 0 ) ] τ > > t 0 ,
N ( t ) = β I 0 2 2 ω · π t 0 8 · exp ( t 0 2 8 τ 2 ) [ 1 + erf ( 2 t 0 t t 0 8 τ ) ] · exp ( t τ ) .
N ( t ) = { β I 0 2 2 ω · τ exp ( 2 t 2 t 0 2 ) t 0 > > τ , β I 0 2 2 ω · π t 0 8 [ 1 + erf ( 2 t 0 t ) ] τ > > t 0 ,
Δ Φ 0 ( t ) = k 0 Δ n ( t ) L eff ,
Δ Φ 0 < Φ 0 ( t ) > = k 0 ( γ I 0 2 + σ r α I 0 ω π t 0 2 ) L eff k 0 n 2 eff I 0 2 L eff ,
n 2 eff = { γ + σ r α τ ω t 0 > > τ , γ + σ r α ω π t 0 2 τ > > t 0 ,
T ( z ) = 1 4 ( z z 0 ) Δ Φ 0 ( 1 + z 2 z 0 2 ) ( 9 + z 2 z 0 2 ) ,
T ( z ) < T ( z , t ) > = 1 1 2 < q 0 ( z , t ) > + 1 3 < q 0 ( z , t ) 2 > 1 4 < q 0 ( z , t ) 3 > + . . . . ,
q 0 ( z , t ) = β I ( t ) L eff 1 + z 2 z 0 2 ,
T ( z ) = 1 1 2 < β I ( t ) + σ ab N ( t ) > L eff 1 + z 2 z 0 2 = 1 1 2 ( β 2 + σ ab α ω π t 0 2 ) I 0 L eff 1 + z 2 z 0 2
1 1 2 2 β eff I 0 L eff 1 + z 2 z 0 2 .
β eff = { β + σ ab α τ ω t 0 > > τ , β + σ ab α ω π t 0 2 τ > > t 0 .
n 2 eff = γ + σ r ( α ω π t 0 2 + C β ω π t 0 I 0 4 ) ,
β eff = β + σ ab ( α ω π t 0 2 + β ω π t 0 I 0 4 ) ,

Metrics