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We experimentally studied enhancement of the third-order nonlinear optical phenomena, i.e., self-phase modulation due to optical Kerr effect and two-photon absorption (TPA) in a small group-velocity (Vg) linedefect guided-mode of AlGaAs-based photonic-crystal slab waveguide. We found that the phase shift ∆φ or nominal Kerr constant n’2 and TPA coefficient β were strikingly enhanced due to small Vg as the band edge was approached, such that they were proportional to (Vg)-2; the nonlinear refractive index n 2 is enhanced proportional to Vg -1. We also observed that owing to this enhancement as well as an extremely small cross-section area, the energy required for inducing π-phase shift was very small, being of an order of a few pJ for 5 ps optical pulse and for a 0.5-mm long sample. Based on those results, we discuss the possibility of developing ultra-fast and ultrasmall all-optical switches that operate due to cross-phase modulation.
©2009 Optical Society of America
1. Introduction
Currently development of ultrafast and ultra-small active components such as all-optical switches and logic devises is highly expected for future optical information processing and telecommunication [1, 2]. One of the best ways to implement ultrafast switches is considered to make use of instantaneous optical Kerr effect (OKE) due to nonlinear refractive index n 2. However, the OKE is relatively weak because of being due to virtual electronic transition [3], unlike some other nonlinear optical phenomena (NOPs) due to real electronic transition. This indicates that, if other conditions of the sample used are similar between the OKE and the above-mentioned NOPs, the sample length becomes much larger for the same required power in the former than in the latter. An attractive way to still develop the ultra-small device may be to make optical pulses to propagate along a confined region with ultra-small cross-section Seff, because the required pulse power can be much reduced, or the sample length L can be made much smaller for the same power. Moreover, as the squared electric-field strength E 2 is inversely proportional to group velocity V g for the same power, utilizing small V g enables further reduction of L [4–6]. From this point of view semiconductor-based photonic-crystal (PhC)-slab waveguides (WGs) of line-defect are very attractive, since in addition to the above properties, those exhibit notable light-propagation characteristics of passing through an abruptly bend corner without appreciable loss [7, 8]. Semiconductor-based PhC WGs are particularly attractive among others, since n 2 in semiconductors is generally large in the near infrared region compared to other materials [3,9,10]. However, it is noted that two-photon absorption (TPA) is detrimental to the OKE, since the power required for inducing π-phase shift increases if depletion of incident power due to TPA occurs; nominal TPA coefficient β is also large in such semiconductors.
In view of the above situation, we experimentally investigated self-phase modulation (SPM) in spectral domain and TPA for the line-defect guided-mode band in AlGaAs-based PhC WGs. We estimated nominal refractive index n’2 and β from the former and the latter, respectively. We also measured V g directly. From the respective dependences on wavelength in a range near the band edge around 1.55 μm, we find that both n’2 and β increase with decrease of V g as the band edge is approached. We discuss this behavior in comparison to the respective theoretical predictions. From all the above wavelength-dependent values we evaluated the magnitude of the two-photon figure of merit F [11,12]. It reveals that F<2 holds even for V g as small as c /20 with c the light velocity in vacuum. From this fact it is concluded that the sample length as small as 0.3 mm should suffice for giving rise to π-phase shift with a few tens of pJ pulse energy without substantial influence of TPA.
2. Theoretical background
Let light wave propagate in the z direction along a PhC WG. For the line-defect band ω(k) the phase shift ∆φ due to change ∆n of the refractive index n 0 is derived as [4]
where ω is angular frequency and k is wave vector. Substituting the relation defined conventionally ∆n=n 2 I for light intensity I into Eq. (1), we have with λ 0 wavelength
In terms of another definition ∆n= n 2|E|2 [3], ∆φ is also expressed as
where ε is the dielectric constant and we used I=V g· (ε/2)|E|2; note that n̅ 2corresponds to the relevant nonlinear optical susceptibility Re[χ (3)], which can be calculated theoretically [3]. Thus we know that ∆φ is enhanced by n 0 -3(c/V g)2, or n 0 -1(n g/n 0)2 for small V g compared to the bulk case of n g=n 0, where n g≡(c/V g). For later use, we describe the wavelength-dependent coupled power Pπ(λ) required for generating ∆φ = π, which is
where n̅ 2(λ) is the wavelength dispersion of n̅ 2, and L eff(λ) [S eff(λ)] denotes the effective length (the effective mode area for third-order nonlinear phenomena) of a sample, which is wavelength dependent. The SPM-induced frequency broadening for optical pulse of the shape I(t) is obtained from δωB=-(d∆φ/dt) using I(t) in Eq. (2) [13].
3. Experimental and Results
3.1 Sample and characterization
We used samples of the air-bridge type of PhC-slab WGs with single (W1)-line-defect, which is surrounded by 10-rows of air-holes on both sides [14]. Those were fabricated in such a way that single-row of air-holes are missing along Γ-K direction in the Brillouin zone (BZ) in a prototype Al0.26Ga0.74As PhC slab with two-dimensional triangular lattice structure. The lattice constant a, thickness d, and hole diameter r are 444, 260, and 120 nm, respectively. The present sample is equipped with a micro-lens of half-disk shape at the input and output edge for efficient coupling of light to outside [16]. Figures 1(a) and 1(b) show the calculated dispersion curves of the line-defect bands, and the corresponding transmittance spectrum observed. The wavelength range used for measurements in this work is shown by the hatched area.
Fig. 1. (a). The calculated W1 line-defect band (dotted curve) in Al0.26Ga0.74As photonic-crystal slab waveguide; the hatched area denotes the range in which the incident laser wavelength is varied. (b) The transmittance spectrum of a sample observed in a range including the guided modes (hatched area).
3.2 Group velocity
By using the optical pulse of 5 ps in duration, we directly measured V g with a time-of-flight method [16]. The result is shown for n g=c/V g in Fig. 2, together with the theoretical one (solid curve) for comparison; the theoretical V g is calculated from the slope of the line-defect band in Fig. 1(a). The agreement between the experimental and the theoretical is good. For later use the calculated dispersion of V g is also presented in Fig. 2.
Fig. 2. Variation of group refractive index n g=c /V g with wavelength in a wavelength range near the band edge of the line-defect guided-mode in a PhC WG. The solid curve denotes the theoretical one; the dotted curve is the dispersion of V g.
3.3 Self-phase modulation
We observed SPM-induced spectral broadening of a short pulse transmitted through a sample [3,10,13]. As excitation pulses we employed repetitive pulses of 5 ps in duration from a mode-locked fiber laser operated at 20 MHz. Otherwise we used an experimental setup basically similar to the previous one [17]. Figure 3 displays a typical example of evolution of output light spectra observed for an input spectrum centered at 1,535 nm for different P in values of 0.3, 1.4, 3.5, and 6.3 W. In Fig. 3 the respective spectra are normalized such that the peak magnitude becomes the same. It is seen that starting with the spectrum (a) with almost the same spectral width with that of the incident light (laser), the spectrum gets gradually broader (b), and then distorted (c, d) with increasing P in. From comparison with the simulation result, the corresponding phase shift of (b), (c) and (d) is π, 1.9 π, and 3.0 π, respectively; for the simulation, see Ref. 17. We know from this result that the power P π ob required for inducing π-phase shift is 1.4 W. Concerning the result shown in Fig. 3, we make a comment. Careful inspection reveals that, as the coupled power is larger, the phase shift is not necessarily proportional to the power in this case. Depletion of incident power due to TPA should be responsible for this phenomenon; notice that as P in becomes large, influence of TPA gets more serious, as will be discussed later.
Fig. 3. A typical example of spectral evolution of an optical pulse of 5.0 ps pulse-duration, transmitted through a PhC WG with varying the coupled power.
Next, we measured variation of Pπ ob with wavelength in a range of the guided-mode band. Figure 4 shows the result together with a theoretical one Pπ th(λ) (solid curve) for comparison; Pπ th (λ) is depicted such that it coincides with the experimental value at 1,528 nm. The result reveals that Pπ th significantly decreases as the band edge is approached, and that agreement between the observed and the theoretical is very good. Notice, however, that decrease of Pπ th turns out more gradually than variation of n g[(λ)]-2. This is mainly becausethe term (A eff/L eff), on which Pπ th(λ) depends in Eq. (4), becomes considerably larger with approaching the band edge, and thereby, total variation of Pπ th becomes moderate to a considerable extent. Note that L eff and A eff are found, by calculation, to be smaller and larger, respectively. In this connection, we remark that for L eff, which is determined by incident power depletion due to both the propagation (scattering) loss and TPA effect, we here adopted L α≡[1-exp(-αL)]/α with α the attenuation (scattering) coefficient, since β was found to be small enough to be neglected, as will be described later; for detail, see the Appendix.
Fig. 4. Required coupled power Pπ(n 2) for yielding π phase shift as a function of wavelength. A theoretical one (solid curve) estimated on the basis of Eq. (4) is also presented for comparison. For detail, see the text.
From Pπ ob we can estimate a nominal Kerr constant n’2 defined, instead of Eq. (2), as
which is conventionally used in the homogeneous system. The wavelength (or V g) dependence of n’2 thus obtained is presented in Fig. 5. In Fig. 5 we also depict for comparison a theoretical curve given by n’2=B· (n g/n 0)2 with B=(2n 0)/(ε·c)=2/(n 0·c),
Fig. 5. Variation of the observed n’2-value with operating wavelength in a range near the band edge; the solid curve denotes the theoretical n’2-value calculated on the basis of Eq. (5), (see the text).
which is derived from Eqs. (2) and (5); we depicted the n’2(λ) curve such that it goes through the data point at 1,528 nm, since the magnitude of n’2(λ) is not known. The comparison reveals that agreement between the experimental and the theoretical is also good.
3.4 Two-photon absorption and its dependence on Vg
When semiconductor is selected for NLO material in the near infrared wavelength region, TPA is often impedimental to instantaneous OKE. So we experimentally examined influence of TPA on the present phase shift. Specifically, we estimated the magnitude of β at several wavelengths in order to compare it with that of n 2’; in some cases this comparison will be made later in terms of F (the two-photon figure of merit). We estimated β in the following way. The variation of I(z) with propagation along the z-axis is expressed as
where the first and the second term on the right-hand side denote attenuation due to scattering and TPA, respectively. It is noted that β is a nominal TPA coefficient defined as β=β TPA·b where β TPA and b stand for TPA coefficient and the modal structure factor given by S eff/A eff with A eff being the effective cross-section area for linear phenomena. Through use of the solution I(L) [output intensity] for Eq. (6), we find the following relation,
where I(0) is the input intensity. Equation (7) reveals that β-value can be extracted from the gradient of the straight line for a plot of the inverse transmittance T 1=I(0)/I(L) versus I(0) [18,19]. Figure 6(a) shows an example of output power I(L)=P(L)/Aeff observed as a function of input power I(0)=P(0)/Aeff at 1,533 nm. Figure 6(b) displays an example of the abovementioned plot. In this case β-value thus extracted is 2.2×10-9 cm / W.
Fig. 6. (a). Variation of output power P(L) as a function of input power P(0) (coupled power to the WG). (b). Plot of inverse transmittance as a function of P(0) for (a).
Similar measurements were made within the input power below 2 W in a wavelength range from 1,532 to 1,548 nm near the band edge. In all cases, the data points are well fitted with a straight line. Figure 7 shows variation of β with wavelength thus obtained. Figure 7 reveals that β increases as the band edge is approached, but the increase turns out much gradual compared to variation of n g 2 with wavelength. Note that β should increase in proportion to (n g / n 0)2 as is the case for n 2’. Therefore, the present result seems to apparently disagree with the theoretical expectation. However this result is not necessarily unreasonable, if we take into consideration the large wavelength dispersion of |Im[χ (3)(ω)]| responsible for TPA. Note that, unlike the case for n 2’, the above dispersion is known to be very large in the present wavelength region just above the wavelength corresponding to half the electronic band gap [9]. More precisely, as is seen in Fig. 7, agreement of the variation (relative) with wavelength between the experimental and the theoretical, i.e., |Im[χ (3)(ω)]|·n g 2 is fairly good except one data point at 1,548 nm.
Fig. 7. Dependence of β on wavelength in a wavelength range near the line-defect band edge; for comparison, the theoretical expectation is also shown.
4. Discussion
Let us discuss first the validity and the significance of the results from a few points of view. It is interesting to compare the present value of n’2 of 1.0×10-12 cm2/W for n g=9.0 obtained at 1,534 nm with the corresponding value of 1.4×10-13 cm2/W in Al0.18Ga0.82As channel WG having n g=n 0=3.5 in Ref. [9]. Postulating that n’2 is proportional to n g 2, the value estimated from n g=3.5 assumed for the PhC WG becomes (1.0×10-12)×(3.5/9.0)2=~1.5×10-12 cm2/W. Therefore the present observed value turns out consistent nicely with that in Ref. [9].
In this experiment we have verified that the third-order nonlinear NOPs or the coefficients are enhanced to great extent in the guided mode having slow-V g in the PhC WG. Specifically we have verified that both n’2 and β become large in proportion to n g 2. This result indicates that although these coefficients are intrinsic to the material, they depend also on n g or V g. Furthermore, concerning V g we have also experimentally shown that the energy required for inducing π-phase shift for a unit sample length is reduced enormously in a PhC WG with a very small Seff like the present one.
Next, the above result indicates that it may be possible to develop the ultra-fast and ultrasmall all-optical switches by using the present type PhC WG. We discuss this problem by considering here, as an example, the Symmetric-Mach-Zehnder(SMZ) switch, schematic of which is shown in Fig. 8 [2, 6]. This switch acts in such a manner as signal pulse (SP) appears at the output port of either cross or bar with or without control pulse (CP). This behavior arises from the π-phase-shift in the upper arm induced by CP through cross-phase modulation (XPM). We choose both pulse durations T 0 of SP and CP as 5 ps, and the wavelength separation ∆λ between CP and SP as 8 nm by setting λ C=1,534 nm and λ S =1,542 nm, respectively, where λ C (λ S) is wavelength of CP (SP). We must take several practical factors into account to ensure short pulse operation. Those include broadening of pulse duration due propagation loss, group velocity mismatching, and spectral broadening due to n 2. Therefore, we introduce, in addition to Leff, the following interaction lengths characteristic of the respective phenomena [20],
Fig. 8. Schematic of SMZ where PCDC, photonic crystal directional coupler; CP and SP are control pulse and signal pulse, respectively.
where P C(0) is the coupled power of CP, and the superscript S(C) represents the signal (control) light. The following inequalities among the above lengths are required to hold,
The first and second inequalities are introduced to ensure the π-phase shift, and the third one is required to keep the pulse width practically unchanged. Note that those inequality relations change depending on T 0, P C and ∆λ.
Let us examine specifically whether or not the above requirements are satisfied for a sample of L=3×10-2 cm. The parameter values at λ C=1,534 nm and λ S =1,542 nm are presented in Table I. Considering that in the XPM case the required power for generating π-phase shift is half of the SPM case [3, 20], n’2 can be replaced with 2n’2. Moreover, as n’2 is proportional to
Table 1. A list of parameters at wavelengths of control(λ C)- and signal(λ S) pulses
n g S· n g C, γ in Eq. (8) is evaluated as ~5.6×101 cm-1·W-1. Thus substitution of P C=1.1 W gives L NL=~1.5×10-2 cm. We arrange those lengths below in inverse order of size.
It follows from this that the inequality relations of Eq. (9) are fulfilled.
Here we mention another requirement related to impediment caused by TPA. In the case of semiconductor, the impediment is devided into two cases. One is that one-photon absorption of TPA-excited free-carriers in the conduction- and valence band gives also rise to refractive index change ∆n, in addition to the one described thus far. In this case the spectra due to this ∆n get blue-shifted, non-symmetric and distorted, so that ultra-fast response is no longer expected [17, 21–23]. The other case, which is possible to occur even when the freecarriers can be ignored because of their fast decay, is that the incident light power is gradually depleted along a WG by TPA, so that π-phase shift is possible o be not reached. Notice that OKE and TPA are the same third-order nonlinear processes. In the case of α ≃ 0 this problem has been successfully discussed in terms of F=(β·λ 0)/n 2 already described [11,12]. The requirement for reaching π-shift is 2 ≥ F. In the present case, in which α can not necessarily be ignored, the situation is complicated. However, the fact that Pπ has been actually observed at all wavelengths examined indicates that the first term dominates the second one on the lefthand side in Eq. (6), or α ≫ βI(0). The above statement is borne out by the fact that 0.75>F is well satisfied as is shown in Fig. 9.
Finally, we briefly mention that the similar group-velocity-dependence should also be expected in the third-order nonlinear-optical processes other than OKE and TPA. In particular, the gain for the signal in stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) should be proportional to n g P × n g S just like the XPM case, where S(P) refers to signal (pump). This dependence for SRS has already been derived and discussed in Refs. [24, 25].
5. Conclusion
Using an optical pulse of 5 ps in duration, we experimentally investigated in detail two important nonlinear coefficients n 2 and β, related to SPM-induced phase-shift and TPA, respectively, in a AlGaAs-based line-defect PhC WG at telecommunication wavelengths. In particular, we examined the dependences of Pπ required for generating π-phase shift, n’2 (and also n 2), and β on V g in a wavelength range near the guided-mode band edge; for this use, we also measured V g in the same range. We find that both n’2, and β increase in proportion to n g 2, while Pπ decreases being inversely proportional to n g 2, as the band edge is approached. Based on all such information, and taking into consideration several important practical parameters involved, we have shown that an ultra-fast all optical switch with an ultra-small size of 0.3×0.3 mm2, operating due to XPM by optical Kerr effect with pump pulse energy of ~2.8 pJ is feasible to be developed, while keeping essentially the pulse duration. We have also discussed the present case in terms of a two-photon figure-of-merit.
Appendix. Effective sample length Leff
We derive here the effective sample length L eff in the case where incident laser pulse intensity I(0) is depleted with propagation, due to both scattering loss and TPA-induced attenuation. We define L eff as
The solution of Eq. (6) is given by
Accordingly, we find
It is noted that for α≠0 and βI(0)Lα≫1 including the case for β=0, L eff= Lα,; namely, L eff is reduced to the conventional effective length [20]. In the general case for α≠0 and β≠0, we know that for 1.7 < βI(0)Lα < 50, ln[1+ βI(0)Lα] is in a range between 1.0 and 4.0. So, from ∆φ = π ≤ (2π/λ 0)n’2 I(0)L eff we see in this case that the inequality relation 2>F, or instead, 2>(F/4) holds for ln[1+ βI(0)Lα]=1 or 4, respectively. This indicates that 2>F is satisfied even for sufficiently large β; notice that this condition is the same one in the case for α=0 [11,12].
On the other hand, if βI(0)Lα is too small, the relation 2>F no longer makes sense, but instead, 2>F’ holds in terms of a new figure of merit F’=(λ 0/n’2)(1/I(0)L a), since (1/β)ln[1+βI(0)Lα]≈ I(0)Lα. This relation does not depend on β, but only on α through Lα. This relation indicates that Lα is better to be longer for a given I(0) under the condition that βI(0)Lα≪1; notice that for α=β=0, the relation is reduced to ∆φ = π < (2 π/λ 0)n2I(0)L or 2 > (λ 0/n 2)(1/I(0)L a).
Acknowledgments
The authors are grateful to Mr. Y. Kitagawa and Dr. N. Ozaki of the University of Tsukuba for help in experiment in the early stage. They express sincere thanks to Drs. Y. Tanaka of Fujitsu Laboratories Ltd and Y. Watanabe of the University of Tsukuba for providing the software to simulate SPM-induced spectral broadening and calculating the effective areas, respectively. We also thank to Drs. Y. Sugimoto of National Institute for Materials Science (NIMS) and H. Kawashima of National Institute of Advanced Industrial Science and Technology (AIST) for help in sample preparation. This work was financially supported by a Grant-in-aid for Scientific Research from the Japanese Ministry of Education, Sciences, Sports, Culture, and Technology and also by NEDO within the framework of NEDO grant.
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