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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 649–659
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Modeling of the dynamic transmission properties of chalcogenide ring resonators in the presence of fast and slow nonlinearities

Kazuhiko Ogusu and Yosuke Oda  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 649-659 (2011)
http://dx.doi.org/10.1364/OE.19.000649


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Abstract

We propose a simple iterative method for calculating the dynamic behavior of ring resonators with fast and slow (cumulative) optical nonlinearities when an optical pulse with an arbitrary-shaped envelope is incident into them. In the case of a slow nonlinearity, the nonlinear phase shift and nonlinear absorption are temporally-integrated over the incident pulse. In this paper, we consider two types of single-ring resonators made out of As2Se3 chalcogenide glass with high nonlinearity and investigate the dynamic properties (especially the effect of the cumulative nonlinearity on optical bistability) using known nonlinear material parameters. It is found that the cumulative nonlinearity suppresses overshoot and ringing after switching, decreases the width of the hysteresis loop between the input and output powers, and shifts its center corresponding to the operating point. The obtained results are useful in developing chalcogenide-based bistable optical devices and the proposed approach is applicable to modeling of a variety of nonlinear optical devices.

© 2011 OSA

1. Introduction

In fact, there are several physical mechanisms that contribute the nonlinear refractive index and nonlinear absorption of materials, depending on the wavelength, pulse width, and peak intensity of an incident light. Moreover ultrafast nonlinearities accompany more or less slow (cumulative) nonlinearities (i.e., thermal), in which case the refractive index change and the absorption change are accumulated over the incident pulse since the decay time is longer than the pulse width [17

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(4), 760–769 (1990). [CrossRef]

19

19. K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express 16(19), 14780–14791 (2008). [CrossRef] [PubMed]

]. It has recently been found that a significant slow nonlinearity with a response time of 15-20 ms is present in chalcogenide glasses, which is presumably attributed not to free-carrier effects or thermal effects, but to photostructural changes inherent in these glasses [20

20. K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008). [CrossRef] [PubMed]

22

22. K. Ogusu and Y. Oda, “Transient absorption in As2Se3 and Ag(Cu)-doped As2Se3 glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. 48(11), 110204 (2009). [CrossRef]

]. Although it is very important to investigate the effect of such cumulative nonlinearities on the performance of nonlinear optical devices, few quantitative studies have been done to date.

In this paper we develop a simple iterative method for calculating the dynamic transmission properties of ring resonators with both instantaneous and cumulative nonlinearities. Moreover we investigate the effect of the cumulative nonlinearity on optical bistability in ring resonators made of As2Se3 chalcogenide glass for its realization. Two types of single-ring resonators with a single coupler and double couplers are investigated in this work. Although the existing iterative method under the assumption of an ideal Kerr nonlinearity is described by a set of simple difference equations [13

13. H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998). [CrossRef]

,23

23. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979). [CrossRef]

,24

24. A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997). [CrossRef]

], we have to solve an integro-differential equation in the presence case. It is found that optical bistability can occur even if such a cumulative nonlinearity is present in the devices. However the hysteresis loop showing the relationship between the input and output optical powers is affected by it.

2. Numerical analysis of nonlinear ring resonators

2.1 Single-coupler ring resonator

A. Iterative method considering n2 and α only

Figure 1 (a)
Fig. 1 Schematic diagram of single-ring resonators and the definition of electric fields for analysis. (a) A single-coupler ring resonator. (b) A double-coupler ring resonator.
shows a schematic diagram of a single-coupler nonlinear ring resonator, which consists of a bus waveguide and a ring of length L(=2πr, r being the ring radius). In order to facilitate understanding of a novel approach proposed in this work, we briefly summarize the conventional iterative method for this configuration [13

13. H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998). [CrossRef]

,23

23. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979). [CrossRef]

,24

24. A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997). [CrossRef]

]. For this purpose, the nonlinearity of the ring waveguide is assumed to be of the Kerr type, i.e., the refractive index is given by
n=n0+n2I=n0+n2n02η0|E|2=n0+n2PSeff,
(1)
where n 0 is the linear refractive index, n 2 is the nonlinear refractive index with an ultrafast (instantaneous) response time, and η 0 is the wave impedance in vacuum. I is the optical intensity inside the waveguide, E is the corresponding optical electric field, and P is the optical power. S eff is the effective mode area of the waveguide. Moreover we define the slowly-varying complex electric field at each position as shown in Fig. 1(a). At the coupling point between the bus waveguide and the ring waveguide, the field amplitudes of incident wave E in(t), transmitted wave E out(t), and circulating cavity wave E c(z,t) satisfy the following equations:
Eout(t)=1γ(1κEin(t)jκEc(L,t)),
(2)
Ec(0,t)=1γ(jκEin(t)+1κEc(L,t)),
(3)
where γ and κ are the fractional intensity loss and intensity coupling coefficient of the coupler, respectively.

Assuming the instantaneous Kerr effect and linear loss of the ring waveguide, the cavity field E c(L,t) at the end of the ring can be expressed by the prior cavity field E c(0,t-τ R) at the entrance as follows:
Ec(L,t)=Ec(0,tτR)exp(αL/2)exp[j(ϕ0+ϕN(tτR))],
(4)
where α is the intensity attenuation coefficient of the ring waveguide and τ R( = n 0 L/c) is the round-trip time of the cavity. ϕ 0( = n 0 k 0 L)is the linear phase shift due to propagation around the ring and ϕ Ν(t-τ R) is the nonlinear phase shift, which is given by
ϕN(tτR)=n0n2k02η00L|Ec(z,tτR+n0z/c)|2dz=n0n2k02η0|Ec(0,tτR)|21exp(αL)α,
(5)
where z is the propagation distance along the ring from the coupling point and k 0 is the propagation constant of free space. In the actual computation, it is convenient to replace ϕ 0 by the detuning from resonance Δϕ 0 = ϕ 0-2qπ, since exp(- 0) has period 2π. Substituting Eq. (4) into Eqs. (2) and (3), we have

Eout(t)=1γ{1κEin(t)jκEc(0,tτR)exp(αL/2)exp[j(ϕ0+ϕN(tτR))]},
(6)
Ec(0,t)=1γ{jκEin(t)+1κEc(0,tτR)exp(αL/2)exp[j(ϕ0+ϕN(tτR))]},
(7)

As seen from Eq. (6), the output field E out(t) is simply an iteration of the cavity field E c(0,t), with respect to the cavity round-trip time τ R . Thus, we can calculate the dynamic properties when an optical pulse with an arbitrary temporal profile is incident into the nonlinear ring using Eqs. (6) and (7).

B. Iterative method considering ultrafast and cumulative nonlinearities

Considering the concept of the iterative scheme, what we have to do is to establish the relation between the two cavity fields E c(0,t-τ R) and E c(L,t). Since these two cavity fields are complex, we decide to express them in polar form as follows:
Ec(0,tτR)=|Ec(0,tτR)|exp(jθ0(tτR)),
(8)
Ec(L,t)=|Ec(L,t)|exp[j(θ0(tτR)+ϕ0+ϕN(tτR))],
(9)
where θ 0(t-τ R) is the phase with respect to the input field E in(t-τ R), i.e., the phase difference between E c(0,t-τ R) and E in(t-τ R), which changes with time as seen from Eq. (3). In the general case where there are a variety of nonlinearities, we must determine the relation between E c(0,t-τ R) and E c(L,t) by directly solving the wave equation that governs the propagation through the nonlinear medium.

In the characterization of third-order nonlinear optical materials, the nonlinear equations of the optical intensity I (∝|E|2) and the phase ϕ N are usually employed for the purpose and the material parameters associated with their optical nonlinearities are defined. Using the slowly-varying envelope approximation, we can fully determine the light propagation within the nonlinear medium by [18

18. 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. 9(3), 405–414 (1992). [CrossRef]

20

20. K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008). [CrossRef] [PubMed]

]
dI(z,t)dz=(α+βI(z,t)+σabN(z,t))I(z,t),
(10)
dϕN(z,t)dz=k0Δn=k0(n2I(z,t)+σrN(z,t)),
(11)
where β is the two-photon absorption coefficient and Δn is the change in the refractive index. N(z,t) is the density (henceforth called the carrier density) of the excited electronic state (for example, a free-carrier state or a higher-lying bound state) induced by one-photon absorption. σab and σr are the changes in absorption coefficient and refractive index per unit carrier density (or absorbed photon density), respectively. The photogenerated carrier density is governed by the following rate equation:
dN(z,t)dt=αI(z,t)ωN(z,t)τ,
(12)
where ω is the photon energy and τ is the decay time of the excited state. If τ is much longer than the incident pulse width τ p, we can neglect the loss term N(z,t)/τ in Eq. (12) and then have

N(z,t)=   tαωI(z,t)dt.
(13)

In this case, the changes in the absorption coefficient and refractive index are accumulated. We must solve Eq. (10) together with Eq. (13) for an initial value I(0,t), which changes with time. However we cannot expect its analytical solutions since that is an integro-differential equation. Therefore we must develop a simple numerical approach for solving Eq. (10).

φN(tτR)=k0n0Δzi=0M1I(iΔz,tτR+iΔt)+I((i+1)Δz,tτR+(i+1)Δt)2                                                                             +k0σrΔzi=0M1N(iΔz,tτR+iΔt)+N((i+1)Δz,tτR+(i+1)Δt)2         .
(15)

Thus, we can calculate a unknown complex value of E c(L,t) at z=L from the known complex value of E c(0,t-τ R) at z=0. Moreover we can determine the transmitted field E out(t) and the cavity field E c(0,t) for the given incident field E in(t) by substituting the obtained E c(L,t) into Eqs. (2) and (3). The obtained cavity field E c(0,t) will be used as an initial value for the computation after τ R. We can entirely calculate the transmission changes when an optical pulse with an arbitrary temporal profile is incident into the nonlinear ring using repeating such a process at a fixed time interval of Δt.

2.2 Double-coupler ring resonator

The application of the proposed approach to a nonlinear double-coupler ring resonator is straightforward. Figure 1(b) shows the configuration of the ring resonator and the definition of each electric field for analysis, where a ring of length L is coupled to two identical waveguides and is divided into a right and a left half. The naming of each port is based on the analogy between the ring resonator and the Fabry-Perot resonator. E r(t) and E t(t) are the transmitted fields at the reflection and transmission ports, respectively. E c1(z,t) and E c2(z,t) are the right-half (0<z<L/2) and left-half (L/2<z<L) cavity fields, respectively. The complex electric fields are connected as follows:
Er(t)=1γ(1κEin(t)jκEc2(L,t)),
(16)
Ec1(0,t)=1γ(jκEin(t)+1κEc2(L,t)),
(17)
for the first coupler at the input port (or reflection port) and
Et(t)=j1γκEc1(L/2,t),
(18)
Ec2(L/2,t)=1γ1κEc1(L/2,t),
(19)
for the second coupler at the transmission port. We must calculate a unknown complex value of E c2(L,t) at z=L from the known complex value of E c1(0,t-τ R) at z=0 as done in the preceding subsection. The presence of the second coupler at z=L/2 forces the amplitude of the circulating cavity field to decrease by [(1-γ)(1-κ)]1/2 with no phase change.

3. Numerical results and discussion

We present the numerical results for the transient properties of two kinds of nonlinear ring resonators shown in Fig. 1. Since the aim of this work is to obtain useful information in realizing bistable optical devices using As2Se3 glass, the practical computation should be performed using as realistic device and material parameters as possible. We also assume a Nd:YAG laser as a light source. All numerical results presented here were calculated for the following values: free-space wavelength λ=1.064 μm, ring radius r=100 μm, effective mode area S eff=2.0 μm 2, linear refractive index n 0=2.818, linear absorption coefficient α=0.621 cm−1, nonlinear refractive index n 2=3.0×10−17 m2/W, two-photon absorption coefficient β=5.0×10−11 m/W, refractive index change per unit photo density σ r=0.89×10−22 cm3, absorption change per unit photo density σ ab=4.46×10−18 cm2. These linear and nonlinear material parameters are the experimental data obtained using the Brewster-angle technique [25

25. 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(7), 909–911 (2006). [CrossRef] [PubMed]

] and z-scan technique [20

20. K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008). [CrossRef] [PubMed]

,21

21. K. Ogusu and K. Shinkawa, “Optical nonlinearities in As2Se3 chalcogenide glasses doped with Cu and Ag for pulse durations on the order of nanoseconds,” Opt. Express 17(10), 8165–8172 (2009). [CrossRef] [PubMed]

]. In this case, the cavity round-trip time τ R given by n 0 L/c is 5.90 ps for r=100 μm and the linear absorption coefficient α=0.621 cm−1 corresponds to a transmission loss of 0.17 dB per round-trip. In order to clarify the effect of the linear loss and the cumulative nonlinearity on optical bistable behavior, the numerical simulations will be performed for the following three cases:

  • Case (i): n 2≠0, α=0, and the other nonlinear parameters=0.
  • Case (ii): n 2≠0, α≠0, and the other nonlinear parameters=0.
  • Case (iii): All linear and nonlinear parameters≠0.

Note that the difference between cases (i) and (ii) is the linear loss of the ring waveguide. Although the nonlinear absorption is caused by the fast two-photon absorption and thecumulative effect, the contribution of the former effect is small in this case.

3.1 Single-coupler ring resonator

Figure 5
Fig. 5 Input-output characteristics of the nonlinear single-coupler ring resonator with γ=0.1 and κ=0.1 for three values of initial detuning Δϕ 0. The incident pulse width is τ P=500 ps.
shows the dependence of the input-output characteristics on the initial detuning Δϕ 0. In this numerical example, a Gaussian pulse with τ P=500 ps is assumed to be incident into the ring resonator with γ=0.1 and κ=0.1. It is found that the width of the hysteresis loop and the switch-off power increase as the magnitude of initial detuning is increased. If the input power does not reach the switching threshold, the ring resonator exhibits linear behavior.

Figure 6 shows the dependence of the input-output characteristics on the pulse width τ P of an incident Gaussian pulse. The ring resonator with γ=0.1, κ=0.1, and Δϕ 0=−0.1π is assumed in this numerical example. For purpose of comparison, the numerical result for case (ii) and τ P=5.0 ns is also presented in the figure. It is confirmed that the width of the hysteresis loop decreases with increasing pulse width. In the absence of the cumulative nonlinearity, the transient hysteresis loop approaches steady-state solution as the pulse width is increased. However such a situation does not occur in the presence of the cumulative nonlinearity. Cumulative nonlinear refraction mainly shifts the operating point (initial detuning) of the device, moving the center of the hysteresis loop on the higher input-power side. Moreover cumulative nonlinear absorption increases the switch-on threshold, decreasing the width of the hysteresis loop. It can be expected that the width of the hysteresis loop decreases with increasing incident pulse width.

3.2 Double-coupler ring resonator

Figure 7
Fig. 7 Temporal change in the power transmitted from the two output ports when a Gaussian pulse with a peak power of 3 W and a pulse width of 500 ps is incident in the nonlinear double-coupler ring resonator with κ=0.1, Δϕ 0=−0.1π, and γ=0 (a) and 0.1 (b). For comparison, the results are given for two cases (ii) and (iii).
shows the nonlinear pulse response at the two output ports when a Gaussian pulse with a peak power of 3 W and a pulse width of 500 ps is incident in the nonlinear double-coupler ring resonators with κ=0.1, Δϕ 0=−0.1π, and γ=0 (a) and 0.1 (b). In order to clarify thecontribution of the cumulative nonlinearity, the output powers, P t and P r, were calculated for two cases (ii) and (iii). Regarding transmission bistability, the switch-on and switch-off occur at the leading edge and the trailing edge of the incident pulse, respectively. The cumulative nonlinearity can suppress overshoot and ringing after switching, but it brings an additional loss. The temporal variation of P t shows that the induced loss increases with time since the nonlinear absorption continues to accumulate. Figure 8
Fig. 8 Input-output characteristics of the nonlinear double-coupler ring resonator with κ=0.1, Δϕ 0=−0.1π, and γ=0 (a) and 0.1 (b) for two values of pulse width τ P. For comparison, the results are given for two cases (ii) and (iii).
shows the input-output characteristics at the transmission port of the double-coupler ring resonators with κ=0.1, Δϕ 0=−0.1π, and γ=0 (a) and 0.1(b). The numerical results were computed for two cases (ii) and (iii), and two pulse widths τ P=500 ps and 5 ns. We can clearly confirm that the effect of the cumulative nonlinearity on the switch-off is greater than that on the switch-on. It is also confirmed that the width of the hysteresis loop decreases with increasing incident pulse width. Moreover it is worth pointing out that we cannot assume an infinite pulse width since the accumulated loss becomes infinite.

4. Conclusions

We proposed a novel approach for calculating the dynamic transmission properties of single-ring resonators with both ultrafast and cumulative nonlinearities when an optical pulse with an arbitrary-shaped envelope is incident into them. Single-coupler and double-coupler ring resonators made of As2Se3 chalcogenide glass with high nonlinearity were considered and the effect of the cumulative nonlinearity on optical bistability was investigated using the proposed numerical method and known nonlinear material parameters. It has been found that we can obtain optical bistability under suitable conditions although the cumulative nonlinearity is considerable intense. Moreover the cumulative nonlinearity can suppress overshoot and ringing after switching. But it decreases the width of the hysteresis loop between the input and output powers, and shifts its center corresponding to the operating point. The proposed approach is easily applicable to higher-order nonlinearities, finite carrier decay time, i.e., the case where the loss term Ν(z,t)/τ in Eq. (12) is not negligible, and so on.

Acknowledgement

This work was supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

References and link

1.

O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters–A tutorial overview,” J. Lightwave Technol. 22(5), 1380–1394 (2004). [CrossRef]

2.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998). [CrossRef]

3.

D. G. Rabus, Integrated ring resonators (Springer, Berlin, 2007).

4.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002). [CrossRef]

5.

T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003). [CrossRef]

6.

T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004). [CrossRef] [PubMed]

7.

V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004). [CrossRef] [PubMed]

8.

Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006). [CrossRef] [PubMed]

9.

D. Sarid, “Analysis of bistability in a ring-channel waveguide,” Opt. Lett. 6(11), 552–553 (1981). [CrossRef] [PubMed]

10.

J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994). [CrossRef]

11.

K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995). [CrossRef] [PubMed]

12.

K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. 32(9), 1537–1543 (1996). [CrossRef]

13.

H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998). [CrossRef]

14.

A. Zakery, and S. R. Elliott, Optical nonlinearities in chalcogenide glasses and their applications (Springer, Berlin, 2007).

15.

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

16.

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(3), 265–267 (2004). [CrossRef] [PubMed]

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(4), 760–769 (1990). [CrossRef]

18.

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. 9(3), 405–414 (1992). [CrossRef]

19.

K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express 16(19), 14780–14791 (2008). [CrossRef] [PubMed]

20.

K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008). [CrossRef] [PubMed]

21.

K. Ogusu and K. Shinkawa, “Optical nonlinearities in As2Se3 chalcogenide glasses doped with Cu and Ag for pulse durations on the order of nanoseconds,” Opt. Express 17(10), 8165–8172 (2009). [CrossRef] [PubMed]

22.

K. Ogusu and Y. Oda, “Transient absorption in As2Se3 and Ag(Cu)-doped As2Se3 glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. 48(11), 110204 (2009). [CrossRef]

23.

K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979). [CrossRef]

24.

A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997). [CrossRef]

25.

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(7), 909–911 (2006). [CrossRef] [PubMed]

26.

A. Yariv, Optical Electronics (Holt, Rinehart and Winston, New York, 1985), p. 151.

OCIS Codes
(190.1450) Nonlinear optics : Bistability
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(230.1150) Optical devices : All-optical devices
(230.4320) Optical devices : Nonlinear optical devices
(230.5750) Optical devices : Resonators

ToC Category:
Nonlinear Optics

History
Original Manuscript: September 20, 2010
Revised Manuscript: December 13, 2010
Manuscript Accepted: December 14, 2010
Published: January 5, 2011

Citation
Kazuhiko Ogusu and Yosuke Oda, "Modeling of the dynamic transmission properties of chalcogenide ring resonators in the presence of fast and slow nonlinearities," Opt. Express 19, 649-659 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-649


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References

  1. O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters–A tutorial overview,” J. Lightwave Technol. 22(5), 1380–1394 (2004). [CrossRef]
  2. B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photon. Technol. Lett. 10(4), 549–551 (1998). [CrossRef]
  3. D. G. Rabus, Integrated ring resonators (Springer, Berlin, 2007).
  4. V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P.-P. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8(3), 705–713 (2002). [CrossRef]
  5. T. A. Ibrahim, R. Grover, L. C. Kuo, S. Kanakaraju, L. C. Calhoun, and P.-P. Ho, “All-optical AND/NAND logic gates using semiconductor microresonators,” IEEE Photon. Technol. Lett. 15(10), 1422–1424 (2003). [CrossRef]
  6. T. A. Ibrahim, K. Amarnath, L. C. Kuo, R. Grover, V. Van, and P. T. Ho, “Photonic logic NOR gate based on two symmetric microring resonators,” Opt. Lett. 29(23), 2779–2781 (2004). [CrossRef] [PubMed]
  7. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. 29(20), 2387–2389 (2004). [CrossRef] [PubMed]
  8. Q. Xu and M. Lipson, “Carrier-induced optical bistability in silicon ring resonators,” Opt. Lett. 31(3), 341–343 (2006). [CrossRef] [PubMed]
  9. D. Sarid, “Analysis of bistability in a ring-channel waveguide,” Opt. Lett. 6(11), 552–553 (1981). [CrossRef] [PubMed]
  10. J. Capmany, F. J. Fraile-Pelaez, and M. A. Muriel, “Optical bistability and differential amplification in nonlinear fiber resonators,” IEEE J. Quantum Electron. 30(11), 2578–2588 (1994). [CrossRef]
  11. K. Ogusu, H. Shigekuni, and Y. Yokota, “Dynamic transmission properties of a nonlinear fiber ring resonator,” Opt. Lett. 20(22), 2288–2290 (1995). [CrossRef] [PubMed]
  12. K. Ogusu, “Dynamic behavior of reflection optical bistability in a nonlinear fiber ring resonator,” IEEE J. Quantum Electron. 32(9), 1537–1543 (1996). [CrossRef]
  13. H. Li and K. Ogusu, “Analysis of optical instability in a double-coupler nonlinear fiber ring resonator,” Opt. Commun. 157(1-6), 27–32 (1998). [CrossRef]
  14. A. Zakery, and S. R. Elliott, Optical nonlinearities in chalcogenide glasses and their applications (Springer, Berlin, 2007).
  15. V. G. Ta’eed, N. J. Baker, L. Fu, K. Finsterbusch, M. R. E. Lamont, D. J. Moss, H. C. Nguyen, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007). [CrossRef] [PubMed]
  16. 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(3), 265–267 (2004). [CrossRef] [PubMed]
  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(4), 760–769 (1990). [CrossRef]
  18. 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. 9(3), 405–414 (1992). [CrossRef]
  19. K. Ogusu and K. Shinkawa, “Optical nonlinearities in silicon for pulse durations of the order of nanoseconds at 1.06 microm,” Opt. Express 16(19), 14780–14791 (2008). [CrossRef] [PubMed]
  20. K. Shinkawa and K. Ogusu, “Pulse-width dependence of optical nonlinearities in As2Se3 chalcogenide glass in the picosecond-to-nanosecond region,” Opt. Express 16(22), 18230–18240 (2008). [CrossRef] [PubMed]
  21. K. Ogusu and K. Shinkawa, “Optical nonlinearities in As2Se3 chalcogenide glasses doped with Cu and Ag for pulse durations on the order of nanoseconds,” Opt. Express 17(10), 8165–8172 (2009). [CrossRef] [PubMed]
  22. K. Ogusu and Y. Oda, “Transient absorption in As2Se3 and Ag(Cu)-doped As2Se3 glasses photoinduced at 1.06 μm,” Jpn. J. Appl. Phys. 48(11), 110204 (2009). [CrossRef]
  23. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30(2), 257–261 (1979). [CrossRef]
  24. A. L. Steele, S. Lynch, and J. E. Hoad, “Analysis of optical instabilities and bistability in a nonlinear optical fibre loop mirror with feedback,” Opt. Commun. 137(1-3), 136–142 (1997). [CrossRef]
  25. 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(7), 909–911 (2006). [CrossRef] [PubMed]
  26. A. Yariv, Optical Electronics (Holt, Rinehart and Winston, New York, 1985), p. 151.

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