IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 48, NO. 5, MAY 2012
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Modeling of Pulse Propagation in Layered Structures With Resonant Nonlinearities Using a Generalized Time-Domain Transfer Matrix Method Peyman Sarrafi and Li Qian
Abstract— We introduce a generalized time-domain transfermatrix (TDTM) method, the only method to our knowledge that is capable of modeling high-index-contrast layered structures with dispersion and slow resonant nonlinearities. In this method transfer matrix is implemented in the time domain, either by switching between time and frequency domains using Fourier transform and its inverse operation, or by replacing the frequency variable (ω) with its temporal operator (−i (d/dt)). This approach allows us to implement the transfer matrix method (which can easily incorporate dispersion, is analytical in nature, and requires less computation time) in the time domain, where we can incorporate nonlinearity of various kinds, instantaneous (such as Kerr nonlinearity), or slow resonant nonlinearity (such as carrier-induced nonlinearity). This generalized TDTM method is capable of incorporate non-analytical forms of dispersion and of nonlinearity, making it a versatile tool for modeling optical devices where dispersion and nonlinearities are obtained phenomenologically. We also provide a few numerical examples to compare our method with the standard finite-difference timedomain (FDTD) method, as well as to examine the range of validity of our method. For pico-second and longer pulses, our results agree with the FDTD simulation results to within 1% and the computation time of our method is more than 100 fold reduced compared to that of FDTD for the longest pulse we used. Index Terms— Nonlinear optical devices, photonic crystals, time domain analysis, transfer function.
I. I NTRODUCTION
N
ONLINEAR optical layered structures such as nonlinear Bragg waveguides [1,2] are widely used for optical signal processing. Though devices based on resonant (carrierinduced) nonlinearities [3-9] offer compactness (micron-size) and low-energy operation, Bragg waveguides made of resonant nonlinear materials have not been analyzed. One reason is that modeling of such devices encounters several difficulties. First, as the mechanisms that induce such nonlinearities can be complex, an analytical relationship between the induced nonlinear polarization and the electric field may not exist. Second, resonant nonlinearity is not instantaneous, as it depends on the history of the input excitation field more than on its current value. As a result, the frequency dependence of the nonlinear susceptibility does not have an analytical form. For the above
Manuscript received September 20, 2011; revised December 22, 2011; accepted January 2, 2012. Date of publication January 6, 2012; date of current version March 2, 2012. The authors are with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S3G4, Canada (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/JQE.2012.2183116
reasons, it is conventional [5, 9] to model nonlinearity in the time domain, i.e., use rate equations to calculate the carrier densities and then relate carrier density to changes in the real and imaginary parts of the susceptibility (i.e. absorption and refractive index, respectively) phenomenologically. However, time-domain methods are in general not conducive to modeling dispersion, which is important for materials with resonant nonlinearities, as they are very dispersive especially near the resonant frequency. A third difficult arises when such nonlinear materials are employed in layered structures, for example, in order to increase sensitivity and reduce switching energy of the device. Conventional modeling approaches, such as Finite Difference Time Domain (FDTD) methods [10] or the Split-Step Fourier (SSF) methods [11], do not model accurately layered structures with high index contrasts and abrupt variations, due to either high numerical discretization errors in FDTD [12] or the violation of the weak dielectric perturbation condition required by the slow-varying envelope approximation (SVEA) [13] in the SSF methods. Later developments of SSF methods led to the bidirectional Beam propagation methods (BPMs) [25], which can model accurately high-index-contrast layered structures and can be generalized to include Kerr nonlinearity [26]. Time-domain BPM has also been developed to analyze pulse propagation in such structures [27]. However, nonlinear bidirectional BPMs[26-27] do not model resonant nonlinearities where an analytical form may not exist. Since no method has been found in the literature so far that is capable of modeling high-index-contrast layered structures with dispersion and resonant nonlinearities, a new method is required to fill in this niche in optical device modeling. Another motivation for developing a new model derives from the consideration of (unnecessary) computation burden of a standard FDTD method in modeling optical devices where the pulse envelop is orders of magnitude longer than the optical cycle. FDTD is a widely used method for modeling microwave and optical devices, to the extent that it has become a standard approach upon which many commercial simulation softwares are based. Its time-domain approach also appeals to the problem at hand, as it is straightforward to include rateequations into its formulism [14]. However, FDTD introduces numerical inaccuracies (including but not limited to numerical dispersion), particularly for layered structures with abrupt and large index changes. To reduce numerical dispersion in FDTD, we need high resolution discretization in space, which in
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 48, NO. 5, MAY 2012
following sections. By generalizing the TDTM method for layered media, we have developed a tool for modeling highindex-contrast layered structures with non-analytical form of dispersion and slow resonant nonlinearities for the first time.
Pulse decomposition
(2) Start with m 1 TDOTM
II. F ORMULATION
d dt ∼ j.± Solve for E m (t) Directly in structure (5)
We consider a one-dimensional layered structure consisting of materials that could possess carrier-induced nonlinearities, such as GaAs, InGaAsP, etc. High-contrast refractive indices, which can be obtained by varying the material composition during growth, are modeled. We consider normal incidence of a pulse that has a carrier frequency near the band-edge of some (or, in some cases, all) of the layers, resulting in large, resonant, carrier-induced nonlinearities in these layers. The electric fields of the transmitted and reflected pulses are calculated using the generalized TDTM method, as detailed below. The starting point of our time-domain approach is to treat this nonlinear problem as a piece-wise linear problem temporally: The input pulse is decomposed into a temporal series of subpulses. We let each subpulse propagate through the entire structure one by one, modeled by the linear transfer-matrix method. After the propagation of each subpulse, the material property, i.e., the real and imaginary parts of the nonlinear susceptibility for every segment of the device, is updated. The whole process is summarize in the flowchart (Fig. 1).
FT-TDTM ∼ 0.+
E m (t)
∼ 0.+
E m (ω) ∼ j.±
Calculate E m (ω) in structure. (3) ∼
Increment m
E mj.±(ω)
∼
ω
−i
E mj.±(t)
m M−1?
Yes
Calculating total response with summing partial responses
No Update carrier density N(x)
Calculate, update n(N), α(N) Fig. 1.
Summary of the TDTM and updating process.
A. Pulse Decomposition turn requires high resolution discretization in time (small time steps) in order to satisfy the stability condition [12]. Hence, the computational burden is considerable when we model device interactions with long pulses (thousands of optical cycles), which is almost always the case for optical devices. One way of reducing computational time by overcoming the stability limitations is applying the ADi-FDTD [15,16]. For propagation of optical pulses with long and slow-varying envelope pulses, the Envelope ADi-FDTD is much more efficient [17], because it allows one to use much bigger FDTD time steps by modifying the stability condition of the Envelop FDTD. However, it is very difficult or, in some cases, impossible to find its stability region for dispersive and lossy materials [18], which is the case in materials with resonant nonlinearities. In short, due to the aforementioned difficulties, no satisfactory method has been presented to simulate optical pulse propagation in layered media having resonant nonlinearities. In this paper, we adopt and generalize a Time Domain Transfer Matrix (TDTM) method for modeling pulse propagation in a resonant nonlinear layered structure. We demonstrate that this method is computationally efficient and can easily include dispersion. Although a TDTM method has been previously applied to model Semiconductor Optical Amplifiers (SOAs) [19], the formulation in [19] only considered pulse propagation in a homogeneous material without modeling reflections. The resulting matrices are much simpler, but the method presented in [19] has very limited applications due to its omission of reflections. In fact, including multiple reflections into the TDTM formulism is not trivial, as will be evident in the
The incident optical pulses can be expressed using a slow˜ varying envelope function, E(t), multiplied by a carrier at the optical frequency, as expressed in iω0 t ˜ ∗ e−iω0 t ˜ + E(t) E(t) = E(t)e
(1)
where E(t) is bounded between t = 0 and t = T . Our model works with the slow-varying envelope rather than the electric field of the pulse, significantly decreasing the numerical computation time compared to FDTD. Pulse decomposition means that we divide the input pulse, expressed by (1), into shorter excitations (subpulses) to be analyzed sequentially. To limit the spectral breadth of the subpulses, we need to avoid introducing abrupt temporal changes when decomposing the input pulse. To do so, we multiply the pulse with bounded and shifted cosine square functions so that the original pulse envelope is the sum of M semi-cosine functions that have smooth but bounded profiles, each representing a time-shifted subpulse (Fig. 2): ˜ E(t) =
(M−1)
E˜ m (t)
(m=1)
where
πM T × t −m × M 2T T M ˜ × t −m× × × E(t) M 2T 1 |x| < 12 (x) ≡ 0 |x| ≥ 12 . E˜ m (t) = cos2
(2)
Field amplitude
Field amplitude
SARRAFI AND QIAN: MODELING OF PULSE PROPAGATION IN LAYERED STRUCTURES
1
561
∼
∼
E mj−1, +(t)
E mj, +(t) Propagation
0.5 ∼ j−1, −
Em 0
0
1
2
3 4 Time (ps) (a)
5
6
7
0.5
0
zj−1 Fig. 3.
1
0
1
2
3 4 Time (ps) (b)
5
6
7
Fig. 2. Decomposition of the incident pulse to small subpulses. (a) Incident pulse envelop and (b) its decomposition to small subpulses.
B. Time Domain Transfer Matrix After the input pulse is decomposed into subpulses, we model the propagation of each subpulse through the material, and determine the amount of change it introduced to the material’s complex susceptibility, using the Time Domain Transfer Matrix (TDTM) method. There are two variants of the TDTM method, and we will present both in the following two sections. C. Fourier-Transform-Based Time-Domain Transfer Matrix Method (FT-TDTM) Transfer Matrix(TM) used in the frequency domain is a common method for analyzing layered structure. Material dispersion can naturally be included in the frequency-domain. In order to keep the advantage of easy inclusion of dispersion in the frequency-domain while carrying out the rest of the calculations in time-domain, we use Fourier Transform (FT) and inverse Fourier Transform (FT−1 ) to convert formulation from the time domain to the frequency domain, and then vice versa, for each step of the matrix multiplication. This approach of using FT and FT−1 to switch between time and frequency domains is also found in the Split-step Fourier method, but our treatment for the nonlinearity is completely different from that of SSF. The procedure of FT-TDTM is as follows: Like all transfer matrix methods, we divide the structure into several sections,
Boundary condition
zj
Schematic plot of a spatial section in transfer matrix.
each with a constant refractive index. For each section, we can then express the relationship between the forward and backward propagating waves analytically, using the boundary conditions. As shown in Fig. 3, each spatial section (including one of the boundaries) is represented by a matrix, T j , and the relationship between the field envelopes of the propagating waves in neighbor sections (indicated by the solid-line arrows) is given as
The required temporal resolution of the decomposition depends on several factors: (1) the rate of change of the pulse envelope; (2) the rate of change of material nonlinearity; and (3) the frequency response range used to model the material. To be computationally efficient and yet not compromising on accuracy, sub-pulses can be generated with asymmetric cosine squared functions, and their lengths can be dynamically chosen during the analysis, depending on the amount of nonlinear absorption experience by the structure when a subpulse enters. Details are discussed in II.D.
∼
E mj, −(t)
(t)
j,+ E˜ m (ω) j,− E˜ m (ω) A (ω + ω0 ) B (ω + ω0 ) 0 e−ik(ω+ω0 )L = 0 e+ik(ω+ω0 )L C (ω + ω0 ) D (ω + ω0 )
Propagation
Tj
j −1,+ E˜ m (ω) . j −1,− ˜ (ω) Em
Boundary
(3a)
Consequently, the relation of any section with the first one in the time domain is: ⎛⎛ ⎞ ⎞ j j,+ 0,+ ˜ ˜ E m (t) E m (t) ⎠ T k⎠ FT = F T −1 ⎝⎝ (3b) j,− 0,− E˜ m (t) E˜ m (t) k=1 where ωo is the carrier frequency, ω is the frequency detuning, which corresponds to ω in the pulse envelope, k is the spatial frequency and a function of ω, L is the length of the segment, and A, B, C, D are coefficients determined by j,± boundary conditions of segment. E˜ m (ω)s s are the forward (+) and backward (−) waves in the j t h spatial section and the m t h time step. It is worth mentioning that (3a) and (3b) are valid for a small time interval only as we assumed the structure behaves linearly during this interval. This assumption is valid when the subpulse is sufficiently short, and therefore its energy sufficiently low, that it does not contribute to a significant change in the carrier density. The salient feature of this method, like in SSF, is the usage of FT and FT−1 to switch between time domain (where modeling of nonlinear effect is easier) and frequency domain (where modeling of dispersion effect is easier). The drawback is the computational burden of FT and FT−1 , which is performed for the number of spatial sections multiplied by the number of time steps. Nevertheless, because of the analytical nature of the TDTM method, the computation is still faster than a bruteforce FDTD. Additionally, this analysis can be made much faster: the FT and FT−1 operations are needed for the very first
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layer only. The rest of the wave profile can be calculated in the time-domain. This point will be made clearer after we explain the time-domain operators at the end of next subsection. D. Time Domain Operator based Transfer Matrix Method (TDOTM) Realizing that switching between time and frequency domain is time consuming, we introduce in this section an alternative TDTM method that uses time domain operators such that the computation remains in the time domain while still making it possible to include dispersion. Following [19, 20], the FT and FT−1 are substituted with derivative time operators. The procedure is described as follows. 0,− First, we find the field reflected by the structure, E˜ m (ω), 0,+ as a function of E˜ m (ω), which is just the FT of the input subpulse, or F T ( E˜ m (t)). The procedure is quite straightforward: the total transfer matrix is obtained using T = j T j , and then reflection can be extracted knowing the backward wave outside 0,+ the last section is zero. Then, each of E˜ m (ω) can be written j,± j,± in the form of G˜ m (ω)F T ( E˜ m (t)), and therefore E˜ m (t) can j,± −1 be written in the form of F T (G˜ m (ω)F T ( E˜ m (t))), where j,± G˜ m (ω)s can be regarded as the impulse response in the t h j spatial section and can be calculated knowing the T j s elements. We can Taylor expand G(ω) and rewrite each of j,± E˜ m (t) as: j,± F T −1 G m (ω) F T E˜ m (t) ⎞ ⎛⎛ ⎞ Np = F T −1 ⎝⎝ an ωn ⎠ F T E˜ m (t) ⎠ (4) n=0
Second, by replacing ω with the time operator −i d/dt, j,± each of E˜ m (t) can then be expressed as the sum of different orders of the time derivatives of E˜ m (t): ⎛⎛ ⎞ ⎞ Np j,± E˜ m (t) = F T −1 ⎝⎝ an ωn ⎠ F T E˜ m (t) ⎠ n=0 Np
=
n=0
an (−i )n
dn dt n
E˜ m (t)
(5)
Therefore, for each forward and backward wave in each spatial section we have a corresponding series of Taylor expansion, which can be approximated by a polynomial numerical fitting of G(ω), to save computation time. In this way, we keep the functions in time domain, although the coefficients are calculated in the frequency-domain. Calculating G(ω) and finding the corresponding polynomial coefficients for each spatial section (or FT and FT−1 calculation as explained in previous subsection) is still time consuming but the computation time can be further reduced. We need only to perform the above process (or FT-TDTM 0,− (t) only, so calculations) for the first layer, to find E˜ m 0,− only G˜ m (t) is Taylor expanded (or only FT and FT−1 for the first layer are taken). The other unknown wave profiles j,± E˜ m (t) ( f or j ≥ 1) can be determined by writing Eq. (3a) in
the time domain: Np n mj −1,+ (t) mj,+ (t) E E j −1,± n d Am,n (−i ) = , mj,− (t) mj −1,− (t) dt n E E n=0
(6)
N p j,± j,± where Tm (ω) = n=0 Am,n ωn . j,± j,± Am,n can be determined by fitting Tm ω to much lower 0,− order polynomials (than that for G m (ω)), and in most cases j,± Am,n can be found analytically because the frequency depenj,± dence of Tm ω is just a function of material dispersion. Analytical formulas for material with first and second order disperj,± sion are given in [19]. In this way, each E˜ m (t) ( f or j ≥ 1) can be calculated sequentially from one section to the next, in a matrix form and in the time domain. For example, if the material dispersion is negligible, the A, B, C, D coefficients in (3a) can be treated as constants, so (3) can be simplified as below: j,+ E˜ m (t) = F T −1 j,− E˜ m (t) ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −ik L ⎟ j −1,+ A B 0 E˜ m (t) ⎟ ⎜ e F T ⎜ ⎟ j −1,− C D 0 e+ik L ⎜ E˜ m (t) ⎟ ⎜ ⎟
⎜ ⎟ Boundary ⎝ Propagation ⎠
(7)
Tj
where k = (ω0 + ω)n/c − i α/2. Here, n and α respectively the real refractive index and absorption. Converting (8) into time-domain, we have: ⎡ −iω0 n L α − 2 L j −1,+ c A Em (t) t − Cn L ⎢e −iω0 n L α − 2 L j −1,− ⎢ mj,+ (t) c B Em (t) t − Cn L E ⎢ +iω +e = 0nL α ⎢ mj,− (t) mj −1,+ (t) t + n L E ⎣ e c + 2 LC E C +iω0 n L α mj −1,− (t) t + n L +e c + 2 L D E
are ⎤ ⎥ ⎥ ⎥. ⎥ ⎦
C
(8) In practice, even with a very wide input pulse bandwidth (say 100 nm) and considerable material dispersion, a third- or fourth-order polynomial would be sufficient, if bulk materials are used for each section. E. Updating Process j,± After determining the pulse envelopes ( E˜ m (t)) in all spatial sections at each time step, the optical property of the entire structure must be updated. In resonant nonlinear materials, the energy of the incident photons is close to energy of the material band-gap, promoting valence electrons to the conduction band, thus changing the carrier density. The evolution of carrier density at each spatial section N(t, z) can be described by the following rate equation [5]: N(t, z) α(N(t, z), ω0 ) ncε0 d N(t, z) | E(t, z)|2 × =− + dt τ hω0 2 (9) where α(N(t, z), ω0 ) is absorption and can be determined experimentally, or according to certain phenomenological
SARRAFI AND QIAN: MODELING OF PULSE PROPAGATION IN LAYERED STRUCTURES
models, e.g., the Bányai-Koch model [21]. The corresponding carrier-induced refractive index change is found through the Kramers-Kronig relation. An analytical solution for above equation can be written as: $ (m+1) T j M α(Nm−1 (t), ω0 ) t t j j τ τ e × Nm = Nm−1 + e T hω0 (m−1) M % $ %2 ncε z j 1 0 %˜j % × % E m (t, z)% dz dt (10) 2 z j − z j −1 z j −1 j One simple way of approximating E˜ m (t, z) over the thickness (z j − z j −1 ) of layer is: zn zn mj (t, z) = E mj,+ t − j mj,− t + j + E E (11) c c Here, possible temporal overlaps between neighbor subj j pulses (e.i. E˜ m (t) and E˜ m+1 (t)) are not taken into account. This overlap is initially introduced in the pulse decomposition process (Eq. (2)) to avoid sharp temporal rising edges of the subpulses. The overlap is further increased due to the multiple reflections of the structure, causing an effective delay in its response. If the length of each sub-pulse is shorter than the length of the structure, the m+1t h subpulse would start to enter the structure while some part of the previous subpulse is still remaining in the structure and changing the carrier density. Without correcting for this effect, the time-domain method cannot be very accurate for these cases. A better approximation of the carrier density is achieved by modifying the time integration limits in (11), so as to include the carrier density evolution from the arrival time of a pulse to the arrival time of the next pulse at every special section, as expressed here: $ (m+1) T + z j −1 j α(Nm−1 (t), ω0 ) M c n t j j −t /τ eτ × Nm = Nm−1 + e hω0 (m−1) T $ zj M 1 j × (12) I˜m (t, z) dz dt z j − z j −1 z j −1 j where n˜ is average refractive index and I˜m is the intensity of the m t h term plus its interference with previous terms: % zη j zη j %%2 ηcε0 % j,− j j,+ Im = + Em t − % Em t + % 2 c c ∗ zη j zη j j,− j,+ + Em t − + Em t + c c m−1 zη zη j,− j,+ + Ep t+ t− · Ep c c p=1
zη j zη j j,− j,+ + Em t + + Em t − c c ∗ m−1 j,− zη j,+ zη p t+ t− · Ep E c c p=1
(13) The average refractive index is used to approximate group index, which is sufficiently accurate in most cases. If a more
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accurate solution is desired, for example, in structures with a very high contrast, a recursive method can be used, at a cost of computation complexity. In the recursive method, we let the j m + 1t h subpulse to propagate before updating Nm and then j t h update Nm by the arrival time of m + 1 and again let the m + 1t h part to propagate considering new optical property of the guiding material. This recursive procedure may be repeated as much as needed. F. Adaptive Pulse Decomposition As explained in II.A, the ideal pulse length depends on the material properties, and its selection is crucial for ensuring accuracy and computational efficiency. As the material properties vary, the length of sub-pulse can be dynamically adapted to achieve maximum efficiency. To do so, instead of using Eq. (2), one can use an asymmetric cosine squared function for the subpulses and determine their lengths using: + m (t) = cos 2 (t − tm ) × π × ((t − tm )/tm+ ) E 2tm+ + π + + cos2 (t − tm ) × + ) ((t − tm−1 )/tm−1 2tm−1 × E(t) + 1 0
