Hamiltonian optics formalism for microring resonator structures with varying ring resonances

Xiaolan Sun; Zhenshan Yang; Xiaohong Liu; Chao Li; Yanhua Dong; Libin Xie; J. E. Sipe

doi:10.1364/OE.19.007176

1. Introduction

Slowing and stopping light in microring resonator structures [1–5] has attracted much attention in recent years. In schemes for slowing light, the maximum delay in a structure of given length is generally limited by the delay-time-bandwidth product (DBP) [6]. In strategies to stop light, the delay time can be increased beyond the limit set by the DBP without inducing additional dispersion [7–9]; however, in such processes another fundamental characteristic parameter more relevant than the maximum delay is the maximum number of pulses that can be simultaneously stopped inside a structure of given length, which is still limited by the DBP [10]. Different schemes have been proposed to increase the DBP and suppress the dispersive distortion [11–13].

In a recent paper [14], we introduced a scheme to enhance the DBP in a micro-ring resonator system by varying the optical ring resonances throughout the structure. By putting different spectral components of a pulse on resonance with different rings, one can obtain a much larger group delay with relatively small additional dispersive distortion. We showed that a SCISSOR (“Side-Coupled Integrated Spaced Sequence of Resonators”) [5] system of 10 rings with varying resonances can yield a delay 8 times larger than a similar structure of identical resonators.

In this paper, we present a detailed study of the above scheme. We develop a quantitative formalism based on Hamiltonian optics to investigate the group delay as well as the dispersive distortion, and we compare this theory to the mathematically more rigorous transfer matrix method in the frequency domain. Although both approaches yield essentially identical results for the average amplitude and phase response (and thus the group delay), the Hamiltonian formalism provides a deeper insight into the physics of this system and also allows us to obtain a detailed picture of the pulse propagation within the microring structure, which is not given by the transfer matrix method.

The remainder of this paper is organized as follows. In Sec. 2, we briefly review the basics of SCISSOR structures; in Sec. 3 we develop the Hamiltonian optics formalism for slow light processes in SCISSOR structures with varying ring resonances; in Sec. 4, we compare the Hamiltonian optics formalism with the more rigorous transfer matrix method, illustrating that the Hamiltonian optics formalism is a good approximation for a broad class of nearly periodic structures; a summary is given in Sec. 5.

2. Review of the SCISSOR structures

A SCISSOR unit [5] is a microring resonator coupled to two channel waveguides, as shown in Fig. 1. Light traveling in the channels and inside the ring couples at the contact points marked by the filled circles through

(\begin{matrix} E_{3}^{up / low} \\ E_{2}^{up / low} \end{matrix}) = (\begin{matrix} σ & i κ \\ i κ & σ \end{matrix}) (\begin{matrix} E_{4}^{up / low} \\ E_{1}^{up / low} \end{matrix}),

where

E_{m}^{up / low}

are the electric field amplitudes at the indicated points and σ and κ are respectively the self- and cross- channel-ring coupling coefficients. Here we adopt a point-contact model for the coupling, taking both σ and κ to be real; energy conservation requires that σ² + κ² = 1.

Fig. 1 Schematic of a SCISSOR unit.

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At frequency ω, the transfer matrix relating the fields on the left to the fields on the right of the SCISSOR unit (see Fig. 1) is easily found to be given by

(\begin{matrix} E_{R}^{+} \\ E_{R}^{-} \end{matrix}) = M (ω) (\begin{matrix} E_{L}^{+} \\ E_{L}^{-} \end{matrix}), M (ω) = (\begin{matrix} α & β \\ β^{★} & α^{★} \end{matrix}),

with

α = \frac{e^{i π \frac{ω}{ω_{B}}}}{2 i σ sin (π \frac{ω}{ω_{r}})} [e^{i π \frac{ω}{ω_{r}}} - σ^{2} e^{- i π \frac{ω}{ω_{r}}}], β = \frac{κ^{2}}{2 i σ sin (π \frac{ω}{ω_{r}})} .

Here ω_r = c/ (n_rR) is the fundamental ring resonance frequency, with n_r and R being the effective refractive index and radius of the ring, respectively; ω_B = πc/ (n_BL) is the fundamental Bragg resonance frequency in a periodic arrangement of such SCISSOR unit cells, with n_B and L being the effective refractive index and length of the channel waveguides of each unit cell, respectively. A systematic way to derive the transfer matrix for microring units (including the SCISSOR here and the AR block in Sec. 4) can be found in, e.g., [5]. The photonic band structure for a periodic sequence of unit cells is determined by

cos (K L) = \frac{1}{2} T r {M (ω)} = ℜ {α},

where K is the crystal wavenumber, Tr stands for the trace of a matrix and ℜ represents the real part of a complex number. A general property of the band structure of periodic SCISSOR systems is the existence of bandgaps associated with the ring resonances m_rω_r and the Bragg resonances m_Bω_B, where m_r, m_B are positive integers. In this paper, we assume that there exists an (m_r, m_B) pair with m_rω_r ≃ m_Bω_B, and the working frequencies are close to these two resonances.

In Fig. 2 we show a typical band structure of the periodic SCISSOR system by numerically solving Eq. (4) using the following parameter values

ω_{B} = 2 π \times 1.973 THz, ω_{r} = 2 π \times 3.942 THz, σ = 0 . 97; m_{r} = 50, m_{B} = 100 .

The frequency range is around the 100th Bragg resonance (100ω_B = 2π × 197.3THz) and the 50th ring resonance (50ω_r = 2π × 197.1THz). In this figure one can identify the two bandgaps associated with the Bragg and the ring resonances, and three photonic bands: the upper, lower bands and the “intermediate band (IB).” The IB lies between 50ω_r and 100ω_B, and can be very narrow if 50ω_r is sufficiently close to 100ω_B; particularly, when 50ω_r = 100ω_B, this band is completely flat, with a zero group velocity for all K. This remarkable feature of the IB can be exploited to stop light inside the structure by parametric manipulation of the ring resonances. A similar scheme has been proposed for quantum well Bragg structures by Yang et al. [8, 10].

Fig. 2 Photonic band structure of the periodic SCISSOR system.

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Moreover, when (m_B/m_r) [κ²/(2πσ)] is small (i.e., weak coupling), as it usually is, the IB is very flat on the ring resonance side, suggesting a large region in K space with low group velocity and low dispersion. [See Eq. (7) and the discussions following it, where Γ = κ²/(2πσ).] This intriguing feature has been used to increase the delay-time-bandwidth product by varying the ring resonances of the unit cells throughout a SCISSOR structure [14], as will be elaborated in later sections.

In this paper we will focus on light propagating inside the SCISSOR structure with frequency components within the IB; it is thus useful to have an analytical solution for the IB dispersion. To this end, we consider a weak coupling SCISSOR structure, of which the cross-coupling coefficient κ is close to zero and the self-coupling coefficient σ is close to unity. Then the band structure of the SCISSOR system around the two closely spaced Bragg and ring resonances (m_Bω_B and m_rω_r) is formally similar to that of a quantum well Bragg structure [15]. To obtain an analytical expression for the IB, we first expand exp (iπω/ω_r) and sin (iπω/ω_r) in Eq. (3) to first order in (ω – m_rω_r)/ω_r

α = e^{i π \frac{ω}{ω_{B}}} {1 - i \frac{Γ}{m_{r}} \frac{m_{r} ω_{r}}{ω - m_{r} ω_{r}}},

with Γ = κ²/(2πσ). Here we have also made the approximation (1 + σ²)/(2σ) ≃ 1 (since σ ≃ 1). Then we expand exp (iπω/ω_B) in Eq. (6) to first order in (ω – m_Bω_B)/ω_B, substitute the result into Eq. (4), and solve for ω to obtain

ω (K) = \frac{cos (K L) - {(- 1)}^{m_{B}}}{cos (K L) - {(- 1)}^{m_{B}} (1 + π \frac{m_{B}}{m_{r}} Γ)} (m_{r} ω_{r} - m_{B} ω_{B}) + m_{B} ω_{B} .

Eq. (7) shows that the IB extends from ω = m_Bω_B at KL = ±π [1 – (−1)^m_B]/2 to ω = 2/[2 + (m_B/m_r)πΓ](m_rω_r – m_Bω_B) + m_Bω_B at KL = ±π [1 + (−1)^m_B]/2. Particularly, when m_Bω_B = m_rω_r, one has ω(K) = m_Bω_B for all K, i.e., the band is completely flat.

3. Hamiltonian optics formalism for inclined SCISSOR systems

Consider a light pulse consisting of crystal momentum components with K on the ring resonance side of the IB region where the group velocity and dispersion are low. As the light propagates through the periodic SCISSOR structure, it can be significantly delayed with small dispersive distortion. However, for pulses in the picosecond regime one cannot directly take advantage of this, since the eigenfrequencies of Bloch functions in this region of K space are restricted to an extremely narrow frequency range. For the photonic band structure in Fig. 2, a pulse would have to be longer than nanoseconds to suffer little dispersion, and the delay of such a pulse propagating through any reasonable finite structure would be small in units of its pulse length.

To solve this problem, one can apply a “quasi-force” on the light pulse. This quasi-force is created by a variation in some parameter(s) of the unit cells throughout the structure. In our structure this parameter is taken to be the ring resonance. If the quasi-force is appropriately designed, an incident pulse with a frequency spectrum centered at the middle of the IB (throughout this paper, we use “middle of the IB” to refer to the middle-point of the IB in the frequency, i.e., ω, domain) will be pushed, in K space, into the “flat” region at the band edge on the ring resonance side. Thus the group delay is increased while the dispersive distortion is suppressed.

In this section we develop a quantitative theory, based on a Hamiltonian optics formalism [16], to analyze the slow light effects in inclined SCISSOR structures (i.e., nearly periodic SCISSOR structures with varying ring resonances, see Fig. 3a for a schematic). The Hamiltonian formalism in optics is in close analogy to the Hamiltonian formalism in mechanics, and has been applied in the study of optical Bloch oscillations and nonuniform photonic crystals [17–20]. It is an approximate theory that requires the definition of a local band structure. Rigorously the concept of a band structure only exists in a periodic structure, but in a broad class of nearly periodic structures the Hamiltonian optics formalism gives results that agree well with those of more rigorous theories, e.g., the transfer matrix method, yielding the group delay and dispersion to a good approximation. Its effectiveness lies in the fact that it is formally compact, simple, and physically intuitive.

Fig. 3 Schematic of: (a) an inclined SCISSOR structure with varying ring resonances; (b) an AR block; (c) a finite SCISSOR sequence with AR blocks on both ends.

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Consider a photonic band of a periodic SCISSOR structure (we call this “uniformly periodic SCISSOR structure”), ω = g(K), where g(K) depends on the parameters of the unit cells. If some of the parameters vary slowly throughout an otherwise periodic structure (as mentioned before we call this “inclined SCISSOR structure”), one can define a “local band structure,” depending on the position z

ω = g (K; z) .

For each K, the “potential energy” ω varies slowly with z, inducing a quasi-force, f (z; K) = −∂g (K; z)/∂z. The concept of this quasi-force is characterized in the Hamiltonian equations for light propagating inside the structure

\frac{d z}{d t} = \frac{\partial g (K; z)}{\partial K},

\frac{d K}{d t} = - \frac{\partial g (K; z)}{\partial z} .

In our current problem, the photonic band is the IB between m_rω_r and m_Bω_B, and the varying parameter is the fundamental ring resonance of the unit cells, ω_r,j, where j is the sequence number of the unit cells. This variation in the ring resonance can be achieved by varying, e.g., the radius or effective refractive index of the rings. According to Eqs. (7) and (8), the local IB is given by

g (K; z) = \frac{cos (K L) - {(- 1)}^{m_{B}}}{cos (K L) - {(- 1)}^{m_{B}} (1 + π \frac{m_{B}}{m_{r}} Γ)} [m_{r} ω_{r} (z) - m_{B} ω_{B}] + m_{B} ω_{B},

where we put z = jL and treat it as a continuous variable. It should be pointed out that when the initial K [i.e., K₀ = K (t = 0)] is very to close to the band edge on either the Bragg resonance side (Bragg edge) or the ring resonance side (ring edge), this continuous model (i.e., Hamiltonian optics formalism) is expected to fail. There are different reasons for this failure at the Bragg edge and the ring edge. For K₀ close to the ring edge, the pulse is almost completely reflected by the first ring, and a continuous model cannot properly account for the delay in just one ring; for K₀ sufficiently close to the Bragg edge, the quasi-force tends to infinity in the continuous model, so the Hamiltonian optics formalism fails as well.

We consider an inclined SCISSOR structure where the ring resonance varies linearly with the sequence number j of the unit cells,

m_{r} ω_{r, j} = m_{r} ω_{r} + (j - 1) \times \frac{m_{B} ω_{B} - m_{r} ω_{r}}{N},

which in the continuum limit is

m_{r} ω_{r} (z) = m_{r} ω_{r} + (z - L) \times \frac{m_{B} ω_{B} - m_{r} ω_{r}}{N L},

with z = jL and N being the total number of unit cells that it takes for the ring resonance to change from m_rω_r (i.e., the m_rth ring resonance of the first ring) to m_Bω_B. Substituting Eqs. (11, 13) into the Hamiltonian equations (9, 10) we have

\frac{d z}{d t} = \frac{Δ Ω}{N} \frac{{(- 1)}^{m_{B} + 1} π \frac{m_{B}}{m_{r}} Γ sin (K L)}{{[cos (K L) - {(- 1)}^{m_{B}} (1 + π \frac{m_{B}}{m_{r}} Γ)]}^{2}} [z - (N + 1) L],

\frac{d K}{d t} = \frac{Δ Ω}{N L} \frac{cos (K L) - {(- 1)}^{m_{B}}}{cos (K L) - {(- 1)}^{m_{B}} (1 + π \frac{m_{B}}{m_{r}} Γ)},

where ΔΩ = m_rω_r – m_Bω_B.

Eq. (15) can be solved analytically by separating K and t

\begin{array}{l} \frac{Δ Ω}{N} t & = & [K L - {(- 1)}^{m_{B}} π \frac{m_{B}}{m_{r}} Γ {(tan \frac{K L}{2})}^{{(- 1)}^{m_{B} + 1}}] \\ - & [K_{0} L - {(- 1)}^{m_{B}} π \frac{m_{B}}{m_{r}} Γ {(tan \frac{K_{0} L}{2})}^{{(- 1)}^{m_{B} + 1}}], \end{array}

where K₀ = K (t = 0). Eq. (16) implicitly determines the evolution of K over time once K₀ is known. By dividing Eq. (15) with Eq. (14) one has

\frac{d K}{d z} = {(- 1)}^{m_{B} + 1} \frac{[cos (K L) - {(- 1)}^{m_{B}} (1 + π \frac{m_{B}}{m_{r}} Γ)] [cos (K L) - {(- 1)}^{m_{B}}]}{π \frac{m_{B}}{m_{r}} Γ L sin (K L)} \frac{1}{z - (N + 1) L},

which can also be solved by separating K and z

\frac{z}{L} = 1 + N {1 - \frac{cos (K_{0} L) - {(- 1)}^{m_{B}}}{cos (K_{0} L) - {(- 1)}^{m_{B}} (1 + π \frac{m_{B}}{m_{r}} Γ)} \frac{cos (K_{0} L) - {(- 1)}^{m_{B}} (1 + π \frac{m_{B}}{m_{r}} Γ)}{cos (K L) - {(- 1)}^{m_{B}}}} .

Eq. (18) gives the relation between the crystal wavenumber K and the position of the pulse traveling inside the structure.

Eqs. (16, 18) can be easily solved numerically to generate the K vs t and K vs z curves, respectively, with the only subtlety being that K evolves continuously instead of being restricted to the first Brillouin zone. That is, we use an extended zone scheme, so for example if K crosses the Brillouin zone boundary at −π/L from the right, it will continue into −π/L – 0⁺ instead of π/L – 0⁺. In Fig. 4, we show the relations between K, z and t, for an inclined structure with the parameter values in (5) and

50 ω_{r, j} = 50 ω_{r} + (j - 1) (100 ω_{B} - 50 ω_{r}) / 12,

i.e., with N = 12 in Eq. (12). Fig. 4a shows the time dependence of K, according to (16) and taking the initial K as K₀ = −0.38/L (corresponding to the middle of the IB, i.e., ω = 2π × 197.2THz in Fig. 2); K as a function of z, determined by (18), is illustrated in Fig. 4b; combining Figs. 4a and 4b gives the time dependence of z in Fig 4c. These figures can be understood as follows. Due to the variation of the ring resonance in the structure, the light pulse feels a quasi-force f(K) in K space, given by the right hand side of Eq. (15)

f (K) = \frac{Δ Ω}{N L} \frac{cos (K L) - {(- 1)}^{m_{B}}}{cos (K L) - {(- 1)}^{m_{B}} (1 + π \frac{m_{B}}{m_{r}} Γ)} .

This force is negative for all K. In the first Brillouin zone, positive K are pushed towards 0 and negative K towards −π/L; when they reach −π/L they jump to π/L and then continue to decrease. In the extended zone scheme, the K are pushed towards −π/L and then continue towards −2π/L. For an incident pulse with negative K components (and thus positive group velocities), this quasi force will push the K-spectrum toward −π into the flat region; after the K pass the Brillouin zone boundary from −π to π, the force will push the K-spectrum through the flat region towards 0, and then to −K₀ (note that −K₀ > 0). These positive K components have negative group velocities, so the final output is reflected rather than transmitted. Since the pulse spends most of its time in the low velocity and low dispersion K-region, one can obtain a large delay with small additional dispersive distortion. More specifically, in Fig. 4c, an incident pulse with a frequency spectrum centered at the middle of the IB and negative K components propagates forward in the structure until the 7th unit cell, and is reflected back since the 50th ring resonance of the 7th unit cell equals the center frequency of the pulse (and after that unit cell, the 50th ring resonance is larger than the center frequency), then it propagates backward until coming out of the structure. The time difference between the incident and reflected pulse at the first unit cell gives the time delay. It is interesting to note that the light pulse stays between the 6th and 7th rings for a long time, as indicated in Fig. 4c, which is expected since the 50th ring resonance of the 7th ring roughly coincides with the center frequency of the pulse.

Fig. 4 The simulation results from the Hamiltonian optics formalism: (a) K as a function of t; (b) K as a function of z; (c) z as a function of t.

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Fig 4a and 4c predict a delay time of T_d = 6.2N/|ΔΩ| ≃ 60ps (for N = 12), which is very close to the exact solution from more rigorous calculations, which will be discussed in the next section. Furthermore, the delay dispersion can be obtained through calculating the delay for different initial K₀ from Eqs. (16, 18). To this end, we first notice from Eq. (18) that z/L equals 1 at both K = K₀ and K = −K₀ – 2π/L. This suggests that if the incident pulse has a spectrum centered at ω (K₀), the reflected pulse is spectrally centered at ω (−K₀ – 2π/L) [note that −K₀ – 2π/L corresponds to −K₀ in the first Brillouin zone]. Thus the total delay time is given by Eq. (16) with K = −K₀ – 2π/L

\begin{array}{l} τ_{Ham} & = & \frac{N}{Δ Ω} [(- K_{0} L - 2 π) - {(- 1)}^{m_{B}} π \frac{m_{B}}{m_{R}} Γ {(tan \frac{- K_{0} L - 2 π}{2})}^{{(- 1)}^{m_{B} + 1}}] \\ - & [K_{0} L - {(- 1)}^{m_{B}} π \frac{m_{B}}{m_{R}} Γ {(tan \frac{K_{0} L}{2})}^{{(- 1)}^{m_{B} + 1}}] \\ = & \frac{2 N}{(- Δ Ω)} [K_{0} L - {(- 1)}^{m_{B}} π \frac{m_{B}}{m_{R}} Γ {(tan \frac{K_{0} L}{2})}^{{(- 1)}^{m_{B} + 1}} + π] . \end{array}

Eq. (21), together with the band structure Eq. (7), ω = ω(K₀), gives the delay dispersion, τ_Ham vs ω, of the Hamiltonian optics formalism. For the inclined structure specified above, this delay dispersion is shown in Fig. 5 as the dashed curve. As we will see in the next section, Eq. (21) actually gives the average delay dispersion instead of the exact one derived by using the transfer matrix method, which is the sum of the average delay dispersion and an oscillatory part. However, as will also be shown later, the dispersive distortion in the reflected pulse (see, e.g., Fig. 8) is essentially determined by the average delay dispersion.

Fig. 5 Delay time as a function of frequency: the Hamiltonian delay τ_Ham for the inclined structure (dashed), the actual delay τ for the inclined structure (solid), and the actual delay for the uniformly periodic structure (dotted).

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Fig. 8 Time dependence of pulse intensities: the input (dotted), transmitted in the 10-unit-cell uniformly periodic system (dashed), reflected in the 10-unit-cell inclined system (solid), transmitted in the 100-unit-cell uniformly periodic system (dash-dotted), and E_out,Ham(t) in Eq. (34) (filled triangles, almost indistinguishable from the solid curve).

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4. Comparison with the coupled-mode transfer matrix method

In this section we compare the results of the Hamiltonian optics formalism in Sec. 3 with those derived from the more rigorous transfer matrix method. Also, we compare the delay and the dispersion of the inclined structure to those of the uniformly periodic structure, showing an obvious performance improvement in the inclined structure.

Before we can compare the two treatments, however, we must address the behavior of light at the boundaries of a finite SCISSOR structure. In the Hamiltonian optics formalism, the immediate reflection at the boundaries is neglected, e.g., by simply assuming that there are infinite number of unit cells on both sides of the finite structure. In practice, the finite SCISSOR structure is usually connected to external waveguides, and one can remove unwanted boundary reflection by adding anti-reflecting (AR) blocks to the ends of the structure (see Figs. 3b and c). A quantitative description of the AR block will be given below in the context of transfer matrix method already used in Sec. 2.

It is easy to see that for a system described by a transfer matrix M

M (ω) = (\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}),

the reflectivity and transmissivity are given by

r = - \frac{M_{21}}{M_{22}}, t = \frac{det M}{M_{22}},

respectively. For an AR block sketched in Fig. 3b, the transfer matrix M_ar is given by

(\begin{matrix} E_{R}^{+} \\ E_{R}^{-} \end{matrix}) = M_{a r} (\begin{matrix} E_{L}^{+} \\ E_{L}^{-} \end{matrix})

and

M_{a r} = \frac{1}{i κ_{a r}} [\begin{matrix} - exp {i \frac{1}{2} π (\frac{ω}{ω_{r, R}^{a r}} + \frac{ω}{ω_{r, L}^{a r}})}, & σ_{a r} exp {i \frac{1}{2} π (\frac{ω}{ω_{r, R}^{a r}} - \frac{ω}{ω_{r, L}^{a r}})} \\ - σ_{a r} exp {- i \frac{1}{2} π (\frac{ω}{ω_{r, R}^{a r}} - \frac{ω}{ω_{r, L}^{a r}})}, & exp {- i \frac{1}{2} π (\frac{ω}{ω_{r, R}^{a r}} + \frac{ω}{ω_{r, L}^{a r}})} \end{matrix}] .

Here

ω_{r, L}^{a r} = \frac{c}{n_{r} R_{a r, L}}, ω_{r, R}^{a r} = \frac{c}{n_{r} R_{a r, R}},

where R_ar,L/R is the radius of the left/right half-ring in the AR block (see Fig. 3b),

ω_{r, L / R}^{a r}

would be the fundamental ring resonance of rings extended by the half-rings of the AR blocks, and σ_ar and κ_ar are the self- and cross- ring-ring coupling coefficients, respectively.

The total transfer matrix for the uniformly periodic SCISSOR structure with AR blocks on both ends is given by multiplying the transfer matrices for all units in the appropriate order

M_{uni, tot} (ω) = M_{a r, B} (ω) M_{uni}^{N_{s}} (ω) M_{a r, A} (ω) .

Here M_uni is the transfer matrix for the SCISSOR unit in the uniformly periodic structure, N_s is the number of SCISSOR units, M_ar_,_A and M_ar_,_B are the transfer matrices for the AR blocks on the left and right ends, respectively (see Fig. 3c). Similarly, the total transfer matrix for the inclined structure is

M_{inc, tot} (ω) = M_{a r, B} (ω) M_{N_{s}} (ω) M_{N_{s} - 1} (ω) \dots M_{1} (ω) M_{a r, A} (ω),

where M_j is the transfer matrix for the jth SCISSOR unit in the inclined structure.

Eqs. (2–3) [transfer matrix for the SCISSOR units] and (22–28) can be numerically solved to obtain the reflectivity and transmissivity. We now consider both a uniformly periodic structure and an inclined structure with AR block at both ends. Both structures consist of N_s = 10 unit-cells, with the same parameters as in (5) and (19) [(19) for the inclined structure only]. The parameters for the left-end AR block are chosen as

ω_{r, L}^{a r} = 2 π \times 3.942 THz, ω_{r, R}^{a r} = 2 π \times 3.866 THz, σ_{a r} = 0.42,

and the right-end AR block is obtained by switching the left and right half-rings of the left-end AR block. These AR Blocks are designed to reduce the surface reflection between the SCISSOR structure and the external waveguides to zero at ω₀ = 2π × 197.2THz, i.e., in the middle of the IB. At this point it is worth noting that we will frequently use ω₀ in the remainder of this paper. For a detailed discussion of minimizing finite-size effects in microring sequences, see, e.g., [21]. For the uniformly periodic structure with ideal AR blocks, light can propagate through the whole system with essentially no reflection. On the other hand, for the inclined structure we are interested in the reflection. Thus we compare the reflection |r|² of the inclined structure with the transmission |t|² of the uniformly periodic structure, which are shown in Fig. 6 as the solid and dashed lines, respectively. For the uniformly periodic structure the transmission is approximately unity at ω₀ as well as frequencies in its spectral neighborhood, due to the AR coating blocks on both ends of the structure; in the same frequency regime, the reflection is close to unity for the inclined structure. Thus, in both structures for an incident pulse with a sufficiently narrow frequency spectrum [E_in(ω)] centered at ω₀, the magnitude of the frequency spectrum of the output pulse [|E_out (ω)|²] is essentially the same as that of the input pulse, i.e., |E_out (ω)|² = |E_in(ω)|². Any distortion in the time domain is mainly due to the phase dispersion in the frequency domain.

Fig. 6 Reflection of the inclined structure (solid) and transmission of the uniformly periodic structure (dashed).

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To investigate the phase properties, we first calculate the group delay τ = dϕ (ω)/dω according to Eqs. (22–28), where ϕ (ω) = Arg {E_out (ω)/E_in (ω)}, i.e., the phase of the transmissivity (reflectivity) of the uniformly-periodic (inclined) structure. We plot them in Fig. 5, using the solid and dotted curves for the inclined and uniformly periodic structures, respectively. We see immediately from these two curves that at ω₀ the group delay for the uniformly periodic structure is about 7ps, and that for the inclined structure is, on average, about 60ps. The latter agrees well with the results of the Hamiltonian formalism.

For the inclined system, the oscillation in τ as a function of the frequency arises because of the discrete variation of the ring resonance. When the frequency is close to one of the ring resonances 50ω_r,j, there is a maximum delay due to the interference between different unit cells inside the structure; the delay decreases when the frequency moves away from that ring resonance, until it comes close to the resonance of the next ring. So the group delay dispersion of the inclined structure includes an ”average delay” part and an “oscillatory delay” part. An intriguing point is that for frequencies not too close to the band edges (i.e., around the middle of the IB) the “average delay” is actually given by the Hamiltonian delay, as illustrated by the dashed curve in Fig. 5. To appreciate the effectiveness of the Hamiltonian formalism, we note that in the frequency region (around ω₀) in which we are interested, the oscillatory part of the dispersion is small in the sense that the dispersive distortion in the output is essentially determined by the average delay dispersion, i.e., the Hamiltonian delay dispersion.

To see this, we first calculate the “Hamiltonian phase” by defining

\frac{d ϕ_{Ham}}{d ω} = τ_{Ham} = \frac{2 N}{(- Δ Ω)} [K_{0} L - {(- 1)}^{m_{B}} π \frac{m_{B}}{m_{R}} Γ {(tan \frac{K_{0} L}{2})}^{{(- 1)}^{m_{B} + 1}} + π] .

Substituting the corresponding K values into Eq. (7) one obtains dϕ_Ham/dω as a function of ω, which can then be integrated to get ϕ_Ham (ω) [up to a constant phase]. We plot ϕ_Ham (ω) in Fig. 7 as the solid line. As a comparison, we also use the transfer matrix method to calculate the actual phase of the reflectivity [see Eq. (23)] of the inclined system, ϕ (ω), plotted as the dashed line in Fig. 7. In the integral for ϕ_Ham (ω), we have taken the initial condition ϕ_Ham(ω = 2π × 197.13THz) = ϕ (ω = 2π × 197.13THz).

Fig. 7 Phase of the reflectivity in the inclined structure: the Hamiltonian phase ϕ_Ham (solid) and the actual phase ϕ (dashed).

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We are now in a position to compare the effects of the Hamiltonian delay dispersion ϕ_Ham(ω) and the actual dispersion ϕ (ω). To do this, we consider a pulse with a frequency spectrum of Δω = 2π × 0.055THz FHWM (corresponding to 8ps FHWM in the time domain) and spectrally centered at ω₀,

E_{in} (ω) = \frac{1}{\sqrt{π} δ ω} exp (- \frac{{(ω - ω_{0})}^{2}}{δ ω^{2}}) exp [i (ω - ω_{0}) t_{0}],

where δω = 2π × 0.0467THz (corresponding to 2π × 0.055 FWHM), t₀ = 13.60ps, and the phase in the exponential factor accounts for the shift of the peak of the input pulse in the time domain from t = 0. The input in the time domain, illustrated as the dotted curve in Fig. 8, is obtained through the Fourier transform of E_in(ω)

E_{in} (t) = \int d ω E_{in} exp (- i ω t) .

The output pulse in the time domain, calculated from the coupled mode transfer matrix method, is given in Fig. 8 as the solid curve, corresponding to the Fourier transform of E_out,T (ω) = E_in (ω) exp [iϕ (ω)]

E_{out, T} (t) = \int d ω E_{in} (ω) exp [i ϕ (ω)] exp (- i ω t) .

We further calculate the output pulse resulting from the Hamiltonian delay dispersion, i.e.,

E_{out, Ham} (t) = \int d ω E_{in} (ω) exp [i ϕ_{Ham} (ω)] exp (- i ω t) .

E_out,Ham(t) is also plotted in Fig. 8 as the filled triangle-points, almost indistinguishable from the solid curve E_out,T(t). Thus we conclude that the dispersive distortion comes mainly from the Hamiltonian delay dispersion. Of course, were the incident pulse spectrally sufficiently broad, there could be a significant difference between the actual pulse distortion and that predicted by Hamitonian optics.

We conclude this section by comparing the slow light performance of the inclined SCISSOR structure and the uniformly periodic structure. We again consider the 10-unit-cell uniformly periodic structure used in Fig. 6. We can easily calculate the phase dispersion of ϕ_uni,10 (ω) from the transfer matrix method and take the Fourier transform as in Eqs. (33) to obtain the transmitted pulse in the time domain, which is shown as the dashed curve in Fig. 8. The delay time is 7ps, only one-ninth the delay in the inclined structure studied above. Also, compared to this 10-unit-cell uniformly periodic structure, the additional dispersive distortion for the inclined structure is small. This becomes more obvious if we compare the pulse distortion for the inclined structure to that of the output of a 100-unit-cell uniformly periodic system, which has the same delay as that of the 10-unit-cell inclined structure and is plotted as the dashed-dotted curve in Fig. 8.

5. Summary

We have demonstrated the delay-time-bandwidth product enhancement in SCISSOR systems by varying ring resonances throughout the structure, and developed a Hamiltonian optics formalism to quantitatively analyze this slow light process. This theory is a good approximation to the more rigorous transfer matrix method, but is more simple and intuitive. We found that in the transfer matrix method, the group delay dispersion includes an average component and an oscillatory component, with the former essentially determining the actual pulse delay as well as dispersive distortion. The effectiveness of the Hamiltonian optics formalism partially lies in the fact that it correctly yields the average part of the dispersion in the transfer matrix method, thus giving a good description of the delay time and pulse distortion in light propagation inside the structure.

Acknowledgments

This work was funded by the National Natural Science Foundation of China ( 61006083), Science and Technology Commission of Shanghai Municipality (STCSM) ( 09520702400 and 10PJ1404300), Innovation Program of Shanghai Municipal Education Commission( 11YZ05) and was partially supported by Shanghai Leading Academic Discipline Project ( S30108).

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Hamiltonian optics formalism for microring resonator structures with varying ring resonances

Abstract

1. Introduction

2. Review of the SCISSOR structures

3. Hamiltonian optics formalism for inclined SCISSOR systems

4. Comparison with the coupled-mode transfer matrix method

5. Summary

Acknowledgments

References and links

Cited By

Figures (8)

Equations (34)

Optics Express