## Abstract

We introduce universal “anti-reflection blocks” for periodic microresonator sequences consisting of 4-port unit cells. When added to the end of a finite periodic sequence, the anti-reflection block can eliminate the reflectivity at a given frequency *ω _{ref}* and significantly reduce it in a frequency range around

*ω*. These anti-reflection blocks are universal: By adjusting only their characteristic parameters, they can be made applicable to any 4-port periodic sequence, regardless of the detailed inside structure of its unit cells, at any frequency within the photonic bands.

_{ref}© 2012 Optical Society of America

## 1. Introduction

Periodic sequences of optical microresonators may have important applications as densely-integrated optical components for slow light propagation [1–9]. The linear optical properties of an infinite periodic structure are characterized by its photonic band structure and corresponding eigen-modes, and have been well studied for a broad class of structures with widely varying geometries. However, any realistic sequence has a finite size, and in applications it is necessary to couple light into the structure from an external medium or waveguide. Often the reflectivity is prohibitively high if “anti-reflection structures” are not added at the ends of the sequence, following in spirit the use of anti-reflection layers in thin film optics. Various strategies [10–13] have been proposed for designing such structures, generally based on numerical minimization procedures. It is the purpose of this paper to generalize such schemes, which have typically used the same unit-cell of the periodic sequence as an anti-reflection layer, to a much broader class of 1D or quasi-1D finite-size periodic photonic structures, whose 4-port unit cell is depicted in Fig. 1. We show that universal anti-reflection blocks, in principle applicable to any finite-size periodic sequence of unit-cells in Fig. 1, can be constructed for any given frequency *ω _{ref}* within the photonic bands; in practice they work well for frequencies near

*ω*.

_{ref}The paper is organized as follows. In Sec. 2 we briefly review some basic properties of the 4-port unit cells, both as the building blocks of the periodic sequences and as the universal anti-reflection blocks for such sequences. The general anti-reflection condition is also derived. In Sec. 3 and 4 we study and compare two specific examples of the universal anti-reflection blocks, the SCISSOR (“side-coupled integrated spaced sequence of resonators”) and half-CROW (“coupled-resonator optical waveguide”) [7] blocks. A brief summary is given in Sec 5.

## 2. General properties of 4-port periodic sequences and anti-reflection blocks

Consider a periodic sequence consisting of identical unit cells illustrated in Fig. 1. Other than the four “+/−” ports, which are assumed to be identical waveguides, the detailed inside structure of the unit cell need not be specified. The linear optical properties of such a structure are described by a 2 × 2 unit-cell transfer matrix *M*(*ω*)

*M*(

*ω*)

*E*

^{±}and

*ξ*on

*ω*is kept implicit. At a frequency

*ω*lying within the photonic bands, the

*ξ*’s are real and, with only rare exceptions such as band edges or “flat bands [9,14,15]” which we will not consider in this paper, there are two real

*ξ*’s for each

*ω*; they (

*ξ*’s) are the Bloch wavevectors (normalized by the reciprocal of the period of the periodic structure) corresponding to the “forward” and “backward” propagating Bloch functions, respectively. For the forward (from left to right) Bloch function, we can write the eigenvector equation in the form

*r*

_{∞}| smaller than 1, where

*r*

_{∞}=

*r*

_{∞}(

*ω*) can be interpreted as the reflection amplitude of the corresponding semi-infinite (in the forward direction) periodic structure [16], with the two “L”-ports of the first unit cell connected to “outside” waveguides identical to the ports. In the rest of this paper,

*unless explicitly stated*, all expressions concerning the termination of periodic sequences deal with the

*left*end, without loss of generality; the generalization to the

*right*end is trivial.

Our goal here is to investigate universal anti-reflection blocks that, with appropriately chosen parameters, are applicable to any optical periodic sequence described above. In contrast to earlier work (see, e.g., Chak and Sipe [11]), we focus on the properties of the anti-reflection blocks; the details of the periodic sequence, of which the only relevant parameter is *r*_{∞}, need not be specified.

The anti-reflection blocks we consider in this paper are also of the abstract 4-port structure as in Fig. 1, with port waveguides assumed to be the same as those of the periodic sequence unit cells. For the reflection of a semi-infinite periodic sequence of blocks preceded by an anti-reflection block to vanish at *ω _{ref}*, it is required that

*α*,

*β*are complex numbers. Substituting Eq. (5) into (4) one has

In the next two sections, we show how to satisfy the conditions (8, 9) using two different types of universal anti-reflection blocks, the SCISSOR and the half-CROW anti-reflection blocks. For our sample calculation we will assume the periodic sequence consists of SCISSOR unit cells, but the generalization to other periodic sequences (such as CROW structures) is straightforward.

## 3. SCISSOR anti-reflection block

The schematic of a SCISSOR unit is shown in Fig. 2(a), with transfer matrix *M*(*ω*) given by [7]

*σ*and

*κ*are respectively the self- and cross- ring-channel coupling constants (assumed to be real in our model);

*ω*=

_{R}*c*/(

*n*) is the fundamental ring resonance, with

_{b}R*R*being the radius of the ring and

*n*the effective index of the ring, and

_{b}*ω*=

_{B}*πc*/(

*n*) would be the first Bragg resonance for an infinite sequence of such rings, with

_{b}L*L*being the overall length of the unit; the effective index of the channel is taken to be the same as the ring.

For the “reference” frequency *ω _{ref}* at which the reflection of a semi-infinite periodic sequence of unit cells (Fig. 1) is to be made zero, Eq. (7) becomes

*r*

_{∞}is the reflection coefficient of the semi-infinite periodic sequence. The modulus of Eq. (11) requires

*κ*

^{2}with a solution (for

*κ*

^{2}> 0)

*σ*

^{2}= 1 −

*κ*

^{2}[7]. It can be proved that the

*κ*

^{2}given in (13) is smaller than unity (for |

*r*

_{∞}| < 1), and thus qualifies as a ring-channel cross coupling constant. Once we have the appropriate

*κ*

^{2}to satisfy the modulus of Eq. (11), we can further adjust

*ω*(or equivalently, the length

_{B}*L*of the anti-reflection block) to satisfy the phase condition

So by properly choosing the coupling constant *κ* and the anti-reflection block length *L*, the anti-reflection block can cancel out reflections induced by the finite size of any periodic sequence at any chosen frequency *ω _{ref}* within the photonic bands. We have treated

*ω*to be fixed, but in principle it can also be varied by changing the radius of the ring.

_{R}To demonstrate the effectiveness of this SCISSOR anti-reflection block, we apply it to both ends of a finite periodic sequence of SCISSOR units [see Fig. 2(b)]. As an example, we take the parameters for the periodic sequence, with 50 unit cells, as

*L*is the overall length of the unit cell in that sequence,

_{ps}*R*is the radius of the ring in the unit cell,

_{ps}*σ*is the self coupling constant of the channel to the ring, and

_{ps}*n*is the assumed index for propagation in both the channel and the ring. The working (vacuum) wavelength range is around

_{b,ps}*λ*=

_{ps}*n*/50 = 1.56

_{b,ps}L_{ps}*μm*; that is, the (circular) frequency at which we design the antireflection block is given by

*ω*= 100

_{ps}*ω*, where

_{B,ps}*ω*is the first Bragg resonance frequency of a periodic sequence of the unit cell blocks. In an infinite periodic sequence there is an “intermediate band ” between the bandgaps associated with the Bragg resonance

_{B,ps}*N*and the ring resonance

_{B}ω_{B,ps}*N*, where

_{R}ω_{R,ps}*ω*is the fundamental ring resonance and

_{R,ps}*N*and

_{B}*N*are positive integers. When

_{R}*N*=

_{B}ω_{B,ps}*N*, this intermediate band is completely flat, indicating a zero group velocity for all wavenumbers. This remarkable feature of the intermediate band has been exploited to stop, store, and release light in quantum well Bragg structures [14], which are mathematically equivalent to SCISSOR structures in the linear, weak coupling regime. The band structure of this infinite periodic SCISSOR sequence around

_{R}ω_{R,ps}*λ*is shown in Fig. 3, from which one can see the intermediate band between 100

_{ps}*ω*/2

_{B,ps}*π*= 192.173THz and 100

*ω*/2

_{R,ps}*π*= 192.099THz.

The transmission spectrum of the 50-SCISSOR-cell finite periodic structure without anti-reflection blocks is plotted in Fig. 4, where the Fabry-Perot-like features with minima near zero signal the high reflectivity at both ends of the structure within the intermediate band. Now we apply the SCISSOR anti-reflection blocks to both ends of the periodic sequence to reduce the reflection at *λ _{ref}* = 1.5603

*μm*(i.e.,

*ω*/2

_{ref}*π*= 192.136THz), which is approximately in the middle of the intermediate band. To obtain the appropriate parameters for the anti-reflection blocks from Eqs. (13) and (14), one has to know

*r*

_{∞}of the periodic sequence, which can be easily obtained either analytically or numerically from the parameters in Eq. (15)

*r*

_{∞}in (16) and parameters in (17) give the required parameters for the anti-reflection blocks

*ω*; however, this neighborhood is small compared to the intermediate band bandwidth. This shortcoming can be overcome by using the half-CROW anti-reflection block introduced in the next section.

_{ref}## 4. Half-CROW anti-reflection block

Figure 6(a) shows the structure of a half-CROW anti-reflection block. The transfer matrix *M*(*ω*) for this unit is [7]

*σ*and

*κ*are the self- and cross-coupling constants between the rings, respectively, and

*ω*=

_{R}*c*/ (

*n*) is the fundamental ring resonance.

_{b}RHere the anti-reflection condition (11) simplifies to

the modulus of which immediately gives Since |*r*

_{∞}| < 1,

*σ*given in (21) qualifies as a self-coupling constant between the rings. Now a natural design would be to choose the radius of the rings in the half-CROW anti-reflection block to be the same as that of the rings in the sequence of unit cells. Unfortunately in general the phase part of (20) will then not be satisfied. But there is a simple way to work around this: One just extends the length of the two ports connected to the periodic sequence by a proper amount Δ

*l*(see Figs. 6b, c). The transfer matrix for the extended half-CROW anti-reflection block [Fig. 6(b)]

*M*′(

*ω*) is

*β*′

^{*}/

*α*′| = |

*r*

_{∞}| if

*σ*= |

*r*

_{∞}|); moreover, if Δ

*l*is properly chosen such that

We now apply this extended half-CROW anti-reflection block to the same finite periodic SCISSOR sequence as that in Sec. 3 [see Fig. 6(c)]. We again set the ring radius and the effective refractive index of the anti-reflection block as

*ω*, as expected, and is still close to unity at most frequencies within the intermediate band. As illustrated in Fig. 7(b), in this frequency range the group delay for the finite periodic sequence with extended half-CROW anti-reflection blocks, defined as

_{ref}*δt*≡

*dϕ*(

*ω*)/

*dω*− 50

*L*/ (

_{ps}*c/n*), with

_{b,ps}*ϕ*(

*ω*) being the phase of the transmission amplitude, is close to the corresponding delay over the same length within an infinite structure, defined as Δ

*t*≡ 50

*L*− 50

_{ps}/v_{g}*L*/ (

_{ps}*c/n*), where

_{b,ps}*v*is the group velocity of light propagating inside the infinite periodic sequence. This suggests that the anti-reflection blocks not only minimize the reflection, but also allow the phase properties of the finite structure to mimic those of the corresponding infinite periodic sequence. This can be understood as follows. Since the anti-reflection block at each end is designed to eliminate the reflection of a semi-infinite periodic sequence, light incident on the left end will enter the finite periodic sequence as a forward Bloch wave, which goes through the right end without inducing any backward Bloch component. So the field inside the finite periodic sequence is purely a forward Bloch wave, just as that in the infinite periodic sequence.

_{g}We conclude this section with a brief comparison between the SCISSOR and the half-CROW anti-reflection blocks. Although both anti-reflection blocks can eliminate the reflection at a chosen frequency *ω _{ref}*, the half-CROW structure is much more effective in minimizing the reflection in the neighborhood of

*ω*. The reason for this can be seen in Fig. 8. Throughout the intermediate band the phase of

_{ref}*r*

_{∞}is constant, and the modulus of

*r*

_{∞}reaches a minimum at the center of the intermediate band and rises slowly with frequency away from band center. The extended half-CROW anti-reflection structure, with the slowly varying phase and uniform modulus of its

*β*

^{★}/

*α*, is better able to match this behavior near the center of the band than is the SCISSOR anti-reflection structure, for which

*β*

^{★}/

*α*varies rapidly with frequency both in modulus and phase.

## 5. Conclusion

We have introduced anti-reflection blocks for a wide variety of 4-port periodic microresonator sequences. Unlike in earlier work [11], our anti-reflection block is not restricted to be of the same structure as the elements of the periodic sequence. The design parameters for the anti-reflection blocks only depend on the reflectivity *r*_{∞} of the semi-infinite periodic sequences, and are always available as long as *r*_{∞} < 1, which is true at any chosen frequency *ω _{ref}* within the photonic bands; this guarantees that the strategy we adopt is universal. We have specifically analyzed two examples of the universal anti-reflection blocks, the SCISSOR and the extended half-CROW structures. For a finite periodic SCISSOR sequence, the half-CROW anti-reflection block is much more effective than the SCISSOR one in minimizing the reflection in a

*neighborhood*of a chosen reference frequency

*ω*. While in general we can expect that the most effective anti-reflection block will be different for periodic sequences constructed of different unit cells, we note that

_{ref}*r*

_{∞}of a periodic sequence of CROW structures has the same qualitative behavior as that of a periodic sequence of SCISSOR structures. Hence we can expect the half-CROW anti-reflection block to have wide applicability in the development of optical microresonator structures for slow light applications.

## Acknowledgments

We gratefully acknowledge financial support from Start-up Research Funds of Liaocheng University.

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