This paper proposes to use slow light effects near the Brillouin zone band edge of one-dimensional gratings for reducing the size of integrated electro-optic (EO) modulators. The gratings are built within the arms of a Mach-Zehnder Interferometer (MZI) for intensity modulation. To overcome the inherent high reflection and low extinction ratio, we introduce various multi-segment grating designs. We use coupled-mode theory and derive transfer matrices to analyze the spectral transmittance and phase delay of each arm of the interferometer. Calculations show that a size-reduction of a factor of 2 or more can be achieved at λ = 1.574µm with an insertion loss of 0.17dB and an amplitude modulation extinction ratio of 18.84dB. The simulated structure is based on a Si slab-waveguide 0.2 μm thick with 30nm deep grating groves on SiO2 substrate.
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Numerous photonic devices utilize integrated Mach-Zehnder interferometer (MZI) to achieve phase-to-intensity conversion, such as electro-optic (EO) modulators [1–3] and bio-sensors [4,5]. For a fixed voltage swing, the waveguide length Lπ for which a π phase-difference occurs between the two MZI arms can be used to characterize the device efficiency . Incorporation of slow light effects in the MZI is a practical and effective way to improve device efficiency and reduce power consumption. The slow light effect has been widely investigated in two-dimensional (2D) photonic crystal (PhC) waveguides in the past few years [7–12]. An EO modulator utilizing slow light effect, having Lπ = 80µm, was demonstrated by forming a 2D PhC waveguide with a defect structure .
A slab waveguide with a periodic grating structure is a one-dimensional (1D) PhC. This differs from a PhC waveguide with light propagating at its defect mode, in that slow light propagation occurs in 1D gratings when the wavelength is near the Brillouin band edge. The 1D grating presents an alternative approach for reducing the EO modulator size. In this paper, we use coupled-mode theory to obtain detailed expressions for the dispersion relation and the transmission spectrum of a 1D grating. Finally we discuss the operation of the MZI EO modulator and show expressions for the arm length reduction.
A major challenge of the 1D grating is that high transmittance is not always obtainable simultaneously with the onset of slow light. In the MZI configuration, the problem becomes even more severe as each arm is expected to transport the same optical power with minimal insertion loss. To achieve these two conditions, we propose and investigate a multi-segment grating design. The multi-segment structure is formed by placing a plain slab waveguide between two grating segments. The purpose of the plain slabs is to form a Fabry-Perot cavity between the two grating segments. We use coupled-mode theory to obtain transfer matrices so as to describe the behavior of slow light propagating in this new structure. A similar approach was used to calculate the transmission and reflection characteristics of phase-shifted gratings [13–15]. In this work, we investigate enhancement of the slow light transmittance, and the behavior of phase delay, as a function of grating spacer length. We further point out the trade-offs between grating design and performance. When the MZI arm length is reduced by a factor of 2 to 3, the group index of the structure is in the range ng < 5. The advantage of a slow light waveguide with a relatively small group index is that it minimizes the coupling loss caused by impedance mismatch .
2. Device structures
A sketch of the grating arm MZI is shown in Fig. 1(a) . The 1D periodic grating can be fabricated by reactive ion etching. We consider only TE-polarized light propagating in the z-direction of a symmetric SiO2/Si/SiO2 slab-waveguide, as shown in Fig. 1(b) where n1 and n2 are the real parts of the indices of refraction of Si and SiO2, respectively, and a and Λ are the groove depth and grating pitch, respectively. The transmitted and the reflected fields are shown in Fig. 1(c) for light incident from left and are represented by A and B at the respective boundaries at z = 0 and z = L. Only first order diffraction is considered.
A phase difference ∆φ between the two arms is introduced by a change ∆n in the refractive index of one of the arms. In the case of slow light propagation, a given value of ∆φ is obtained in a shorter propagation distance simply because of the larger effective index represented by the propagation constant difference between the two arms, Δβ. The larger the real value of β, the slower the propagation group speed of light. We explore slow light propagation in a semiconductor grating structure for light energy near the Brillouin zone boundary of the 1D periodic structure.
The grating period Λ for the grating structure satisfies the Bragg condition Λ = 2λ0/Neff, where Neff is the real part of the effective index of the waveguide, and λ0 is the incident wavelength in vacuum. The groove width is half of the grating period. The groove depth and profile are two important parameters in a grating design, as will be discussed in detail below.
3. Dispersion and spectral transmission
Coupled-mode theory is widely used in analyzing mode coupling in parallel waveguides and grating structures. In the case of a 1D grating, mode coupling occurs between forward and backward-traveling waves. For a multi-mode waveguide, because higher order modes are often truncated in numerical calculations, both spectral dispersion and transmission spectra obtained by coupled-mode theory, are approximate .
As light propagates along a periodic grating, partial reflection occurs at each groove which represents an abrupt discontinuity. At the Bragg condition, the forward- and backward- propagating waves are in-phase so the coupling gives rise to a standing-wave. For wavelengths near the Bragg condition, the forward- and backward-waves are slightly out of phase, leading to a slow-moving traveling wave interference pattern. When this happens, a stop-band or gap develops in the transmission spectrum. In momentum space, or k-space, the slow mode occurs when the propagation constant is close to the Brillouin zone boundary. Because slow light effects occur only near the Brillouin zone band edge for a 1D grating, the spectral bandwidth is limited.
The real part of the dielectric constant, εr, can be written as εr(x,z) = εr(0)(x,z) + Δεr(x,z), where εr(0)(x,z) is the index of refraction of the bulk semiconductor silicon, and Δεr is the periodic index perturbation. As a periodic function, Δεr can be expanded in a Fourier sum of odd harmonics. According to coupled-mode theory, the electric-field component Ey is expanded in terms of all modes supported by the unperturbed waveguide. Expressions of Δεr and Ey are shown in Eq. (1) as
Considering only the 1st diffraction order (p = 1) and the fundamental mode of the optical waveguide, the effective propagation constant βc and dispersion relation of a 1D grating are shown in Eqs. (2) and (3), where |K| denotes the magnitude of coupling coefficient .
The two solutions of βC correspond to the two system modes in the grating. To obtain the complete dependence of βC on ω, |K| and the fundamental mode propagation constant β1, Eqs. (2) and (3) are to be solved simultaneously for each frequency ω. Since the mathematical representation for Δεr(x,z) depends on geometric details, |K| differs from one groove profile to another. The expressions of |K| for rectangular- and triangular-groove are obtained, for example, in Eq. (4):
Since a cosine function appears in the integration of |K| for triangular grooves, the coupling strength is weaker in the case of triangular grooves. The transmission spectrum of 1D grating with rectangular grooves is shown in Eq. (5) , where L is the grating length and δβ is defined as β1 – π/Λ:
The transmission spectrum of a grating structure is determined by groove depth and grating length, (expressed as the number of grating periods). A SiO2/Si/SiO2 symmetric slab-waveguide with a core thickness of 0.2μm is considered in this work. The refractive index of the core is taken as nc = 3.42, and that of the top and bottom cladding indices as n2 = 1.45. The period of the 1st order grating is Λ = 0.2851μm for an incident vacuum wavelength λ0 = 1.55μm.
The transmission spectrum for groove depth of 10nm, 20nm, and 30nm are calculated from Eq. (5) and graphically displayed in Fig. 2 (a) . It is clear that the groove depth and its control during fabrication strongly affects the grating transmittance and can be used to obtain the desired transmittance value.
The effective attenuation distance Lp can be used to describe how far a lightwave at a particular vacuum wavelength propagates into the grating waveguide. It is a useful measure for evaluating the relation between transmittance and grating length. Lp is mathematically defined by exp(-zβCimg) as the distance into the grating at which the optical power has decayed to 1/e of the incident value. Thus, Lp = 1/βCimg, where βCimg is the imaginary part of βC. From the above equations we find an attenuation distance Lp = 5.14μm when light of vacuum wavelength λ0 = 1.55μm propagates in a silicon slab grating having groove depth of 30nm. This attenuation length corresponds to a grating with 18 periods (18Λ). Transmittances for gratings with periods 10Λ, 18Λ, and 50Λ are obtained from Eq. (5) and are plotted in Fig. 2(b). Near the Bragg wavelength, the transmittance decreases as the number of grating periods increases until a stopband forms as is the case of the 50-period (50Λ)-grating. Further increasing the number of periods leads to broadening of the stop-band. It can be inferred from the discussion that a 1D grating is subject to a length-constraint if a certain value of transmittance is expected.
3.2 Dispersion relation
The normalized dispersion relation for grating depth of 10nm, 20nm, and 30nm is plotted in Fig. 3 for the real part of the propagation constant βC. Mode propagation direction can be determined from the sign of group velocity vg = dω/dβC. As is indicated in Fig. 3, deeper grooves lead to a wider bandgap, owning to stronger light scattering from deeper grooves. It is also seen that deeper grooves exhibit larger radii of curvature near the Brillouin zone edge or band edge, where dω/dβC is smaller, indicating a lower group velocity.
The effects of groove profiles on βC are shown in Fig. 4 , where only the forward mode is plotted. The dispersion relations reveal a wider bandgap and a greater |βCimg| for the rectangular groove as a result of larger coupling coefficient |K|. When a refractive index change ∆n occurs in the waveguide core, the dispersion relation shifts accordingly as a result of a detuned Bragg wavelength. When ∆n = 0, ∆n = −0.01, the two corresponding dispersion curves for triangular and rectangular grating of 30 nm depth are shown in Fig. 4.
Due to the irregular dispersion characteristics near the band edge, it is possible to obtain a larger Δβ at the Brillouin zone edge, as shown in Fig. 4(a) where it can also be seen that ∆βC real(ω) depends strongly on ω and approaches ∆β in slab-waveguides as ω moves away from band edge.
4. MZI length reduction utilizing slow light effect
For devices based on MZI, Δφ is converted into intensity modulation through interference at the output. The best extinction ratio is achieved by comparing the transmitted amplitudes when Δφ = 0 and Δφ = π. In the following calculations, a relatively large and uniform Δn of −0.01 is assumed in the modulation arm. The large Δn allows the device to be a few tens of microns, and display a relatively narrow stop band, however, the grating cannot be too long because of the finite propagation length.
The length reduction effect is easier to understand from the viewpoint of ΔβC. The wavelength dependence of ΔβC is shown in Fig. 5(a) , which is obtained by measuring the dispersion shift in Fig. 4(a) at each wavelength. The singularities in Fig. 5(b) are the result of the finite number of data points in the numerical computation. It is seen from Fig. 5(a) that ΔβC is about 0.122μm−1 at λ = 1.573μm for a grating with rectangular grooves 30nm deep. For the unperturbed slab-waveguide, Δβ is about 0.0385μm−1. Therefore, a silicon grating with rectangular grooves reduces Lπ by a factor of 3, when compared to a plane slab waveguide, as estimated from Lπ = π/ΔβC. Further calculations also show that a length reduction of 3 still holds for a MZI with a Δn of the order of −0.001. Of course, smaller Δn values generally result in overall longer Lπ.
Based on the Δβ calculation results shown in Fig. 5(a), the π phase shift length Lπ is calculated and shown in Fig. 5(b). Since |ΔβC| is symmetric in wavelength, only those wavelengths longer than Bragg wavelength are plotted in Fig. 5(b). The length reduction ratio is defined as Lπ(slab)/Lπ(rectangular) and is also shown in Fig. 5(b). Lπ (slab) refers to the silicon slab waveguide and Lπ (rectangular) to the silicon slab waveguide with one surface structured with a rectangular groove grating. It is evident that the reduction ratio degrades rapidly as the wavelength moves away from the band edge, and thus only a narrow wavelength range offers significant length reduction.
5. MZI with a multi-segment grating structure
As shown in Fig. 5(b), each length reduction-ratio corresponds to a particular operating wavelength. The transmission spectrum oscillates drastically with wavelength in the slow light region, while the transmittance for a particular wavelength is strongly affected by the grating length. As a result, the transmittance for a specific wavelength can be very low, leading to a high insertion loss for the corresponding reduction ratios. Additionally, as a refractive index shift is introduced in the modulation arm of the MZI, the transmittance between the modulation arm and the reference arm can differ by a large value. Exceptions occur at certain wavelengths, for which the transmittance of the modulation arm and the reference arm happen to be large simultaneously. However, in general, low insertion loss and high extinction ratio are not simultaneously attainable in a MZI having a uniform or single grating waveguide in each branch. To address this problem, we propose the multi-segment grating design that incorporates a number of equal-length grating segments and spacers, as sketched in Fig. 6 .
Let’s take the simplest two-segment configuration (M = 2) as an example. The two grating segments separated by a plain waveguide, referred to as spacer in this paper, constitutes a resonant cavity whose performance can be understood in analogy to a Fabry-Perot cavity. In this case, however, the plane mirror reflectors of the Fabry-Perot cavity are replaced with distributed Bragg reflectors. Suppose the change of phase on reflection is φr at each spacer/grating interface, then the round-trip phase delay in the cavity can be described as 2φr + 2β1D, where β1 is the propagation constant in the spacer and D is the spacer length. If we choose a D that satisfies the phase condition of 2φr + 2β1D = 2π, then a maximum transmittance can be achieved in this grating waveguide. On the other hand, since the grating waveguide transmittance depends on the choice of D, it is possible to obtain equal power transmission in both the reference Tr and modulation Tm arms (Tr = Tm) for discrete values of spacer lengths, D. If the spacer length for the modulation and reference arms are individually optimized, then the condition Tr = Tm = 1 can be achieved. In the case of multiple-segments (M > 2), several coupled cavities can be formed along the grating and we can follow the same procedure to determine the spacer length.
To analyze the performance of a multi-segment grating, we combine coupled-mode theory with complex transition matrices to describe the transmitted and reflected optical wave. These are similar to the transfer matrices representing a dielectric slab used when calculating transmission and reflection coefficients for a plane wave incident on a dielectric film slab. In our grating case, coupled-mode theory is used to obtain the transition matrix representing each segment. In subsection 5.1, transfer matrix construction is discussed in details. In subsection 5.2, 5.3 and 5.4, we summarize calculations for various multi-segment grating designs and discuss the results.
5.1 Matrix construction
To simplify the derivation process, we first calculate an auxiliary scattering matrix (S-matrix) and convert it to the corresponding transmission matrix (T-matrix). The S-matrix and T-matrix variables are defined in Fig. 7 .
If we compare the parameters defined in Fig. 1(c) with those in Fig. 7 (a), simple relations are established as follows: a1 = A(0), a2 = B(L), b1 = B(0), b2 = A(L). By using coupled-mode theory to calculate the complex amplitudes A(0), B(L), B(0), and A(L), the S-matrix parameters can be obtained as follows :
The corresponding T-matrix of a grating segment (T g) can be constructed by utilizing the following conversion relation:
The T-matrix for a spacer (Tf) is as follows
With Tg and Tf calculated from Eq. (7) and (8), it is easy to calculate the total T-matrix for an M-segment configuration, the corresponding transmission coefficient t, and reflection coefficient r are obtained from Eqs. (9) and (10):
5.2 M = 1,2,3 for fixed grating length
The length reduction ratio is taken to be 2.4, as an example for which the operating wavelength is λ = 1.574µm. At this wavelength, a group index of ng = 14.9 is obtained from Fig. 3. For a single grating segment, M = 1, the grating length for π-phase shift is L = 120Λ. The calculated transmittance of the reference arm and modulation arm are Tr = 0.0516 and Tm = 0.3397, respectively. The small Tr and large difference |Tr – Tm|, gives rise to a high insertion loss (12.87dB) and low extinction ratio (−4.46dB).
In the multi-segment grating structure, we fix the grating length at 120Λ and plot Tr and Tm versus spacer length D. Figure 8 shows plots of Tr and Tm for a two-segment grating (M = 2) structure, where each grating segment has a length of 60Λ. The Tr and Tm curve in Fig. 8 shows periodicity of D ≈1.1Λ. Since the total length of the grating structure is L = 2 × 60Λ + D, a small spacer length is desirable for short MZI. We choose D between 0 and 1.1Λ.
Figure 8 shows two intersections at D = 0.90Λ and at D = 0.97Λ. Of the two intersections, D = 0.97Λ corresponds to a slightly higher transmittance, and is closer to π phase shift. The resulting parameters are Tr = 0.2980, Tm = 0.2446, and Δφ = 0.933π. Δφ is the phase difference between the two arms calculated according to tan−1(timg/treal). The MZI insertion loss is 5.36dB and extinction ratio is 8.83dB. Here, the insertion loss is calculated as 10log10(Tr) and the extinction ratio is calculated as 10log10[2Tr/(Tr 2 + Tm 2 + 2TrTmcos∆φ)1/2]. Following the same procedure, we evaluate an M = 3 grating waveguide. The best performance is achieved at D = 0.98Λ, which gives a total device length of L = 3 × 40Λ + 2 × 0.98Λ. In this case, calculations yield Tr = 0.4790, Tm = 0.4411, and Δφ = 1.089π, which corresponds to a loss of 3.20dB and an extinction ratio of 8.56dB. Compared to the M = 2 case, the M = 3 option yields a higher transmittance and thus a lower loss.
The above results indicate that the multi-segment grating waveguide results in a substantial performance improvement for both loss and extinction ratio. The insertion loss of the M = 3 design exhibits a loss reduction of 9.67dB, and an extinction ratio improvement of 13.29dB when compared to the MZI with a single grating segment and of approximately the same total length.
As shown in Fig. 8, Tr is Lorentzian function and always exhibits a single, narrow line shape whereas the Tm varies more gradually. Further investigation shows that Tm becomes even broader as L increases. Therefore, it is possible to manipulate the Tm curve to intercept the Tr curve at a higher transmittance point. Further discussion is presented in the following subsection.
5.3 M = 2,3 for unfixed grating length
In section 5.2, we demonstrated a performance enhancement attainable to a multi-segment grating waveguide with a fixed grating length of L = 120Λ. As discussed earlier, the transmittance of an M = 2 design can be evaluated simply by invoking F-P cavity theory in which the planar end-mirrors are replaced by a pair of distributed feed-back gratings. For a given grating segment length, the spacer length D determines the round trip phase delay δ = 2φr + 2β1D of the cavity. The transmittance of an F-P cavity is Figure 9 plots the transmittance T 1 when the length of the grating segment changes from 0 to 80Λ. For the modulation arm, it is obvious that T 1 increases monotonically to 1 when the length of the grating segment increases from 60Λ to 80Λ. In other words, the reflectivity R 1 goes to 0, which makes the effect of δ negligible in Eq. (11). As a result, Tm approaches 1 regardless of the choice of D.. For the M = 2 design, since each grating segment has a length of L/2, one segment change from 60Λ to 80Λ corresponds to a variation of total grating length from 120Λ to 160Λ. As for Tr, this changes very little, since T 1 in Eq. (11) is relatively insensitive to the length of the grating segment. However, because φr changes with grating segment length, the peak in a Tr vs. D plot will appear at different positions.
We now explore the trade-off between grating segment length and the transmittance. As we increase the grating segment from 60Λ to 80Λ in the M = 2 design, i.e. a total length of approximately 160Λ, the Tm and Tr curves intersects at a point where the transmittance is high for both arms, as shown in Fig. 10 . Tm is nearly a constant across a wide range of spacer lengths, D, as seen in Fig. 10 because it is insensitive to its phase delay δ, whereas Tr reaches its maximum when δ = 2φr + 2β1D = 2π, i.e. the resonant condition of the F-P cavity. As the segment length varies, the spacer length D also varies to give rise to the maximum Tr. It is worth noting that L cannot be picked randomly from 120 Λ to 160Λ because the phase delay difference between the modulation and reference arm Δφ needs to hold at π or close to π to obtain a good extinction ratio for the MZI.
Our calculations show that for the M = 2 design, a grating length of 156Λ with D = 0.97Λ gives Tr = 0.9907, Tm = 0.9516, and Δφ = −0.9786π, which correspond to a loss of 0.04dB and an extinction ratio of 14.16dB. Following the same argument and analysis, for the M = 3 design, a grating length of 140Λ with D = 0.30Λ gives Tr = 0.9616, Tm = 0.9630, and Δφ = 1.008π. The corresponding loss is 0.17dB, and the extinction ratio is 18.84dB. The length reduction ratio is 1.84 and 2.06 for M = 2 and M = 3 design, respectively.
5.4 Unequal Ds for M = 2 design
In the discussion of section 5.2 where the size reduction ratio of 2.4 is fixed, we studied how the transmittance may vary with the spacer length D, which is the same for both the modulation and reference arms. In this subsection we propose and analyze the transmittance when the modulation and reference arms are optimized separately by varying D separately in order to achieve Tm = Tr ≈1. We denote Dr,m as the spacer length that gives the maximum transmittance for both Tr and Tm.
The adoption of different Ds could result in significant departure of Δφ from π for a given length, which would degrade the extinction ratio of the MZI. The phase shift Δφ consists of two parts: one is the phase difference caused by the light traveling in the cavity region, i.e. within the spacer; the other part arises from wave propagation in the grating segments. Within the cavity, due to separate optimization, both arms meet the phase condition for maximum transmittance. Therefore, the phase difference in the spacer between the two arms is zero. Our matrix calculations show that the phase difference due to a single grating segment is not proportional to the grating length. More specifically, in the case under study, the phase difference obtained from a grating segment of 60Λ is less than half that from a grating segment of 120Λ. Therefore, the total phase difference in the M = 2 configuration with two 60Λ segments is less than π, which is achieved in its M = 1 counterpart at L = 120Λ.
The M = 2 structure, for example, can be used to illustrate this idea. According to Fig. 8, the Tr and Tm reach their maximum value at Dr = 0.94Λ and Dm = 0.18Λ, respectively. However, Δφ is only 0.8463π, which limits the extinction ratio to 6.13dB. To improve Δφ in this design, the grating length is allowed to extend at the expense of the reduction ratio, similar to the approach discussed in section 5.4.
As a summary, Table 1 below lists the designs and results from sections 5.2 to section 5.4.
In this paper, we have explored the slow light effect in a MZI structure consisting of 1D periodic gratings. The modeled structure consists of a slab silicon waveguide in a SiO2/Si/SiO2 structure in which the Si slab is 0.2mm thick and supports various grating structure on one surface. As an example, we have considered in details rectangular gratings with 30nm-deep grooves, subject to a refractive index change in the modulation arm of ∆n = −0.01. We have demonstrated that this design gives a length reduction factor over 2 due to the slow light effects in the grating. To enhance the transmittance of the MZI in the slow light region, we have proposed a multi-segment grating design and have analyzed its performance by combining coupled-mode theory with a transfer matrix representation of the grating segments. A substantial reduction in the insertion loss and a substantial enhancement in the extinction ratio have been obtained. Tradeoffs between the reduction ratio, insertion loss and extinction ratio for different design variations are also presented in the paper. For a fixed reduction ratio of 2.4, we obtained a design with insertion loss of 0.17dB and extinction ratio of 6.13dB by matching the transmittance in the modulation arm and the reference arm. By trading reduction ratio for performance, an M = 3 design with a reduction ratio of 2.06 displays an insertion loss of 0.17dB and an extinction ratio of 18.84dB.
This research was supported in part by Nation Science Foundation (NSF) and in part by O’Brien & Gere. We also like to thank Dr. Daniel Guidotti for his many suggestions on the manuscript.
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