A new integrated-optic coupler was proposed for coupling a guided wave to a free-space wave propagating vertically from the waveguide surface. The coupler consists of a grating coupler in a cavity formed by two distributed Bragg reflectors, and has a small aperture. The cavity was designed to eliminate both transmission and reflection of the incident guided wave, resulting in 100 % radiation by a several-micron aperture. Design consideration was theoretically discussed based on the coupled mode analysis. Predicted performance was simulated and confirmed by the finite difference time domain method.
© 2008 Optical Society of America
A grating coupler (GC) is an integrated-optic component formed by a refractive index modulation in a thin layer on a waveguide for coupling a guided wave and a free-space wave [1,2] with a surface-coupling scheme. Besides the output or input coupling of the guided wave, GC can simultaneously provide a variety of functions such as focusing [3–7], polarization splitting , switching , guided-mode selecting , etc. A key issue for practical applications is the realization of high coupling efficiency. Blazed gratings , parallelogramic gratings [12,13], and combinations with high-reflection substrates [10,14,15] have been discussed and demonstrated to give a one-beam-coupling scheme for the high efficiency. Another concern will be a small coupling aperture. For example, there arises a new important application to the optical-interconnect system-in-packaging for a future ultrahigh- performance signal-processing unit. An integration of GCs of a few hundreds micron coupling length was proposed and reported so far [16–18] for a wavelength-division-multiplexed signal transmission from an array of vertical-cavity surface-emitting laser (VCSELs) to an array of photodetectors. A shorter coupling length enables a narrower channel width since the incident light from VCSEL has circular symmetry. A several-micron coupling length will be able to provide a transmission density of several Tbps/mm. Thus the reduction of the coupling length becomes an important issue.
The effective coupling length Leff of GC is usually given by the reciprocal of a radiation decay factor α. Although larger α gives shorter Leff, α depends on and is therefore limited by the refractive-index modulation depth and the groove depth. Recently, short GCs in semiconductor waveguides have been reported [15,19–21] with use of the large difference in the refractive index between the core material and the air. However, such large index difference is not available in dielectric waveguides. In this paper, we propose a new integrated-optic coupler consisting of one GC and two distributed Bragg reflectors (DBRs), namely a resonance GC (RGC), in order to realize much smaller aperture as well as higher efficiency in comparison to conventional GCs. Theoretical considerations are given on the basis of the coupled mode analysis, and a design example is shown with some simulation results by the finite-difference time-domain (FDTD) method.
2. Configuration and design concept
A basic configuration of the proposed RGC is depicted in Fig. 1. GC of a length LGC is integrated between a front and a rear DBRs with phase-shifting spaces in a single-mode waveguide. We discuss a two-dimensional model without a channel structure in order to avoid nonessential complexity in this paper, and consider a coupling of an incident transverse-electric (TE) guided mode coming from the left-hand side. The incident wave is partially reflected by the front DBR of a length LBF, partially coupled out by GC to radiation waves, and reflected by the rear DBR of a length LBR. The grating period λGC of GC is determined to give vertical radiations by the first order diffraction, causing a reflection to backward propagating guided wave by the second order diffraction. The guided modes and the radiation are illustrated in Fig. 2. The complex field amplitudes of the guided modes propagating forward and backward along z-axis are denoted by A(z) and B(z), respectively. The rear end of GC was chosen as the origin of z. LBR is determined to eliminate A(z) in z > lR + LBR sufficiently. The phase-shifting space lR between GC and the rear DBR is determined so that the wave guided backward from the rear DBR is radiated by GC in phase with the radiation of the forward guided wave. The coupling efficiency of the front DBR and the space lF between GC and the front DBR are chosen so that the transmission of the wave guided backward from GC is canceled by the reflection of A(z) by the front DBR. In other words, B(z) in z < - LGC - lF - LBF is eliminated. Thus the proposed RGC generates neither transmission to z > lR + LBR nor reflection to z < - LGC - lF - LBF, providing 100% radiation even if GC has too small α and LGC to give high efficiency in a usual one-way propagation scheme.
3. Design consideration based on coupled mode analysis
Two radiation modes are degenerated with respect to the z-propagation constant βν, and the complex amplitude is denoted as aν i(z) with a subscript i = a or s for air or substrate radiation, respectively. Coupled mode equations of two guided modes A(z) and B(z) and the radiation modes aν i(z) in GC can be written  as
where βA and βB are the propagation constants of A(z) and B(z) along z-direction and should be βB = - βA. KGC is the grating vector size defined by 2π/λGC. Since TE guided mode is considered, coupling coefficients κν i,A and κν i,B are defined by
where Eν i and Eg are the normalized electric fields of the radiation and guided modes, respectively. ϵ0, ω, and Δϵ are the permittivity in the vacuum, an angular frequency, and the dielectric constant increments representing GC structure, respectively. By substituting aν i(z) in Eqs. (1)–(2) with those calculated from Eq. (3) under an approximation of A(z) ≈ const. and B(z) ≈ const. and executing integrals with respect to βν, we can rewrite the coupled mode equations as
α, κ□□, and Δ□□ are a radiation decay factor, a guided-mode coupling coefficient, and a phase-mismatching factor, respectively, and are given by
A set of boundary conditions of A(0) = 1 and B(0) = rBR gives a solution of A(z) and B(z) expressed by
where sinc(x) is a function defied by sin(x)/x. In the phase-matched case of ΔGC = 0, ζ becomes 0 from Eq. (12) because α = κGC is deduced from Eqs. (4), (7) and (8). In this condition, Eqs. (10) and (11) are rewritten as
It can be seen from Eq. (3) that the increase of the radiation is proportional to A(z) + B(z). Therefore, rBR should be a positive real number and as large as possible in order to obtain a high-efficiency radiation. rBR is determined by the reflection of the rear DBR and the space lR as discussed in the next paragraph.
Coupled mode equations in both DBRs can be written  as
where κBA, and ΔDBR are a coupling coefficient, and a phase-mismatching factor, respectively, and are given by
By applying a set of boundary conditions of A(0) = 1 and B(lR + LBR) = 0 to the rear DBR, rBR is expressed by
In the phase-matched case of ΔDBR = ΔGC =0, Eq. (19) is rewritten as
In order to give a large positive real number of rBR as discussed above, κBA LBR should be large and
With respect to the front DBR, A(z) and B(z) at z = - LGC - lF are written as A(- LGC - lF) = A(- LGC) exp(j βA lF) and B(- LGC - lF) = B(- LGC) exp(-j βA lF) with use of A(- LGC) and B(-LGC) given by Eqs. (10) and (11). Therefore, A(- LGC - lF - LBF) and B(- LGC - lF - LBF) are expressed by
The condition of B(-LGC - lF - LBF) = 0 with ΔDBR = 0 is rewritten from Eq. (23) as
Therefore lF and LBF should be given by
4. Design example and simulation results
A model structure is depicted in Fig. 3. An operation wavelength of 850 nm is considered. The waveguide consists of 0.65 μm-thick Ge:SiO2 guiding core layer on 1.64 μm-thick SiO2 buffer layer on Si substrate. The refractive indices of the core layer, buffer layer and substrate are 1.54, 1.46 and 3.75, respectively. The refractive index modulation required for GC and DBR is formed by embedded Si-N teeth of 50 nm height and a refractive index of 2.01 in the guiding core layer. The effective refractive index of TE0 mode is 1.516, and the grating periods of GC and DBR are 0.5607 and 0.2804 μm, respectively. α = κ□□ = 8.70 mm-1 and κBA = 140 mm-1 are theoretically predicted. LBR was chosen to be 20 μm, resulting in |rBR | = 0.993 at the wavelength 850 nm. LBF was chosen from Eq. (27) to be 8.55 μm for LGC of 5 μm.
A calculated example of wavelength dependences of a normalized power transmission PT = (1-|B(0)|2) / |A(- LGC - lF - LBF)|2, a normalized power reflection PR = |B(- LGC - lF - LBF)|2/ |A(- LGC - lF - LBF)|2 and a normalized power radiation into air, namely an output coupling efficiency, Pout = η0 (1- PT - PR) were summarized in Fig. 4. η0 represents a power distribution ratio to the output coupling against all the radiation in GC and is given by |κν a,A|2/ (|κν a,A|2+|κν s,A|2) = |κν a,B|2/ (|κν a,B|2+|κν s,B|2). The SiO2 buffer layer thickness was optimized to maximize η0 by an interference effect  caused by a substrate reflection. η0 was calculated to be 0.78 in the current case. An output coupling efficiency as high as η0 is predicted at the coupling wavelength 850 nm with full width at half maximum of 4 nm. The maximum efficiencies are 12 times higher than a value of 0.065 estimated from η0 (1 - 2 αLGC) for GC of the same α and LGC without DBRs.
Wavelength dependences of PT, PR and Pout were simulated by FDTD method, and are summarized in Fig. 5. The simulated dependences were similar to those shown in Fig. 4 except for a couple of points. The most essential difference is in the maximum efficiency. The maximum efficiency was estimated to be lower than the value predicted in Fig. 4 by 0.1. The difference was much smaller for a case of longer LGC such as 10 μm. We think the reason of the difference must be a scattering at the refractive-index boundary of GC. In other words, the smaller the number of the grating teeth, the larger a deformation of the diffraction wavefront from a plane. Even so, the output coupling efficiency is expected to be higher than 0.6.
A new integrated-optic coupler named RGC was proposed for coupling a guided wave into a vertically propagating free-space wave. RGC consists of GC located inside a cavity formed by two DBRs. It was designed to eliminate both transmission and reflection of the incident guided wave, resulting in 100% radiation efficiency by a several-micron aperture in a dielectric waveguide. The design consideration was discussed by the coupled mode analysis. A design example using Si substrate was demonstrated with FDTD method, where the output efficiency was predicted almost the same as the power distribution ratio of 0.78, whereas the efficiency can be improved by employing a highly reflective substrate. Experimental work is now under study.
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