## Abstract

Input-grating coupling characteristics have been studied for narrow Gaussian beam incidence and finite-length grating coupler with an electromagnetic full-vector field model based on the finite-difference time domain (FDTD) method. Analytic analysis based on perturbation theory has been compared to the FDTD technique. The influences of the variation in grating period, modulation depth, corrugation, and beam size have been investigated. Certain aspects of the calculated results have been confirmed with experiments.

©2004 Optical Society of America

## 1. Introduction

In optical and heat-assisted magnetic data storage area density is limited by light confinement. To surmount diffraction-limited focused spot, near-field recording is introduced [1]. Flying a structure with sub-wavelength light confinement in a close proximity to a storage medium achieves high-density recording [2]. In heat-assisted magnetic recording localized light absorption in the medium raises the temperature of the storage layer, reducing the coercity of the magnetic medium and realizing the magnetic recording. One way to integrate optical components for light confinement into a magnetic head employs optical planar waveguide. Light confinement is achieved by both mode confinement normal to the film and two-dimensional devices built on it.

To realize above idea as a first step we need to launch light efficiently from free space into the waveguide. Grating couplers are one of means. There has been extensive study both in theoretical analysis and in practical design of efficient coupling gratings [3–8]. For inherently periodic structures of infinite extent, rigorous coupled-wave analysis has been widely used [5,9–11]. In such basic analytic analyses, the incident beam is typically a plane wave and the phase-matching condition for resonant coupling is assumed. With miniaturized devices such as magnetic slides a grating coupler has a finite length, perhaps a few tens of light wavelengths, and the incident beam on the grating has a broad angular distribution. To evaluate such structures with aperiodic features and of finite extent illuminated by a narrow beam of light, a variety of techniques have been proposed, including an extended coupled-wave analysis [12, 13], the finite-element methods [14], the boundary variation method [8], finite-difference time domain (FDTD) [15], and spectral collocation [16]. In the FDTD and spectral collocation techniques, Maxwell’s equations are directly solved in the time domain. These techniques are quite general for handling complicated geometric structures, but they need large computational memory.

Grating couplers are well understood for mm sized incident beams and gratings, and widely used in integrated optics. Good coupling efficiency has been obtained. For an incident beam less than a hundred wavelengths, only a few studies have been reported [13, 17]. Pascal *et al*. [13] has discovered that the relative coupling efficiency depends on the beam size and also on the beam position from the groove edge.

In this paper we use the FDTD technique to analyze light coupling of narrow Gaussian beam into a planar waveguide and investigate the influence of groove depth on coupling efficiency. Studies are mainly concentrated on a 50-µm sized focused beam but light coupling with narrower Gaussian beams is also addressed. We present experimental results to validate certain aspects of our modeling. In this studied light wavelength is in the visible range (632 nm), which is interested in heat-assisted data storage.

## 2. Theoretical analysis

Figure 1 shows the waveguide structure studied. It consists of a 100-nm thick Ta_{2}O_{5} core layer on a SiO_{2} glass substrate. To have good light confinement the core layer was chosen to have high index of refraction and cladding layer has low index. The layer thickness of core is designed such that only the fundamental transverse electric mode (TE_{0}) can propagate in the waveguide and the width of the mode field intensity is minimized for excitation wavelength λ=632 nm. At 632 nm, the refractive index of the core layer is 2.09 and that of the SiO_{2} layer is 1.47. A grating is etched on the surface of the core layer (a) or on the substrate (b) for coupling light into the waveguide. A linearly polarized beam of light is incident on the grating for exciting the waveguide mode. The state of polarization is parallel to the film plane. The incident beam has a Gaussian (transverse) profile and the full width of the beam at 1/e^{2} intensity is 2w_{0}=50 µm.

The simulation is performed in a two-dimensional computation domain. The incident beam is assumed to be uniform along the y-axis and has a Gaussian profile along the z-axis. The beam waist is centered on the surface of the film, and a distance z_{c} from the grating edge. For simplicity, the groove is assumed to be rectangular with a 50% duty cycle.

To validate our simulation we compare the TE_{0} mode field distribution, calculated with the FDTD technique, with that from analytic solution. It was found that the mode field distribution obtained with the numerical calculation is consistent with that from analytic modeling. Based on the phase wavefront along propagation, we determined that the propagation constant of the TE_{0} mode n_{eff}=1.637, which is in agreement with that from the analytic solution, 1.645.

#### 2.1 Groove depth dependence of coupling efficiency, resonant angle, and angular width

Figure 2 shows the coupling efficiency vs. angle of incidence at various groove depths for two grating couplers with differing groove period (Λ). In Fig. 2(a) Λ is set to have an angle of incidence θ=45°, based on the well-known grating equation:

where m is the diffraction order. For m=1, Λ=0.68 µm.

It is interesting to see that the coupling efficiency η, the resonant angle θ_{0} for optimal coupling, and the angular width are dependent on the groove depth d. In particular, at *d*=100 nm, θ_{0}=33°, which is an 11° deviation from the Bragg condition. For the 50 µm incident beam, η is maximized at the groove depth *d*=20-30 nm and η is only ~12%. At *d*=30 nm, the angular width of η is ~0.79°, which is much broader than the angular width of the incident beam, 0.01°. This indicates that the angular width is dominated by finiteness of the grating coupler.

To improve the efficiency, we can reduce Λ such that only the zero-th order transmitted beam and the 1^{st} order diffracted beam propagates in the SiO_{2} cladding layer. For example, if Λ=0.36 µm, based on Eq. (1), the angle of incidence θ for the optimal coupling is θ_{0}=-6.5°. Fig. 2(b) shows the calculated η vs. θ at various groove depths. Once again, it is seen that η, θ_{0}, and the angular width are dependent on the groove depth. At *d*=70 nm, θ_{0} for optimal coupling is deviated by 9° from the Bragg condition. Indeed, η increases from ~12 % to ~20%. It is also seen that η is less sensitive to *d* for Λ=0.36 µm.

#### 2.2 Coupling efficiency in the configuration of substrate corrugation

We may also place the grating coupler at the core/substrate interface, referred to the substrate-corrugation, as shown in Fig. 1(b). Figure 3 shows the comparison of η vs. *d* for three cases. Case (a) uses substrate-corrugation with Λ=0.36 µm, case (b) uses “surface-corrugation” for which the grating coupler is on the film surface, and Λ=0.36 µm, case (c) also uses surface-corrugation, but Λ=0.68 µm. It is evident that case (a) yields the highest η, ~55%.

According to the perturbation theory [4,5], at the optimal angle of incidence θ_{0}, η is a convolution of the incident Gaussian beam distribution with the mode field, we may write

Here *f _{c}* (≤1) is the coupling coefficient,

*α*is the damping coefficient of the electric field amplitude while propagating in the grating region, ${w}_{1}=\frac{{w}_{0}}{\mathrm{cos}\left({\theta}_{0}\right)}$, and erf(x) is the error function of x. Maximum η can be obtained if

which yields *η*=*f _{c}*×0.81. For a given waveguide and beam size the optimal α can be reached by tuning the groove parameters (depth, duty cycle, etc.). η is thus determined by

*f*.

_{c}With the FDTD technique we can also calculate α. In this calculation we assume that a TE_{0} mode propagates in the waveguide and through a grooved region. By observing the mode field decay in the core layer, *α* value is obtained. Figure 4 shows the calculated α vs. *d*. In the calculation the grating has a rectangular groove profile with Λ=0.36 µm and a 50% duty cycle. Two curves are displayed in the figure. Curve (a) corresponds to surface-corrugation, and curve (b) does substrate-corrugation. In the configuration of surface-corrugation, α increases with *d* at *d*<60 nm, has a little dip around 75 nm, and then increases rapidly. Perturbation analysis gives α∝*d*
^{2} for shallow grooves, and oscillates with *d* for deep grooves [6]. Our calculation does show *α*∝*d*
^{2} for *d*<40 nm, and has a little dip at *d*≈75 nm, but there is no evident oscillation afterward. In the substrate-corrugation α increases with *d* quadratically for shallow grooves, gradually saturates, and then decreases slightly.

From Fig. 4 it is evident that *α* vs. *d* differs substantially between the surface- and substrate-corrugation. At the same groove depth, the magnitude of *α* for the surface-corrugation is greater than that for the substrate-corrugation, which results in more efficient coupling for the substrate-corrugation. For the 50 µm incident beam and near the normal incidence, from Eq. (3), the optimal α=0.0272 µm^{-1}, which corresponds to *d*=30 nm for the surface-corrugation and 47 nm for the substrate-corrugation, as seen in Fig. 4, which agrees with the results in Fig. 3.

The difference in *α* between the surface- and substrate-corrugation can be understood based on the interference of radiated optical rays into the substrate from the grooved region. Figure 5 shows how a guided beam radiates into the substrate and free space each time it meets the grooved region. The phase difference (ϕ) between consecutive radiated rays into the substrate can be approximated as follows

Here *k*
_{0} denote the wave number in free space, *d _{c}* is the average core layer thickness in the groove region, θ

_{c}is the angle of incidence of the guided ray to the grooved interface, θ

_{s}is the angle of radiated beam exiting into the substrate, n

_{c}is the refractive index of the core layer, n

_{s}is the refractive index of the substrate, and ϕ

_{gh}is Goos-Hanchen phase-shift of the guided mode ray striking at the film surface.

For the surface-corrugation at Λ=0.36 µm and *d*=30 nm, the effective refractive index n_{eff}=sin(θ_{0})+λ/Λ=1.58, *θ _{c}*=sin

^{-1}(

*n*/

_{eff}*n*)=49.3°,

_{c}*d*=85 nm, n

_{c}_{s}sin(θ

_{s})=sin(θ

_{0})=sin(-10°), ϕ

_{gh}=-46.05°, we have ϕ=1.57 π. For the substrate-corrugation at Λ=0.36 µm and

*d*=50 nm, n

_{eff}=1.55, θ

_{c}=47.9°,

*d*=75 nm, n

_{c}_{s}sin(θ

_{s})=-0.208, ϕ

_{gh}=-80.4°, we have ϕ=1.14 π, which means that the those radiated rays are almost out of phase. Due to the destructive interference among radiated rays, the radiation into the substrate is much lower for the substrate-corrugation than that for the surface-corrugation.

From Fig. 4 α has a dip at *d*≈75 nm. This phenomenon can be also illustrated based on Eq. (4). At this groove depth, θ_{0}=-16°, Goos-Hanchen phase-shift ϕ_{gh}≈0 and ϕ≈π. The destructive interference among radiated rays causes little light escaped from the grooved region, causing reduced α.

#### 2.3 Optimal beam position for efficiency

The coupling efficiency η is also dependent on the beam position *z _{c}* relative to the groove edge. Figure 6 shows η vs. z

_{c}for

*d*=30 nm and 70 nm in the configuration of surface-corrugation. In the figure the solid and dot-dashed lines represent the results obtained from Eq. (2) using the values

*f*and α obtained numerically. The individual data points are obtained with the FDTD technique. When the incident beam is centered at the groove edge,

_{c}*z*=0.

_{c}For the case of surface-corrugation with Λ=0.36 µm, η is maximized at *d*=30 nm. From Eq. (3), *z _{c}* for maximum η is

*z*=0.735

_{opt}*w*

_{0}/cos(

*θ*

_{0}). In our case w

_{0}=25 µm, θ

_{0}≈10° at

*d*=30 nm, so z

_{opt}=18.7 µm, which is consistent with our numerical calculation shown in Fig. 9. For

*d*=70 nm, assuming

*α*=0.0755 µm

^{-1}(see Fig. 4), the perturbation analysis gives z

_{opt}=10.5 µm. This is slightly larger than the value found with the FDTD technique, z

_{opt}=9.6 µm. However, we do not expect the perturbation analysis to be valid at

*d*=70 nm, for which the grooves extend through 70% of the core layer.

η vs. *z _{c}* differs somewhat between the perturbation analysis and the FDTD method, as seen in Fig. 6. The FDTD method yields a faster fall-off of η vs.

*z*. When the beam is on the grooved region,

_{c}*z*<z

_{c}_{opt}, the two analyses give the same result; when the beam is far away from the groove edge and on the un-grooved region,

*z*>z

_{c}_{opt}, the two analyses differ. That difference may be due to the finite-length effect of grating coupling. With increasing

*z*and

_{c}*z*>z

_{c}_{opt}, the length of the incident beam on the grooved region decreases, the finite-length effect becomes more manifested, which is not accounted for in the perturbation analysis. For shallow grooves, α is small, and the difference is more apparent as seen in Fig. 6.

#### 2.4 Light coupling for narrower Gaussian beam

To see how the incident beam size affects the coupling efficiency, we also calculate η for incident beam narrower than 50-µm. Figure 7 shows the calculated results for beam width 2w_{0}=79λ, 40λ, and 20λ in surface-corrugation. It is seen that the optimized η drops from 22% to 16% as the beam size decreases from 79λ to 20λ. It is also seen that η vs. angle detuning is broadened rapidly as the beam size decreases. The angular width is 0.79° for 2w_{0}=79λ, 1.51° for 2w_{0}=40λ, and 3.51° for 2w_{0}=20λ. Apparently Δθ is dominated by finiteness of grating length that the incident beam interacts with, since the spread of the incident beam is only ~0.04° even at 2w_{0}=20λ.

## 3. Experiments

The optical waveguides were prepared by evaporation of Ta_{2}O_{5} on a thermally oxidized silicon wafer. The thickness of the core layer (Ta_{2}O_{5}) is 100 nm and the SiO_{2} layer is ~2 µm. The refractive index of the core layer at 633 nm is 1.997. Those waveguides support only the fundamental mode TE_{0}. The coupling grating was etched into the core layer from film surface by electron beam lithography. Several gratings were etched on the same wafer with the same period, Λ=0.48 µm, but differing in groove depth. Figure 8 shows the topography of the grooved region and groove profile for *d*=15 nm and 57 nm, obtained with an atomic-force-microscope (AFM). For *d*=15 nm the groove profile is nearly rectangular and has a 50% duty cycle.

Figure 9 shows schematic diagram for characterizing the grating couplers. Light exiting from a He-Ne laser is expanded and gently focused onto the film surface with an achromatic lens. The full width of the focused beam at 1/e^{2} intensity is measured to be 50 µm. The sample is mounted on a goniometer for tuning the angle of incidence. To measure the coupled light inside the waveguide, the wafer is cleaved at a distance of ~2-3 mm away from the grating edge. A photodiode is placed in contact with the cleaved edge and an index-matching fluid is applied between the cleaved edge and the detector. To account for the loss of beam propagation from the groove edge to the detector, the 1/e beam propagation length in the film is measured to be 9. 4 mm.

Figure 10 shows the measured η as a function of d. The filled circles are for grooves with a nearly 50% duty cycle, and the hollow squares are for grooves with a duty cycle less than 50%, as determined by atomic-force-microscopy measurements of groove topography. Also shown is the calculated result, represented by the solid curve, assuming a rectangular groove profile with a 50% duty cycle. It is seen that ~15% efficiency is obtained experimentally. This is lower than the theoretical value of 20%. Light scattering resulting from irregularities of groove profiles and roughness in the grooved region may cause this difference.

Figure 11 shows the angle of incidence (θ_{0}) for optimal η as a function of d. Again, the solid curve in the figure is theoretically obtained while the individual data points are the experimental data. It is evident that θ_{0} decreases with *d* for *d*<60 nm, and gradually approaches a plateau for *d*>60 nm. The theoretical θ_{0} is close to the experiment. The difference between the two is less than 1° at all depths studied. According to the perturbation analysis it is expected that sin(*θ*
_{0}) is linearly dependent on *d* for shallow grooves. As seen in the insert of Fig. 11, this is indeed found to be true for *d*<50 nm. For *d*>50 nm sin(*θ*
_{0}) vs. *d* obviously deviates from the linear relationship.

Figure 12 shows normalized η as a function of angle of incidence Δθ from θ_{0} for *d*=30 nm. The solid curve represents experimental data while the scattered data are calculated, assuming a rectangular groove profile with a 50% duty cycle. It is seen that the calculated η vs. Δθ is almost the same as the experiment and the full-width-at-half-maximum angular width is ~1.0°.

## 4. Conclusions

Characteristics of input grating couplers for an incident beam with a 50-µm diameter have been investigated by solving Maxwell’s equations numerically with the FDTD method. Numerically obtained results are consistent with the perturbation analysis for shallow grooves, but differ for deep grooves. The input coupling efficiency, the angle of incidence for optimal coupling, and angular width are all groove depth dependent. The coupling efficiency drops and the resonant curve is broadened as the beam size decreases. Certain finite-length grating effects are observed. For the waveguide studied the substrate-corrugation yields greater coupling efficiency than the surface-corrugation. Experiments have been carried out on Ta_{2}O_{5} planar waveguides and confirmed these depth-dependent characteristics.

## Acknowledgments

The authors gratefully thank Kalman Pelhos, Christophe Mihalcea, and Keith Mountfield for the preparation of the waveguides and grating couplers used in our experiment. This work was performed as part of the INSIC HAMR ATP program, with the support of the U.S. department of Commerce, National Institute of Standards and Technology, Advanced Technology Program, Cooperative Agreement Number 70NAB1H3056.

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