We propose a novel beam-steering device based on a slow-light waveguide amplifier. In this paper, we present the idea of this steering technique and show its modeling characteristics. Giant steering of the radiation beam is obtained by tuning the wavelength of input light, which is coupled into the Bragg reflector waveguide. A tunable deflection-angle range can be over 40 degrees. High beam coherency and flat intensity distribution enable us to obtain an ultra-large number of resolution-points over 1,000 for few-millimeter long devices.
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Mechanical beam-steering has been commonly used for scanning and beam-steering systems in sensing and imaging applications [1,2]. Polygonal mirror scanners have been widely used to provide a large deflection angle and hence high-resolution, however it is bulky and the steering speed is limited. Special electro-optic effect crystals have been studied for electrical beam steering and recently beam steering based on photonic crystal lasers was demonstrated [3,4]. In any beam-steering techniques, the maximum beam-steering angle θr-max needs to be much larger comparing to the beam divergence angle θdiv. Number of resolution point N ( = θr-max / θdiv), which can be counted in far-field pattern observation, is requested to be over 1,000 for practical applications. Except using polygonal mirror, approaches mentioned above all met difficulty in getting high N of larger than 100. Besides, high operation voltage, large size and cost are also remained problems. An optical phased-array has been another approach for achieving high-resolution beam steering [5,6]. However, there are difficulties in increasing the number of the inter-elements of an array and in eliminating sidelobes in far-field patterns for those passive devices. Therefore, there is an eager need for a small and high-resolution beam-steering device which can provide a large beam steering angle and a small divergence angle at the same time.
Slowing light is an interesting phenomenon that reveals a new route to control light [7,8]. It can promote stronger light-matter interaction and introduce larger nonlinearity as well as dispersion [9–11]. In recently years, slow-light has been attracting much interest for building optical functional devices such as tunable optical delay lines, switches, modulators and amplifiers in different types of waveguides [12–17]. Merits are highlighted at its compact size, various functions and nice ability in integration. Recently we found that a high quality beam steering technique is prospective on a slow-light waveguide amplifier  with a Bragg reflector waveguide . If we excite a slow-light mode along the Bragg reflector waveguide, a beam of high spatial coherency can be obtained from the waveguide surface. This output would be similar to a virtually imaged phased array  which provides large angular dispersion. In our proposal here, an amplifier gain is added in the slow light waveguide to compensate the radiation loss. It enables us to make a longer radiation window which results in an ultra-high number of resolution points.
In this paper, we propose a novel beam steering device composed of a slow light amplifier with a Bragg reflector waveguide and a tunable laser. Modeling of beam steering characteristics will be presented. Although the coupling efficiency of external tunable light is limited and inconstant for different wavelengths, we are still able to get a sound performance of beam steering with a beam steering angle over 40° and resolution-points over 1,000. We believe our proposal will be able to fill the blank of ultra-compact but high performance beam steering devices.
Our proposed beam-steering device is based on a slow-light waveguide amplifier. A schematic view of the device is shown in Fig. 1 . The structure in the vertical direction is as same as a conventional 980 nm one-λ cavity VCSEL. An active region and an oxidization confinement layer are sandwiched by two distributed Bragg reflectors (DBR) as mirrors. Light is input onto the coupling region at the left side of the device, which is formed by etching. Etching depth, input angle and input position are designed to maximize the coupling efficiency . Slow-light mode is then excited along the waveguide and radiates light through the top mirror. The top-mirror reflectivity, which depends on the number of DBR pairs, is important to increase the effective radiation window length and provide enough radiation output. In addition, high reflectivity can decrease the necessary gain in the active region and suppress the amplified spontaneous emission (ASE). The bottom mirror is composed of 40 pairs of GaAs/AlAs DBR, providing full reflection. Electrodes are placed for current injection to the amplifier. Oxidization confinement layer is designed to guide the light mode laterally and confine the area of active region that provides gain. The un-oxidized aperture width is around a few microns for single-mode condition. In this device, the radiation window length L can be simply designed by lithography process. A typical length is from several hundreds of microns to a few millimeters. The radiation through the top mirror results in a decay of slow light intensity distribution. The loss can be compensated by adding gain in the active region with current injection. By adjusting the injecting current, we expect to get a near-uniform distribution of the radiation along propagation direction.
3. Beam steering
Due to the insufficient reflectivity of the top-DBR mirror (around 99%), a portion of the light is radiated from the surface when the slow-light is propagating inside the waveguide. As illustrated in Fig. 2 , the direction of the radiated light θr is determined by the angle θi of a traveling slow light mode following Snell’s Law (sinθr = nwg × sinθi) with a refractive index nwg of the slow light waveguide. It is around 3.5 for 980 nm light and has small dispersion. It is noted that θi is determined by the slow-light propagation constant β ( = k × sinθi = 2πnwg × sinθi / λ), which is highly dispersive and hence strongly dependent on the wavelength of an incident light. The relationship between β and λ is shown in Fig. 3 . Large dispersion can be observed especially near the cut-off condition. Therefore, there is a possibility of beam-steering of the radiated light if we are able to tune the incident light wavelength. The calculated slow-down factor is also plotted in Fig. 3, which is defined as the ratio of the group velocity of slow-light versus that in conventional semiconductor waveguides. Above calculations and later results in this paper are all for TE mode. TM mode is not considered because its propagation loss is much higher than TE mode.
It is easier to use an equivalent refractive index neq ( = β × λ/ 2π) to illustrate the travelling condition of a slow-light mode. Replacing β by neq, the deflection angle θr can be given by:
Now we are able to get a full view of the analytical relation between θr and λ. It is illustrated as dashed line in Fig. 4 . Nevertheless, this analytical calculation is an approximation excluding the field penetration in DBRs. We need to take this difference into consideration. Therefore, we carried out a full-vectorial numerical simulation on the radiation field from the slow-light amplifier using a film-mode-matching method  (FIMMWAVE, Photon Design Co.) applying a simulation structure more close to the real one. By knowing the neq of the slow-light mode from simulation, we are able to get a full view of numerical relation between θr and λ. The result is illustrated in Fig. 4 by solid line. The curve is similar to what is given from the analytic formula (Eq. (3). The maximum deflection angle θr-max of larger than 80° can be obtained if we are able to tune the wavelength of over 50 nm from the cut-off wavelength. During this simulation, the number of top-DBR pairs is designed to be 20. However, we found that a change in the pair number will not bring noticeable influence to θr. Therefore we believe the wavelength is the only factor dominating the deflection angle.
In addition to the maximum deflection angle θr-max, a beam divergence angle θdiv is another feature parameter in beam steering. The number of resolution-points N, which is defined as the ratio of θr-max and θdiv, is used to evaluate the figure of merits in beam steering. Our proposed beam steering technique shows its distinct advantage in getting an ultra-small θdiv and hence an ultra-large N. It comes from the high coherent property of the radiation beam and possibility in increasing the beam width. It is similar to phased-array technologies [5,6]. In Fig. 5 , we show the electric field amplitude distribution along the amplifier waveguide for different wavelengths. First, we can confirm from this graphical view that θr increases when λ is tuned away from λc. At the same time, we are able to observe very clear parallel lines, which indicate the constant front of the radiation beam. For this high coherency beam, by increasing the beam width w ( = L × cosθr), it is possible to get a very small θdiv ( = λ / w).
In Fig. 6 , we show θdiv and N as function of L, supposing a wavelength tuning range of 25 nm (which later we will show is the acceptable tuning range using fiber coupling scheme). We are able to obtain an N over 1,000 for a 3 mm long device, which is already an order larger than that of electrical beam steering devices reported previously. N will increase linearly by further lengthening L. In the above discussions, we assumed the radiation distribution is uniform, which means the radiation loss is just compensated by a necessary gain in the active region. This optimal gain coefficient go is in function of wavelength and the number of top-DBR pairs (namely the top-mirror reflectivity R). Values after simulations are 2,500~3,000 cm−1, 850~1000 cm−1 and 320~400 cm−1, for the number of top-DBR pairs of 20, 25 and 30, respectively. Please note that go can be obtained at the same range of an injection current density (from a few hundreds to a few thousands A/cm2) as VCSELs with the same top mirror design. However, in real devices, we may meet a difficulty in injecting a precise current for getting such go. That is to say, the intensity distribution on the waveguide surface may not be ideally uniform. Any non-uniformity here will influence beam divergence and thus increase θdiv. The intensity distribution is changing exponentially, and in most cases decaying. The non-uniform intensity distribution results in broadening in θdiv. An electrical field distribution on the waveguide surface can be expressed by E(z) = exp(-α·go·z/2), where α is gain coefficient shortage in percentage. By calculating a far-field light angular profile referring to Fraunhofer diffraction theory, we are able to get different divergence angles for these non-uniform intensity distributions. In Fig. 7 we show θdiv (a full width at half-maximum) for devices with radiation window lengths of 1 and 3 mm, considering the gain shortage. We compared the results for different top-mirror designs. From the figure we can see, merely by increasing L we may not be able to decrease θdiv as much as we want. A restraining of the gain shortage is important. On the other hand, by increasing the top-DBR number, it is easier to sharpen θdiv. Therefore, a high reflectivity top-mirror is preferable especially for long devices.
Gain coefficient is mainly dependent on the injection current. Different active region designs will introduce different current-gain relations. For example, of a conventional 980nm laser with In0.2Ga0.8As/GaAs 80Å thick quantum well active region, an analytical relation between its material gain g (cm−1) and current density J (A/cm2) can be approximately given by: g = 1300 × ln(J/50) [22,23]. By a good control of the inject current, it is possible to minimize gain shortage and hens get a small θdiv.
The calculations in this paper are under small-signal linear limit where the effect of gain saturation is not included and amplified spontaneous emission (ASE) is not taken into consideration. The ASE is severer for large gain in the active region, thus slow light waveguides with higher reflectivity are more suitable to reduce its effect. Although ASE may be a factor that limits the maximum length of the radiation window, few-millimeter long amplifiers have been reported [24,25] and can already meet the requirement for getting N over thousand in our purpose.
4. Coupling and device gain
At the present stage, to provide a large tuning range of the input light, we choose to use an external tunable light source and couple light into the waveguide as shown in Fig. 1. How to obtain a good coupling into the slow-light mode is a key issue in reducing unwanted leaky modes and offering high radiation output. The coupling efficiency is calculated by comparing the excited slow-light mode power to the input beam power. We have to fix the incident position d and incident angle θ of the input light in real devices when operating beam steering. d and θ are determined based on the tilt coupling scheme  designed for Bragg reflector waveguides. The optimal values are wavelength dependent but have good tolerance. Also, the number of top-DBR pairs on the coupling region (x pairs) and waveguide (y pairs) are influential. In the simulation, a Gaussian beam with TE mode is assumed to be coupled into the waveguide and the following parameters are used: x = 9, y = 20, d = 0.8μm, θ = 45°. We will find that the above parameters are suitable for large wavelength range as to be discussed.
As illustrated in Fig. 8(a) , after inputting light, there will be some leaky modes excited at the beginning of the waveguide due to imperfect coupling. However, as the slow-light travels along the waveguide, a fundamental slow light mode will dominate. We define Lw as the length of a transition region, where other leaky modes cannot be neglected. As shown in Fig. 8(b), clutter light is observed at the beginning of the waveguide, while clear constant radiation can be observed above the waveguide away from the beginning part. Intensity distribution above the waveguide is shown in Fig. 8(c). Ripples indicate the light interferences between a fundamental slow light mode and other leaky modes. Transition region can be defined as the only region with an interference amplitude of larger than 20% of the average radiation intensity. Its wavelength dependence is shown in the inserted figure in Fig. 8(c). Lw is shorter than 40 μm when λ is larger than 955 nm. Considering the typical device length of over hundreds of microns, we think the clutter light will not distort the beam quality noticeably.
Input coupling efficiency and device gain as function of wavelength are shown in Fig. 9 . The device gain here is defined as the ratio of radiation power on waveguide surface to the input Gaussian beam power. The coupling efficiency is calculated inside the coupling region, so it is independent on the device gain. When increasing the amplifier length, the device gain will proportionally increase. A gain of over 10 dB can be obtained for a 1 mm long device. Radiation spectrum may not be uniform in intensity due to the variation in coupling efficiency and also device gain, but the device can still meet requirements for broad applications for example some types of sensors and scanners. It is expected to both improve the coupling efficiency as well as device gain after further optimizing the waveguide structure and coupling conditions. Even if we only allow a wavelength tuning range of 25 nm using this fiber coupling scheme, we are still able to get a giant steering angle of 40 degrees (see Fig. 4) and resolution-points over 1,000 for a 3 mm long device (see Fig. 6). In our device, the wavelength of the output changes with beam steering. We believe it is not a big issue for the application of laser scanners such as laser radars and bar-code readers, since the spectrum is within the wavelength range of detectors. In the near future, we look forward to integrating a widely-tunable MEMS VCSEL  with our proposed device. A lateral coupling scheme has been demonstrated to realize their monolithic integration . We believe it can bring us better coupling, compacter device size and sound tunability. Also, electrical steering can be possible.
We proposed a novel beam-steering technique based on a slow-light waveguide amplifier and tunable light source. We carried out the modeling under small-signal linear limit where the effect of gain saturation is not included and amplified spontaneous emission is not taken into consideration. Wavelength-tunable light is coupled into the waveguide and excites the slow-light mode. Due to the reduced reflectivity of the top-mirror, radiation takes place from the surface. By tuning the wavelength, a giant change in deflection angle can be obtained. For a 25 nm tuning, we are able to get a beam-steering range of over 40°. The radiation light is highly coherent which brings the device an ultra-large number of resolution-points. A near-uniform intensity distribution along the slow light amplifier can be realized with balancing the radiation loss and the gain under small-signal linear limit. The number of resolution-points can be over 1,000 for a 3 mm long device and even larger by lengthening the device. Both coupling efficiency and device gain can be further improved by optimizing the waveguide structure and coupling conditions.
This work was supported by Grant-in-Aid for Scientific Research (S) from the Ministry of Education, Culture, Sports, Science and Technology of Japan (#22226008).
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