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Abstract
Utilizing the cutoff characteristic of zero-order TE mode of the W type waveguide, we propose a cutoff modulator with tunable filtering characteristic in this paper. Simulation results show that: for the modulator based on GaAlAs/GaAs epitaxial layers, its cutoff wavelength can be shifted from 1.62 µm to 1.42 µm when the refractive index of the core layer is decreased by -0.01 in the modulation region of 800 µm; and the extinction ration at the wavelength of 1.55 µm reaches 40 dB. Compared with other traditional cutoff modulators, it has the advantages of more compact size, simpler structure and the potential for high intensity integration.
©2007 Optical Society of America
1. Introduction
Classified by their structures, the optical amplitude modulators can be divided into several different categories: the electro-absorption modulator, the Mach-Zehnder interferometer, directional coupler, the micromechanical modulator, and so on. The cutoff modulator is an attractive candidate among them, for its merits such as: simple structure, low insertion loss, monotonic modulating characteristic with respect to the driving signal. In the past years, it has already been fabricated on nearly all material systems: LiNbO3 [1–4], GaAs/GaAlAs [5], Si/Ge [6], Polymer [7]. However, these cutoff modulators fabricated on the bulk materials are wavelength independent [5,7], which means the cutoff is equal for all bands. Wavelength dependent cutoff modulator is first proposed by T.Tinker utilizing thermo-optical effect of the silicon-based photonic crystal [8]. The operation principle of the device is just like a tunable high-pass filter: its cutoff wavelength is moved from one side of the input optical wavelength to the other side of it by applying a control signal. However, to shift the cutoff wavelength by 60 nm, the photonic crystal modulator needs a refractive index increase of 0.15 by heating the silicon to 650 °C. Such a high temperature limits its practical application. In this paper, we propose a cutoff modulator with tunable high-pass filtering characteristic based on the W type five-layer symmetric slab waveguide, its cutoff wavelength can be moved by up to 200 nm only with a refractive index change of -0.01. Since its special cutoff manner, the modulator in this paper has a more compact size and a simpler structure than other traditional cutoff modulators. It is very fit for constructing large-scale high-density variable optical attenuator (VOA) array or gate switch array in the optical communication system.
In the section 2 of the paper, we study the cutoff behavior of zero-order TE mode of the W type five-layer symmetric slab waveguide. A criterion is given in this section for judging whether the zero-order TE mode of a W waveguide has the non-zero cutoff frequency. In section 3, we design a cutoff modulator working at the telecommunication wavelength of 1.55 µm based on the W type epitaxial layers. Its tunable filtering characteristic is validated by simulation, and then its performance is compared in detail with various cutoff modulators reported previously. At last, we reach a conclusion in section 4.
Fig. 1. Refractive index distribution of the w type five-layer symmetric slab waveguide and the cross-section of the cutoff modulator.
2. Cutoff characteristic of the W type five-layer symmetric slab waveguide
Five-layer symmetric slab waveguide with its refractive index profile resembling the letter W (see Fig. 1) has many special characteristics, such as large cross section for single mode operation, high confinement factor, and mode filtering [9]. Utilizing these properties, a lot of practical devices were reported in the past: low loss waveguide [10] and phase modulator [11], large cross section asymmetric switch based on carrier injection effect [12] and MMI type electro-optical switch [13]. However, there is still a special property of the W waveguide which hasn’t attracted any attention. It is the cutoff characteristic of its zero-order TE mode. Unlike other symmetric slab structures, the cutoff frequency of the zero-order TE mode is not zero if the structure of a W waveguide satisfies a certain condition. As we want to employ the cutoff characteristic, this condition should be studied at first.
Refractive index distribution of the W waveguide is shown in Fig. 1, two confinement layers (n 2) are inserted between the core (n 1) and the cladding layers (n 3), their refractive indices satisfy the inequation: n 1>n 3>n 2. The eigenvalue equation for the TE mode of the W waveguide is [9]:
Where N is the mode order, the definitions of normalized variables v, u, w, t and c are: v 2=a 2 k 2(n 2 1-n 2 3, u 2=a 2(k 2 n 2 1-β 2), c 2=(n 2 1-n 2 2)(n 2 1-n 2 3)≅(n 1-n 2)/(n 1-n 3), t 2=a 2(β 2-k 2 n 2 2)=v 2 c 2-u 2, w 2=a 2(β 2-k 2 n 2 3)=v 2-u 2. Then at cutoff, we have: w=0, u=vc and t 2=v 2 c(c 2-1), so the normalized cutoff frequency vc for the zero-order TE mode of the W waveguide satisfies the equation:
Obviously, zero is a solution of Eq. (2), but this does not imply the cutoff frequency for the zero-order TE mode of the W waveguide is zero. Eq. (2) can have another positive solution besides zero under a certain condition, that is to say, f (v) intersects with the abscissa axis at the origin and another point on the positive axis. Function f (v) is depicted in Fig. 2 for different values of b/a and c. When Eq. (2) has two solutions, the practical cutoff frequency vc would be the positive solution rather than zero.
Fig. 2. Intersecting points of the function with the abscissa axis for different structural parameters: (a) b/a=2; (b) c=2
To judge whether the zero-order TE mode of a W waveguide has non-zero cutoff frequency, we can examine the first order derivative of the function f (v) :
We note that f′(v) is a increasing function when v≥0, so if f′(0)≥0, f′(v) is always positive on the entire positive abscissa axis. With its first order derivative being positive, f (v) is also an increasing function on the positive abscissa axis. Together with f (0)=0, we can conclude that f (v)>0 for v>0 and Eq. (2) has only one solution of zero. On the other hand, if f ′(0)<0, starting from the origin, f (v) first decreases and then increases. It ultimately intersects with the abscissa axis again as shown in Fig. 2. So the necessary and sufficient condition for the cutoff frequency of the zero-order TE mode being non-zero in a W waveguide is:
Since we want to use the cutoff characteristic of the zero-order TE mode, the criterion in Eq. (4) should be well satisfied when we determine the structure of the W waveguide.
Fig. 3. Plots of cutoff wavelength of the zero-order TE mode in the W type GaAs/GaAlAs epitaxial layers versus Al contents in the confinement layer and cladding layer respectively. (a) cutoff wavelength versus Al content in the cladding layer, Al concentration in the confinement layer is 0.4, and b/a=2; (b) cutoff wavelength versus Al content in the confinement layer, Al concentration in cladding layer is 0.1, and 2a=0.8 µm.
3. Cutoff modulator on the W type GaAs/GaAlAs epitaxial layers
The practical W type waveguide can be formed by the epitaxial growth of GaAs/GaAlAs layers [10–13]. Controlling Al content and thickness of every epitaxial layer, we can adjust cutoff wavelength of the zero-order TE mode. Variations of the cutoff wavelength with Al concentrations in the confinement and cladding layers for various values of b/a and a are shown in Fig. 3. The material dispersions of GaAs and GaAlAs have been involved in the calculation. From Fig. 3, we can derive the manner how various parameters of the epitaxial layers affect the value of the cutoff wavelength, which is shown in table.1. It can guide us to set the structure of epitaxial layers and then get the cutoff wavelength we need.
Table 1. Influence of modulating various parameters of the W type GaAs/GaAlAs epitaxial layers on the cutoff wavelength of its zero-order TE mode.
After the GaAs/GaAlAs epitaxial layers are established, the inherent cutoff wavelength can be changed by the free carrier injection effect. The modulation mechanism can be understood with Fig. 2(a): Increasing the value of c can shift the position of normalized cutoff frequency vc positively. Since the confinement effect of the double heterostructure to the injected carriers, the refractive index decreases locally in the core layer while n 2 and n 3 maintain unchanged [14]. From the definition of c, decreasing n 1 means to increase c, so the cutoff frequency is increased. Plots of the cutoff wavelength of the zero-order TE mode versus refractive index decrease Δn in the core layer are shown in Fig. 4. As Δn is induced by the free carrier injection effect, its wavelength dispersion (Δn∝λ 2) has already been taken into account in all calculations and simulations throughout this paper. And all the given values of Δn are uniformly specified at the wavelength of 1.31 µm.
Fig. 4. Cutoff wavelength of the zero-order TE mode versus refractive index decrease Δn in the core layer. Labelling parameters give the different values of Al content in the cladding layer. In the calculation, Al content in the confinement layer is 0.35, b/a=2, 2a=0.7 µm
In this paper we plan to design a cutoff modulator working at the telecommunication wavelength of 1.55 µm. By calculation we get the W type slab waveguide consisting of a 2.5 µm GaAl0.14As0.86 as lower cladding layer, a 0.35 µm GaAl0.35As0.65 lower confinement layer, a 0.7 µm GaAs core layer, a 0.35 µm GaAl0.35As0.65 up confinement layer, and a 0.6 µm GaAl0.14As0.86 as up cladding layer. Their refractive indices at the wavelength of 1.55 µm are 3.305, 3.208, 3.37, 3.208 and 3.305 respectively. Its cutoff wavelength can be shifted from 2.01 µm to 1.53 µm with a refractive index decrease of -0.01 in the core layer, which is plotted as the blue curve in Fig. 4.
The cutoff modulator then consists of a rib etched on the well-designed epitaxial layers and an electrode covering the rib for the carrier injection. The waveguide width, etched depth, and the size of the carrier injection area are 4 µm, 1.1 µm, 4 µm×800 µm respectively. The cross-section of the rib waveguide is shown in Fig. 1. The etched depth should be deep enough to avoid the cutoff in horizontal direction when the carriers are injected. Since the cutoff in this direction is wavelength independent [5, 7], it will ruin the filtering characteristic of the device. We must make sure the cutoff only occur in the perpendicular direction when etching the rib. And then the waveguide width should only support the fundament mode. Increasing the electrode length can enhance both the light attenuation and the injected current proportionally, the electrode of 800 µm is a trade-off between them.
The simulation result employing Beam propagation method(BPM) in Fig. 5 gives the wavelength response of the modulator, the commercial software utilized in our simulation is the BeamPROP developed by the Rsoft company. High-pass filtering characteristic of the modulator is clear in Fig. 5, and the high-pass cutoff wavelength shifts from 1.62 µm to 1.42 µm when the refractive index in the core layer is decreased by -0.01 (a typical refractive index change induced by the free carrier injection effect). If we keep on decreasing the refractive index, the cutoff wavelength can be shifted even further. Shifting range of the cutoff wavelength is then limited by the level of injected current density that we can reach. The efficiency is fifty times bigger than that of the photonic crystal [8], which is 60 nm versus 0.15. We note that the cutoff wavelength of the final rib waveguide is lower than that of the original unetched slab. The point should be taken into well account when we design a cutoff modulator. Figure 6 shows the modulating characteristic of the device at two key telecommunication wavelengths: 1.31 µm and 1.55 µm. Because of the high-pass filtering property, cutoff is selective for the wavelength of 1.55 µm. In Fig. 6, light of 1.55 µm is attenuated by 41 dB as the corresponding refractive index change at this wavelength reaches -0.014. The propagation losses of the device without injected current (loss of straight waveguide) are -0.5 dB/cm and -0.004 dB/cm for the wavelength of 1.55 µm and 1.31 µm respectively. Here, the propagation loss is caused by the substrate leakage, it is bigger for the light whose wavelength is closer to the cutoff wavelength. The device can be used in the optical communication systems. When it works, light of 1.55 µm is filtered out while light of 1.31 µm keeps unaffected.
Fig. 5. Wavelength response of the cutoff modulator for a set of different refractive index decrements Δn in the core layer.
Fig. 6. The simulated modulating characteristic at wavelength of 1.31µm and 1.55µm. With the same injected current, Δn for the wavelength of 1.55 µm is 1.4 times bigger than that for 1.31 µm. The abscissa gives the value of Δn at 1.31 µm.
Figure 7 gives the influence of the waveguide width on the filtering performance. Increasing the waveguide width means to increasing the cutoff wavelength. This can be understood from the basic propagation theory of optical waveguide: A narrower waveguide has the smaller propagation constant, and can be cutoff more easily. As w→∞, cutoff wavelength of the device then reaches its limit, which is that of the unetched W slab. We can also adjust the filtering characteristic of the device by setting proper waveguide width after the epitaxial layers is fixed.
Fig. 7. Wavelength response of the cutoff modulator for a set of different waveguide widths. The refractive index decrement in the core layer Δn is -0.01
According to the classical Drude model, the relationship between the refractive index decrease Δn and the free carriers concentration N is:
Where subscripts e and h refer to electron and hole. m* is the effective mass. Moreover, λ is optical wavelength, q is the electron charge, c is the velocity of light in vacuum, ε 0 is the free space permittivity, n 0 is the refractive index of the intrinsic GaAs, and N e=N h=N. The exact values of these parameters can refer to reference 15. Calculation result shows the free carriers concentration in the core layer should reach 2.47e18/cm2 in order to reduce the refractive index by -0.01 at the wavelength of 1.31 µm. The formula about the injected current density is [14]:
In Eq. (6), d is the thickness of the core, which 0.7µm for our device, τ is the carrier life. With Eq. (6), we can estimate that the injected current density is about 2.8 kA/cm2, and then the corresponding injected current is about 90 mA. The operation speed of the devices using the free carrier injection effect is determined by the carrier recombination time and the device capacitance. Generally, it is on the scale of 10 ns [16].
Up to now, the absorption loss is not involved in our analyse. Expression of the absorption coefficient induced by the free carriers is:
µh and µe are the mobility of hole and electron repectively. Δα then can be expressed in terms of Δn:
the value of the first section in Eq. (8) is 3.36e4. As the formula of loss is 10lg(eΔαL), then the absorption loss for our device is 116.8Δn dB. L is the length of injected area, which is 800 µm here. So the absorption loss is proportional to Δn and is independent to the wavelength. The absorption loss is only -1.16 dB for a Δn of -0.01, it is negligible compared with the propagation loss caused by the mode cutoff. One can easily get the propagation characteristic of the device including the free carrier absorption by simply adding the absorption loss to previous simulation results.
Fig. 8. Intensity plots of the light propagation at the wavelength of 1.55 µm. x,y and z denote horizontal direction, vertical direction and propagation direction respectively. Form the top down, Δn are 0,-0.005,-0.01 respectively. The monitor value is normalized propagation power.
Besides the high-pass filtering characteristic, the cutoff modulator on the W waveguide has the advantages of a more compact size and the potential for high density integration compared with the traditional cutoff modulators. The length of the electrode in this work is only 800 µm, much shorter than those previously reported by A. Neyer (3.2 mm in Ti-diffusion y-cut LiNbO3) [1], N.H.Zhu(18 mm in z-cut LiNbO3) [2], P. R. Ashley(4 mm in Ti-diffusion x-cut LiNbO3) [3], Retzon Chen(5 mm in proton-exchanged x-cut LiNbO3) [4], and also Retzon Chen (2.5 mm in GaAs/GaAlAs heterostructure) [5]. Furthermore, the traditional cutoff modulators need to operate on the very edge of cutoff, so a waveguide should be divided into the lossless propagation region and the active region of weak mode confinement. Additional processes are needed to realize two different confinement effects on a single waveguide, such as two steps reactive ion etching with shadow mask [7] or electron beam deposition with multiple masks [3]. And tapers are needed to connect different sections, which increase the length of device additionally. The cutoff modulator we proposed in this paper does not need to work on the near-cutoff region. The refractive index change induced by free carrier injection is bigger enough to bring the mode from the lossless propagation state to the deep cutoff state, as shown in Fig. 6. So the modulator in this paper has a more compact size and a simpler fabrication process. Another problem for the traditional cutoff modulators is the crosstalk, which is caused by their cutoff manner. For the cutoff modulators in Si/Ge [6] and polymer [7], the cutoff is in the horizontal direction, the energy leaking out laterally can be coupled into the adjoining waveguides [6], which aggravates the crosstalk performance. So two neighbor waveguides shouldn’t be placed too closely, and then the integration density is limited. One advantage of using the W waveguide is that the cutoff is in the vertical direction: the energy penetrates through the lower cladding layer and then be absorbed by the substrate. Adjacent waveguides will not be affected. The point is clearly shown in Fig. 8.
4. Conclusion
In summary, we propose a cutoff modulator with tunable high-pass filtering characteristic using the W type GaAs/GaAlAs layers. Compared with the cutoff modulator of the same ability on photonic crystal, its high-pass cutoff wavelength can be shifted in a larger range with a much smaller refractive index change. And it has a more compact size and a simpler structure than other traditional cutoff modulators. In this paper, the cutoff modulator works on the band of 1.55 µm. In fact, we can also design the cutoff modulator working on other optical wavelengths by setting proper epitaxial layers. Next, we will try to fabricate the practical cutoff modulator according to our design in this paper, and then verify its tunable filtering characteristic.
In this paper, we concentrate our attention on the cutoff characteristic of TE mode. However, the TM mode also has the analogous cutoff characteristic. Of course, their exact cutoff wavelengths are different from each other. Using the difference, we can design a single-polarization waveguide at a given band. This will be our future work.
Acknowledgments
This work is supported by the Key Fund of Natural Science Foundation of China (No 60436020) and the Major State Basic Research Development Program of China (No 2007CB163405).
References and links
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