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Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces
Optics Communications 283 (2010) 4078–4084
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces Wei Jia ⁎, Liyong Jiang, Kai Chen, Xiangyin Li School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
a r t i c l e
i n f o
Article history: Received 19 December 2009 Received in revised form 12 May 2010 Accepted 25 May 2010 Keywords: Power beam splitters Genetic algorithms Photonic crystal waveguide
a b s t r a c t In this work, we use the finite-difference time-domain method in conjunction with a genetic algorithm to design photonic crystal power beam splitters based upon a typical planar photonic crystal waveguide with corrugated surfaces or gratinglike surfaces covered behind the termination. Considering a power detector placed at different locations in the output field, we have obtained several beam splitters designs with different splitting angles. These beam splitters have high splitting efficiency and power intensity in the propagation direction. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals (PCs) [1,2] have attracted much attention because of their ability to manipulate light at the wavelength scale. As such, they have been recognized as a promising candidate for the realization of high-density optical integration [3]. The study of the beam splitter with low energy loss has become an important aspect of PC-based optical devices. There are three main types of the beam splitter, such as power beam splitters [4–14], polarization beam splitters [15–17], and frequency beam splitters [18,19]. In this paper, we focus our attention on the design of power beam splitters. In most previous works, the design of power beam splitter was mainly based on the self-collimation effect [4–10] or PC waveguide (PCW) structures [11–14]. In the first case, the light is transmitted along the direction of self-collimation effect and separated with several branches in the output area. Hence, we can call such splitter as open-type power splitter. The most important advantage of such splitter is long transmission distance with non-contact mode when light is spitted and coupled from one PC device to another, while they also exist some apparent draws, for examples, the applicable frequency band is relatively narrow, the coupling efficiency is too low, the structure is too complex to fabricate, and the splitting angle is hard to control. Using similar definition, the later case can be named as closed-type power splitter because the light is confined and transmitted inside the PCW. Such splitter can solve most problems existed in the first case, while how to realize long transmission distance is a litter
difficult. Considering these situations, very recently, our group has proposed a new kind of open-type power beam splitter which is based upon a planar PCW with corrugated surfaces added behind the termination [20]. By optimizing the structure parameters of the corrugated surfaces using a genetic algorithm (GA) in conjunction with the finite-difference time-domain method (FDTD), such PCW can produce a clear splitting effect with splitting angle of 30° in the output area. The physical mechanism behind this phenomenon is quite in similar with the directional-emission effect [20–27], that is, the splitting effect is the interference of surface modes on the corrugated surfaces and the main output light of PCW. More importantly, such open-type power beam splitter has high coupling efficiency, large frequency band and the splitting angle can be easily controlled. So it can combine the advantages of former two kinds of splitters and is believed to be a competitive candidate in the market-place of this area. As an extended work, in the present study, we will give a more detailed research about such new power splitter and introduce the gratinglike surface behind the termination to realize splitting effect for comparison. We still employ a combined method mixed by the GA and FDTD to design these structures. As a global optimization algorithm, GA [28] has been successfully applied in designing various kinds of photonics devices [20,27,29–33], including large band-gap photonic crystals, photonic crystal fibers, spot-size converters, photonic crystal filters, and micro cavity, directional emitters. By using such combined method, it is believed the gratinglike surface will be more efficient than corrugated surface in controlling the splitting effect. 2. Model and methods
⁎ Corresponding author. E-mail addresses: [email protected] (W. Jia), [email protected] (L. Jiang), [email protected] (X. Li). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.05.050
We consider a finite PC with size of 8a(x-direction) × 24a(ydirection), where a is the lattice period. This PC is created by a square
W. Jia et al. / Optics Communications 283 (2010) 4078–4084
lattice of cylinders with dielectric constant n = 3.4 (e.g., InGaAsP–InP at a wavelength of 1.55 μm) and radius r = 0.18a. A row of cylinders are removed along the plane x = 0 to form a single-mode waveguide that supports a guided mode with frequency between ω = 0.30 × 2πc/a and ω = 0.44 × 2πc/a, propagating in the plane normal to the cylinders with the electric field parallel to them. For source frequencies within the band-gap range, input light waves are well confined within the waveguide, and any subsequent transmission from the waveguide is not affected by the width of the PC. In this work, we focus our beam splitter designs on two models. Fig. 1(a) shows the model of a PCW with a corrugated surface added behind the termination. We will use the GA to optimize the parameters of corrugated surface for obtaining the best possible design. The to-be optimized parameters include the diameter of even rods (r1), the diameter of odd rods (r2), the displacement of even rods along x (r3), the displacement of odd rods along x (r4), the lattice length defined as the center to center distance between even and odd rods (r5), and the distance of each sub-column to the surface of PCW (r6). We define the union of all parameters as S_1 = (r1, r2, r3, r4, r5, r6). Fig. 1(b) shows the model of a PCW with a gratinglike surface added behind the termination. The to-be optimized parameters in the gratinglike surface are the diameter of rods (t1), the displacement of rods along x (t2), the lattice length defined as the center to center distance between two rods (t3), and the distance of each sub-column to the surface of PCW (t4). And the union of the parameters is defined as S_2 = (t1, t2, t3, t4). For each model, a 2D point source (i.e. line source), which emits Gaussian continuous wave with frequency of ω and width of λ, was placed at the location of x =−11a and y = 0. The distance between the detectors and the output of the added surfaces is R, while θD can represent the off-axis degree of detectors. The directed power of detector, normalized to the input power, obtained as time average by the FDTD can be written as: PD ðω0 Þ =
2 t ∫t21 jΩðω0 ; t Þjdt t2 −t1
ð1Þ
Where 8 → > 6a * > Re ∫0 ED ðt Þ × HD ðt Þ ⋅ d l > > > > > h i → > Ωðω0 ; t Þ = > λ > > Re ∫ ES ðt Þ × HS* ðt Þ ⋅d l′ < 0 > > > > > > > t1 = N1 Δt; > > > > :
½
1 t2 = N2 Δt; Δt = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c 1 = Δx + 1 = Δy2
ð2Þ
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Here, ES and HS represent the electric and magnetic field intensity of light source, ED and HD represent the electric and magnetic field intensity of detector, t1 and t2 represent the beginning and stop time of detector, Δt represents the time step of FDTD, and Δx and Δy represent the spatial interval of FDTD. In our study, to satisfy convergence of the FDTD technique, each periodic lattice in the computational domain is divided into 25 × 25 grid cells (Δx = Δy = 0.04a), and the beginning and stop time of detector is set as 300Δt and 600Δt, respectively. So Ω(ω0, t) and PD(ω0) can represent the instantaneous and time-average normalized power of detector. Not that, in the following calculations, the FDTD method is combined with perfectly matched layers, which are absorption boundaries, to calculate the propagation of the electromagnetic wave. In this work, we use a simple GA to optimize the above proposed parameters. To clarify the implementation process of the GA, we give a detailed description now. To initiate the GA, the to-be optimized parameters are encoded by binary strings and directly stored in a vector, called a chromosome. The initial population P0 is created by randomly yielding Nc chromosomes. Each chromosome therefore compactly represents a corrugated surface or a gratinglike surface to be simulated. For example, the ith chromosome to represent the corrugated surface in Fig. 1(a) can be expressed by the following L-length vector → Si = ðC1 ; C2 ; C3 ; ⋯; CL Þ = ðSr1 ; Sr2 ; Sr3 ; ⋯; Sr6 Þ;
ði = 1; 2; 3; ⋯; Nc Þ
ð3Þ
Here, Ck = 0 or 1, (k =1, 2, 3, ⋯, L) can represent a binary bit; Sr1, Sr2, Sr3, ⋯, Sr6 are six binary strings representing the information of r1, r2, r3, ⋯, r6. Specially, in this paper, we set the searching space for the above six variables as (0, a), (0, a), (0, 2a], (0, 2a], (a, 2a], and (0, 2a], respectively. Considering the requirement in practical fabrication, we assume the precision for all variables is 0.01a. The chromosome for gratinglike surface in Fig. 1(b) is similar and the searching space for t1, t2, t3, t4 is set as (0, a), (0, 2a), (a, 2a], and (0, 2a], respectively. Besides, according to many previous works and our experience, if the scale of corrugated surface or gratinglike surface is large than that of PCW in y-direction, it will only bring tiny difference for the final directional-emission effect or splitting effect when the number of rod is added along y-direction. Hence, with the definition of searching space for r5, we set a total number of 30 rods in ydirection for both corrugated surface and gratinglike surface. After encoding, each chromosome will be further decoded to determine the values of all variables. We utilize the following equation as decoding scheme ri =
BTDðSri Þ × a; ði = 1; 2; 3; ⋯; 6Þ 100
ð4Þ
Fig. 1. Schematic diagram of two kinds of power beam splitters. (a) PCW with an added corrugated surface. (b) PCW with an added gratinglike surface. For each model, the left part is a planar PCW, a being the lattice constant, the middle part is an added surface with several to-be optimized parameters. The right part is a detector to characterize the transmission properties in the direction along angle of θD.
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Fig. 2. (a) Statistic information on searching process about PCW with an added corrugated surface. (b) Statistic information on searching process about PCW with an added gratinglike surface.
Here, ‘BTD’ is a function to translate the binary string to decimal value. After decoding, the fitness value Fi of each chromosome will be evaluated by the fitness function, which is represented by Eq. (1) in our study. The GA then works by in turn using three genetic operators, Selection, Crossover, and Mutation, to evolve the population. Selection creates a new generation Ps by allotting more positions in the new population to those chromosomes with favorable fitness values, and eliminating those with poor values. Crossover is the basic operator for producing new chromosomes in the GA. Like its counterpart in nature, Crossover produces new individuals that have some parts of both parent's genetic material. All off individuals creates a new generation Pc. In natural evolution, mutation is a random process where one allele of a gene is replaced by another to produce a new genetic structure.
Similarly, in the GA, the role of Mutation is often seen as providing a guarantee that the probability of searching any given string will never be zero and acting as a safety net to recover good genetic material that may be lost through the action of Selection and Crossover. Based on the generation Pc, a new generation Pm is produced via the step of Mutation. The above three genetic operators is repeated until the algorithm terminates, typically after a particularly “fit” solution is found, or more generally, after a pre-defined number of generations. The best chromosome in the final generation will be decoded to provide the optimum designs. Moreover, for the model in Fig. 1(b), the search space for the four variables is set as (0, a), (0, 2a], (a, 2a], and (0, 2a], respectively. Other steps are similar as mentioned in the above flowchart for Fig. 1(a). 3. Optimization results and discussion
Table 1 The optimum parameters of corrugated surfaces for three different splitting angle. Splitting angle
r1
r2
r3
r4
r5
r6
10° 20° 30°
0.52a 0.22a 0.18a
0.42a 0.17a 0.84a
0.49a 0.45a 0.34a
1.99a 0.55a 1.56a
0.84a 0.81a 0.78a
1.42a 1.05a 1.46a
Table 2 The optimum parameters of gratinglike surfaces for three different splitting angle. Splitting angle
t1
t2
t3
t4
10° 20° 30°
0.62a 0.88a 0.86a
1.66a 1.25a 0.72a
0.92a 1.16a 1.03a
0.61a 0.78a 0.68a
In this section, we will present several GA projects for beam splitters design. In each GA project, we used a population size of 200 chromosomes for each generation, and allowed the program to run for 80 generations. We initially assumed the frequency of the Gaussian continuous wave is equal to 0.39 × 2πc/a. For each model, different GA projects were executed considering the detector placed along the angle of 10°, 20°, and 30° respectively. And the R for each project is set as 40a. Fig. 2(a) and (b) shows the statistic information on searching process for each GA project. Our GA is performed as expected, and we got a general increase of fitness as the algorithm progressing. The maximum fitness was obtained at 63, 59, 69, 66, 55, and 69 generations for different projects, respectively. For two models, the optimum parameters are listed in Tables 1 and 2 respectively. Fig. 3(a) and (b) shows the spectrum of transmission versus wavelength when
Fig. 3. (a) Directed power versus wavelength for three optimum structures with an added corrugated surface, (b) Directed power versus wavelength for three optimum structures with an added gratinglike surface.
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the normal incidence Gaussian wave is launched into the above optimum PC structures. From Fig. 3(a) and (b), it can be seen that the maximum detector power of every optimum PC structure is beyond 0.4359 when the frequency of incidence Gaussian wave is 0.39 × 2πc/ a. We define an acceptable bandwidth within which the detector power is larger than 30% of the maximum power. With this definition, from Fig. 3(a) and (b), it is found that the acceptable bandwidth for these optimum structures are all applicable in [0.382 × 2πc/a, 0.402 × 2πc/a]. In order to give a directly description about the splitting effect of our designed beam splitters, we have firstly calculated the power distribution of the output beam in the output field for each optimum
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design as shown in Figs. 4 and 5. In particular, the working frequency in Fig. 4 is set to be 0.39 × 2πc/a while the working frequency in Fig. 5 is set to be 0.382 × 2πc/a to verify the acceptable bandwidth as indicated in Fig. 3. For case of model 1, Fig. 4(a), (b), and (c) gives the power distribution of three designs with detector placed at angle of 10°, 20°, and 30° respectively. From the figures, it can be seen that all optimum structures show apparent splitting effects. In addition, though the power detector is predefined at location of R = 40a in GA project, the output beams can propagate a longer distance. The effective beams terminate at around R = 70a. The maximum detector power is found to be 42% of the source power. For case of model 2, as shown in Fig. 4(d), (e), and (f), we find that the optimum structures
Fig. 4. Electric-field intensity distributions for different structures: (a), (b), and (c) represent PCW with an added optimum corrugated surfaces for detector angle of 10°, 20°, and 30° respectively, (d), (e), and (f) represent PCW with an added optimum gratinglike surfaces for detector angle of 10°, 20°, and 30° respectively, The right part of each sub-figure shows the time-average power distribution detected along the y-direction at the position of a surface-launched of the entire structure on the left. The frequency equals to 0.39 × 2πc/a for all sub-figures.
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also have obvious splitting effects, whereas the power of the splitting beams seem to be greatly enhanced in comparison with case of model 1. To gain further insight into the effect of the system size on the emission quality, the right part of each sub-figure in Fig. 4 are shown time-average power distribution detected along the y-direction at the position of a surface-launched of the entire structure on the left. According to the amplitude of the light, it can be believed that the splitting effect is mainly influenced by the interference of surface waves excited along the corrugated surfaces or gratinglike surfaces. On the other hand, from the right part of Fig. 4, we can find the main power of surface waves is mainly distributed in the area of [−12a, 12a], equals to the width of PCW in y-direction. So it is believed that, in order to produce a beam splitting effect, the length of corrugated
surface and gratinglike surface is at least 24a. Furthermore, from Fig. 5, it is found that when the optimum structures work with frequency of 0.382 × 2πc/a, there still exit clear beam splitting effects for all designs, despite the intensity of output light is apparently weaker than those in Fig. 4. This can verify the effectiveness of the acceptable bandwidth as indicated in Fig. 3. In order to understand the splitting effect more clearly, as shown in Fig. 6, we calculate the angular intensity distribution of the output light for each optimum structure, which is obtained by placing a number of angular detectors in the output field (for example in Fig. 4a). Specially, we set R = 40a for all calculations. When a corrugated surface is added behind the termination of PCW, obviously, the transmission power is split into two beams with
Fig. 5. Electric-field intensity distributions for different structures: (a), (b), and (c) represent PCW with an added optimum corrugated surfaces for detector angle of 10°, 20°, and 30° respectively, (d), (e), and (f) represent PCW with an added optimum gratinglike surfaces for detector angle of 10°, 20°, and 30° respectively. The frequency equals to 0.382 × 2πc/a for all sub-figures.
W. Jia et al. / Optics Communications 283 (2010) 4078–4084
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Fig. 6. The corresponding angular intensity distribution of the output light for each optimum structure as indicated in (a)–(f). The frequency equals to 0.39×2πc/a for all sub-figures.
angular separations of 20°, 40°, and 60° in Fig. 6(a), (b), and (c), respectively. The divergence angle of each beam is about 10°, 20°, and 30°, respectively. Similarly, Fig. 6(d), (e), and (f) present the angular intensity distributions for PCW with gratinglike surfaces added behind the termination. In order to verify the reasonableness of our design and the application in practice, the size effect of the proposed system on the quality of splitting beam should be investigated. The time-averaged transmission coefficients of power (normalized to the incident power) along the beam are calculated by placing the detector with width of 6a at different positions, e.g., the location of transmission detector in Fig. 4(a) varies from R = 25a to R = 70a with step of 5a. The calculated results are given in Fig. 7(a) and (b), from which the transmission inevitably decreases with the increase of R, but the light intensity of the splitting beam still reached 33.5% when R = 70a, and
the light intensity of PCW with gratinglike surfaces is better than PCW with corrugated surfaces. The efficiency of single beam for three PCW with gratinglike surfaces can reach 46.9%, 44.7% and 48.8% at 40a away from the exit. At last, we want to give some comments on the common characters of our designed results. Firstly, for each model, it seems that a clear splitting effect may happen only when the predefined divergence angle is in range of 10° to 30°. For example, compared with Fig. 4(b) and (e), in Fig. 4(a) and (d), there seems a focusing effect happened before the main beam is split, while in Fig. 4(c) and (f), there seems exist some more lateral lobes in each splitting beam. Second, for each splitting angle, the final optimized design for the second model has obviously better performance as well as simpler structure than that for the first model. On the other words, the grantinglike surface has dominant advantage to realize open-type
Fig. 7. (a) Normalized transmission coefficients versus detector location for three optimum beam splitters in Fig. 6(a)–(c). (b) Normalized transmission coefficients versus detector location for three optimum beam splitters in Fig. 6(d)–(f). The frequency equals to 0.39 × 2πc/a.
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splitting effect than the corrugated surface. By decreasing the width of the detector, it is also believed to be an ideal candidate to realize highperformance focusing effects. 4. Conclusion In summary, we have proposed two kinds of new power beam splitters based upon typical planar PCW with corrugated surfaces or gratinglike surfaces added behind the termination of PCW. To design such power beam splitters, we use a combined method by the FDTD and the GA. By globally optimizing the parameters in two types of added surfaces, we have realized several beam splitters with different splitting angles. It is found the splitting efficiency for case of PCW with added gratinglike surfaces is obviously higher than that for PCW with added corrugated surfaces, while the former also has a simpler structure which is more convenient for fabrication. The advantages of our proposed models are the controllable splitting properties and their design feasibility. The power beam splitters obtained in this study may find more important applications in integrated optical circuits. Acknowledgements This work was supported by the Independent research fund of Nanjing University of Science and Technology (NJUST) (No. 2010ZYTS059, No. AE88030, No. AB39103), and the doctorial creative fund of NJUST. References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059–2062. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486–2489. [3] C.M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, Kluwer Academic Publishers, Netherlands, 2001.
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Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces Wei Jia ⁎, Liyong Jiang, Kai Chen, Xiangyin Li School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
a r t i c l e
i n f o
Article history: Received 19 December 2009 Received in revised form 12 May 2010 Accepted 25 May 2010 Keywords: Power beam splitters Genetic algorithms Photonic crystal waveguide
a b s t r a c t In this work, we use the finite-difference time-domain method in conjunction with a genetic algorithm to design photonic crystal power beam splitters based upon a typical planar photonic crystal waveguide with corrugated surfaces or gratinglike surfaces covered behind the termination. Considering a power detector placed at different locations in the output field, we have obtained several beam splitters designs with different splitting angles. These beam splitters have high splitting efficiency and power intensity in the propagation direction. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals (PCs) [1,2] have attracted much attention because of their ability to manipulate light at the wavelength scale. As such, they have been recognized as a promising candidate for the realization of high-density optical integration [3]. The study of the beam splitter with low energy loss has become an important aspect of PC-based optical devices. There are three main types of the beam splitter, such as power beam splitters [4–14], polarization beam splitters [15–17], and frequency beam splitters [18,19]. In this paper, we focus our attention on the design of power beam splitters. In most previous works, the design of power beam splitter was mainly based on the self-collimation effect [4–10] or PC waveguide (PCW) structures [11–14]. In the first case, the light is transmitted along the direction of self-collimation effect and separated with several branches in the output area. Hence, we can call such splitter as open-type power splitter. The most important advantage of such splitter is long transmission distance with non-contact mode when light is spitted and coupled from one PC device to another, while they also exist some apparent draws, for examples, the applicable frequency band is relatively narrow, the coupling efficiency is too low, the structure is too complex to fabricate, and the splitting angle is hard to control. Using similar definition, the later case can be named as closed-type power splitter because the light is confined and transmitted inside the PCW. Such splitter can solve most problems existed in the first case, while how to realize long transmission distance is a litter
difficult. Considering these situations, very recently, our group has proposed a new kind of open-type power beam splitter which is based upon a planar PCW with corrugated surfaces added behind the termination [20]. By optimizing the structure parameters of the corrugated surfaces using a genetic algorithm (GA) in conjunction with the finite-difference time-domain method (FDTD), such PCW can produce a clear splitting effect with splitting angle of 30° in the output area. The physical mechanism behind this phenomenon is quite in similar with the directional-emission effect [20–27], that is, the splitting effect is the interference of surface modes on the corrugated surfaces and the main output light of PCW. More importantly, such open-type power beam splitter has high coupling efficiency, large frequency band and the splitting angle can be easily controlled. So it can combine the advantages of former two kinds of splitters and is believed to be a competitive candidate in the market-place of this area. As an extended work, in the present study, we will give a more detailed research about such new power splitter and introduce the gratinglike surface behind the termination to realize splitting effect for comparison. We still employ a combined method mixed by the GA and FDTD to design these structures. As a global optimization algorithm, GA [28] has been successfully applied in designing various kinds of photonics devices [20,27,29–33], including large band-gap photonic crystals, photonic crystal fibers, spot-size converters, photonic crystal filters, and micro cavity, directional emitters. By using such combined method, it is believed the gratinglike surface will be more efficient than corrugated surface in controlling the splitting effect. 2. Model and methods
⁎ Corresponding author. E-mail addresses: [email protected] (W. Jia), [email protected] (L. Jiang), [email protected] (X. Li). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.05.050
We consider a finite PC with size of 8a(x-direction) × 24a(ydirection), where a is the lattice period. This PC is created by a square
W. Jia et al. / Optics Communications 283 (2010) 4078–4084
lattice of cylinders with dielectric constant n = 3.4 (e.g., InGaAsP–InP at a wavelength of 1.55 μm) and radius r = 0.18a. A row of cylinders are removed along the plane x = 0 to form a single-mode waveguide that supports a guided mode with frequency between ω = 0.30 × 2πc/a and ω = 0.44 × 2πc/a, propagating in the plane normal to the cylinders with the electric field parallel to them. For source frequencies within the band-gap range, input light waves are well confined within the waveguide, and any subsequent transmission from the waveguide is not affected by the width of the PC. In this work, we focus our beam splitter designs on two models. Fig. 1(a) shows the model of a PCW with a corrugated surface added behind the termination. We will use the GA to optimize the parameters of corrugated surface for obtaining the best possible design. The to-be optimized parameters include the diameter of even rods (r1), the diameter of odd rods (r2), the displacement of even rods along x (r3), the displacement of odd rods along x (r4), the lattice length defined as the center to center distance between even and odd rods (r5), and the distance of each sub-column to the surface of PCW (r6). We define the union of all parameters as S_1 = (r1, r2, r3, r4, r5, r6). Fig. 1(b) shows the model of a PCW with a gratinglike surface added behind the termination. The to-be optimized parameters in the gratinglike surface are the diameter of rods (t1), the displacement of rods along x (t2), the lattice length defined as the center to center distance between two rods (t3), and the distance of each sub-column to the surface of PCW (t4). And the union of the parameters is defined as S_2 = (t1, t2, t3, t4). For each model, a 2D point source (i.e. line source), which emits Gaussian continuous wave with frequency of ω and width of λ, was placed at the location of x =−11a and y = 0. The distance between the detectors and the output of the added surfaces is R, while θD can represent the off-axis degree of detectors. The directed power of detector, normalized to the input power, obtained as time average by the FDTD can be written as: PD ðω0 Þ =
2 t ∫t21 jΩðω0 ; t Þjdt t2 −t1
ð1Þ
Where 8 → > 6a * > Re ∫0 ED ðt Þ × HD ðt Þ ⋅ d l > > > > > h i → > Ωðω0 ; t Þ = > λ > > Re ∫ ES ðt Þ × HS* ðt Þ ⋅d l′ < 0 > > > > > > > t1 = N1 Δt; > > > > :
½
1 t2 = N2 Δt; Δt = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 c 1 = Δx + 1 = Δy2
ð2Þ
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Here, ES and HS represent the electric and magnetic field intensity of light source, ED and HD represent the electric and magnetic field intensity of detector, t1 and t2 represent the beginning and stop time of detector, Δt represents the time step of FDTD, and Δx and Δy represent the spatial interval of FDTD. In our study, to satisfy convergence of the FDTD technique, each periodic lattice in the computational domain is divided into 25 × 25 grid cells (Δx = Δy = 0.04a), and the beginning and stop time of detector is set as 300Δt and 600Δt, respectively. So Ω(ω0, t) and PD(ω0) can represent the instantaneous and time-average normalized power of detector. Not that, in the following calculations, the FDTD method is combined with perfectly matched layers, which are absorption boundaries, to calculate the propagation of the electromagnetic wave. In this work, we use a simple GA to optimize the above proposed parameters. To clarify the implementation process of the GA, we give a detailed description now. To initiate the GA, the to-be optimized parameters are encoded by binary strings and directly stored in a vector, called a chromosome. The initial population P0 is created by randomly yielding Nc chromosomes. Each chromosome therefore compactly represents a corrugated surface or a gratinglike surface to be simulated. For example, the ith chromosome to represent the corrugated surface in Fig. 1(a) can be expressed by the following L-length vector → Si = ðC1 ; C2 ; C3 ; ⋯; CL Þ = ðSr1 ; Sr2 ; Sr3 ; ⋯; Sr6 Þ;
ði = 1; 2; 3; ⋯; Nc Þ
ð3Þ
Here, Ck = 0 or 1, (k =1, 2, 3, ⋯, L) can represent a binary bit; Sr1, Sr2, Sr3, ⋯, Sr6 are six binary strings representing the information of r1, r2, r3, ⋯, r6. Specially, in this paper, we set the searching space for the above six variables as (0, a), (0, a), (0, 2a], (0, 2a], (a, 2a], and (0, 2a], respectively. Considering the requirement in practical fabrication, we assume the precision for all variables is 0.01a. The chromosome for gratinglike surface in Fig. 1(b) is similar and the searching space for t1, t2, t3, t4 is set as (0, a), (0, 2a), (a, 2a], and (0, 2a], respectively. Besides, according to many previous works and our experience, if the scale of corrugated surface or gratinglike surface is large than that of PCW in y-direction, it will only bring tiny difference for the final directional-emission effect or splitting effect when the number of rod is added along y-direction. Hence, with the definition of searching space for r5, we set a total number of 30 rods in ydirection for both corrugated surface and gratinglike surface. After encoding, each chromosome will be further decoded to determine the values of all variables. We utilize the following equation as decoding scheme ri =
BTDðSri Þ × a; ði = 1; 2; 3; ⋯; 6Þ 100
ð4Þ
Fig. 1. Schematic diagram of two kinds of power beam splitters. (a) PCW with an added corrugated surface. (b) PCW with an added gratinglike surface. For each model, the left part is a planar PCW, a being the lattice constant, the middle part is an added surface with several to-be optimized parameters. The right part is a detector to characterize the transmission properties in the direction along angle of θD.
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Fig. 2. (a) Statistic information on searching process about PCW with an added corrugated surface. (b) Statistic information on searching process about PCW with an added gratinglike surface.
Here, ‘BTD’ is a function to translate the binary string to decimal value. After decoding, the fitness value Fi of each chromosome will be evaluated by the fitness function, which is represented by Eq. (1) in our study. The GA then works by in turn using three genetic operators, Selection, Crossover, and Mutation, to evolve the population. Selection creates a new generation Ps by allotting more positions in the new population to those chromosomes with favorable fitness values, and eliminating those with poor values. Crossover is the basic operator for producing new chromosomes in the GA. Like its counterpart in nature, Crossover produces new individuals that have some parts of both parent's genetic material. All off individuals creates a new generation Pc. In natural evolution, mutation is a random process where one allele of a gene is replaced by another to produce a new genetic structure.
Similarly, in the GA, the role of Mutation is often seen as providing a guarantee that the probability of searching any given string will never be zero and acting as a safety net to recover good genetic material that may be lost through the action of Selection and Crossover. Based on the generation Pc, a new generation Pm is produced via the step of Mutation. The above three genetic operators is repeated until the algorithm terminates, typically after a particularly “fit” solution is found, or more generally, after a pre-defined number of generations. The best chromosome in the final generation will be decoded to provide the optimum designs. Moreover, for the model in Fig. 1(b), the search space for the four variables is set as (0, a), (0, 2a], (a, 2a], and (0, 2a], respectively. Other steps are similar as mentioned in the above flowchart for Fig. 1(a). 3. Optimization results and discussion
Table 1 The optimum parameters of corrugated surfaces for three different splitting angle. Splitting angle
r1
r2
r3
r4
r5
r6
10° 20° 30°
0.52a 0.22a 0.18a
0.42a 0.17a 0.84a
0.49a 0.45a 0.34a
1.99a 0.55a 1.56a
0.84a 0.81a 0.78a
1.42a 1.05a 1.46a
Table 2 The optimum parameters of gratinglike surfaces for three different splitting angle. Splitting angle
t1
t2
t3
t4
10° 20° 30°
0.62a 0.88a 0.86a
1.66a 1.25a 0.72a
0.92a 1.16a 1.03a
0.61a 0.78a 0.68a
In this section, we will present several GA projects for beam splitters design. In each GA project, we used a population size of 200 chromosomes for each generation, and allowed the program to run for 80 generations. We initially assumed the frequency of the Gaussian continuous wave is equal to 0.39 × 2πc/a. For each model, different GA projects were executed considering the detector placed along the angle of 10°, 20°, and 30° respectively. And the R for each project is set as 40a. Fig. 2(a) and (b) shows the statistic information on searching process for each GA project. Our GA is performed as expected, and we got a general increase of fitness as the algorithm progressing. The maximum fitness was obtained at 63, 59, 69, 66, 55, and 69 generations for different projects, respectively. For two models, the optimum parameters are listed in Tables 1 and 2 respectively. Fig. 3(a) and (b) shows the spectrum of transmission versus wavelength when
Fig. 3. (a) Directed power versus wavelength for three optimum structures with an added corrugated surface, (b) Directed power versus wavelength for three optimum structures with an added gratinglike surface.
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the normal incidence Gaussian wave is launched into the above optimum PC structures. From Fig. 3(a) and (b), it can be seen that the maximum detector power of every optimum PC structure is beyond 0.4359 when the frequency of incidence Gaussian wave is 0.39 × 2πc/ a. We define an acceptable bandwidth within which the detector power is larger than 30% of the maximum power. With this definition, from Fig. 3(a) and (b), it is found that the acceptable bandwidth for these optimum structures are all applicable in [0.382 × 2πc/a, 0.402 × 2πc/a]. In order to give a directly description about the splitting effect of our designed beam splitters, we have firstly calculated the power distribution of the output beam in the output field for each optimum
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design as shown in Figs. 4 and 5. In particular, the working frequency in Fig. 4 is set to be 0.39 × 2πc/a while the working frequency in Fig. 5 is set to be 0.382 × 2πc/a to verify the acceptable bandwidth as indicated in Fig. 3. For case of model 1, Fig. 4(a), (b), and (c) gives the power distribution of three designs with detector placed at angle of 10°, 20°, and 30° respectively. From the figures, it can be seen that all optimum structures show apparent splitting effects. In addition, though the power detector is predefined at location of R = 40a in GA project, the output beams can propagate a longer distance. The effective beams terminate at around R = 70a. The maximum detector power is found to be 42% of the source power. For case of model 2, as shown in Fig. 4(d), (e), and (f), we find that the optimum structures
Fig. 4. Electric-field intensity distributions for different structures: (a), (b), and (c) represent PCW with an added optimum corrugated surfaces for detector angle of 10°, 20°, and 30° respectively, (d), (e), and (f) represent PCW with an added optimum gratinglike surfaces for detector angle of 10°, 20°, and 30° respectively, The right part of each sub-figure shows the time-average power distribution detected along the y-direction at the position of a surface-launched of the entire structure on the left. The frequency equals to 0.39 × 2πc/a for all sub-figures.
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also have obvious splitting effects, whereas the power of the splitting beams seem to be greatly enhanced in comparison with case of model 1. To gain further insight into the effect of the system size on the emission quality, the right part of each sub-figure in Fig. 4 are shown time-average power distribution detected along the y-direction at the position of a surface-launched of the entire structure on the left. According to the amplitude of the light, it can be believed that the splitting effect is mainly influenced by the interference of surface waves excited along the corrugated surfaces or gratinglike surfaces. On the other hand, from the right part of Fig. 4, we can find the main power of surface waves is mainly distributed in the area of [−12a, 12a], equals to the width of PCW in y-direction. So it is believed that, in order to produce a beam splitting effect, the length of corrugated
surface and gratinglike surface is at least 24a. Furthermore, from Fig. 5, it is found that when the optimum structures work with frequency of 0.382 × 2πc/a, there still exit clear beam splitting effects for all designs, despite the intensity of output light is apparently weaker than those in Fig. 4. This can verify the effectiveness of the acceptable bandwidth as indicated in Fig. 3. In order to understand the splitting effect more clearly, as shown in Fig. 6, we calculate the angular intensity distribution of the output light for each optimum structure, which is obtained by placing a number of angular detectors in the output field (for example in Fig. 4a). Specially, we set R = 40a for all calculations. When a corrugated surface is added behind the termination of PCW, obviously, the transmission power is split into two beams with
Fig. 5. Electric-field intensity distributions for different structures: (a), (b), and (c) represent PCW with an added optimum corrugated surfaces for detector angle of 10°, 20°, and 30° respectively, (d), (e), and (f) represent PCW with an added optimum gratinglike surfaces for detector angle of 10°, 20°, and 30° respectively. The frequency equals to 0.382 × 2πc/a for all sub-figures.
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Fig. 6. The corresponding angular intensity distribution of the output light for each optimum structure as indicated in (a)–(f). The frequency equals to 0.39×2πc/a for all sub-figures.
angular separations of 20°, 40°, and 60° in Fig. 6(a), (b), and (c), respectively. The divergence angle of each beam is about 10°, 20°, and 30°, respectively. Similarly, Fig. 6(d), (e), and (f) present the angular intensity distributions for PCW with gratinglike surfaces added behind the termination. In order to verify the reasonableness of our design and the application in practice, the size effect of the proposed system on the quality of splitting beam should be investigated. The time-averaged transmission coefficients of power (normalized to the incident power) along the beam are calculated by placing the detector with width of 6a at different positions, e.g., the location of transmission detector in Fig. 4(a) varies from R = 25a to R = 70a with step of 5a. The calculated results are given in Fig. 7(a) and (b), from which the transmission inevitably decreases with the increase of R, but the light intensity of the splitting beam still reached 33.5% when R = 70a, and
the light intensity of PCW with gratinglike surfaces is better than PCW with corrugated surfaces. The efficiency of single beam for three PCW with gratinglike surfaces can reach 46.9%, 44.7% and 48.8% at 40a away from the exit. At last, we want to give some comments on the common characters of our designed results. Firstly, for each model, it seems that a clear splitting effect may happen only when the predefined divergence angle is in range of 10° to 30°. For example, compared with Fig. 4(b) and (e), in Fig. 4(a) and (d), there seems a focusing effect happened before the main beam is split, while in Fig. 4(c) and (f), there seems exist some more lateral lobes in each splitting beam. Second, for each splitting angle, the final optimized design for the second model has obviously better performance as well as simpler structure than that for the first model. On the other words, the grantinglike surface has dominant advantage to realize open-type
Fig. 7. (a) Normalized transmission coefficients versus detector location for three optimum beam splitters in Fig. 6(a)–(c). (b) Normalized transmission coefficients versus detector location for three optimum beam splitters in Fig. 6(d)–(f). The frequency equals to 0.39 × 2πc/a.
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splitting effect than the corrugated surface. By decreasing the width of the detector, it is also believed to be an ideal candidate to realize highperformance focusing effects. 4. Conclusion In summary, we have proposed two kinds of new power beam splitters based upon typical planar PCW with corrugated surfaces or gratinglike surfaces added behind the termination of PCW. To design such power beam splitters, we use a combined method by the FDTD and the GA. By globally optimizing the parameters in two types of added surfaces, we have realized several beam splitters with different splitting angles. It is found the splitting efficiency for case of PCW with added gratinglike surfaces is obviously higher than that for PCW with added corrugated surfaces, while the former also has a simpler structure which is more convenient for fabrication. The advantages of our proposed models are the controllable splitting properties and their design feasibility. The power beam splitters obtained in this study may find more important applications in integrated optical circuits. Acknowledgements This work was supported by the Independent research fund of Nanjing University of Science and Technology (NJUST) (No. 2010ZYTS059, No. AE88030, No. AB39103), and the doctorial creative fund of NJUST. References [1] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059–2062. [2] S. John, Phys. Rev. Lett. 58 (1987) 2486–2489. [3] C.M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, Kluwer Academic Publishers, Netherlands, 2001.
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