Similar role of waveguide bends in photonic crystal circuits and ...

PHYSICAL REVIEW B 67, 115208 共2003兲

Similar role of waveguide bends in photonic crystal circuits and disordered defects in coupled cavity waveguides: An intrinsic problem in realizing photonic crystal circuits Sheng Lan,* Kyozo Kanamoto, Tao Yang, Satoshi Nishikawa,† Yoshimasa Sugimoto, Naoki Ikeda, Hitoshi Nakamura, Kiyoshi Asakawa, and Hiroshi Ishikawa The Femtosecond Technology Research Association (FESTA), 5-5 Tokodai, Tsukuba 300-2635, Japan 共Received 24 June 2002; revised manuscript received 18 November 2002; published 26 March 2003兲 We investigate, experimentally and theoretically, the influence of disordered defects on the transmission properties of optical delay lines based on coupled cavity waveguides 共CCW’s兲. Also, the modification of transmission for a line-defect waveguide in a two-dimensional 共2D兲 photonic crystal 共PC兲 upon bending is examined by numerical simulation. We reveal that the effect of waveguide bending on the transmission properties of PC circuits is very similar to that of disordered defects on the transmission properties of CCW’s. It is shown that the broad impurity band of a one-dimensional CCW will evolve into sharp defect modes upon increasing the disorder. Using a symmetric Mach-Zehnder structure formed in a 2D PC as an example, we show that a PC circuit containing multiple bends and branches of different properties generally gives rise to sharp resonant modes whose linewidth is inversely proportional to the size of the PC circuit. This might be an intrinsic problem in realizing PC circuits. DOI: 10.1103/PhysRevB.67.115208

PACS number共s兲: 42.70.Qs, 42.60.Da

I. INTRODUCTION

Photonic crystals 共PC’s兲 formed by periodic modulation of dielectric constant or refractive index act as a promising platform for the control and manipulation of the propagation of photons.1 Their potential applications have been successfully demonstrated in the fabrication of lasers, light emitting diodes, waveguides, waveguide bends, filters, and delay lines.2–7 Also, some very basic passive components such as waveguides, waveguide bends, and waveguide intersections have received intensive study because of their importance in the construction of ultracompact PC circuits.8 –15 Particularly, waveguide bends with negligible reflection are of great interest because they are crucial for realizing PC circuits. So far, much effort has been devoted to the optimization and improvement of waveguide bends by using various designs.16 –19 The one-dimensional 共1D兲 scattering theory is generally applied to the analysis of waveguide bends.4,5,18 Another very important passive device for all-optical signal processing, e.g., optical time domain demultiplexing, is optical delay line. Recently, we have proposed the use of coupled cavity waveguides 共CCW’s兲, which are formed by the coupling of periodically placed PC defects,20,21 for the construction of efficient optical delay lines for ultrashort pulses.22,23 In addition, we have generalized the criteria for designing CCW’s with quasiflat impurity bands and analyzed the transmission of ultrashort pulses through impurity bands of various types.23,24 Based on these theoretical studies, we have designed and fabricated optical delay lines based on CCW’s.7 The basic structure consists of cylindrical air holes made in a silicon on silicon dioxide (Si/SiO2 ) ridge waveguide. In experiments, we have observed impurity bands located at ⬃1.55 ␮m and demonstrated the tunability of impurity bands as well as the efficient delay for ultrashort pulses.7 However, while the designed delay lines exhibit quasiflat impurity bands with high transmission 共⬃0.9兲, the experimentally observed impurity bands consist of sharp resonant peaks with lower transmission 共⬃0.15兲. Obviously, the existence of these sharp peaks are undesirable for the transmis0163-1829/2003/67共11兲/115208共7兲/$20.00

sion of short pulses, especially for those with pulse width comparable to the linewidth of the sharp peaks.23 Fluctuation in the size and position of air holes inevitably introduced in the fabrication process may be responsible for the appearance of sharp peaks as well as the degradation of transmittance. Relying on scanning electron microscope 共SEM兲 measurements, we can evaluate the control of size and position for air holes in the fabrication process. In addition, the finitedifference time-domain 共FDTD兲 simulation serves as a powerful tool for the analysis of light propagation in PC structures.25 Thus, it is possible to investigate how the fluctuation in air holes affects the transmission properties of delay lines. In addition, we have noticed that the study of waveguide bending so far is generally based on the 1D scattering theory.4,5,18 Another way of treating waveguide bend is to consider it as a disordered defect in a line-defect waveguide. On the other hand, a line-defect waveguide can be considered as the limit of a CCW with the strongest coupling. Thus, it is believed that the role of waveguide bends in PC circuits should be very similar to that of disordered defects in CCW’s. In this article, we present a detailed investigation of the effect of defect fluctuation on the transmission properties of CCW’s. Then, we extend the analysis to waveguide bends in two-dimensional 共2D兲 PC’s, showing that there exists a strong similarity between the effect of defect fluctuation and that of waveguide bending. Based on this, we indicate that some general and important features of PC circuits can be predicted from the study of defect fluctuation in CCW’s. This paper is organized as follows. In Sec. II, we analyze the effect of defect fluctuation on the transmission properties of impurity bands. Then, we study numerically in Sec. III the effect of waveguide bending in a 2D PC and indicate its similarity to the effect of disordered defects in CCW’s. In Sec. IV, we show that some important properties of PC circuits with multiple bends can be predicted from the analysis of disorders in 1D CCW’s. Finally, we summarize our find-

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FIG. 2. Measured transmission spectra for several samples with different structure parameters.

FIG. 1. 共a兲 Measured impurity band for the fabricated delay line (N⫽11, n⫽4, m⫽2). 共b兲 Calculated impurity bands without 共dashed curve兲 and with 共solid curve兲 consideration of fluctuation in air holes.

ings in the conclusion and point out the intrinsic problems of PC circuits. II. EFFECT OF DEFECT FLUCTUATION ON THE TRANSMISSION PROPERTIES OF IMPURITY BANDS

The structure, fabrication, and characterization of the optical delay lines have been described in detail in Ref. 7. In this paper, we will focus on the effect of size fluctuation, which is inevitably introduced in the fabrication, on the transmission properties of the fabricated delay lines. This issue naturally arises from the comparison of the measured and the calculated transmission spectra of the optical delay lines, as shown in Fig. 1. In both cases, the transmission spectra have been normalized by using the spectrum of a ridge waveguide without air holes as reference. The measured spectrum presented in Fig. 1共a兲 shows an impurity band of ⬃20 nm with a central wavelength located at ⬃1.535 ␮m. Clearly, we can observe 4 –5 resonant peaks with different transmittances in the impurity band. The maximum peak transmittance is ⬃0.15. In Fig. 1共b兲, we present two calculated spectra obtained by using different parameters. First, we perform FDTD simulation on a perfect structure with a lattice constant of 414 nm which is close to the average value for the fabricated sample according to the SEM measurements. The spectrum is shown by the dashed curve. It can be seen that the location of the calculated impurity band is in good agreement with that of the measured one. However, the top of the spectrum is much flatter and no sharp peak is observed. Also, it should be noted that the impurity band has a maximum transmittance of ⬃0.90. The high transmittance in the absence of disorder indicates that the radiation loss in our structure can be neglected because of the broad bandwidth 共or the short lifetime of photons in each

defect兲. To be more precise, we have measured by SEM the diameter and position for all air holes in the sample and used them in the FDTD simulation. The calculated result is shown in Fig. 1共b兲 by the solid curve. It is found that the maximum transmittance is almost the same as that in the measured spectrum. Additionally, several resonant peaks are resolved in the spectrum. More interestingly, a one-one correspondence in resonant peaks is found between the measured and calculated spectra, as indicated by the dotted lines in Fig. 1. According to our theoretical study,23 the presence of sharp peaks in impurity bands will severely distort the short pulses whose widths are similar to the linewidth of the sharp peaks. For short pulses whose widths are much wider than the linewidth of the sharp peaks, the distortion in the main output pulse can be small. However, its intensity is markedly attenuated. Thus, the sharp resonant peaks are not good for transmitting short pulses. Therefore, it is necessary to find out the origin for these sharp peaks. It is thought that the fluctuation in the size and position of the air holes is responsible for the appearance of sharp peaks as well as the degradation of transmission. Obviously, the effect of fluctuation will become more and more pronounced in the samples with large N, where N is the total number of the defects. In order to see the influence of fluctuation, we have intentionally fabricated several samples with different N. Figure 2 shows the measured transmission spectra for these samples (N⫽2, 11, 30; n⫽4, m⫽2, here n is the number of air holes between two neighboring defects and m is the number of air holes at the two ends of the delay lines兲. It can be seen that the transmittance is decreased and the bandwidth is reduced when N is increased from 2 to 11. As N is further increased to 30, the transmission is so weak that the impurity band cannot be resolved from the background noise. It implies that in this case the inhomogeneous broadening of defect modes induced by the fluctuation in the size and position of air holes exceeds the width of the impurity band. Consequently, the impurity band is completely destroyed, leading to the strong localization of photons and extremely weak transmission 共or strong reflection兲. This is confirmed by examining another sample with wider bandwidth (N⫽30,n ⫽2,m⫽1). As shown in Fig. 2, the impurity band of this CCW survives for the similar fluctuation although the maxi-

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SIMILAR ROLE OF WAVEGUIDE BENDS IN PHOTONIC . . .

FIG. 3. 共a兲 Deviation in the diameter and lattice constant of the air holes with respect to the average value evaluated by SEM measurements. 共b兲 An example of larger air hole 共indicated by the arrow兲 resulting from over exposure in the electron beam lithography. 共c兲 Deviation of the defect size with respect to the average value.

mum transmittance is only ⬃0.05. In Ref. 7, we have presented the fluctuation in the diameter and position 共or lattice constant兲 of the air holes in one of the samples (N⫽11,n⫽4,m⫽2) based on the SEM measurements. For the sake of completeness, we show it again in Fig. 3共a兲. For both diameter and lattice constant, the deviation is evaluated with respect to the average value. From the viewpoint of disordered systems, the fluctuation in size cor-

responds to the Anderson disorder while the position fluctuation belongs to the Lifshutz disorder.26 From Fig. 3共a兲, we can see several air holes with large deviation in diameter and position, e.g., the No. 2, No. 7, and No. 32 air holes. Figure 3共b兲 gives an example of larger air holes. They are found to appear at the boundary of two exposure areas in the electron beam lithography. Thus, the origin for such larger air holes is considered to be the over exposure to the electron beam. Since the electric field is primarily confined at defect regions in CCW’s we think that only the air holes adjacent to defects 共e.g., the No. 2 and No. 7 air holes兲 play an important role in determining the transmission characteristics of the impurity band. The fluctuation in other air holes affects only the transmission of normal pass bands and has little influence on the impurity band. More precisely, it is the size of the defects that determines the frequency of the defect modes and thus the inhomogeneous broadening of the defect modes. Apparently, our delay lines can be approximated as 1D CCW’s consisting of Si and air layers, as shown at the bottom of Fig. 3共b兲. The size for each defect in the sample extracted from the SEM measurements is plotted in Fig. 3共c兲. It can be seen that, except the No. 1 and No. 9 defects, the size fluctuation for the other defects in the sample is within ⫾1%. Therefore, the No. 1 and No. 9 defects are expected to play a crucial role in determining the transmission properties of the sample. In order to understand the role of defects with large deviation in size, the influence of defect fluctuation on the transmission characteristics of impurity bands has been investigated in detail, using a 1D CCW composed of Si and air layers and the transfer-matrix method 共TMM兲. From Fig. 3共c兲, we can see that the fluctuations of most defects are within ⫾1%. Thus, we can further simplify our study by assuming that the fluctuations within this small range can be neglected, focusing on the defects with large deviation. In addition, the complexity of the problem is reduced by considering only the size fluctuation 共i.e., the Anderson disorder兲. The reason is that the position of air holes is easier to control as compared to their size from the viewpoint of practical fabrication, as manifested in Fig. 3共a兲. In addition, a simplified model such as this is sufficient for the understanding of waveguide bending to be discussed in the next section. Let us consider a 1D CCW composed of Si and air layers with equal thickness of 0.50a, where a is the lattice constant. In addition, it has the same structure parameters 共N, n, and m兲 as those of the delay line. The defects are created by periodically increasing the thickness of Si layers from 0.50a to 0.75a. In the absence of any fluctuation, a quasiflat impurity band with unit transmittance is obtained, as shown in Fig. 4. If we introduce a size fluctuation of 4% into the No. 5 defect 共or randomly into any defect兲, i.e., changing the defect size to 0.78a, it is found that the transmittance over the entire band is reduced to about ⫺10 dB. However, it is noted that the impurity band remains to be flat. Surprisingly, the addition of the same disorder in a second defect, e.g., in the No. 11 defect, leads to several sharp resonant peaks in the impurity band. It is remarkable that all the sharp peaks have transmittances close to unit. We think that the two disordered defects act as the two mirrors of a Fabry-Perot cavity. The

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FIG. 4. Modification of the impurity band for a 1D CCW caused by randomly introducing one and two disordered defects.

confinement provided by the two mirrors is responsible for the formation of sharp peaks. Therefore, it is understood that the large deviation in the size of the No. 1 and No. 9 defects may be responsible for the sharp peaks observed in Fig. 1共a兲. The introduction of the third disordered defect results in many such sharper peaks with markedly attenuated transmittance 共not shown in Fig. 4兲. III. SIMILARITIES BETWEEN WAVEGUIDE BENDS IN PC CIRCUITS AND DISORDERED DEFECTS IN CCW’s

Line-defect waveguides are one of the basic components for the construction of PC circuits. In fact, a line-defect waveguide can be considered as the special case of a CCW with n⫽0, as depicted in Fig. 5共a兲. In other words, a linedefect waveguide is actually the limit of a CCW with the strongest coupling of defects. Although in this case the methods for conventional CCW’s considering only the coupling between the nearest neighbors may not be applicable, all defects in the waveguide are considered to be identical or indistinguishable. In other words, the coupling strength is almost uniform over the entire waveguide. Once a straight waveguide is bent, the coupling strength between the defect located at the bending corner and its neighbors will be modified, depending strongly on the match between the field pattern in the defect and the bending angle. This property has been addressed by Yariv et al. and they suggested that it could be employed to build high-efficiency waveguide bends.20 Very recently, we also demonstrated by numerical simulation that broadband waveguide intersections could be realized by use of this feature.27 It implies that the bending of a line-defect waveguide is very similar to the introduction of a disordered defect into a 1D CCW discussed above. This similarity is schematically shown in Figs. 5共b兲 and 5共c兲 for a double-bends structure. Actually, this is another way of treating waveguide bends which is different from the 1D scattering theory commonly used so far. The modification of the coupling strength between the defect located at the bending corner and its neighbors can be considered as the deviation of the defect mode. If it is true, the waveguide bends in PC

FIG. 5. 共a兲 A line-defect waveguide can be viewed as the limit of a 1D CCW with the strongest coupling strength (n⫽0). Each defect of the CCW is a removing air hole represented by an air hole in a dashed circle. 共b兲 A double-bends structure in which the coupling strength for the defects located at the bend corners 共denoted as dark holes兲 and their neighbors is modified. 共c兲 A line-defect waveguide with two disordered defects 共denoted as dark holes兲 which is equivalent to a double-bends structure.

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FIG. 6. 共a兲 Change of the transmission spectrum of a straight line-defect waveguide upon a single bending. 共b兲 A comparison of the transmission spectrum of a single bend and a double-bends structure.

circuits should behave similarly as the disordered defects in 1D CCW’s. In order to confirm this, we have studied by numerical simulation the single and double waveguide bends in a 2D PC with a triangular lattice of air holes made in a dielectric material 共e.g., GaAs with ␧⫽11.56). The radius of air holes is chosen to be r⫽0.30a, where a is the lattice constant of the triangular lattice. The transmission spectra of the TM mode 共magnetic field normal to the 2D plane兲 for the single and double bends obtained by FDTD simulation are shown in Figs. 6共a兲 and 6共b兲, respectively. To accurately determine the transmittance, the transmission spectra were obtained by scanning a continuous wave source as we did in the experimental measurements. The computed mode of a stripe waveguide of 3 1/2a was employed as the input wave. In Fig. 6共a兲, the transmission spectrum for the straight line-defect waveguide along ⌫-J direction is provided as a reference. Apparently, the absolute transmittance depends on the configuration of the connection 共i.e., coupling efficiency兲 between the line-defect waveguide and the stripe waveguide 关e.g., the value of d in Fig. 5共a兲兴 and also on the width of the stripe waveguide 共w兲.28 Here, we use the same input structure (d ⫽a,w⫽3 1/2a) for the straight waveguide and waveguide bends in order to eliminate the influence of coupling efficiency on bending efficiency. What we really concern is the

relative change of the transmittance 共increase or decrease兲 upon bending, not the absolute transmittance. The corresponding impurity band within the band gap ranges from 0.265c/a to 0.335c/a which can be divided into three regions.29 In the regions 关 0.265c/a,0.290c/a 兴 and 关 0.305c/a,0.335c/a 兴 , the straight waveguide contains a single even mode. In the central region of 关 0.290c/a,0.305c/a 兴 , an odd mode coexists with the even mode. It can be seen that the transmittance over the entire band is reduced upon the bending. However, it is noted that the reduction of transmittance is different in different regions. Apparently, the decrease of transmittance in the region containing the odd mode can be ascribed to the excitation of the odd mode in the bent waveguide. The large mode mismatch between the even and odd modes causes large reflection at the bending corner. In the other two regions where only the even mode exists, the difference in the reduction of transmittance is caused by the difference in field distribution. More concretely, the confinement of light in the transverse direction is different in these two regions. While it is weak in the low-frequency region, it is really tight in the highfrequency region. Apparently, the weak confinement in the transverse direction is more favorable for achieving high bending efficiency. Neglecting the details originating from the 2D feature of the waveguide, it is observed that the bending of a line-defect waveguide is very similar to the introduction of a disordered defect in a 1D CCW which is illustrated in Fig. 4. In both cases, the transmittance is reduced. It is interesting to note, however, that another bending of the line-defect waveguide back to the original direction does not necessarily lead to a further decrease in transmittance. Instead, in the regions with a single even mode, an enhancement instead of a further degradation of transmission is observed, as shown in Fig. 6共b兲. In the low-frequency region, the enhancement is observed nearly for all frequencies. In contrast, the enhancement in the high-frequency region occurs only in a very narrow frequency region 共a resonant frequency兲. It is accompanied by the suppression of transmission at other frequencies. Although the modification of transmission spectrum depends on the distance between the two bends which is 10a in our case, we can always observe the enhancement of transmission at certain frequencies which belong to the intrinsic modes of the Fabry-Perot cavity formed by the two bends. Again, the phenomena observed in the double bends are extremely similar to those found in the 1D CCW’s with two disordered defects. It further confirms that in principle the bending of a waveguide is equivalent to the introduction of a disordered defect into the waveguide. In the frequency region of a single mode, the bending efficiency or the backward reflectance is determined by the field distribution which is frequency dependent over the impurity band. IV. ANALYSIS OF PC CIRCUITS CONTAINING MULTIPLE BENDS

In general, a complex PC circuit contains a number of waveguide bends. In addition, these bends may be different from each other. In the PC circuit, they act as mirrors of

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FIG. 7. Evolution of the transmission spectrum for a 1D CCW with increasing disorder.

different natures. In other words, these bends may have different reflectivity at a specified frequency. As mentioned above, the similarities between the waveguide bends in PC circuits and the disordered defects in 1D CCW’s offer us the opportunity to investigate some important issues in PC circuits. The physical picture of a PC circuit with multiple bends can be imagined as a 1D CCW with randomly distributed disordered defects. If we suppose that all the bends have similar properties in the frequency region of interest, then the situation corresponds to a 1D CCW in which the deviation of defects forms a Gaussian distribution. In order to understand the effects of multiple bends on the transmission characteristics of the PC circuit, we have studied by numerical simulation the influence of disordered defects on the transmission properties of a 1D CCW. The investigation is carried on the 1D CCW used previously with structure parameters of (N⫽11,n⫽2,m⫽1). We randomly introduced two size fluctuations with Gaussian distributions characterized by the standard deviation ␴ for the defects in the 1D CCW. The calculated transmission spectra using TMM are shown in Fig. 7. As compared with the results shown in Fig. 4, we found that a small fluctuation could lead to sharp resonant peaks with markedly reduced transmittance.

FIG. 8. Transmission spectrum for the symmetric MachZehnder structure with two types of resonant modes denoted as A and B.

FIG. 9. Intrinsic resonant modes for the symmetric MachZehnder structure. 共a兲 A type. 共b兲 B type.

In order to gain a deep insight into PC circuits, let us inspect a very special case of PC circuits. It is a symmetric Mach-Zehnder 共SMZ兲 structure formed in the 2D PC described above. The radius of the air holes is chosen to be r ⫽0.37a. The SMZ structure consists of an input and an output line-defect waveguides (4a for each兲, four arms along the ⌫-X direction (11a for each兲 and two arms along the ⌫-J direction (6a for each兲. Its transmission spectrum is presented in Fig. 8. As expected, we observe a series of sharp resonant modes instead of a broad band. Due to the perfect symmetry of the structure, the peak transmittance for the resonant modes is high. A detailed investigation reveals that these resonant modes can be roughly classified into two types 共noted as A and B types兲, according to the field distribution in the structure. The computed field distributions for the two types of modes are presented in Figs. 9共a兲 and 9共b兲. It can be seen that the electric field is concentrated on the two arms along the ⌫-J direction for the A-type mode while it is mainly distributed on the four arms along the ⌫-X direction for the B-type mode. This simple and symmetric PC circuit has only one input and one output port. In fact, it can be regarded as a cluster of defect. The sharp peaks in the transmission spectrum are actually the resonant modes of this cluster. An important fea-

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ture that needs to be addressed here is that the linewidth of the resonant modes is inversely proportionally to the size of the structure. It implies that only extremely narrow resonant modes can be obtained in PC circuits with large size. Moreover, if we consider different properties of waveguide bends and branches, the high transmission of the narrow resonant modes may not be guaranteed. In experiments, the sharp defect resonant modes mentioned above may not be observed in the presence of large radiation loss. This is because that the radiation loss is inversely proportional to the group velocity which is extremely small at the resonant peaks. In this case, relatively broad bands may be seen as a consequence of significant attenuation of absolute transmittance. Our research results indicate that a PC-SMZ structure may not work as an interferometer unless the reflection at each bend and branch can be suppressed to be negligible. V. CONCLUSION

In summary, we have investigated the effect of defects with large deviation on the transmission properties of the

ACKNOWLEDGMENTS

This work was supported by the New Energy and Industrial Technology Development Organization 共NEDO兲 within the framework of the Femtosecond Technology Project.

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*Electronic mail: [email protected] †

experimentally fabricated optical delay lines. The problem has been generalized to the influence of disordered defects on the impurity bands of 1D CCW’s. We have revealed that this issue is very similar to the waveguide bending in PC circuits. It has been shown that some general and important features of PC circuits with multiple waveguide bends can be derived from the investigation of disorders in 1D CCW’s. Based on numerical simulation, we find that narrow resonant modes instead of a broad band can be obtained in PC circuits with multiple bends and branches of slightly different properties. More importantly, we show that the linewidth of these resonant modes is inversely proportional to the size of PC circuits and their high transmittances are not guaranteed.

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