Equalization in photonic bandgap multiwavelength ... - OSA Publishing

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J. Opt. Soc. Am. B / Vol. 28, No. 7 / July 2011

Calò et al.

Equalization in photonic bandgap multiwavelength filters by the Newton binomial distribution Giovanna Calò, Antonio Farinola, and Vincenzo Petruzzelli* Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Via Re David n. 200, 70125 Bari, Italy *Corresponding author: [email protected] Received January 19, 2011; revised April 21, 2011; accepted May 18, 2011; posted May 18, 2011 (Doc. ID 141397); published June 14, 2011 In this paper, the design criteria of a multiwavelength photonic bandgap (PBG) filter are shown. The spectral behavior of different defective structures, such as structures with a single defect and with multiple defects, is investigated. The filter behavior is analyzed in connection with the variation of different design parameters, i.e., the number of defects within the PBG structure, the defect length, and the defect position. In particular, the introduction of defects spatially distributed according to Newton binomial coefficients along the PBG structure allows the equalization of the transmission channels. The design and the electromagnetic simulations of the proposed structures were performed using proprietary codes based on the bidirectional beam propagation method with the method of the lines. The binomial distribution of multiple defects significantly increases the passband of each transmission channel with respect to the case of a single defect. For example, the PBG waveguide with a single centered defect having length Ld ¼ 340 μm exhibits 15 transmission channels in the PBG with free spectral range FSR ¼ 1:6 nm and 3 dB bandwidth Δλ3 dB ¼ 0:35 nm. Conversely, the PBG waveguide with four binomially spaced defects exhibits the same number of channels and the same free spectral range, whereas the 3 dB bandwidth of each channel is increased to Δλ3 dB ¼ 0:9 nm. © 2011 Optical Society of America OCIS codes: 230.5298, 130.3120.

1. INTRODUCTION Photonic crystals (PhCs) are one of the most intriguing, promising, and versatile technologies thanks to their ability of manipulating light propagation. By tailoring the lattice properties and by suitably locating defects in the PhC, virtually every device for the transmission and routing of optical data can be realized. The capability of reflecting and trapping the light in a wavelength-selective manner is the base mechanism for the implementation of mirrors, waveguides, filters, resonators, etc. [1]. Moreover, the inclusion of nonlinear or active materials into the PhC structures allows the wavelength tunability [2,3], the enhanced light emission [4], and the switching behavior [5] to be achieved. Among the other components, the multichannel wavelength filters play an important role in the optical communication systems. Different configurations of multichannel filters have been proposed in the literature such as fiber Bragg grating filters based on phase-sampling or chirping effects [6,7] or one-dimensional PhCs, which behave as multichannel filters thanks to the introduction of defects within the periodic lattice [8,9]. In fact, the PhC transmission spectrum is characterized by a photonic bandgap (PBG) in which the transmission of light is inhibited. The introduction of defects in the periodic PhC lattice permits localized states within the PBG and therefore allows the transmission at specific wavelengths. Fundamental requirements for the transmission channels of optical communication filters are the band flatness in the passband, the narrow transition band, and the high extinction in the stop band [10]. Efficient techniques for tailoring the stop band and passband characteristics were proposed by using phaseshifted Bragg gratings [11,12], stacked Bragg reflectors [13], and PhC gratings with a phase-only sampling approach [14]. 0740-3224/11/071668-12$15.00/0

In this paper, we propose an alternative, simple, and efficient design technique to realize optimized PBG multichannel filters. The design criterion refers to periodic gratings in which the introduction of multiple defects allows the creation of transmission channels within the PBG. The number of channels and their spectral behavior are linked to the defect length, to the number of defects, and to their position in the periodic grating. Here we report the analysis of different PBG structures in which an increasing number of defects is inserted. A simple configuration exploits uniformly spaced defects in the PBG structure. A further, more performing configuration is proposed by positioning the defects according to the Newton binomial distribution. The binomial space distribution of multiple defects within the PBG lattice allows us to improve the band flatness of the transmission channels.

2. NUMERICAL MODEL Figure 1 shows a three-dimensional schematic of the multiwavelength PBG filter. It is an index-confined one-dimensional PBG waveguide made of an Si3 N4 layer grown on the SiO2 substrate and patterned with a Bragg grating. The refractive indices of the considered materials are core (Si3 N4 ) refractive index nc ¼ 2 and substrate (SiO2 ) refractive index ns ¼ 1:45 at wavelength λ ¼ 1:55 μm. The PBG structure is made of a slab waveguide in which the thickness of the core is periodically varied in order to form a Bragg grating. Along the propagation direction z, the thickness of the core assumes two different values d1 ¼ 0:480 μm and d2 ¼ 0:420 μm. In the following, we will call “alternating layers” along the z longitudinal propagation direction, the two three-layered waveguides having thicknesses d1 and d2 , respectively. The lengths of the two alternating layers of the grating l1 ¼ 0:112 μm and l2 ¼ 0:332 μm were chosen in order © 2011 Optical Society of America

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Vol. 28, No. 7 / July 2011 / J. Opt. Soc. Am. B

to achieve the PBG near the telecommunication third window wavelength λ ¼ 1:55 μm. In the analyzed structure the defects denote a geometrical perturbation of the grating periodicity. Referring to Fig. 1, the defect is given by substituting, in the grating unit cell, the second waveguide with another one having the same thickness d2 and different length Ld . The PBG waveguide was analyzed by a proprietary code based on the bidirectional beam propagation method together with the method of lines (BBPM–MoL) [15–17]. The BBPM– MoL algorithm takes into account the forward (þz) and the backward (−z) propagation along the longitudinal z direction of the waveguide structure. Since the propagating guided wave encounters refractive index discontinuities along the grating structures, it is transmitted and reflected at each interface between the alternating layers. This mechanism is taken into account at each iteration step, both in the backward and in the forward propagation, by imposing the continuity of the electromagnetic field components at the interfaces between different dielectric layers. The overall electromagnetic field in the structure is given by the superposition of all the reflected and the transmitted waves at each iteration step. The algorithm proceeds until the convergence criterion is met, i.e., the change of the electromagnetic field in the output section is less than a given tolerance ε (in the analyzed cases ε ¼ 10−5 ) [9]: Ei − E i−1 < ε; Ei where Ei and Ei−1 are the electric field moduli calculated at the generic ith and (i − 1)th iterations, respectively. The number of layers was optimized to achieve a good compromise between the PBG performances and the computational effort required for the electromagnetic simulations. To this aim, we analyzed the reflectance and the transmittance spectra for different values of the number of layers N. In particular, the reflectance R in the PBG increases with the number of layers, whereas the bandwidth of the PBG decreases. Figure 2 shows the reflectance R at the Bragg wavelength λB ¼ 1:5427 μm as a function of the number of layers N. From Fig. 2 we can see that the maximum reflectance, which can be considered a figure of merit for the periodic PBG structure,

Fig. 1. (Color online) Schematic of the index-confined onedimensional PBG waveguide.

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increases with the number of layers. The aim is to achieve a maximum reflectance R greater than 0.8. However, we must also consider that the computational effort increases with the number of layers. Owing to the particular convergence criterion applied in the BBPM simulations, some instabilities in the simulation results occur when R is greater than 0.9. On the basis of the aforesaid considerations, we have chosen a total number of layers N ¼ 256, which gives a maximum reflectance R ¼ 0:88. Being N ¼ 256 a power of 2, it also allows an easy application of the binomial defect distribution, which will be discussed later on. It is worth mentioning that the reflectance R and the transmittance T spectra do not change significantly if an odd or an even number of layers N are considered.

3. SINGLE DEFECT IN THE PBG WAVEGUIDE In order to localize multiple states, i.e., transmission channels with transmittance T ≅ 1, within the PBG, we first considered the introduction of a single defect in the periodic grating. The occurrence of resonances in the PBG can be described in terms of Fabry–Perot resonances. This analysis is straightforward in the case of a single defect placed in the middle of the periodic grating. In fact, this PBG structure can be seen as a Fabry–Perot resonator made of two identical mirrors that delimit a cavity (i.e., the defect) of length Ld . The two mirrors are characterized by a wavelength-dependent reflection coefficient. The resonances of the system can be approximately calculated as [18] 2π n L þ Φr ¼ sπðs ¼ 0; 1; 2; …Þ λ eff d

ð1Þ

where λ is the wavelength corresponding to the resonance, neff is the effective refractive index pertaining to the defect region, and Φr is the phase of the reflection coefficient of the mirrors. In the following, we will refer to a more general form of Eq. (1) that applies also in the case of cavities delimited by two different Bragg reflectors: Φdef ðλÞ þ Φr1PBG ðλÞ þ Φr2PBG ðλÞ ¼ 2sπðs ¼ 0; 1; 2; …Þ; ð2Þ where Φr1PBG ðλÞ and Φr2PBG ðλÞ are the two phases of the reflection coefficient for the two Bragg mirrors and Φdef ðλÞ is the total phase shift experienced along the cavity in the whole forward and backward optical paths. As an example, we can choose the length Ld of the cavity that assures the resonance condition at the Bragg wavelength λB ¼ 1:5427 μm by considering the phase coherence condition (2). To this aim, Fig. 3 shows (a) the phase ΦrPBG of the reflection coefficient and (b) the corresponding values of the reflectance R as a function of the wavelength for various periodic PBG structures having different values of the number of layers N. We can observe in Fig. 3 that the phase of the reflection coefficient r of the PBG structure without defect is equal to ΦrPBG ≅ −2:39 rad at the Bragg wavelength (solid vertical line) for all the values of the number of layers N. On the other hand, for the wavelength value λB ¼ 1:5427 μm, the reflectance R decreases by reducing the number of layers N. In Fig. 3 odd values of the number of layers are reported,


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Reflectance R

0.8

0.6

0.4

0.2

0 0

50

100

150

200

250

Number of Layers N Fig. 2. Reflectance R at the Bragg wavelength λB ¼ 1:5427 μm as a function of the number of layers N.

but the behavior is exactly the same if the corresponding even values of N are considered (N ¼ 30, 58, 64, 86, 128, and 192). Exploiting Eq. (2) and the results shown in Fig. 3, the phase coherence condition becomes at the Bragg wavelength λB ¼ 1:5427 μm: Φdef ðλB Þ þ Φr1PBG ðλB Þ þ Φr2PBG ðλB Þ ¼ 2sπðs ¼ 0; 1; 2; …Þ; ð3Þ where Φr1PBG ðλB Þ ¼ Φr2PBG ðλB Þ ≅ −2:39 rad are the two phases of the grating reflection coefficients and Φdef ðλB Þ ¼ −4πneff ðλB ÞLd =λB is the total phase shift experienced along the cavity in the whole forward and backward optical paths. The value of the phase shift Φdef ðλB Þ was calculated considering the defective region as a homogeneous layer having a refractive index equal to the refractive effective index of the fundamental mode propagating in the defect waveguiding region. The value of the defect length equal to Ld ¼ 9:91 μm gives a total phase shift Φdef ðλB Þ ≅ −ð1:5 þ 44πÞrad that satisfies the condition of Eq. (3) at the Bragg wavelength λB ¼ 1:5427 μm. The wavelength dependence of Φdef ðλÞ for the chosen value of the defect length Ld ¼ 9:91 μm is shown in Fig. 3(a) (dashed line).

4. EFFECT OF THE DEFECT POSITION A. Single Defect The introduction of one or more geometrical defects in the periodic grating allows the localization of transmission channels in the PBG. The resulting transmission spectrum depends on both the length and the position of the defects within the periodic grating. A simple but significant example is given by Fig. 4, which shows (a) the transmittance spectra T and (b) the phase of the reflection coefficient ΦrPBGþdef for the PBG structure with 256 layers and a single defect. The defect has length Ld ¼ 9:91 μm, and it is placed in different positions p ¼ 30, 64, 86, 128, 170, 192, and 226. As expected, we can see from Fig. 4 that the resonance occurs for the same value of the wavelength λB , but the best performance (transmittance value T ≅ 1 and R ≅ 0) is achieved when the defect is placed in the central

128th layer of the PBG waveguide. This happens because, only for the case of p ¼ 128, the reflectance R ¼ 0:5 is the same for both the PBG mirrors at the edges of the cavity [see Fig. 3(b)]. In Fig. 4(b) it is evident that the phase ΦrPBGþdef of the reflection coefficient of the PBG structure with a defect assumes two different values at the Bragg wavelength. In particular, ΦrPBGþdef ðλB Þ is equal to about 0:8 rad for p ≤ N=2 ¼ 128, whereas ΦrPBGþdef ðλB Þ ≅ −2:46 rad for p > N=2 ¼ 128. This behavior, obviously valid for a generic N number of waveguiding periodic layers, can be justified by examining, in Fig. 5, the curves of the moduli of the incident jE ib jn (solid curve) and the reflected jErb jn (dashed curve) backwardpropagating electric field normalized to the input incident electric field. The moduli of the electric field were calculated, at the Bragg wavelength, in the middle of the transversal section of the waveguiding film along the z propagation direction. The curves in Fig. 5 refer to the PBG structure with number of 4 3 2 N=191

Φ rPBG [rad]

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(b) Fig. 3. (a) Phase ΦrPBG of the reflection coefficient and (b) reflectance spectra R of different PBG waveguides with number of layers N ¼ 29, 57, 63, 85, 127, and 191, as a function of the wavelength; the dashed line refers to the total phase shift Φdefect due to both the forward and the backward propagations in the defective region 9:91 μm long.

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backpropagation is equal to 0 only at the end of the whole structure. In this case, the ΦrPBGþdef value can be approximately calculated as

Trasmittance T

p=128

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ΦrPBGþdef ðλB Þ ¼ ΦrPBG ðλB Þ − Φr;2 ðλB Þ þ Φdef ðλB Þ ≅ 0:79 rad;

170, 86

where ΦrPBG ðλB Þ ¼ −2:39 rad is the phase shift of a PBG structure without defect, Φr;2 ðλB Þ ¼ 2ð−2:34Þ rad ¼ −4:68 rad is the contribution of the substituted pth layer (second layer of the unit cell), and Φdef ðλB Þ ≅ −ð1:5 þ 44πÞ rad is the contribution of the defect in the whole forward and backward propagations. The aforesaid discussion suggests that the number of the defects and their position within the PBG structure are further design parameters that strongly influence the filter spectrum.

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B. Multiple Defects In order to make a simple analysis of the filter behavior, we first consider two defects, having length Ld ¼ 9:91 μm,

4 226

64

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Fig. 4. (a) Transmittance spectra and (b) phase ΦrPBGþdef of the reflection coefficient for the PBG waveguides with number of layers N ¼ 256 and one defect having length Ld ¼ 9:91 μm placed at the layer positions p ¼ 30, 64, 86, 128, 170, 192, and 226.

1.5

1

|Eib|n, |Erb|n

layers N ¼ 256 and a single defect placed at (a) p ¼ 192 and (b) p ¼ 64. For p ¼ 192 the modulus of the incident electric field of the backpropagation is almost equal to 0 at z ¼ 28:35 μm, which is in correspondence of the central layer N=2 ¼ 128, thus allowing us to consider at λB the whole structure as subdivided into two separated PBG structures. The former, referred to as PBG1 , is made of 128 alternating waveguiding layers without defect; the latter (PBG2 ) is made of 128 layers having a defect at p ¼ 64. The output transmitted field of PBG1 is directly incident at the input port of PBG2 , whereas the backward incident field at the output port (at z ¼ 28:35 μm) of PBG1 is practically null. This happens because the reflected field outgoing from the PBG2 is practically equal to 0. Therefore, considering at λB the two structures as separated, the phase of the reflection coefficient of the whole structure approaches the value of that of PBG1 , and it is equal to about ΦrPBGþdef ðλB Þ ¼ ΦrPBG ðλB Þ ¼ −2:39 rad. The same does not happen for the case of Fig. 5(b), for which the electric incident field of the

10

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30

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z [µm] (b) Fig. 5. Moduli of the incident jE ib jn (solid curve) and the reflected jErb jn (dashed curve) backward electric field normalized to the input incident field, calculated in the middle of the transversal section of the waveguiding film along the PBG structure at the Bragg wavelength for the defect placed at (a) p ¼ 192 and (b) p ¼ 64. Defect length Ld ¼ 9:91 μm.

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Fig. 7. Transmittance T for the grating with N ¼ 256 alternating layers and two defects having length Ld ¼ 9:91 μm, for different values of the position index m: m ¼ 12 (solid curve), m ¼ 42 (dashed curve), m ¼ 64 (dotted curve), m ¼ 70 (dotted–dashed curve), m ¼ 98 (dotted–solid curve).

alternating layers, this condition occurs when m ¼ 64, and it gives the best performance in terms of band flatness of the transmission channel. The mechanism of the phase of the reflection coefficient of the structures in Fig. 4(b) explains the resonance of the signal at the Bragg wavelength in the case of PBG structure with two binomially spaced defects, i.e., m ¼ 64. In general, by considering the symmetrical position of the defects of Fig. 6, the condition of Eq. (3) for the wavelength λB can be satisfied only by assuming a distance between the two defects equal or larger than two times the length of the PBG structure placed at the edges of the defects (i.e., m ≥ 64 according to the indexing convention in Fig. 6(a) and N t2 ≥ 2N t1 ). As shown in Fig. 6, the two defects (Def 1 and Def 2 ) are delimited by two reflectors. The former [Refl1 in Fig. 6(b)] is a PBG structure with N r1 ¼ N t1 layers; the latter [Refl2 1.555

Wavelength λ [µm]

symmetrically positioned within the grating according to Fig. 6. As Fig. 6(a) shows, each of the two defects is positioned in one half of the grating. Moreover, the two defects are placed in correspondence of the p1 ¼ ðM − mÞ and p2 ¼ ðM þ mÞ layers, respectively. The even values of the index m ¼ 2; 4; … M establish the defect position and M ¼ N=2 is the position of the central layer of the grating. Figure 7 shows the transmittance T spectra, for a grating with N ¼ 256 alternating layers with two defects having length Ld ¼ 9:91 μm, for different values of the position index m: m ¼ 12 (solid curve), m ¼ 42 (dashed curve), m ¼ 64 (dotted curve), m ¼ 70 (dotted–dashed curve), m ¼ 98 (dotted–solid curve). As we can see from Fig. 7, when m ¼ 12 (solid curve) and therefore when the two defects are placed at the p1 ¼ 116 and p2 ¼ 140 layers, two transmission channels (T ≅ 1) are induced in the PBG at the wavelengths λ1 ¼ 1:537 μm and λ2 ¼ 1:548 μm. As the two defects are moved far from the center of the grating, i.e., m ¼ 42 and p1 ¼ 86 and p2 ¼ 170 (dashed curve), the two transmission channels become closer to each other (λ1 ¼ 1:539 μm and λ2 ¼ 1:546 μm) and the minimum transmittance between the two channels is about equal to T ¼ 0:7. When m ¼ 64 (i.e., p1 ¼ 64 and p2 ¼ 192) the two channels are no longer separated. In this case, we can say that the two defects induce a single channel within the PBG characterized by a larger bandwidth, given by the superposition of the two transmission channels. When m > 64, we still have a single channel, but the value of its transmittance strongly decreases with m. To better describe this behavior, Fig. 8 shows the calculated values (circles and triangles) of the wavelengths corresponding to the two transmission channels (T ≅ 1) as a function of the position index m. As the position index m is increased, the two maxima tend to move toward the Bragg wavelength (λB ¼ 1:5427 μm). For m ≥ 64 and m < 100, only a single transmission channel is induced, whereas for m > 100 the maximum transmittance of the channel decreases (T < 0:4) until the latter completely disappears. It is worth mentioning that the case m ¼ 64 corresponds to the Newton binomial distribution of defects within the PBG structure, which will be analyzed later on. For the Newton binomial distribution of the defects, we intend that the distance between multiple defects inserted in the grating follows the binomial rule. In the case of two defects and of N ¼ 256

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Defect position index m Fig. 6. (Color online) (a) Schematic of the position of two defects within the PBG structure and schemes of the two reflectors (b) Refl1 and (c) Refl2 delimiting the Def 1 defective region.

Fig. 8. Wavelengths of the two transmission channels (T ≅ 1) as a function of the position index m for the grating with N ¼ 256 alternating layers and two defects having length Ld ¼ 9:91 μm: calculated values (circles and triangles) and Bragg wavelength λB ¼ 1:5427 μm (dashed line).

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in Fig. 6(c)] is a PBG with N r2 ¼ ðN t1 þ N t2 þ 1Þ waveguiding layers and a defect placed at p ¼ N t2 þ 1. Figure 9 shows (a) the transmittance spectrum T of different reflectors Refl1 (solid curves with N r1 ¼ 29, 57, 63, and 85) and Refl2 (dotted curves with N r2 ¼ 170 and p ¼ 84, N r2 ¼ 192 and p ¼ 128, N r2 ¼ 198 and p ¼ 140, N r2 ¼ 226 and p ¼ 196) and (b) the corresponding phases Φr of the reflection coefficient r. Figure 9(b) shows the whole phase shift of the defect (dashed line) as a function of the wavelength, too. The resonance conditions for the whole structure occur when the phases of the two reflectors are equal. Moreover, the maximum overall transmittance T ≅ 1 is achieved for equal values of the reflectance of both the reflectors. In particular, for m < 64 two resonances (T ≅ 1) appear. As an example, for m ¼ 42 (N r2 ¼ 170 and p ¼ 184) the two points Q of Fig. 9(a) satisfy both the phase and the reflectance condition. For m ≥ 64, the phase coherence condition occurs at the Bragg wavelength, but the T ≅ 1 condition is achieved 1

Reflectance R

0.8 Nr2=226, p=196

Nr2=170, p=84

Nr2=192, p=128

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0.4 Nr1=85 Q

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Q P

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Wavelength λ [µ m] (a) 4

57 63 N =85 r1

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170, 84

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3 192, 128

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Wavelength λ [µ m] (b) Fig. 9. (a) Transmittance spectra T and (b) reflection coefficient phase Φr for the PBG waveguide structures without (solid curves, N r1 ¼ 29, 57, 63, and 85) and with (dotted curves, N r2 ¼ 170 and p ¼ 84, N r2 ¼ 192 and p ¼ 128, N r2 ¼ 198 and p ¼ 140, N r2 ¼ 226 and p ¼ 196) defect Ld ¼ 9:91 μm long.

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only for m ¼ 64 (N r1 ¼ 63, N r2 ¼ 192, and p ¼ 128). This is because, at the wavelength λB , the reflectance R ¼ 0:17 of Refl1 for N r1 ¼ 63 assumes the same values of Refl2 for N r2 ¼ 192 and p ¼ 128 [point P in Fig. 9(a)] and the reflection coefficient phases of Refl1 and Refl2 are equal, Φr ðλB Þ ¼ −2:39 rad. This occurrence is in good agreement with the discussion reported in Subsection 4.A. In fact, having the defect a position p ¼ 128 greater than N r2 =2 ¼ 96, the Refl2 has phase Φr ðλB Þ ¼ −2:39 rad. Summing up, the best performance of the overall structure, in terms of the transmittance spectrum (maximum T value and a single resonance), is achieved for m ¼ 64 that corresponds to the binomially spaced defect configuration. A different way to justify the results obtained for the binomially spaced defect criterion can be found by observing the curves of Fig. 10, which shows the moduli of the incident jEib jn (solid curve) and the reflected backward jErb jn (dashed curve) propagating electric fields normalized to the input incident electric field. These were calculated at the Bragg wavelength in the middle of the transversal section of the waveguiding film along the PBG structures: (a) N ¼ 256 alternating layers and two defects binomially spaced at the positions p ¼ 64 and p ¼ 192, (b) N ¼ 128 and a defect placed in the middle at p ¼ 64. We can see in Fig. 10(a) that the modulus of the reflected backward-propagating electric field jErb jn assumes an almost-null value at z ¼ 28:35 μm that is at the central layer N=2 ¼ 128, thus separating the whole structure into two identical PBG structures with N ¼ 128 and a single defect placed at p ¼ 64. The output transmitted signal of the first PBG structure becomes the input signal of the second one, whereas the reflected field outgoing from the second to the first structure is practically null. In fact, we can observe that both the jEib jn and jErb jn values in Fig. 10(b) are practically coincident with those corresponding of Fig. 10(a) for z > 28:35 μm. The overall PBG structure experiences a single localized state centered at λB exactly as each of the two cascaded substructures. By generalizing, the PBG structures, which can be considered as a cascade of two or more shorter PBG substructures, each having a single defect placed in the middle of the substructures, show a single localized state in correspondence of the Bragg wavelength. It is evident that, for the case of two or more binomially spaced defects, the whole structure can always be considered as the cascade of two or more substructures having only one defect in the middle, thus assuring the presence of a single localized state at λB . These results suggest that, by varying the length and the position of one or multiple defects, the number of the transmission channels and their bandwidth can be engineered. In the following, we will apply the aforesaid considerations to the case of multiple channels in the PBG. This condition can be achieved by properly choosing the defect length. Moreover, we will consider two different criteria for the positioning of multiple defects, according to the uniform and to the Newton binomial distributions.

5. EFFECT OF THE DEFECT LENGTH When the defect length is increased, multiple wavelengths falling in the PBG range can satisfy Eq. (2). For example, Fig. 11 compares the transmittance T spectrum of the periodic

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1

Transmittance T

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0

1.53

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Wavelength λ [µm] Fig. 11. Transmittance T spectra of the periodic nondefective structure (dotted curve) and of the defective structure with a single defect having length Ld ¼ 100 μm (solid curve). The two dotted–dashed lines mark the PBG band edges (λ ¼ 1:531 μm and λ ¼ 1:555 μm) of the periodic nondefective grating.

Fig. 10. Moduli of the incident jEib jn (solid curve) and the reflected jE rb jn (dashed curve) backward electric field normalized to the input incident field, calculated in the middle of the transversal section of the waveguiding film at the Bragg wavelength along the PBG structure with (a) N ¼ 256 and p ¼ 64 and 192, (b) N ¼ 128 and p ¼ 64. Defect length Ld ¼ 9:91 μm.

nondefective structure (dotted curve) with the one pertaining to a defective structure with a single defect having arbitrarily chosen length Ld ¼ 100 μm (solid curve) in the case of N ¼ 256. As Fig. 11 shows, the defect induces five transmission channels (having T > 0:75) within the wavelength range delimited by the band edges (λ ¼ 1:531 μm and λ ¼ 1:555 μm) of the PBG, which are marked by the two dotted–dashed lines in Fig. 11. These transmission channels exhibit a 3 dB bandwidth Δλ3 dB ≅ 1:2 nm and a wavelength distance between two resonances, i.e., free spectral range (FSR) or channel spacing, equal to Δλc ¼ 4:8 nm. Moreover, each of the transmission channels in Fig. 11 is characterized by a narrow bandwidth and by an approximately triangular shape with respect to the wavelength. According to the Fabry–Perot resonator, the defective structure behaves as a 100 μm cavity delimited by two periodic gratings (i.e., mirrors), each having 128 alternating layers. Table 1 compares the resonance wavelengths λFP calculated

by Eq. (2) with the ones (λBBPM ) calculated by the BBPM simulations of the complete structure (i.e., the PBG structure with 256 alternating layers and a centered defect). In Eq. (2), the phases Φr1 ¼ Φr2 of the mirror reflection coefficient were obtained by the BBPM simulations of the 128-layer periodic grating. Comparing the resonant wavelengths reported in Table 1, a good agreement appears. Obviously, the defect length is a relevant design parameter that allows the multiwavelength filter spectrum to be optimized according to the specific application, in terms of the number of channels and FSR. In particular, Fig. 12(a) reports the number of channels N c in the PBG, characterized by a transmittance value T > 0:75, as a function of the defect length Ld . In addition, Fig. 12(b) shows the FSR as a function of the defect length Ld . In all the simulated cases of Fig. 12, a single defect was considered centered at the layer p ¼ 128 within the PBG structure. As expected, from Fig. 12 we can infer that the number of transmission channels and therefore the localized states in the PBG increase with the defect length. Accordingly, lower values of the FSR and a denser allocation of states in the PBG are achieved by increasing the defect length Ld .

6. UNIFORMLY SPACED DEFECTS IN THE PBG WAVEGUIDE Aiming at optimizing the transmittance spectrum of the multiwavelength filter, we considered multiple defects, each having Table 1. Transmission Channel Wavelengths Calculated by the BBPM Simulations, λBBPM and by the Fabry–Perot Resonances Given by Eq. (2), λFP , for the PBG Structure with a Central Defect Having Length Ld 100 μm λBBPM ðμmÞ

λFP ðμmÞ

1.5389 1.5437 1.5485 1.5532 1.5578

1.5393 1.5441 1.5489 1.5537 1.5582

Calò et al.


1

14

0.8 12

Transmittance T

Number of channels Nc

16

10 8 6 4 2 0

0.6

0.4

0.2 50

100

150

200

250

300

350

0

Defect length Ld [µm]

1.53

1.54

12 10

r% ¼ 8

4 2

100

150

200

250

300

350

Defect length Ld [µm] (b) Fig. 12. (a) Number of channels N c and (b) FSR as a function of the defect length Ld .

length Ld ¼ 100 μm, uniformly spaced within the grating. As an example, Fig. 13 shows the transmittance T spectrum (solid curve) for the PBG structure with three uniformly spaced defects placed at the layers p ¼ 64, 128, and 192. The two band edges of the PBG pertaining to the periodic nondefective grating are marked with the dotted–dashed lines. As we can see from Fig. 13, the presence of uniformly spaced multiple defects, on one hand, tends to increase the 3 dB bandwidth of each channel (Δλ3 dB ¼ 2 nm), leaving unchanged the channel spacing (Δλc ≅5 nm) with respect to the case of a single defect; on the other hand, it induces a ripple in the transmission bandwidth of each channel. This behavior is more apparent in Fig. 14, which shows a portion of the transmittance spectra, pertaining to one of the channels of the defective PBG structure, calculated for an increasing number of uniformly spaced defects each having length Ld ¼ 100 μm. Table 2 reports the percent variation r% of the ripple in the transmission channels, the number of peaks N p in the ripple, the maximum transmittance T MAX value, the 3 dB bandwidth Δλ3 dB , and the overall length Lf of the filter (i.e., the grating length plus the defect lengths) for the increasing number of defects. Moreover, the percent ripple is calculated as

T1 − T2 · 100; T1

where T 1 and T 2 are, respectively, the maximum and the minimum transmittance values in the passband ripple. As we can see from Table 2, the increase of the number of defects from 1 to 6 widens the channel bandwidth from Δλ3 dB ¼ 1:2 nm to Δλ3 dB ¼ 4:2 nm. Conversely, no significant change in the channel spacing was evaluated. In all the considered cases, the maximum transmittance is higher than 0.89. The percent ripple assumes values in the range from 16% to 22%, thus giving a relatively poor uniformity of the transmission spectrum of each channel. It is worth mentioning that the number of channels within the PBG is unchanged with respect to the case shown in Fig. 11 of a single defect having length Ld ¼ 100 μm and centered in the periodic grating. We verified that, for the uniformly distributed defects, the number of channels is only affected by the defect length, whereas the number of defects influences the wavelength spectrum shape. In particular, the number of peaks in the ripple of the transmittance spectra equals the number of defects. This

Transmittance T

6

50

1.56

Fig. 13. Transmittance T spectrum (solid curve) of the defective structure with three uniformly spaced defects, placed at the layers p ¼ 64, 128, and 192, having length Ld ¼ 100 μm. The two dotted– dashed lines mark the PBG band edges (λ ¼ 1:531 μm and λ ¼ 1:555 μm) of the periodic nondefective grating.

14

0 0

1.55

Wavelength λ [μm]

(a)

FSR [nm]

1675

1

1 defect 2 defects

0.8

4 defects 6 defects

0.6

0.4

0.2

0 1.54

1.542

1.544

1.546

Wavelength λ [µm]

Fig. 14. Transmittance T spectrum (solid curve) of one of the transmission channels for the defective structure with uniformly spaced defects having length Ld ¼ 100 μm.

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Calò et al.

Table 2. Percent Variation of the Ripple in the Transmission Channels r%, Number of Peaks Np in the Ripple, Maximum Transmittance T MAX Value, 3 dB Bandwidth Δλ3 dB, and the Overall Length Lf of the Filter for the PBG Structure with Uniformly Spaced Defects Having Constant Length Equal to Ld 100 μm Number of Defects

r%

Np

T MAX

Δλ3 dB ðμmÞ

Lf ðμmÞ

1 2 3 4 5 6

22 16 18 18 18

2 3 4 5 6

0.95 0.90 0.92 0.91 0.90 0.89

1.2 2.4 3.2 3.6 4.0 4.2

156.720 256.608 356.496 456.384 556.272 656.160

phenomenon can be analyzed by the well-known tight binding (TB) approximation [19,20]. As described in [19,20], multiple defects in the PBG waveguiding structure behave as coupled resonant cavities. Thanks to the presence of the coupled cavities, light can propagate in the PBG structure from one cavity to the other. This behavior is due to the interactions between the neighboring evanescent cavity modes. Moreover, when two, three, or more coupled cavities are considered, the resonant modes split in multiple resonances. According to the TB approach, a single defect in a periodic structure sustains an individual localized mode E Ω ðzÞ that satisfies the Maxwell equations. In the case of two coupled defects, the eigenmode satisfying the Maxwell equations can be calculated as a superposition of the individual evanescent defect modes E Ω ðzÞ and EΩ ðz − z0 Þ, where z0 denotes the spatial position of the second cavity. According to the TB approach, the single defect mode is split into two eigenfrequencies, which correspond to two resonance wavelengths λ1 and λ2 : λ1;2

sffiffiffiffiffiffiffiffiffiffiffiffiffi 1 α1 ¼ λ0 ; 1 β1

ð4Þ

where λ0 is the resonance wavelength of the single cavity mode and α1 and β1 are the first-order coupling parameters given by the overlap integrals of the two coupled cavity modes [20]. Similarly, for a system with three coupled defects, the eigenfrequency is split into three resonant frequencies, whose corresponding wavelengths are given by the following equations: λ1;3

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ffi 1 2α1 pffiffiffi ; ¼ λ0 1 2β1

ð5Þ

λ2 ¼ λ0 ; where the second-order coupling terms are neglected. The wavelength position of the ripple peaks varies with the defect positions. Figure 8 is an example of this behavior in the case of two defects. In fact, the overlap integrals α1 and β1 in Eqs. (4) and (5) depend on the distance between the coupled cavities. Therefore, a shift of the resonance wavelengths is induced by varying the distance between the defects. Although the uniformly distributed defects increase considerably the passband of each channel, the ripple can cause a nonuniform attenuation of the transmitted signals in the passband. In order to improve the performance of each transmission channel in terms of band flatness, we also investigated

the effect of different positions of the defects along the PBG structure according to the Newton binomial distribution.

7. BINOMIALLY SPACED DEFECTS As shown in Section 4, an improvement, in terms of band flatness, in the spectral behavior of each channel can be achieved by suitably choosing the position of multiple defects within the PBG grating. A general design criterion is suggested from the analysis of the dotted curve of Fig. 7, which corresponds to two defects positioned within the grating according to the binomial coefficients. As already pointed out, this disposition improves the band flatness of each channel, thus suppressing the passband ripple. This concept somehow recalls the problem approached in the antenna theory concerning the suppression of the sidelobes in the radiation pattern of antenna arrays. In fact, a linear antenna array, i.e., an array with antenna elements arranged along a straight line, exhibits sidelobes in the radiation pattern that induce a detriment of the array directivity. Sidelobes can be suppressed by binomial antenna arrays in which the feeding currents of each antenna are chosen according to the Newton binomial series coefficients. The introduction into the PBG grating of D ¼ N D − 1 defects divides the PBG structure into N D periodic sections. The lengths of the N D sections, defined by the D defects, are chosen according to the Newton binomial rule. Figure 15 shows the binomial coefficients Bn arranged in the well-known Pascal triangle in which each row corresponds to a different number N D of grating sections. As an example, the insertion of two defects into the PBG grating divides the structure into three sections, therefore, in order to follow the binomial rule, the third row of the Pascal triangle of Fig. 15 must be considered. Accordingly, the central section must have a double length with respect to the other two. More generally, the length of the periodic grating sections and consequently the position of the defects are determined on the basis of the number of layers of the PBG grating. Considering a total number of alternating layers N t and N D grating sections (i.e., D ¼ N D − 1 defects), the total number of alternating layers N t is linked to the number of layers ls composing the shorter grating section by the following equation: N t ¼ ls

ND X

Bn ;

n¼1

where Bn are the binomial coefficients given by Bn ¼

ND − 1 : n−1

ð6Þ

Calò et al.


1

ND

Binomial coefficients Bn

1 2 3 4 5 6 7

1

1

1

1 2 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1

ΣB

n

n= 1

1 2 4 8 16 32 64

Fig. 15. Pascal triangle rule for the position of the defects.

The number of layers ls in the shorter grating sections can be easily obtained inverting Eq. (6). Moreover, the number of layers in the other grating sections is proportional to ls according to the binomial coefficient Bn . As shown in Fig. 15, the sum of the N D Newton binomial coefficients along the rows of the Pascal triangle is always a power of 2; thus, the total number N t of alternating layers must be itself a power of 2. Figure 16 schematizes the position of an increasing number of defects, from 1 to 6, according to the binomial rule for a PBG grating having 256 alternating layers. The number displayed below each of them is the number of alternating layers for each section. The different sections are delimited by the defects (white rectangles). As an example, Fig. 17 shows the transmittance T spectrum (solid curve) for the PBG structure with three defects, having equal lengths Ld ¼ 100 μm, binomially spaced at the layers p ¼ 32, 128, and 224. The two band edges of the PBG, pertaining to the periodic nondefective grating, are marked with the dotted—dashed lines. Comparing the results in the case of the uniformly spaced defect, reported in Fig. 13, with the analogous ones in the case of the binomially spaced defects, shown in Fig. 17, we can see that, by binomially spacing the defects, the bandwidth of each transmission channel assumes lower values (Δλ3 dB ¼ 3:2 nm in the case of uniform defect distribution and Δλ3 dB ¼ 2 nm in the case of binomial defect position), but the ripple in the transmission bandwidth is suppressed only in the case of binomially spaced defects. Figure 18 shows a portion of the transmittance spectra of one of the channels of the PBG grating with binomially spaced

Transmittance T, Reflectance R

ND

1677

0.8

0.6

0.4

0.2

0

1.53

1.54

1.55

Wavelength λ [µm]

1.56

Fig. 17. Transmittance T spectrum (solid curve) of the defective structure with three binomially spaced defects, placed at the layers p ¼ 32, 128, and 224, having length Ld ¼ 100 μm. The two dotted– dashed lines mark the PBG band edges (λ ¼ 1:531 μm and λ ¼ 1:555 μm) of the periodic nondefective grating.

defects. The different curves refer to an increasing number of defects from 1 to 6. Figure 18 shows that the 3 dB bandwidth becomes wider (from 1.2 to 3:3 nm) by increasing the number of defects according to the binomial rule, whereas we verified that the central wavelength of each channel remains unchanged. Conversely, the maximum value of the transmittance T MAX slightly decreases from 0.94, in the case of a single defect, to 0.88 for six binomial defects. The number of the binomially distributed defects is therefore a fundamental parameter to increase the passband of each channel, whereas the number of transmission channels within the PBG is mainly due to the length of the defects. This behavior is apparent in Table 3, which reports the number of channels N c , the FSR, and the 3 dB bandwidth Δλ3 dB for different lengths of the defects Ld . In particular, Table 3 compares the results pertaining to a PBG grating with a single defect placed in the middle of the structure with the ones pertaining to four defects, having equal length, distributed along the grating according to the binomial rule. As can be inferred from Table 3, for all the considered defect lengths, the insertion of multiple binomially spaced defects does not 1

1 defect 2 defects 4 defects

Transmittance T

0.8

6 defects 0.6

0.4

0.2

0 1.54

Fig. 16. (Color online) Position of an increasing number of defects according to the binomial rule for a PBG grating having 256 alternating layers: (a) one defect, (b) two defects, (c) three defects, (d) four defects, (e) five defects, (f) six defects.

1.542

1.544

Wavelength λ [µm]

1.546

Fig. 18. Transmittance T spectrum (solid curve) of one of the transmission channels for the defective structure with binomially spaced defects having length Ld ¼ 100 μm.

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Calò et al.

Table 3. Number of Channels N c , FSR, and 3 dB Bandwidth Δλ3 dB for Different Lengths of Defects Ld in the Cases of a Periodic Grating with a Single and with Four Binomially Spaced Defects 1 Central Defect

4 Binomially Spaced Defects

Ld ðμmÞ

Nc

FSR (nm)

Δλ3 dB ðnmÞ

Nc

FSR (nm)

Δλ3 dB (nm)

100 105 120 149 175 200 240 300 340

5 5 5 7 9 9 10 12 15

4.8 4.6 4.2 3.5 3.0 2.6 2.2 1.8 1.6

1.2 1.2 1.2 0.9 0.8 0.7 0.6 0.4 0.35

5 5 5 7 9 9 10 12 15

4.8 4.6 4.2 3.5 3.0 2.6 2.2 1.8 1.6

2.8 2.9 2.4 2.0 1.8 1.6 1.3 1.0 0.9

change both the number of channels and the FSR, whereas it increases the 3 dB bandwidth of each channel. The great advantage in the PBG structure with multiple defects, distributed according to the Newton coefficients, is the equalization of each channel and the consequent uniform transmittance of the signal in the passband. Conversely, the increase in the channel bandwidth is slightly narrower in the case of binomially distributed defects (e.g., Δλ3 dB ¼ 3:3 nm for six defects) with respect to the uniform distribution (e.g., Δλ3 dB ¼ 4:2 nm for six defects).

8. CONCLUSIONS The design criteria to achieve an increase in the passband and an improvement of the band flatness of the transmission channels in a multiwavelength optical filter have been proposed. The filter is based on an index-confined one-dimensional PBG waveguide in which multiple defects are included. The inclusion of a single defect in the one-dimensional grating permits the localization of multiple states within the PBG and therefore of the transmission channels having maximum transmittance. Moreover, the number of channels N c within the PBG increases with the defect length from N c ¼ 5 to N c ¼ 15 when the length of the defect changes from Ld ¼ 100 μm to Ld ¼ 340 μm. The introduction of multiple defects within the onedimensional grating strongly modifies the transmittance spectrum in terms of the shape and of the bandwidth of each channel. Multiple defects, uniformly spaced within the grating, induce an increase of the bandwidth of each channel from Δλ3 dB ¼ 1:2 nm for a single defect to Δλ3 dB ¼ 4:2 nm for six defects. However, a poor band flatness is achieved since the uniform distribution of the defects causes a ripple in the passband of the order of 20%. The optimal design criterion, which gives the equalization of the passband of each channel, was achieved considering the distribution of multiple defects within the periodic grating according to the Newton binomial coefficients. This design criterion improves the multiwavelength filter performance in terms of increased 3 dB bandwidth and band flatness, without changing the number of channels, the FSR, and the central wavelength of each channel with respect to the case of a single defect centered within the periodic grating.

ACKNOWLEDGMENTS The work has been supported by the Photonic Interconnect Technology for Chip Multiprocessing Architectures (“PHOTO-

NICA”) project under the Fondo per gli Investimenti della Ricerca di Base 2008 (“FIRB”) program, funded by the Italian government and by the project “Regional laboratory for synthesis and characterization of new organic and nanostructured materials for electronics, photonics, and advanced technologies” funded by the Apulia Region. The research has been conducted in the framework of the European Cooperation in Science and Technology (“COST”) Action MP0805.

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