Abstract

A method for synthesizing bandpass photonic crystal filters for wavelength division multiplexing (WDM) systems is presented. The proposed method permits the calculation of the physical dimensions of the crystalline structures given the desired frequency response of the filter in terms of bandwidth, in-band ripple, minimum out-of-band attenuation, and central frequency. The method, explained in detail for Chebyshev frequency responses, is equivalent circuit based. The resulting devices are very compact, have a high out-of-band attenuation, and are suitable for high density photonic integrated circuits. The validity of the proposed method is confirmed through contrasting the simulation concluded from the finite-difference time-domain (FDTD) method by the design of a third-order Chebyshev filter having a center frequency of 1THz, a flat bandwidth of 4GHz, and ripples of 0.5 dB in the passband.

© 2011 OSA

1. Introduction

Photonic crystals (PCs) have inspired a lot of research due to their ability to control the propagation of light [15]. Up to now, many optical devices based on PCs have been designed and fabricated, such as the waveguides, the optical switches, the power splitters and so on [68]. Among these PCs devices, the filters are the essential components of photonic integrated circuits and optical communication systems.

High-Q-factor optical resonant filters, utilizing a single-defect mode in PCs, have been demonstrated experimentally, and the transmission spectra of such filters are Lorentzian [9,10]. For optical resonant filters to be used in WDM optical communication systems, the transmission characteristics need to be improved, so as to get steep roll-off and flattened passband. This requires higher order filters which can be created by coupling multiple resonators. A third-order filter [11,12] and N-coupled-resonators filter [13] have been designed for improving the filtering performance. With this in sight, it is desirable to have a robust and systematic method for synthesizing PCs filters that can directly relate the characteristics of the prescribed filter transfer function to the PCs parameters.

Several methods for synthesizing coupled-resonators filters based on PCs have been proposed. An approach based on the coupled-mode theory in time [14] was adopted for analyzing and designing of many types of filters [1518] including the coupled-resonators PCs filters [12]. The method is simple and high efficient in analyzing the PCs structure, but difficult in synthesizing the filters with desired frequency response. The most promising approach was proposed in [19], in which the coupled-microring resonators were represented by an equivalent bandpass oscillator circuit with impedance inverters as coupling elements. Microwave filter-synthesis techniques were then applied to synthesize Butterworth and Chebyshev filters, for which closed-form design formulas could be obtained. A similar approach based on the LC filter circuit synthesis technique was also used in [20] to design symmetric resonant add-drop filters.

In this paper, we adopt a similar approach as those in [19] and [20] and establish the equivalence of a coupled-resonators PCs filter with a baseband LC ladder network within the narrowband approximation. The design of bandpass photonic crystal filters involves two main steps. The first one is to select an appropriate lowpass prototype. The choice of the type of response, including passband ripple and the number of reactive elements, will depend on the required specifications. Then, the realizable lumped-element circuit for bandpass filters can be formed by immittance and frequency transformations. Having obtained a suitable lumped-element filter design, the next main step in the design of coupled-resonators PCs bandpass filters is to find an appropriate PCs realization which approximates the lumped-element filter. The calculations of equivalent LC’s values of general structure in microwave frequency range, such as microstrip or metal waveguide, are relatively simple, but it is difficult for PCs’. However, the circuit parameters of lowpass prototype derived by the first step can be rewritten as the form of external quality factor Qe and coupling coefficient kij which of PCs’ can be relatively easily extract. Figure 1 shows the schematic diagram of the filter in the waveguide-cavities-waveguide form. The coupling between every two cavities is characterized as coupling coefficient, and the coupling between waveguide and cavity is characterized as external quality factor. So we need to find out the structures of which the parameters are close to the results calculated before, then the structures are those we are desired. In this paper, we aim to provide the process of synthesizing from the given specifications to the values of external quality factors and coupling coefficients, and the method to extract Qeand kij from the realistic photonic crystal filters’ structure. At last, we design a third-order Chebyshev filter by the method above and simulate it with the FDTD method; the results illustrate the effectiveness of the proposed approach.

 

Fig. 1 Basic structure of filter for developing the theoretical method

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2. Filters synthesis

We first summarize the synthesis procedures for bandpass filters in the form of lumped-element circuit from the given specifications of the filter in terms of frequency response (that is bandwidth, maximum reflection in band, out-of-band attenuation) and the type of filter (i.e., Butterworth, Chebyshev, elliptic, Bessel). This step, valid in both the microwave and the optical domains, requires simple closed-form formulae [11,12]. On many occasions, an amplitude-squared transfer function for a lossless passive filter network is defined as

|S21(jΩ)|2=11+ε2Fn2(Ω)
where ε is a ripple constant, Fn(Ω)represents a filtering or characteristic function, and Ω is a frequency variable. For our discussion here, it is convenient to let Ω represent a radian frequency variable of a lowpass prototype filter which has a cutoff frequency at Ω=Ωc for Ωc=1(rad/s). For a given function of Eq. (1), the insertion loss response of the filter can be computed by

LA(Ω)=10log1|S21(jΩ)|2dB

Here we chose the Chebyshev function which is defined as Eq. (3) as filtering function, the Chebyshev response that exhibits the equal-ripple passband and maximally flat stopband is depicted in Fig. 2(a) .

 

Fig. 2 (a) Chebyshev low-pass response. (b) Low-pass prototype filters for Chebyshev filters. (c) Band-pass filters using frequency transformations and immittance inverters

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Fn(Ω)={cos(ncos1Ω)|Ω|1cosh(ncosh1Ω)|Ω|1

The ripple constant ε is related to a given passband ripple LAr in dB by

ε=10LAr101

For the required passband ripple LArdB, the minimum stopband attenuation LAsdB atΩ=Ωs, the degree of a Chebyshev lowpass prototype, which will meet this specification, can be found by

ncosh1100.1LAs1100.1LAr1cosh1Ωs

Filter syntheses for realizing the transfer functions usually result in the so-called lowpass prototype filters. A lowpass prototype filter, in general, is defined as the lowpass filter whose element values are normalized to make the source resistance or conductance equal to one, denoted byg0=1, and the cutoff angular frequency to be unity, denoted by Ωc=1(rad/s). For Chebyshev lowpass prototype filters the element values for the two-port networks shown in Fig. 2(b) may be computed by using the following formulas:

g0=1.0
g1=2γsin(π2n)
gi=1gi14sin[(2i1)2nπ]sin[(2i3)2nπ]γ2+sin2[(i1)nπ]i=2,3,n
gn+1={1.0fornoddcoth2(β4)forneven
whereβ=ln[coth(LAr17.37)],γ=sinh(β2n).

To obtain frequency characteristics and element values for practical filters based on the lowpass prototype, we may apply frequency transformations and immittance inverters. The frequency transformation is required to map a Chebyshev response in the lowpass prototype frequency domain Ω to that in the frequency domain ω in which a practical bandpass filter response is expressed. The frequency transformation will have an effect on all the reactive elements accordingly, with no effect on the resistive elements. The immittance inverters have the ability to shift impedance or admittance levels depending on the choice of K or J parameters. Making use of these properties enables us to convert a filter circuit to an equivalent form that would be more convenient for implementation with microwave or optical structures.

J0,1=Y0FBWω0Cp1Ωcg0g1
Ji,i+1=FBWω0ΩcCpiCp(i+1)gigi+1i=1ton1
Jn,n+1=FBWω0CpnYn+1Ωcgngn+1
{Cpi=(ΩcFBWω0)giγ0Lpi=1/(ω02Cpi)i=1ton
whereω0=ω1ω2denotes the center angular frequency, ω1 and ω2 indicate the passband-edge angular frequency, and FBW=(ω2ω1)/ω0 is defined as the fractional bandwidth.

So far, the desired filter structure can be obtained as long as we find the proper microwave or optical structure of which the circuit parameters are close to those calculated above. Figure 3(a) shows a PCs structure of the bandpass filter, which uses a cascaded resonators formed by introducing point defects in PCs. The coupling between the resonators can be treated as if the resonators interact through a waveguide, although the length of the waveguide may be shorter than the resonance wavelength of the filter. Based on this assumption, the filter to be designed can be equivalent to the circuit in Fig. 3(b). The proper resonator Q factors (or equivalently, the decay rates) can be determined from the standard LC filter circuit design theory, by treating the resonators as lumped elements [22], and the phase shift that occurs in the short waveguide between the resonators will be determined by both the center frequency of the filter and the effective index of the waveguide. In the case ofϕ=π/2, the transmission line can be transformed to the J inverter depicted in Fig. 2(c).

 

Fig. 3 (a) Structure of the second-order coupled-resonator filter in a PC, where 1-D defect resonators are embedded in a 2-D PC waveguide. (b) Equivalent circuit of the structure in Fig. 3.(a), where the coupling between the resonators is treated as if it occurs through a waveguide

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Theoretically, the value of inductance and capacitance of Fig. 3(b) can be obtained by space integral on the each resonator, and phase shift ϕ=βL, β is the propagation constant of the waveguide, and L is the length between two reference planes as shown in Fig. 3(a).

The phase shift ϕ is a critical parameter in the design of the multi-resonators reflection filter. It can be defined as

ϕ=neffωLc
where neff is the effective refractive index [22].

However it is difficult to calculate the LC’s values by the method above; the phase shift will be determined by both the center frequency of the filter, the effective index of the waveguide and the choice of reference planes. This implies that the structure of the filter is neither easily controllable nor easily measurable. Here we can use the external quality factor and the coupling coefficient to represent the characteristic of the filter, their definitions are as follows:

Qe1=g0g1ΩcFBW
Qen=gngn+1ΩcFBW
ki,i+1=FBWΩcgigi+1|i=1~n1

For filters based on PCs structure, the external quality factor and the coupling coefficient can be extracted as [21]:

Qe=2ω0Δω3dB
kij=±(ω0jω0i+ω0iω0j)(ωj2ωi2ωj2+ωi2)2(ω0j2ω0i2ω0j2+ω0i2)2
where ω0 and Δω3dB are the resonant frequency and bandwidth for a doubly loaded resonator respectively; ω=0i(LiCi)1/2 and ω=0j(LjCj)1/2 are the two self-resonant frequencies, ωi and ωj are the characteristic frequencies of the physical structure of two coupled resonators corresponding to odd and even modes which are shown in Fig. 4 .

 

Fig. 4 Two states of two coupled resonators: (a) even mode. (b)odd mode

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It should be remarked that the interaction of the coupled resonators is mathematically described by the dot operation of their space vector fields, which allows the coupling to have either positive or negative sign. A positive sign would imply that the coupling enhances the stored energy of uncoupled resonators, whereas a negative sign would indicate a reduction. Therefore, the electric and magnetic couplings could either have the same effect if they have the same sign, or have the opposite effect if their signs are opposite.

In this paper, we specify that the positive sign is taken when the frequency of odd mode is greater than that of even mode; conversely, the negative sign is taken. With the external quality factor and the coupling coefficient, the transmission characteristics of the filter can be expressed as [21]:

S21=|S+2S+1|=21qe1qen[A]n11
With
[A]=[q]+p[U]j[k]
where [q] is an n × n matrix with all entries zero, except for q11 = qe1 and qnn=qen, qeiandQei are the scaled external quality factor and external quality factor respectively:
qei=Qei(Δω/ω)fori=1,n
ω0 and Δωare the center frequency and the bandwidth of filter respectively. p=j(ω0/Δω)(ω/ω0ω0/ω), and [U] is the n × n unit or identity matrix, [k] is the so-called general coupling matrix, which is an n × n reciprocal matrix (i.e., kij=kji).

3. Examples

To demonstrate the robustness of the proposed circuit-based method for synthesizing photonic crystal filters, we present two examples in this section. The first example is the analysis of a two-order filter whose transmission spectra is obtained by the coupled resonator method (CRM) described in Section 2 and the FDTD method. The second example is the synthesis of a Chebyshev filter based on photonic crystal from the given specifications.

Figure 5(b) shows the transmission spectra of two coupled-resonators filter shown in Fig. 5(a) by the CRM and the FDTD method respectively. For length between the two resonators L=4a (‘a’ is the lattice constant), length from resonator to waveguidel=3a, and radius of rods/defectr0/rd=0.2a/0.1a, as shown in Fig. 5(a), the external quality factors areQe1=Qe2=560, and the coupling coefficients arek12=k21=0.0035. The two results are in accordance with each other as illustrated in Fig. 5(b).

 

Fig. 5 (a) Schematic representation of the structure of the coupled-resonators. (b) Transmission characteristic of the filter obtained by coupled-resonators method and FDTD method

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In order to design the PCs-based higher order filter, the resonators are separately designed in a 2-D PCs waveguide to have the determined center frequency and the proper Qe factors, and then, the coupling coefficient can be extracted by treating every two resonators as a whole. Table 1 shows the different radius with its resonance frequencies and the coupling coefficients calculated by Eq. (19).

Tables Icon

Table 1. Coupling coefficients of two resonatorsa

At last, we design a coupled-resonators bandpass photonic crystal filter for terahertz application by the method presented in this paper. The specifications for the filter under consideration are

  • • Center frequency 1.0THz
  • • Fractional bandwidth 0.4%
  • • Minimum stop-band attenuation 20dB at 1.006THz
  • • Pass-band ripple 0.5dB

In this example, the design process is performed in two steps. First step, the frequency of the specifications should be normalized to the center frequency, and the order of the designed filter could be obtained by the Eq. (5). Here, it is worked out by using a n =3 Chebyshev low-pass prototype. Then, based on the values calculated by the Eq. (6-9), the external quality factors and the coupling coefficients can be obtained by the Eq. (15-17), which are shown as following:

  • Qe1=Qe3=480
  • k1,2=k2,3=0.053,k1,3=0.0084

Second step, we should choose a PCs topology structure of which the two parameters are closed to the results obtained above. First, the rods in air type PCs is adopted, the radius and the dielectric constant of the rods in the background 2-D PCs are set to 0.2a (herea=0.1mm) and 11.56, respectively. By the plane-wave expansion technique, the normalized frequency of the TE bandgap of the PCs is 0.28547 to 0.41987, which is corresponding to 0.8564THz and 1.2596THz. Then, it is found that the resonance frequency of 1THz is attained when the three defect rods r1=r2=r3 =0.095a. For the resonators with a Qe factor of 495(resonator 1 and resonator 3), the length between waveguide and resonator is determined to be 3a. At first, we set the three resonators in a straight line with 4aspacing, and extract the coupling coefficients of them. But the results are largely different from the desired ones. After extensive calculations, it has been found thatk1,2, k2,3and k1,3 factors of 0.049, 0.049 and 0.0078 are attained when the second resonator is placed above the waveguide of which the topology structure is shown in Fig. 6(a) . It seemed to be a little different between the design results and the desired ones, but after some fine tuning the characteristic of the designed filter would be very close to the given parameters. Here, we merely tuned the radius of the second defect rod to 0.1a. At last, the calculated result is plotted as real line and compared to the result obtained by 2-D FDTD method with dotted line in Fig. 6(b). It indicates that the method presented in this paper is valid for synthesizing of coupled-resonators bandpass photonic crystal filters.

 

Fig. 6 (a) Schematic diagram of the proposed filter, which is composed of 3 resonators. (b) Transmission spectra of the desired filter by the synthesized method (real line) and the simulation result by the FDTD method (dotted line).

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4. Conclusion

We present a circuit-based approach for synthesizing serially coupled-resonators bandpass photonic crystal filters. It is shown that a chain of serially coupled-resonators can be represented by an equivalent baseband LC ladder network in the narrowband approximation. By introducing the external quality factor and the coupling coefficient, the circuit model allows the standard analog-filter-realization techniques to be directly applied to synthesize the serially coupled-resonators filters. Examples are provided to illustrate the application of the technique to synthesize the standard Chebyshev filters. And it indicates that the method can be extended to other filters such as Butterworth, Generalized Chebyshev and Elliptic Function.

References and links

1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of Ligh, (Princeton Univ. Press, Princeton, 2008).

2. M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004). [CrossRef]  

3. M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. 19(12), 1970–1975 (2001). [CrossRef]  

4. M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999). [CrossRef]  

5. M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28(24), 2506–2508 (2003). [CrossRef]   [PubMed]  

6. X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005). [CrossRef]  

7. M. Belotti, J. F. Galisteo Lòpez, S. De Angelis, M. Galli, I. Maksymov, L. C. Andreani, D. Peyrade, and Y. Chen, “All-optical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities,” Opt. Express 16(15), 11624–11636 (2008). [PubMed]  

8. H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003). [CrossRef]  

9. J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996). [CrossRef]  

10. M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002). [CrossRef]  

11. R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15(3), 401–403 (2003). [CrossRef]  

12. D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. 23(5), 1923–1928 (2005). [CrossRef]  

13. X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007). [CrossRef]  

14. H. A. Haus, Wave and Fields in Optoelectronics (Englewood Cliffs, NJ: Prentice-Hall, 1984).

15. S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3(1), 4–11 (1998). [CrossRef]   [PubMed]  

16. C. Chen, X. Li, H. Li, K. Xu, J. Wu, and J. Lin, “Bandpass filters based on phase-shifted photonic crystal waveguide gratings,” Opt. Express 15(18), 11278–11284 (2007). [CrossRef]   [PubMed]  

17. K. Fasihi and S. Mohammadnejad, “Highly efficient channel-drop filter with a coupled cavity-based wavelength-selective reflection feedback,” Opt. Express 17(11), 8983–8997 (2009). [CrossRef]   [PubMed]  

18. Y. Akahane, T. Asano, H. Takano, B.-S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express 13(7), 2512–2530 (2005). [CrossRef]   [PubMed]  

19. A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20(2), 296–303 (2002). [CrossRef]  

20. M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999). [CrossRef]  

21. J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications (John wiley & sons, INC. 2001).

22. H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–212 (1992). [CrossRef]  

References

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of Ligh, (Princeton Univ. Press, Princeton, 2008).
  2. M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004).
    [CrossRef]
  3. M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. 19(12), 1970–1975 (2001).
    [CrossRef]
  4. M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999).
    [CrossRef]
  5. M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28(24), 2506–2508 (2003).
    [CrossRef] [PubMed]
  6. X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005).
    [CrossRef]
  7. M. Belotti, J. F. Galisteo Lòpez, S. De Angelis, M. Galli, I. Maksymov, L. C. Andreani, D. Peyrade, and Y. Chen, “All-optical switching in 2D silicon photonic crystals with low loss waveguides and optical cavities,” Opt. Express 16(15), 11624–11636 (2008).
    [PubMed]
  8. H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003).
    [CrossRef]
  9. J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996).
    [CrossRef]
  10. M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002).
    [CrossRef]
  11. R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15(3), 401–403 (2003).
    [CrossRef]
  12. D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. 23(5), 1923–1928 (2005).
    [CrossRef]
  13. X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007).
    [CrossRef]
  14. H. A. Haus, Wave and Fields in Optoelectronics (Englewood Cliffs, NJ: Prentice-Hall, 1984).
  15. S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3(1), 4–11 (1998).
    [CrossRef] [PubMed]
  16. C. Chen, X. Li, H. Li, K. Xu, J. Wu, and J. Lin, “Bandpass filters based on phase-shifted photonic crystal waveguide gratings,” Opt. Express 15(18), 11278–11284 (2007).
    [CrossRef] [PubMed]
  17. K. Fasihi and S. Mohammadnejad, “Highly efficient channel-drop filter with a coupled cavity-based wavelength-selective reflection feedback,” Opt. Express 17(11), 8983–8997 (2009).
    [CrossRef] [PubMed]
  18. Y. Akahane, T. Asano, H. Takano, B.-S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express 13(7), 2512–2530 (2005).
    [CrossRef] [PubMed]
  19. A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20(2), 296–303 (2002).
    [CrossRef]
  20. M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
    [CrossRef]
  21. J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications (John wiley & sons, INC. 2001).
  22. H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–212 (1992).
    [CrossRef]

2009 (1)

2008 (1)

2007 (2)

X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007).
[CrossRef]

C. Chen, X. Li, H. Li, K. Xu, J. Wu, and J. Lin, “Bandpass filters based on phase-shifted photonic crystal waveguide gratings,” Opt. Express 15(18), 11278–11284 (2007).
[CrossRef] [PubMed]

2005 (3)

2004 (1)

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004).
[CrossRef]

2003 (3)

M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28(24), 2506–2508 (2003).
[CrossRef] [PubMed]

H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003).
[CrossRef]

R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15(3), 401–403 (2003).
[CrossRef]

2002 (2)

M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002).
[CrossRef]

A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20(2), 296–303 (2002).
[CrossRef]

2001 (1)

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. 19(12), 1970–1975 (2001).
[CrossRef]

1999 (2)

M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
[CrossRef]

1998 (1)

1996 (1)

J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996).
[CrossRef]

1992 (1)

H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–212 (1992).
[CrossRef]

Akahane, Y.

Altug, H.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004).
[CrossRef]

Andreani, L. C.

Asano, T.

Belotti, M.

Chen, C.

Chen, J. C.

J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996).
[CrossRef]

Chen, Y.

CHENG, B.

X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005).
[CrossRef]

Chutinan, A.

M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002).
[CrossRef]

Costa, R.

R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15(3), 401–403 (2003).
[CrossRef]

De Angelis, S.

Dodabalapur, A.

M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999).
[CrossRef]

Fan, S.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004).
[CrossRef]

M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28(24), 2506–2508 (2003).
[CrossRef] [PubMed]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
[CrossRef]

S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Channel drop filters in photonic crystals,” Opt. Express 3(1), 4–11 (1998).
[CrossRef] [PubMed]

J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996).
[CrossRef]

Fasihi, K.

Galisteo Lòpez, J. F.

Galli, M.

GONG, Q.

X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005).
[CrossRef]

Haus, H.

Haus, H. A.

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
[CrossRef]

J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996).
[CrossRef]

H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–212 (1992).
[CrossRef]

HU, X.

X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005).
[CrossRef]

Imada, M.

M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002).
[CrossRef]

Joannopoulos, J.

Joannopoulos, J. D.

M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28(24), 2506–2508 (2003).
[CrossRef] [PubMed]

M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999).
[CrossRef]

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
[CrossRef]

J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996).
[CrossRef]

Khan, M. J.

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
[CrossRef]

Kim, S.

Koshiba, M.

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. 19(12), 1970–1975 (2001).
[CrossRef]

Lai, Y.

H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–212 (1992).
[CrossRef]

Lee, Y. H.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003).
[CrossRef]

Li, H.

Li, X.

Li, X. C.

X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007).
[CrossRef]

Lim, H.

Lin, J.

Lin, J. T.

X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007).
[CrossRef]

Liu, A. Q.

X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007).
[CrossRef]

LIU, Y.

X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005).
[CrossRef]

Maksymov, I.

Manolatou, C.

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
[CrossRef]

Martinelli, M.

R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15(3), 401–403 (2003).
[CrossRef]

A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20(2), 296–303 (2002).
[CrossRef]

Meier, M.

M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999).
[CrossRef]

Mekis, M.

M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999).
[CrossRef]

Melloni, A.

R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15(3), 401–403 (2003).
[CrossRef]

A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20(2), 296–303 (2002).
[CrossRef]

Mochizuki, M.

M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002).
[CrossRef]

Mohammadnejad, S.

Noda, S.

Y. Akahane, T. Asano, H. Takano, B.-S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express 13(7), 2512–2530 (2005).
[CrossRef] [PubMed]

M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002).
[CrossRef]

Notomi, M.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003).
[CrossRef]

Park, D.

Park, I.

Peyrade, D.

Ryu, H. Y.

H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003).
[CrossRef]

Slusher, R. E.

M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999).
[CrossRef]

Soljacic, M.

Song, B.-S.

Takana, Y.

Takano, H.

Tanaka, T.

M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002).
[CrossRef]

Villeneuve, P.

Villeneuve, P. R.

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
[CrossRef]

J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996).
[CrossRef]

Vuckovic, J.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004).
[CrossRef]

Wu, J.

Xu, J.

X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007).
[CrossRef]

Xu, K.

X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007).
[CrossRef]

C. Chen, X. Li, H. Li, K. Xu, J. Wu, and J. Lin, “Bandpass filters based on phase-shifted photonic crystal waveguide gratings,” Opt. Express 15(18), 11278–11284 (2007).
[CrossRef] [PubMed]

Yanik, M. F.

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004).
[CrossRef]

M. F. Yanik, S. Fan, M. Soljacić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28(24), 2506–2508 (2003).
[CrossRef] [PubMed]

ZHANG, D.

X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005).
[CrossRef]

Appl. Phys. B (1)

X. HU, Q. GONG, Y. LIU, B. CHENG, and D. ZHANG, “Fabrication of two-dimensional organic photonic crystal filter,” Appl. Phys. B 81, 779–781 (2005).
[CrossRef]

Appl. Phys. Lett. (1)

H. Y. Ryu, M. Notomi, and Y. H. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. 83(21), 4294–4296 (2003).
[CrossRef]

Appl. Phys., A Mater. Sci. Process. (2)

M. Mekis, M. Meier, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Lasing mechanism in two dimensional photonic crystal lasers,” Appl. Phys., A Mater. Sci. Process. 69(1), 111–114 (1999).
[CrossRef]

X. C. Li, J. Xu, K. Xu, A. Q. Liu, and J. T. Lin, “A side-coupled photonic crystal filter with sidelobe suppression,” Appl. Phys., A Mater. Sci. Process. 89(2), 327–332 (2007).
[CrossRef]

IEEE J. Lightw. Technol. (2)

M. F. Yanik, H. Altug, J. Vuckovic, and S. Fan, “Submicrometer All-Optical Digital Memory and Integration of Nanoscale Photonic Devices without Isolator,” IEEE J. Lightw. Technol. 22(10), 2316–2322 (2004).
[CrossRef]

M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing With Photonic Crystal Waveguide Coupler,” IEEE J. Lightw. Technol. 19(12), 1970–1975 (2001).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. J. Khan, C. Manolatou, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Mode-coupling analysis of multipole symmetric resonant add/drop filters,” IEEE J. Quantum Electron. 35(10), 1451–1460 (1999).
[CrossRef]

H. A. Haus and Y. Lai, “Theory of cascaded Quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–212 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

R. Costa, A. Melloni, and M. Martinelli, “Bandpass resonant filters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett. 15(3), 401–403 (2003).
[CrossRef]

J. Lightwave Technol. (4)

D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. 23(5), 1923–1928 (2005).
[CrossRef]

J. C. Chen, H. A. Haus, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Optical filters from photonic band gap air bridges,” J. Lightwave Technol. 14(11), 2575–2580 (1996).
[CrossRef]

M. Imada, S. Noda, A. Chutinan, M. Mochizuki, and T. Tanaka, “Channel drop filter using a single defect in a 2-D photonic crystal slab waveguide,” J. Lightwave Technol. 20(5), 873–878 (2002).
[CrossRef]

A. Melloni and M. Martinelli; “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. 20(2), 296–303 (2002).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Other (3)

H. A. Haus, Wave and Fields in Optoelectronics (Englewood Cliffs, NJ: Prentice-Hall, 1984).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystal: Molding the Flow of Ligh, (Princeton Univ. Press, Princeton, 2008).

J. S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications (John wiley & sons, INC. 2001).

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Figures (6)

Fig. 1
Fig. 1

Basic structure of filter for developing the theoretical method

Fig. 2
Fig. 2

(a) Chebyshev low-pass response. (b) Low-pass prototype filters for Chebyshev filters. (c) Band-pass filters using frequency transformations and immittance inverters

Fig. 3
Fig. 3

(a) Structure of the second-order coupled-resonator filter in a PC, where 1-D defect resonators are embedded in a 2-D PC waveguide. (b) Equivalent circuit of the structure in Fig. 3.(a), where the coupling between the resonators is treated as if it occurs through a waveguide

Fig. 4
Fig. 4

Two states of two coupled resonators: (a) even mode. (b)odd mode

Fig. 5
Fig. 5

(a) Schematic representation of the structure of the coupled-resonators. (b) Transmission characteristic of the filter obtained by coupled-resonators method and FDTD method

Fig. 6
Fig. 6

(a) Schematic diagram of the proposed filter, which is composed of 3 resonators. (b) Transmission spectra of the desired filter by the synthesized method (real line) and the simulation result by the FDTD method (dotted line).

Tables (1)

Tables Icon

Table 1 Coupling coefficients of two resonators a

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

| S 21 ( j Ω ) | 2 = 1 1 + ε 2 F n 2 ( Ω )
L A ( Ω ) = 10 log 1 | S 21 ( j Ω ) | 2 d B
F n ( Ω ) = { cos ( n cos 1 Ω ) | Ω | 1 cos h ( n cos h 1 Ω ) | Ω | 1
ε = 10 L A r 10 1
n cos h 1 10 0.1 L A s 1 10 0.1 L A r 1 cos h 1 Ω s
g 0 = 1.0
g 1 = 2 γ sin ( π 2 n )
g i = 1 g i 1 4 sin [ ( 2 i 1 ) 2 n π ] sin [ ( 2 i 3 ) 2 n π ] γ 2 + sin 2 [ ( i 1 ) n π ] i = 2 , 3 , n
g n + 1 = { 1.0 for n odd cot h 2 ( β 4 ) for n even
J 0 , 1 = Y 0 F B W ω 0 C p 1 Ω c g 0 g 1
J i , i + 1 = F B W ω 0 Ω c C p i C p ( i + 1 ) g i g i + 1 i = 1 to n 1
J n , n + 1 = F B W ω 0 C p n Y n + 1 Ω c g n g n + 1
{ C p i = ( Ω c F B W ω 0 ) g i γ 0 L p i = 1 / ( ω 0 2 C p i ) i = 1 to n
ϕ = n e f f ω L c
Q e 1 = g 0 g 1 Ω c F B W
Q e n = g n g n + 1 Ω c F B W
k i , i + 1 = F B W Ω c g i g i + 1 | i = 1 ~ n 1
Q e = 2 ω 0 Δ ω 3 d B
k i j = ± ( ω 0 j ω 0 i + ω 0 i ω 0 j ) ( ω j 2 ω i 2 ω j 2 + ω i 2 ) 2 ( ω 0 j 2 ω 0 i 2 ω 0 j 2 + ω 0 i 2 ) 2
S 21 = | S + 2 S + 1 | = 2 1 q e 1 q e n [ A ] n 1 1
[ A ] = [ q ] + p [ U ] j [ k ]
q e i = Q e i ( Δ ω / ω ) for i = 1 , n

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