Analysis of tuning in a photonic crystal multichannel filter containing coupled defects

Optik 124 (2013) 2028–2032

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Analysis of tuning in a photonic crystal multichannel filter containing coupled defects Tsung-Wen Chang a , Chien-Jang Wu b,∗ a b

Graduate Institute of Electro-Optical Engineering, Chang Gung University, Tao-Yuan 333, Taiwan Institute of Electro-Optical Science and Technology, National Taiwan Normal University, Taipei 116, Taiwan

a r t i c l e

i n f o

Article history: Received 21 January 2012 Accepted 14 June 2012

Keywords: Photonic crystals Multichannel filter Temperature tuning Voltage tuning

a b s t r a c t We investigate the tuning properties in a photonic crystal multichannel filter made of coupled defects. The tuning in defect modes is achieved by the temperature change as well as the applied bias voltage. The temperature tuning arises from the thermal expansion and thermal–optical effects in the constituent layers. It is found that the resonant peaks are red-shifted as the temperature increases. In the voltage tuning, the defect modes are blue-shifted as the positive bias is applied, whereas they are red-shifted when the bias voltage is negative. The position of peak wavelength is further shown to linearly decrease with the applied voltage. The analysis of tunable features reveals that such a multichannel filter can be used not only a tunable device but a temperature- or voltage sensor. © 2012 Elsevier GmbH. All rights reserved.

1. Introduction A useful application of a one-dimensional photonic crystal (1D PC) is the design of a multilayer transmission filter also called a multilayer Fabry-Perot resonator [1,2]. In general, this filter has a structure of (AB)N D(AB)N , where (AB)N is the defect-free 1D PC, A, B, D are dielectric materials and D is known as a defect layer. In the transmission spectrum, there exists a single resonant transmission peak inside the photonic band gap (PBG). This peak elucidates the existence of an additional photonic defect physically analogous to the impurity or defect state in the electronic band gap of solids. From the spectral utility viewpoint, the efficiency of using PBG is quite limited because there is only a single channel for transmission. In order to enhance the spectral efficiency, it is necessary to have a filter that can filter multiple frequencies. Based on the use photonic quantum well (PQW) in the 1D PC, the function of a multichannel filter can be realized and its structure is (AB)N (CD)M (AB)N , where N and M are the stack numbers of the two PCs made of (AB) and (CD) bilayers, respectively [3–6]. The central part, (CD)M , plays the role of PQW and the number of transmission channels is equal to M. The most important design idea is that one of the pass bands of (CD)M must be completely overlap with one of the PBGs of the host PC of (AB)N . This appears to be a strict condition because the thicknesses and the refractive indices of both (AB)N and (CD)M are altogether closely related due to the required overlapping.

∗ Corresponding author. Tel.: +886 2 77346724; fax: +886 2 86631954. E-mail address: [email protected] (C.-J. Wu). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.06.023

In addition to the use of PQW to design a multichannel filter, another version, which can avoid the above strict condition and thus has a more flexible design, has been proposed by Jiang et al. [7]. This kind of multichannel filter has a structure of (AB)N (ABAC)M (ABA)(BA)N , which is also obtained from the original PC, (AB)N A. Here, C is the impurity or defect layer. The multichannel feature can be ascribed to the number of coupled defect. Indeed, the number of filtering channel in this filter is equal to M, the number of defect layers. Filter design based on the photonic crystal has been of interest to the optical and photonic communities. Recently, photonic crystal filters with tunable feature have attracted much attention because they are of more practical use in the optical signal processing. To make filters tunable, different defect materials like lithium niobate (LiNbO3 ) or liquid crystals (LCs) have been incorporated [8–11]. Tuning can also be achieved by the temperature such as in the semiconducting defect for the permittivity of a semiconductor can be varied by the temperature [12,13]. The purpose of this paper is to investigate the tunable properties in a multichannel filter. We shall consider the multichannel filter containing the coupled defects like (AB)N (ABAC)M (ABA)(BA)N and investigate temperature tuning and voltage tuning. The incident region is air and the transmitted region is the substrate, as illustrated in Fig. 1. Two ways of tuning, tuned by temperature and voltage, will be involved in this filter. In temperature tuning, the defect layer C is taken to be Bi4 Ge3 O12 (BGO), whose refractive index is dependent on the temperature [14,15]. In addition, we also incorporate the thermal expansion effect. In voltage tuning, the layer C is replaced by LiNbO3 , whose refractive index can be changed by the externally applied bias voltage.

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Fig. 1. The model structure for a multichannel filter, (AB)N (ABAC)M (ABA)(BA)N . The transmittance T will be the quantity of interest calculated for the purpose of investigating tunable properties.

The format of this paper is as follows: Section 1 is introductory part. In Section 2, we describe the theoretical basis that will be used in our calculation. The numerical results and discussion will be presented in Section 3, where the first part discusses the temperature tuning whereas the issue of voltage tuning is given in the second part. The summary is finally concluded in Section 4. 2. Basic equations To study the tunable properties in a multilayer filter, we shall use the transmittance spectrum calculated from the transfer matrix method (TMM) [16]. According to the TMM, transmission coefficient t and reflection coefficient r can be determined by matrix elements of the total system characteristic matrix Msys . In general, if p is the number of layers of the total system, each with a particular value of index of refraction n and of thickness d, then the first and the last boundaries are related by



EI



HI

 = MI MII · · ·Mp

E(p+1) H(p+1)

 .

(1)

The characteristic matrix of the entire system is the resultant of the product of the individual 2 × 2 matrices, that is,



Msys = MI MII · · ·Mp =

m11

m12

m21

m22



.

(2)

In the considered multichannel filter, (AB)N (ABAC)M (ABA)(BA)N , Msys takes the form

 Msys =

m11

m12

m21

m22

 N

M

N

= (MA MB ) (MA MB MA MC ) (MA MB MA )(MB MA ) ,

(3)

where Mp (p = A, B, or C) is the characteristic matrix for each single layer. Let the time part for any field be exp(jωt). Mp can be expressed as



Mp =

cos(k0 lp )

jZM sin(k0 lp )

jZp−1 sin(k0 lp )

cos(k0 lp )



,

(4)

where lp = np dp , np , and dp are respectively the optical length, the refractive index, and the physical thickness of layer p, and k0 = ω/c = 2/ is the wave number of free space. In addition, in the case of normal incidence, Zp =

Z0 , np

where Z0 =

(5)



0 /ε0 = 120  is the wave impedance of free space.

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Fig. 2. The wavelength-dependent transmittance for an ideal PC of Air/(AB)N /Air. Here, N = 20 and the design wavelength 0 = 500 nm are used.

With the matrix elements in Eq. (3), reflection coefficient r and transmission coefficient t can be determined by the following equations, r=

Zsub m11 + m12 − Z0 Zsub m21 − Z0 m22 , Zsub m11 + m12 + Z0 Zsub m21 + Z0 m22

(6)

t=

2Zsub , Zsub m11 + m12 + Z0 Zsub m21 + Z0 m22

(7)

where Zsub = Z0 /nsub is the wave impedance of the substrate. In this work, the whole system is assumed to be immersed in the air such that nsub = 1 will be used. Once the transmission coefficient, Eq. (7), has been obtained, the transmittance can be readily to be calculated, namely T = |t|2

(8)

In what follows, the wavelength-dependent transmittance will be calculated to demonstrate the properties of multichannel as well as tuning in the photonic crystal filter. 3. Numerical results and discussion Let us begin with a defect-free PC, Air/(AB)N /Air, where A and B are respectively the high- and low-index layers, and N is the number of periods. We would like to design this PC to have a PBG in the visible region. To achieve this goal, we take A as Si with nA = 3.45, B as SiO2 with nB = 1.45, and the thicknesses of both layers are equal to quarter-wavelength, i.e., nA dA = nB dB = 0/ 4, where 0 = 500 nm. With these material parameters, the wavelength-dependent transmittance T of this ideal photonic crystal is depicted in Fig. 2, where it is clear that there is a wide PBG covering the visible region. The purpose of this paper is to engineer this PBG to produce multiple defect modes and then to investigate how they are affected by the change in temperature or applied voltage. Now we consider a multichannel transmission filter that has a structure of Air/(AB)4 (ABAC)M ABA(BA)4 /Air, where the defect layer C is taken to be Bi4 Ge3 O12 (BGO). The index of refraction of BGO at room temperature (298 K) is nC = 2.13 and it is taken as a half-wave layer, nC dC = 0 /2. The calculated wavelength-dependent transmittance is shown in Fig. 3. It can be seen that the number of transmission peaks (the defect modes) is equal to M, the number of coupled defects. The distribution of resonant peaks is centrosymmetric about 0 . For m = odd, the central peak appears just at 0 . It is worth mentioning that the results shown in Fig. 3 are calculated at the room temperature of 298 K. We now turn our attention to the temperature dependence of these defect modes. The temperature tuning in the resonant peaks arises from two factors. One is that the thicknesses of the constituent layers are a function of temperature due to the thermal expansion. The other is that the index of refraction of the each layer can be varied as the temperature changes due to the

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Fig. 4. The calculated transmittance spectra for a single-peak filter of Air/(AB)4 (ABAC)1 ABA(BA)4 /Air at four different temperatures, T = 298, 310, 330, and 350 K, respectively. Fig. 3. The calculated defect modes, at 298 K, inside the PBG for the filter, Air/(AB)4 (ABAC)M ABA(BA)4 /Air, at M = 1, 2, 3, and 4 (from top to bottom), respectively.

thermal–optical effect. For the thermal expansion, the thickness d of each layer is written as, d(T ) = d0 [1 + ˛(T − 298)],

(9)

where d0 is set to be the thickness at T = 298 K (used in the calculation of Fig. 3) and ˛ is the thermally linear expansion coefficient which is 2.6 × 10−6 , 5.5 × 10−7 , and 6.3 × 10−6 K−1 for Si, SiO2 , and BGO, respectively [15,17]. For the thermal–optical effect, the temperature dependence of index of refraction n of each layer is n(T ) = n0 + ˇ(T − 298),

(10)

where n0 is the index of refraction at T = 298 K and ˇ is the thermooptical coefficient which is given by 1.86 × 10−4 , 1 × 10−5 , and 3.9 × 10−5 K−1 for Si, SiO2 , and BGO, respectively [14,15]. By incorporating both temperature effects in Eqs. (9) and (10), the calculated transmittance spectra for a single-defect filter Air/(AB)4 (ABAC)1 ABA(BA)4 /Air at four different temperatures, T = 298, 310, 330, and 350 K, are plotted in Fig. 4. It is seen that the transmittance peak inside the PBG is red-shifted as the temperature increases. The corresponding peak wavelengths for these four temperatures are 500, 518.5, 548.8, and 578 nm, respectively. The shifting behavior can be qualitatively explained by the condition of constant phase when resonance occurs. Since the phase is C = 2nC dC /, it is clear that  will be increased when nC dC is increased by increasing the temperature. The same shifting behavior is also obtained for the double-peak filter, Air/(AB)4 (ABAC)2 ABA(BA)4 /Air. In this case, the effect of temperature on the resonant peaks is shown in Fig. 5. In addition to the red-shift for the two peaks, we see that the spacing between two peaks is also slightly widened as the temperature increases. We now switch our attention to the issue of voltage tuning. In this case, the defect layer C in the multichannel transmission filter Air/(AB)4 (ABAC)M ABA(BA)4 /Air is replaced by the electro-optical material like LiNbO3 (LNO). The index of refraction of LNO is dependent on electric field and wavelength, namely [10] nLNO (, E) = nLNO −

1 3 · LNO · E, n 2 LNO

(11)

where the electro-optical coefficient is LNO = 3.09 × 10−11 m/V, and the wavelength-dependent nLNO is given by ne = ne () =



4.5820 −

0.099169 − 0.0219502 0.044432 − 2

1/2 ,

(12)

where is in unit of ␮m. Here, the electric field in Eq. (11) is E = V/H, where V is the applied voltage and H is the height of the sample. In Fig. 6, we plot the zero-voltage transmittance spectra for Air/(AB)4 (ABAC)M ABA(BA)4 /Air at M = 1, 2, 3, and 4, where nA = 3.45, nB = 1.45, and the thicknesses of A and B are equal to quarter-wavelength, i.e., nA dA = nB dB = 0/ 4, where 0 = 500 nm. The thickness of LNO is dC = 0/ 2 and H = 0.4 mm is taken [10]. It can be seen the multiple channels (defect modes) are very similar to those in Fig. 3. To investigate the voltage tuning, we take, for example, for the single defect case, M = 1, i.e., Air/(AB)4 (ABAC)1 ABA(BA)4 /Air. The configuration of applied bias voltage is the same as that in Ref. [10]. The resonant transmission peak at different bias voltages is shown

Fig. 5. The calculated transmittance spectra for a double-peak filter of Air/(AB)4 (ABAC)2 ABA(BA)4 /Air at four different temperatures, T = 298, 310, 330, and 350 K, respectively.

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Fig. 8. The peak wavelength in Fig. 6 as a function of the applied voltage in a filter, Air/(AB)4 (ABAC)1 ABA(BA)4 /Air.

Fig. 6. The multiple channels inside the PBG for the filter, Air/(AB)4 (ABAC)M ABA(BA)4 /Air, at M = 1, 2, 3, and 4 (from top to bottom), respectively. Here, C = LNO and zero voltage is applied.

in Fig. 7. It is seen that peak wavelength p is blue-shifted as the positive bias voltage increases. Conversely, it will be red-shifted when the negative bias voltage is applied. The dependence of peak wavelength p on the applied voltage is plotted in Fig. 8. It can be seen that there is a linear dependence for p versus V. The relation can be written as p (V ) = −0.046V + 500.624 (nm)

(13)

where V is in unit of kV. The tuning rate is thus given by 0.046 nm/kV. If we consider the case of M = 2, in which there are two resonant peaks, as shown in the second panel of Fig. 6. The left and right peak wavelengths are denoted as p1 and p2 , respectively. Then the dependence of applied voltage is shown in Fig. 9. Both peak wavelengths are a decreasing function of the applied voltage. In addition, the spacing between two peaks remains nearly unchanged.

Fig. 9. The peak wavelengths, p1 and p2 , as a function of the applied voltage in a filter, Air/(AB)4 (ABAC)2 ABA(BA)4 /Air.

4. Conclusion The temperature and voltage tuning in a multichannel filter based on the use of one-dimensional photonic crystal has been theoretically investigated in this work. In the temperature tuning, the thermal–optical and thermal expansion have been incorporated and it is found that the resonant transmission peaks will be red-shifted as the temperature increases. In the voltage tuning, the shifting in the transmission peaks is dependent on the bias voltage. For a positive bias voltage, the peak wavelengths are blue-shifted as the voltage increases. Conversely, they are red-shifted when the negative bias is applied. The analysis of tunable properties is informative to the applications of one-dimensional photonic crystals. A tunable and multichannel filter could be of particular use in the optical signal processing. Acknowledgements C.-J. Wu acknowledges the financial support from the National Science Council (NSC) of the Republic of China (R.O.C., Taiwan) under Contract No. NSC-100–2112-M-003–005-MY3 and from the National Taiwan Normal University under NTNU100-D-01. References

Fig. 7. The transmission channel inside the PBG in a filter, Air/(AB)4 (ABAC)1 ABA(BA)4 /Air, at different bias voltages, V = 0, 10, 20, and 30 kV (from top to bottom), respectively.

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