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Design of an optical filter using photonic band gap material
Optik 114, No. 3 (2003) 101–105 ª 2003 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik
S. P. Ojha et al., Design of an optical filter using photonic band gap material
101
International Journal for Light and Electron Optics
Design of an optical filter using photonic band gap material S. P. Ojha, Sanjeev K. Srivastava, N. Kumar, S. K. Srivastava Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi 221005, India
Abstract: The article presents a design of an optical filter using Photonic Band Gap (PBG) materials in the visible and infrared region of the electromagnetic spectrum. This idea is based on the famous Kronig-Penny model in the band theory of solids. The suggested filter may work at any range of the wavelength that can be achieved by choosing the appropriate values of the controlling parameters. This structure is also able to pass the light emitted by the Ruby laser and this acts as a monochromator. Also, we have studied the anomalous behavior of refractive index for the structure having alternate layers of air and GaAs. Key words: Optical filters – PBG materials – monochromator – anomalous dispersion
1. Introduction During the last decade a tremendous interest in new purpose-built materials called the Photonic Band Gap (PBG) material has been generated and emerged as a new multidisciplinary field of study [1–7]. These materials are composed of thin dielectric materials, semiconductors or metals surrounded by air or other materials of lower refractive index in order to confine the light wave. These structures exhibit frequency bands where electromagnetic waves cannot propagate. Furthermore, the structure can be designed for a different frequency ranges simply by scaling their dimensions [8]. The ability of PBG materials to manipulate photons has many potential applications in efficient optical filters [9–10], thresholdless semiconductor laser [11], endlessly single mode optical fibers [12] etc. These photonic devices have great advantages over conventional electronic devices. They can offer very high speed of operation, tolerance to temperature fluctuation, increased lifetime and the capability to care high repetition rates. All these devices work on the principle of photonic band edge and they are extremely compact in structure. Conventional gratings have index modulations of few percent whereas PBG materials have large index contrasts in their indices to the
Received 28 January 2002; accepted 20 March 2003. Correspondence to: S. P. Ojha Fax: ++91-542-310925 E-mail: [email protected]
extent of 4 : 1 [13]. Wide stop bands are achieved due to this constrast. Fabriacation of optical filters in the near infrared regions of the wavelength was suggested by Ojha et al. [14] which was based on the weak guidance approxin1 n2 1. Chen et al. [15] also mation such that n1 suggested the optical filtering properties by photonic band gap air bridges. In this article we suggest the design of an optical filter in the visible and infrared region by using the periodic refractive index profile of the material with rectangular symmetry. In our previous work we have suggested the optical filter in the ultraviolet region using the same idea [16]. We must emphasize that the proposed filter may work at any region of the electromagnetic spectrum by choosing the suitable values of the lattice parameters. This principle has been exploited for constructing monochromatic sources. We also studied the anomalous behavior of refractive index in case of PBG materials having alternate layers of air and GaAs.
2. Theoretical analysis It is well known that when electrons move through a periodic lattice, allowed and forbidden energy bands are obtained. The same idea may be applicable to the case of optical radiation if the electron waves are replaced by optical waves and the lattice periodicity structure is replaced by a periodic refractive index pattern. One expects allowed and forbidden bands of frequencies instead of energies. By choosing a linearly periodic refractive index profile in the filter material one obtains a given set of wavelength ranges that are allowed or forbidden to pass through the filter material. Selecting a particular x-axis through the material, we shall assume a periodic step function for the index of the form [13, 17] n1 ; 0 x a ; ð1Þ nðxÞ ¼ n2 ; b x 0 ; where n1 ðx þ tdÞ ¼ n1 and n2 ¼ ðx þ tdÞ ¼ n2 . Here t is the translation factor, which takes the values t ¼ 0; 1; 2; 3; . . . and d ¼ a þ b is the period of the lattice 0030-4026/03/114/03-101 $ 15.00/0
102
S. P. Ojha et al., Design of an optical filter using photonic band gap material
A31 ¼ eiaðakÞ ; A33 ¼ eibðbkÞ ;
A32 ¼ eiaðaþkÞ ; A34 ¼ eibðbþkÞ ;
A41 ¼ iða kÞ eiaðakÞ ;
Fig. 1. Periodic variation of the refractive index profile in the form of a rectangular structure.
with a and b being the width of the two medium having refractive indices ðn1 Þ and ðn2 Þ respectively. The periodic structure of the material in the form of rectangular symmetry is shown in the fig. 1. In this case the one-dimensional wave equation for the spatial part of the electromagnetic eigen mode wk ðxÞ is given by 2
2
d wk ðxÞ n ðxÞ þ dx2 c2
w2k
wk ðxÞ ¼ 0
ð2Þ
where nðxÞ is given by eq. (1). Assuming that nðxÞ is constant in n1 and n2 regions, eq. (2) for wave equation may be written as d2 wk ðxÞ n21 w2k þ 2 wk ðxÞ ¼ 0 ; dx2 c
0xa
d2 wk ðxÞ n22 w2k þ 2 wk ðxÞ ¼ 0 ; c dx2
b x 0 :
ð3aÞ
ð3bÞ Now making use of Bloch’s theorem, wk ¼ uk ðxÞ eikx and applying boundary conditions as given below u1 ðxÞjx¼0 ¼ u2 ðxÞjx¼0
ð4aÞ
u01 ðxÞjx¼0
ð4bÞ
¼
u02 ðxÞjx¼0
u1 ðxÞjx¼a ¼ u2 ðxÞjx¼b
ð4cÞ
u01 ðxÞjx¼a
ð4dÞ
¼
u02 ðxÞjx¼b
we get four equations having four unknown constants. To obtain a nontrivial solution for the equations, the determinant of the coefficients of the unknown constants should be zero, which is given as A11 A12 A13 A14 A 21 A22 A23 A24 ð5Þ ¼0 A31 A32 A33 A34 A41 A42 A43 A44 where A11 ¼ A12 ¼ A13 ¼ A14 ¼ 1 ; A21 ¼ iða kÞ;
A22 ¼ iða þ kÞ;
A23 ¼ iðb kÞ;
A24 ¼ iðb þ kÞ ;
A42 ¼ iða þ kÞ eiaðaþkÞ ;
A44 ¼ iðb þ kÞ eibðbþkÞ ; A43 ¼ iðb kÞ eibðbkÞ ; n w n w 1 k 2 k and b ¼ , k is the wave number a¼ c c related to frequency w. On solving eq. (5) we get n wa n wa n2 wb 1 n1 n2 1 1 cos þ sin cos c c 2 n2 n1 c n2 wb sin ¼ cos kða þ bÞ : ð6Þ c On abbreviating the L.H.S. as Fl, eq. (6) may be written as Fl ¼ cos kða þ bÞ
ð7Þ
where the L.H.S. ðFl Þ in terms of wavelength ðlÞ can be written as 2pn1 a 2pn2 b 1 n1 n2 Fl ¼ cos þ cos l l 2 n2 n1 2pn1 a 2pn2 b sin sin : ð8Þ l l From eq. (6) we can write the expression for the dispersion relation and effective index of refraction that are given as n wa 1 n2 wb 1 1 cos kðwÞ ¼ cos cos d c c 1 n1 n2 n1 wa n2 wb sin þ sin ð9Þ 2 n2 n1 c c c n wa n2 wb 1 cos1 cos cos neff ¼ wd c c 1 n1 n2 n1 wa n2 wb sin þ sin : 2 n2 n1 c c ð10Þ Now our aim is to find the allowed and forbidden values of the wavelength bands and dispersion curves for structures using the different materials.
3. Results and discussion To find the forbidden and allowed values of wavelengths, the characteristic eq. (7) has been evaluated numerically. Here we have considered three cases for the sake of illustration. In these cases we have alternate regions of (air-GaAs), (air-Ge) and (air-PbTe) respectively. These results are employed for the design of an optical filter and a monochromator as mentioned below:
S. P. Ojha et al., Design of an optical filter using photonic band gap material
Fig. 2. Variation of Fl with the wavelength l for in the visible and infrared region for n1 ¼ 1.0, n2 ¼ 3.6, a ¼ 0.68 mm and b ¼ 0.12 mm.
Design of the filter: For the proposed filters the width of the two alternate regions ‘a’ and ‘b’ are taken as 0.68 mm and 0.12 mm respectively in all the three cases, according to the Yablonovitch structure ða=d ¼ 0:85 and b=d ¼ 0:15Þ. The values of refractive indices n1 and n2 in these cases have been chosen as n1 ¼ 1:0 (air), n2 ¼ 3:6 (GaAs); n1 ¼ 1:0 (air), n2 ¼ 4:0 (Ge) and n1 ¼ 1:0 (air) and n2 ¼ 5:4 (PbTe) respectively. Using these values, eq. (7) is plotted against the wavelength l and the curves are shown in the figures 2–4 respectively. The photonic bands (in terms of wavelength) so obtained are shown in the tables 1–3 for the visible and infrared regions. Because of the existence of the cosine function on the right-hand side of the eq. (7), the upper and lower limiting values will obviously be þ1 and 1 respectively. The portion of the curve
103
Fig. 4. Variation of Fl with the wavelength l for in the visible and infrared region for n1 ¼ 1.0, n2 ¼ 5.4, a ¼ 0.68 mm and b ¼ 0.12 mm.
lying between these limiting values will yield the allowed ranges of l. Those outsides will show the forbidden ranges of transmission. On examining the graphs and the corresponding tables it is found that the width of the allowed bands generally decreases as one goes towards the higher wavelength side for the fixed values of the parameters ða; b; n1 and n2 Þ. Actually these allowed bands give the different ranges of wavelength, which can transmit through the filter. So by choosing suitable values of these parameters one can get the desired range of transmission (or reflection). From the observation of graphs it is found that the structure having alternate layers of (air-GaAs) and (air-Ge) have three bands in the visible region and two bands in the near infrared region while the structure having alternate layers of air and lead telluride gives four bands in the visible region and two bands in the near infrared region and the bandwith of air-telluride structure is smaller than that of (air-GaAs) and (air-Ge) structure. Thus it is inferred that structure having (air-PbTe) in their alternate regions acts as multi-band wavelength selector in the visible region and double band wavelength selector in the infrared regions. It is also observed that the (air-PbTe) structure passes greater percentage of light wave incident on it whereas (air-Ge)
Table 1. Photonic bands for n1 ¼ 1.0, n2 ¼ 3.6, a ¼ 0.68 mm and b ¼ 0.12 mm. Allowed bands
Fig. 3. Variation of Fl with the wavelength l for in the visible and infrared region for n1 ¼ 1.0, n2 ¼ 4.0, a ¼ 0.68 mm and b ¼ 0.12 mm.
Allowed Range (in A) Bandwith (in A)
1.
4000–4385
385
2.
4502–5077
575
3.
6119–7035
916
4.
7958–9828
1870
5.
12462–16647
4185
104
S. P. Ojha et al., Design of an optical filter using photonic band gap material
Table 2. Photonik bands for n1 ¼ 1.0, n2 ¼ 4.0, a ¼ 0.68 mm and b ¼ 0.12 mm. Allowed bands
Allowed Range (in A) Bandwith (in A)
1.
4151–4574
423
2.
4729–5304
575
3.
6333–7142
809
4.
8587–10452
1865
5.
12773–16851
4078
a)
Table 3. Photonik bands for n1 ¼ 1.0, n2 ¼ 5.4, a ¼ 0.68 mm and b ¼ 0.12 mm. Allowed bands
Allowed Range (in A) Bandwith (in A)
1.
4066–4356
290
2.
4501–4858
357
3.
5870–6533
663
4.
6753–7642
889
5.
10543–13070
2527
6.
13500–17934
4434
passes the least among these three proposed filter structures. The proposed filter structures can act as a narrow band wavelength selector as well as a wide band selector in the same region. This property of the filter structures depends on the thickness of the two alternate regions for the selected material. For the larger values of the lattice parameters we get the narrow band wavelength selector with the larger number of bands while for the smaller values of the parameters we find the wide band wavelength selector with the lesser number of pass bands. Monochromator: It is possible that the above filter can also work as a nearly single wavelength selector by controlling the thickness of the two alternate regions without changing the refractive index or any other parameter. To illustrate this we have chosen the values of lattice parameters ‘a’ and ‘b’ as 680 mm and 200 mm for (air-GaAs) and 684 mm and 202 mm for the (air-Ge) structure respectively. Using these values, we obtain Fl versus l curves, whose amplified portion is depicted in figs. 5a–5b respectively. The allowed range of transmitted wavelengths can be seen in these figures, which are 6942.10–6943.40 A (with the difference of approximately 1.3 A) and 6942.60–6943.90 A (with the difference of approximately 1.3 A) respectively, i.e. only this range of wavelength (<1.5 A) can pass through the filter. This means that the filter act just as a monochromator and transmits the wavelength corresponding to the wavelength of a solid-state ruby laser system which is 6943 A.
b) Fig. 5. Monochromator which transmits the principle wavelength of solid-state ruby laser for a) n1 ¼ 1.0, n2 ¼ 3.6, a ¼ 680 mm and b ¼ 200 mm. b) n1 ¼ 1.0, n2 ¼ 4.0, a ¼ 684 mm and b ¼ 202 mm.
Anomalous refractive index: For the anomalous behavior of refractive index we have considered the case of PBG material having air and GaAs in their alternate regions. For the sake of numerical illustrations, we choose n1 ¼ 1:0 and n2 ¼ 3:6 and the values of lattice parameters are chosen as a ¼ 85% of d and b ¼ 15% of d. Now substituting these values in eq. (9) and eq. (10) we plot the two curves one for the dispersion wd and other for the effective index of rekd Vs c wd which are depicted in figs. 6a fraction neff Vs c and 6b respectively. From the slope of the dispersion curve (fig. 6a) it is noticed that the group velocity dw , approaches zero at the band edges. At these Vg ¼ dk points, we find that standing wave is formed in the similar manner as in the band theory of solids. Further it is observed that the phase velocity ðVp Þ as given by w Vp ¼ , tends towards infinity at the band edges k wd has approximate where kd approaches zero and c values 4.0, 5.1 and 8.23 respectively. Now the effective index of refraction is defined as c neff ¼ , this implies that neff ! 0 at the band edges Vp because Vp ! 1 at these points. From the observation
S. P. Ojha et al., Design of an optical filter using photonic band gap material
105
ing to the practical need by the choice of relevant parameters and can be used as a narrow band filters in astronomy. These can also be employed for the design of glare suppression filters and for the construction of single wavelength selectors. Such filters can be fabricated using the fabrication techniques of optical coatings. More theoretical and experimental results are needed to compare our results. Acknowledgement. One of the authors Mr. Sanjeev K. Srivastava is thankful to UGC-India for the financial assistance. S. K. Srivastava is also thankful to CSIR, New Delhi and AICTE, New-Delhi, (Project No., 2043), for the financial assistance.
a)
b) Fig. 6. a) plot of anomalous dispersion relation for n1 ¼ 1:0, n2 ¼ 3.6, a ¼ 85% of d and b ¼ 15% of d; b) Plot of effective wd for n1 ¼ 1.0, n2 ¼ 3.6, a ¼ 85% index of refraction against c of d and b ¼ 15% of d.
of curve as mentioned in the fig. 6b, we obtain that neff is less than one at the frequencies near the band edges wd has approximate values 4.05, 5.1 and and at which c 8.23 respectively. This modified refractive index photonic band gap material finds two major applications in laser acceleration of electrons and novel refractive elements.
4. Conclusions It is concluded that the suggested filter can be used to pass bands in visible and infrared region. However, the idea is very general, and filter can be designed accord-
References [1] Yablonovitch E: Photonic cystalas. J. Mod. Opt. 41 (1994) 173–194 [2] Pendry J: Photonic band structures, J. Mod. Opt. 41 (1994) 209–229 [3] Joannopoulos JD, Meade RD, Winn JN: Photonic crystals: Modeling the flow of light. Princeton University Press, Princeton 1995 [4] Shawn LY, Fleming JG, Robin L, Sigalas MM, Biswas R, Ho KM: Complete three-dimensional photonic band gap in a single cubic structure. J. Opt. Soc. Am. B 18 (2001) 32–35 [5] Temelkuran B, Mehmet B, Ozbay E, Kavanaugh JP, Sigalas MM, Tuttle G: Quasimetallic silicon micromachined photonic crystals. Appl. Phys. Lett. 78 (2001) 264–266 [6] Taniyama H: Waveguide structures using one-dimensional photonic crystal. J. Appl. Phys. 91 (2002) 3511–3515 [7] Hache A, Poirier L: Long-range superluminal pulse propagation in a coaxial photonic crystal. Appl. Phys. Lett. 80 (2002) 518–520 [8] Kosaka H, Kawashima T, Tomita A, Sato T, Kawakami SK: Photonic-crystal spot-size conveter. Appl. Phys. Lett. 76 (2000) 268–270 [9] Foresi J, Villeneuve P, Ferrera J, Thoen E, Steinmeyer G, Fan S, Joannopoulos J, Kimerling L, Smith H, Ippen E: Photonic band gap microcavities in optical waveguides. Nature 390 (1997) 143–145 [10] Villeneuve P, Abrams D, Fan S, Joannopoulos J: Singlemode waveguide microcavity for fast optical switching. Opt. Lett. 21 (1996) 2017–2019 [11] Villeneuve PR, Fan S. Joannopoulos JD, Lim KY, Petrich GS Kolodjeski LA, Reif R: Air-bridge microcavities. Appl. Phys. Lett. 67 (1995) 167–170 [12] Birks T, Knight J, Russel J: Endlessly single-mode photonic crystal fiber. Opt. Lett. 22 (1997) 961–963 [13] Dowling JP, Bowden CM: Anomalous index of refraction in photonic band gap materials. J. Mod. Opt. 41 (1994) 345–351 [14] Ojha SP, Choudhary PK, Khastgir P, Singh ON: Operating characteristics of an optical filter with linearly periodic refractive index pattern in the filter material. Jpn. J. Appl. Phys. 34 (1992) 281–285 [15] Chen JC, Haus A, Fan S, Villeneuve PR, Joannapoulos JD: Optical filters from photonic band gap air bridges. Lightwave Technol. 14 (1996) 2575–2580 [16] Srivastava SK, Ojha SP, Ramesh KS: Design of an ultraviolet filter based on photonic band gap materials. Microw. Opt. Technol. Lett. 33 (2002) 308–314 [17] John S, Wang J: Quantum Optics of localized light in a photonic band gap. Phys. Rev. B 43 (1991) 1272–1289
S. P. Ojha et al., Design of an optical filter using photonic band gap material
101
International Journal for Light and Electron Optics
Design of an optical filter using photonic band gap material S. P. Ojha, Sanjeev K. Srivastava, N. Kumar, S. K. Srivastava Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi 221005, India
Abstract: The article presents a design of an optical filter using Photonic Band Gap (PBG) materials in the visible and infrared region of the electromagnetic spectrum. This idea is based on the famous Kronig-Penny model in the band theory of solids. The suggested filter may work at any range of the wavelength that can be achieved by choosing the appropriate values of the controlling parameters. This structure is also able to pass the light emitted by the Ruby laser and this acts as a monochromator. Also, we have studied the anomalous behavior of refractive index for the structure having alternate layers of air and GaAs. Key words: Optical filters – PBG materials – monochromator – anomalous dispersion
1. Introduction During the last decade a tremendous interest in new purpose-built materials called the Photonic Band Gap (PBG) material has been generated and emerged as a new multidisciplinary field of study [1–7]. These materials are composed of thin dielectric materials, semiconductors or metals surrounded by air or other materials of lower refractive index in order to confine the light wave. These structures exhibit frequency bands where electromagnetic waves cannot propagate. Furthermore, the structure can be designed for a different frequency ranges simply by scaling their dimensions [8]. The ability of PBG materials to manipulate photons has many potential applications in efficient optical filters [9–10], thresholdless semiconductor laser [11], endlessly single mode optical fibers [12] etc. These photonic devices have great advantages over conventional electronic devices. They can offer very high speed of operation, tolerance to temperature fluctuation, increased lifetime and the capability to care high repetition rates. All these devices work on the principle of photonic band edge and they are extremely compact in structure. Conventional gratings have index modulations of few percent whereas PBG materials have large index contrasts in their indices to the
Received 28 January 2002; accepted 20 March 2003. Correspondence to: S. P. Ojha Fax: ++91-542-310925 E-mail: [email protected]
extent of 4 : 1 [13]. Wide stop bands are achieved due to this constrast. Fabriacation of optical filters in the near infrared regions of the wavelength was suggested by Ojha et al. [14] which was based on the weak guidance approxin1 n2 1. Chen et al. [15] also mation such that n1 suggested the optical filtering properties by photonic band gap air bridges. In this article we suggest the design of an optical filter in the visible and infrared region by using the periodic refractive index profile of the material with rectangular symmetry. In our previous work we have suggested the optical filter in the ultraviolet region using the same idea [16]. We must emphasize that the proposed filter may work at any region of the electromagnetic spectrum by choosing the suitable values of the lattice parameters. This principle has been exploited for constructing monochromatic sources. We also studied the anomalous behavior of refractive index in case of PBG materials having alternate layers of air and GaAs.
2. Theoretical analysis It is well known that when electrons move through a periodic lattice, allowed and forbidden energy bands are obtained. The same idea may be applicable to the case of optical radiation if the electron waves are replaced by optical waves and the lattice periodicity structure is replaced by a periodic refractive index pattern. One expects allowed and forbidden bands of frequencies instead of energies. By choosing a linearly periodic refractive index profile in the filter material one obtains a given set of wavelength ranges that are allowed or forbidden to pass through the filter material. Selecting a particular x-axis through the material, we shall assume a periodic step function for the index of the form [13, 17] n1 ; 0 x a ; ð1Þ nðxÞ ¼ n2 ; b x 0 ; where n1 ðx þ tdÞ ¼ n1 and n2 ¼ ðx þ tdÞ ¼ n2 . Here t is the translation factor, which takes the values t ¼ 0; 1; 2; 3; . . . and d ¼ a þ b is the period of the lattice 0030-4026/03/114/03-101 $ 15.00/0
102
S. P. Ojha et al., Design of an optical filter using photonic band gap material
A31 ¼ eiaðakÞ ; A33 ¼ eibðbkÞ ;
A32 ¼ eiaðaþkÞ ; A34 ¼ eibðbþkÞ ;
A41 ¼ iða kÞ eiaðakÞ ;
Fig. 1. Periodic variation of the refractive index profile in the form of a rectangular structure.
with a and b being the width of the two medium having refractive indices ðn1 Þ and ðn2 Þ respectively. The periodic structure of the material in the form of rectangular symmetry is shown in the fig. 1. In this case the one-dimensional wave equation for the spatial part of the electromagnetic eigen mode wk ðxÞ is given by 2
2
d wk ðxÞ n ðxÞ þ dx2 c2
w2k
wk ðxÞ ¼ 0
ð2Þ
where nðxÞ is given by eq. (1). Assuming that nðxÞ is constant in n1 and n2 regions, eq. (2) for wave equation may be written as d2 wk ðxÞ n21 w2k þ 2 wk ðxÞ ¼ 0 ; dx2 c
0xa
d2 wk ðxÞ n22 w2k þ 2 wk ðxÞ ¼ 0 ; c dx2
b x 0 :
ð3aÞ
ð3bÞ Now making use of Bloch’s theorem, wk ¼ uk ðxÞ eikx and applying boundary conditions as given below u1 ðxÞjx¼0 ¼ u2 ðxÞjx¼0
ð4aÞ
u01 ðxÞjx¼0
ð4bÞ
¼
u02 ðxÞjx¼0
u1 ðxÞjx¼a ¼ u2 ðxÞjx¼b
ð4cÞ
u01 ðxÞjx¼a
ð4dÞ
¼
u02 ðxÞjx¼b
we get four equations having four unknown constants. To obtain a nontrivial solution for the equations, the determinant of the coefficients of the unknown constants should be zero, which is given as A11 A12 A13 A14 A 21 A22 A23 A24 ð5Þ ¼0 A31 A32 A33 A34 A41 A42 A43 A44 where A11 ¼ A12 ¼ A13 ¼ A14 ¼ 1 ; A21 ¼ iða kÞ;
A22 ¼ iða þ kÞ;
A23 ¼ iðb kÞ;
A24 ¼ iðb þ kÞ ;
A42 ¼ iða þ kÞ eiaðaþkÞ ;
A44 ¼ iðb þ kÞ eibðbþkÞ ; A43 ¼ iðb kÞ eibðbkÞ ; n w n w 1 k 2 k and b ¼ , k is the wave number a¼ c c related to frequency w. On solving eq. (5) we get n wa n wa n2 wb 1 n1 n2 1 1 cos þ sin cos c c 2 n2 n1 c n2 wb sin ¼ cos kða þ bÞ : ð6Þ c On abbreviating the L.H.S. as Fl, eq. (6) may be written as Fl ¼ cos kða þ bÞ
ð7Þ
where the L.H.S. ðFl Þ in terms of wavelength ðlÞ can be written as 2pn1 a 2pn2 b 1 n1 n2 Fl ¼ cos þ cos l l 2 n2 n1 2pn1 a 2pn2 b sin sin : ð8Þ l l From eq. (6) we can write the expression for the dispersion relation and effective index of refraction that are given as n wa 1 n2 wb 1 1 cos kðwÞ ¼ cos cos d c c 1 n1 n2 n1 wa n2 wb sin þ sin ð9Þ 2 n2 n1 c c c n wa n2 wb 1 cos1 cos cos neff ¼ wd c c 1 n1 n2 n1 wa n2 wb sin þ sin : 2 n2 n1 c c ð10Þ Now our aim is to find the allowed and forbidden values of the wavelength bands and dispersion curves for structures using the different materials.
3. Results and discussion To find the forbidden and allowed values of wavelengths, the characteristic eq. (7) has been evaluated numerically. Here we have considered three cases for the sake of illustration. In these cases we have alternate regions of (air-GaAs), (air-Ge) and (air-PbTe) respectively. These results are employed for the design of an optical filter and a monochromator as mentioned below:
S. P. Ojha et al., Design of an optical filter using photonic band gap material
Fig. 2. Variation of Fl with the wavelength l for in the visible and infrared region for n1 ¼ 1.0, n2 ¼ 3.6, a ¼ 0.68 mm and b ¼ 0.12 mm.
Design of the filter: For the proposed filters the width of the two alternate regions ‘a’ and ‘b’ are taken as 0.68 mm and 0.12 mm respectively in all the three cases, according to the Yablonovitch structure ða=d ¼ 0:85 and b=d ¼ 0:15Þ. The values of refractive indices n1 and n2 in these cases have been chosen as n1 ¼ 1:0 (air), n2 ¼ 3:6 (GaAs); n1 ¼ 1:0 (air), n2 ¼ 4:0 (Ge) and n1 ¼ 1:0 (air) and n2 ¼ 5:4 (PbTe) respectively. Using these values, eq. (7) is plotted against the wavelength l and the curves are shown in the figures 2–4 respectively. The photonic bands (in terms of wavelength) so obtained are shown in the tables 1–3 for the visible and infrared regions. Because of the existence of the cosine function on the right-hand side of the eq. (7), the upper and lower limiting values will obviously be þ1 and 1 respectively. The portion of the curve
103
Fig. 4. Variation of Fl with the wavelength l for in the visible and infrared region for n1 ¼ 1.0, n2 ¼ 5.4, a ¼ 0.68 mm and b ¼ 0.12 mm.
lying between these limiting values will yield the allowed ranges of l. Those outsides will show the forbidden ranges of transmission. On examining the graphs and the corresponding tables it is found that the width of the allowed bands generally decreases as one goes towards the higher wavelength side for the fixed values of the parameters ða; b; n1 and n2 Þ. Actually these allowed bands give the different ranges of wavelength, which can transmit through the filter. So by choosing suitable values of these parameters one can get the desired range of transmission (or reflection). From the observation of graphs it is found that the structure having alternate layers of (air-GaAs) and (air-Ge) have three bands in the visible region and two bands in the near infrared region while the structure having alternate layers of air and lead telluride gives four bands in the visible region and two bands in the near infrared region and the bandwith of air-telluride structure is smaller than that of (air-GaAs) and (air-Ge) structure. Thus it is inferred that structure having (air-PbTe) in their alternate regions acts as multi-band wavelength selector in the visible region and double band wavelength selector in the infrared regions. It is also observed that the (air-PbTe) structure passes greater percentage of light wave incident on it whereas (air-Ge)
Table 1. Photonic bands for n1 ¼ 1.0, n2 ¼ 3.6, a ¼ 0.68 mm and b ¼ 0.12 mm. Allowed bands
Fig. 3. Variation of Fl with the wavelength l for in the visible and infrared region for n1 ¼ 1.0, n2 ¼ 4.0, a ¼ 0.68 mm and b ¼ 0.12 mm.
Allowed Range (in A) Bandwith (in A)
1.
4000–4385
385
2.
4502–5077
575
3.
6119–7035
916
4.
7958–9828
1870
5.
12462–16647
4185
104
S. P. Ojha et al., Design of an optical filter using photonic band gap material
Table 2. Photonik bands for n1 ¼ 1.0, n2 ¼ 4.0, a ¼ 0.68 mm and b ¼ 0.12 mm. Allowed bands
Allowed Range (in A) Bandwith (in A)
1.
4151–4574
423
2.
4729–5304
575
3.
6333–7142
809
4.
8587–10452
1865
5.
12773–16851
4078
a)
Table 3. Photonik bands for n1 ¼ 1.0, n2 ¼ 5.4, a ¼ 0.68 mm and b ¼ 0.12 mm. Allowed bands
Allowed Range (in A) Bandwith (in A)
1.
4066–4356
290
2.
4501–4858
357
3.
5870–6533
663
4.
6753–7642
889
5.
10543–13070
2527
6.
13500–17934
4434
passes the least among these three proposed filter structures. The proposed filter structures can act as a narrow band wavelength selector as well as a wide band selector in the same region. This property of the filter structures depends on the thickness of the two alternate regions for the selected material. For the larger values of the lattice parameters we get the narrow band wavelength selector with the larger number of bands while for the smaller values of the parameters we find the wide band wavelength selector with the lesser number of pass bands. Monochromator: It is possible that the above filter can also work as a nearly single wavelength selector by controlling the thickness of the two alternate regions without changing the refractive index or any other parameter. To illustrate this we have chosen the values of lattice parameters ‘a’ and ‘b’ as 680 mm and 200 mm for (air-GaAs) and 684 mm and 202 mm for the (air-Ge) structure respectively. Using these values, we obtain Fl versus l curves, whose amplified portion is depicted in figs. 5a–5b respectively. The allowed range of transmitted wavelengths can be seen in these figures, which are 6942.10–6943.40 A (with the difference of approximately 1.3 A) and 6942.60–6943.90 A (with the difference of approximately 1.3 A) respectively, i.e. only this range of wavelength (<1.5 A) can pass through the filter. This means that the filter act just as a monochromator and transmits the wavelength corresponding to the wavelength of a solid-state ruby laser system which is 6943 A.
b) Fig. 5. Monochromator which transmits the principle wavelength of solid-state ruby laser for a) n1 ¼ 1.0, n2 ¼ 3.6, a ¼ 680 mm and b ¼ 200 mm. b) n1 ¼ 1.0, n2 ¼ 4.0, a ¼ 684 mm and b ¼ 202 mm.
Anomalous refractive index: For the anomalous behavior of refractive index we have considered the case of PBG material having air and GaAs in their alternate regions. For the sake of numerical illustrations, we choose n1 ¼ 1:0 and n2 ¼ 3:6 and the values of lattice parameters are chosen as a ¼ 85% of d and b ¼ 15% of d. Now substituting these values in eq. (9) and eq. (10) we plot the two curves one for the dispersion wd and other for the effective index of rekd Vs c wd which are depicted in figs. 6a fraction neff Vs c and 6b respectively. From the slope of the dispersion curve (fig. 6a) it is noticed that the group velocity dw , approaches zero at the band edges. At these Vg ¼ dk points, we find that standing wave is formed in the similar manner as in the band theory of solids. Further it is observed that the phase velocity ðVp Þ as given by w Vp ¼ , tends towards infinity at the band edges k wd has approximate where kd approaches zero and c values 4.0, 5.1 and 8.23 respectively. Now the effective index of refraction is defined as c neff ¼ , this implies that neff ! 0 at the band edges Vp because Vp ! 1 at these points. From the observation
S. P. Ojha et al., Design of an optical filter using photonic band gap material
105
ing to the practical need by the choice of relevant parameters and can be used as a narrow band filters in astronomy. These can also be employed for the design of glare suppression filters and for the construction of single wavelength selectors. Such filters can be fabricated using the fabrication techniques of optical coatings. More theoretical and experimental results are needed to compare our results. Acknowledgement. One of the authors Mr. Sanjeev K. Srivastava is thankful to UGC-India for the financial assistance. S. K. Srivastava is also thankful to CSIR, New Delhi and AICTE, New-Delhi, (Project No., 2043), for the financial assistance.
a)
b) Fig. 6. a) plot of anomalous dispersion relation for n1 ¼ 1:0, n2 ¼ 3.6, a ¼ 85% of d and b ¼ 15% of d; b) Plot of effective wd for n1 ¼ 1.0, n2 ¼ 3.6, a ¼ 85% index of refraction against c of d and b ¼ 15% of d.
of curve as mentioned in the fig. 6b, we obtain that neff is less than one at the frequencies near the band edges wd has approximate values 4.05, 5.1 and and at which c 8.23 respectively. This modified refractive index photonic band gap material finds two major applications in laser acceleration of electrons and novel refractive elements.
4. Conclusions It is concluded that the suggested filter can be used to pass bands in visible and infrared region. However, the idea is very general, and filter can be designed accord-
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