Multiple narrow bandpass optical filters basedon one-dimensional rugate

photonic structures of two periodicitiesZhao Lu

Cornell University, Ithaca, New York 14853, USA (zhaolu@cornell.edu)

Received October 8, 2010; revised January 13, 2011; accepted January 14, 2011;posted January 19, 2011 (Doc. ID 136353); published February 15, 2011

A theoretical analysis of a design of multiple narrow bandpass filters based on one-dimensional (1D) rugate photonicstructures with a period jump defect is presented. The optical properties, including transmittance and energy den-sity distributions, are numerically calculated using the propagation matrix method. Our results show that multipleresonance transmissionmodes are producedwhen the period jump defect is introduced into the 1D rugate film. Bothblueshift and redshift of the stop band of the rugate structure and wavelengths of resonant modes are observed,depending on the change of period jump. The number, the wavelengths, the band intervals, and the intensitiesof multiple resonance transmission modes are tunable by adjusting structure parameters of the rugate structure.Experimental feasibility of the proposed multiple narrow bandpass optical filters using the technique of glancingangle deposition is also discussed. © 2011 Optical Society of AmericaOCIS codes: 350.2460, 310.6860, 260.2110.

Photonic crystals (PCs) are optical structures of a peri-odic modulation of the index of refraction that showphotonic bandgaps in which the propagation of light isforbidden [1,2]. Because of the relatively easy fabri-cation process and optical properties similar to three-dimensional PCs under some conditions, one-dimensional (1D) PCs have received considerable atten-tion during the past decade [3–5]. Interference filters, asone of the important applications in optics, have beenrealized using either 1D discrete multilayer or gradi-ent-index optical materials [6,7]. With the introductionof defects into 1D PCs, single or multiple narrow band-pass (multichannel) filters based on the discrete multi-layer heterostructures or negative materials have beenextensively investigated [8–13]. Recently, single narrowbandpass optical filters based on 1D rugate photonicstructures that possess a continuously sinusoidal indexof refraction with a phase shift defect have been illu-strated using the technique of glancing angle deposition(GLAD) [14,15]. Because GLAD is able to fabricate rugatefilms with defects at micro- or nanoscale in a one-stepprocess using a single optical material [16,17] and, in con-trast to discrete multilayer structures, rugate films havebetter mechanistic properties, reduced optical losses,and smaller sensitivity to different angles of incidence[18], an interesting and important question would behow to design a multiple narrow bandpass filter basedon 1D rugate films with a defect.The purpose of this study is to theoretically demon-

strate a design of multiple narrow bandpass filters basedon 1D rugate films by introducing a novel defect—a per-iod jump defect. We will show how structure parametersof the 1D rugate film with the period jump defect affectthe number, the wavelengths, the band intervals, and theintensities of multiple resonance transmission modes.Figure 1 illustrates the refraction index profile of a sand-wiched 1D rugate structure. The middle rugate segmentintroduced as a defect has a period that is different fromthe end segment. Mathematical expression of the refrac-tion index is described by

nðzÞ ¼

8>>>><>>>>:

na þ np

2 sin�2πpAzþ α0

�; z∈ ½z0; z1�

na þ np

2 sin�2πpBðz − z1Þ þ α0

�; z∈ ½z1; z2�

na þ np

2 sin�2πpAðz − z2Þ þ α0

�; z∈ ½z2; z3�

: ð1Þ

The average index, the perturbation index, the initialphase angle, the original period, and the modified periodof the 1D rugate structure are denoted by na, np, α0, pA,and pB, respectively. The original period is related to thedesign wavelength λ0 via pA ¼ λ0=2na, and the modifiedperiod is associated with the original period throughpB ¼ ð1þ δÞpA, where the change of the original periodis parameterized by δ. The average and perturbation in-dices of a dielectric optical material are assumed to be2.0 and 0.52 [15], respectively; the design wavelength λ0is selected as 550 nm. The numbers of cycles of the endand defect segments of the 1D rugate structure are repre-sented by NA and NB, respectively; the positions z1, z2,and z3 are equal to NApA, NApA þ NBpB, and 2NApAþNBpB, respectively. The period jump (Δp ¼ pB−pA ¼ δpA) may be positive or negative; for instance,two special cases are shown in Figs. 1(a) (Δp ¼−0:5pA) and 1(b) (Δp ¼ þ0:5pA). To study optical prop-erties of multiple resonance transmission modes pro-duced by the 1D rugate structure with the period jump

Fig. 1. Refraction index profile of a 1D rugate structure with aperiod jump defect in the center; (a) Δp ¼ −0:5pA, (b)Δp ¼ þ0:5pA.

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defect, an efficient propagation matrix method is appliedto numerically calculate transmittance and electromag-netic energy densities [19–22]. The monochrome lightis assumed to be normally incident on the surface ofthe 1D rugate structure from air and exits from a glasssubstrate.Figure 2 shows the effect of the period change of the

defect segment on transmittance for a rugate structurewith NA ¼ NB ¼ 10. When the modified period (pB) issmaller than the original period (pA), i.e., δ < 0, a blue-shift of the stop band and multiple resonant modes closeto the right band edge are observed; when δ > 0, a red-shift of the forbidden band and multiple resonant modesclose to the left band edge are found.To investigate the effect of the number of cycles

(NB) of the defect segment on transmittance, two caseswith fixed δ ¼ −0:5 and þ0:5 are illustrated in Figs. 3(a)and 3(b), respectively. When NB increases from 2 to 50,we find that the number of multiple resonant modesincreases and the band intervals between the neighborresonant modes become narrower in both Figs. 3(a)and 3(b); however, the number of multiple resonantmodes for the large modified period (δ ¼ 0:5) is morethan that for the small modified period (δ ¼ −0:5).

To examine the energy profiles at resonant modes andthe effect of the number of cycles (NA) of the end seg-ment of the rugate structure, energy density distributionsfor three special resonant modes labeled as I, II, and III inFig. 3(b) against the thickness of the rugate structure areplotted in Fig. 4. When NA ¼ 10, we find that the energydensities within the defect segment for three different re-sonant modes are nearly constant, while within the endsegment of the rugate structure they are decayed expo-nentially. When NA is increased to 20, a similar pattern ofthe three energy density distributions is observed, but theflat peak energy density within the defect segment ismuch higher (near 70-fold) than that of NA ¼ 10.

The multiple resonant modes produced by the designof the 1D rugate structure shown in Fig. 1 because of aperiod jump defect may be understood as follows. Theoptical length of the end segment of the 1D rugate struc-ture in Fig. 1 is NAλ0=2, while it is NBð1þ δÞλ0=2 for theperiod jump defect segment. When certain conditions ofoptical length are satisfied, resonant modes appear. Theblueshift and redshift of the stop band and wavelengthsat resonant modes can be explained by the relationshipbetween the modified period and the design wavelength,i.e., pB ¼ ð1þ δÞλ0=2na ¼ λ�0=2na. When δ < 0, the modi-fied design wavelength, λ�0, is smaller than the original de-sign wavelength, λ0; therefore, a blueshift occurs. Whenδ > 0, a redshift happens because of λ�0 > λ0. Our resultsin Figs. 2 and 3 show that the number, the wavelengths,and the band intervals of the resonant modes are mainlycontrolled by structure parameters δðpBÞ and NB, whilethe intensities are determined by NA. These findings de-monstrate that tunable multiple narrow bandpass opticalfilters can be made by adjusting the structure parametersof the rugate structure.

To understand the pattern of the flat peak energy den-sity distribution in Fig. 4, group velocities in Fig. 5 cor-responding to transmittance in the three middle panels ofFig. 3(b) over wavelength are also evaluated [23]. It indi-cates that almost the same group velocities for NA ¼ 10are found for the three special resonant modes [I, II, andIII in Fig. 3(b)]. For NA ¼ 20, the magnitudes of groupvelocities at these resonant modes become much smallerand are still nearly the same.

Multiple narrow bandpass transmission filters havebeen demonstrated both theoretically and experimen-tally using multilayer photonic structures that are

Fig. 2. Transmittance of five different changes of the modifiedperiod, δ ¼ −0:2, −0:1, 0, 0.1, 0.2. The equal number of cycles ofthe end and defect segments is assumed, NA ¼ NB ¼ 10.

Fig. 3. Transmittance for (a) δ ¼ −0:5 and (b) δ ¼ þ0:5, with fixed NA ¼ 10.

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composed of dual-periodic dielectric materials with de-fect layers in previous studies [8–10]. By increasingthe number of periods of defect layers, multiple transmis-sion resonant modes are produced from those studies. Inthe present study, by introducing a novel period jump de-fect segment into a simple rugate film (Fig. 1), we areable to provide a new way to control the shifting beha-viors of resonant wavelengths (either blueshift or red-shift can be achieved through modifying the period ofthe defect rugate segment) and the number of resonantmodes and band intervals between the neighbor resonantwavelengths. This new feature is particularly useful andconvenient for the design of multiple narrow bandpassfilters. Since we assume that rugate structures shownin Fig. 1 have the same average and perturbation indicesas the optical material (TiO2) used in fabricating the sin-gle narrow bandpass optical filters [15], experimental

fabrication of the proposed multiple narrow bandpassoptical filters using GLAD will be feasible.

In summary, we have presented a theoretical analysisof a design of multiple narrow bandpass filters based on1D rugate structures with a period jump defect. The stopband and resonant wavelengths may be blueshift or red-shift, depending on the change of the period jump. Thenumber, the wavelengths, the band intervals, and the in-tensities of multiple resonance transmission modes aretunable by adjusting structure parameters of 1D rugatestructures. In addition, the high flat peak energy densitiesshown in Fig. 4 may also have potential applications fornonlinear optics. Although only normal incidence is con-sidered for simplicity in the present analysis, it would beinteresting to examine how different angles of incidenceimpact multiple resonant modes. This study provides afirst step to designing multiple narrow bandpass opticalfilters using 1D rugate structures, and more complex de-signs of 1D rugate structures with a combination of phaseshift and period jump defects are necessary to meet var-ious customized needs.

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Fig. 4. Energy density distributions with respect to the physi-cal thickness of the rugate structure at three resonance trans-mission modes labeled as I, II, and III in Fig. 3(b) for NA ¼ 10.The energy density distributions for NA ¼ 20 at the same threeresonant modes are denoted by I′, II′, and III′.

Fig. 5. Group velocities of the three middle panels shown inFig. 3(b).

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