1. Introduction

This paper proposes multilayer-type edge and band rejection wavelength filters that have extremely steep cut-off characteristics.

Wavelength selective filters are key components used in various fields of research and industry of photonic engineering such as image sensing, optical fiber communication and fluorescent spectroscopy. Wavelength filters include band rejection filters that suppress specific wavelength range components, and edge filters that transmit either a shorter-wave or longer-wave portion at a specific cut-off wavelength. These filters often require a higher extinction ratio (ratio of the passband transmission to the stopband transmission) and a narrower transition bandwidth (intermediate wavelength range between the passband and stopband). Such characteristics are particularly important, for example, for channel separation in multichannel optical communication systems, and selective rejection of high-power pumping laser signals in laser Raman spectroscopy.

Dielectric multilayer structures are widely used to realize such sharp wavelength selectivity. A multilayer consisting of a high index layer (refractive index: n₁, thickness: d₁) and low index layer (refractive index: n₂, thickness: d₂) exhibits an optical stopband around the wavelength region expressed by: n₁d₁+n₂d₂=mλ/2, where m is an integer [1,2]. This stopband is also called photonic bandgap (PBG) [3].

Such stopbands are used for band rejection filters, and the boundary between the stopband and the passband is used for edge filters. For the case of normal incidence, it is well known that the spatial decay constant at a mid-gap frequency is proportional to the width of the stopband in the frequency range. Therefore, to widen the stopband is equivalent to increasing the extinction ratio. For a given pair of dielectric materials, the widest and deepest stopband is formed when the layers satisfy the quarter-wave condition, i.e., n₁d₁=n₂d₂=λ/4. In other words, if the materials and the maximum allowed number of layers are fixed, band rejection or edge filters that have higher performance than the above configuration cannot be created by the conventional flat alternating multilayer structure.

In the following sections we demonstrate that by imposing a proper periodic perturbation onto the layer structure, band rejection or edge filters that have extinction ratio and transition bandwidth characteristics exceeding the above limitation can be realized. We focus on two types of perturbations: one is modifying the layer shape from flat to wavy, and the other is to separate one of the layers to form a periodic array of rods.

2. Principle of operation

2.1. Example structure I. Autocloned multilayer

Figure 1 shows a schematic of several examples of dielectric multilayer filters. Figure 1(a) is the conventional flat layers. Multilayer with wavy layer perturbation as shown in Fig. 1(b) can be fabricated by a sputtering-based deposition process called the “Autocloning method” [4,5,6]. Let us designate the substrate surface by the x-y plane and the surface normal direction by z. Refractive index of the conventional multilayer is modulated only along the z direction. On the other hand, in the proposed structure the index changes in two directions (x and z). In this sense, it can be classified into a two-dimensional photonic crystal (PhC).

The sharp-cut characteristics of the above multilayer PhC arise from their deep PBG, and can be explained by the dispersion relation of light. Figure 2 shows the dispersion relation in a one- and two-dimensional empty lattice (the real space lattice with uniform refractive index), respectively. The horizontal and the vertical axes correspond to the Bloch wavenumber (k) and the normalized optical frequency, respectively. Throughout this paper we assume that the x- and z- components of k are an integer multiple of 2 π/a_x and a non-zero complex number, respectively (that means we only deal with the dispersion relation along the Γ-Z direction). Figure 2(a) is simply a folded representation of the dispersion relation of light in free space propagating into the z direction. In Fig. 2(b) the obliquely propagating modes (H) and evanescent modes (D) appear in addition to the vertically propagating modes (indicated by “V”), due to the existence of the horizontal periodicity. Their field profiles are sinusoidal with respect to x. The proposed sharp-cut filter makes use of the coupling of the above three modes.

 

Fig. 1. Example structures of dielectric multilayer filters. (a) Conventional flat layer structure. Refractive index modulation exists only into the z direction. (b) Proposed structure. This kind of wavy multilayer can be fabricated by the Autocloning method. (c) Another example of proposed structures. An array of rods is embedded in a background material.

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Fig. 2. Dispersion relation of light in an empty lattice. (a) One-dimensional lattice. (b) Two-dimensional lattice with the aspect ratio of a_x/a_z =1.1. Black and purple lines indicate the dispersion of the vertically (“V”, field profile is constant along x) and the obliquely propagating modes (“H”, fields have sinusoidal variation along x), respectively. “D” indicates the evanescent modes.

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The fundamental coupling of the lowest order evanescent mode (D₁) and the oblique and vertical modes occurs near the Γ point (as indicated by the dotted circle) when the aspect ratio (a_z/a_x) of the lattice is around 1. The optimum lattice geometry for the practical structure may slightly differ from the square depending on their manner of refractive index modulation. However, the condition for coupling the above modes can be roughly expressed as:

where n_g is an average refractive index for the polarization of interest.

Figure 3 shows the calculated dispersion relations for the actual refractive index profile. Nb₂O₅ (n₁=2.28) and SiO₂ (n₂=1.47) are assumed as constituent materials. The thickness of each film satisfies the relation n₁d₁=n₂d₂. The volume average of the refractive index, which corresponds to n_g for E-polarization (electric field parallel to y), is n_g=(n₁d₁+n₂d₂)/(d₁+d₂)=1.78752. Figure 3(a) is the dispersion curve of light propagating normally in the one-dimensional lattice. Figure 3(b) represents the curves of E-polarized modes in the wavy layer. Figure 3(c) is a magnified view of the “M₂” stopband. In Fig. 3(b) and (c) only the even symmetric modes whose electric field distribution is even with respect to the mirror plane are shown. This is because the odd ones cannot be excited by a perpendicular light incidence. This representation of a dispersion characteristic, sometimes called as a complex photonic band diagram, is useful for discussing the behavior of not only the ordinary propagating but also the spatially-decaying Bloch modes [7, 8]. We used the method described in Ref. [9] to calculate the diagram of the former modes, and the method in Ref. [10] for the latter modes, respectively. Both calculations use the conventional FDTD (Finite-Difference Time-Domain) method based on the Yee’s algorithm [11].

 

Fig. 3. Complex photonic band diagram of the multilayer for E-polarization (E parallel to y). Consistuent film materials are Nb₂O₅ (n₁=2.28) and SiO₂ (n₂=1.47). Film thicknesses satisfy n₁d₁=n₂d₂. (a) Flat multilayer. “N” denotes the stopbands (PBG of Normal layers). (b) Wavy multilayer (“M” denotes the PBG of modified layers). Aspect ratio is a_x/a_z=1.05. (c) Magnified view of the “M₂” stopband of (b). α_max represents the maximum decay constant. F_3dB and F_5dB refer to the normalized frequency at which the intensity of the field decays at a rate of 3dB/lattice and 5dB/lattice, respectively. F_c is the low-frequency edge of the stopband.

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Black and blue parts denote Re(k) (k is the complex Bloch wavenumber) of the propagating and decaying Bloch modes, respectively. Re(k) refers to the propagation constant of the ordinary meaning (phase shift per a film pair). Red lines represent the Im(k) of the decaying Bloch modes, which corresponds to the rate of evanescent decay [12]. As shown in the figures, several PBGs open as an effect of the finite index difference. In the quarter-wave stack of flat layers, PBGs open where the normalized frequency is an odd multiple of 1/2n_g [2] and their depth is proportional to the PBG width. N₁ in Fig. 3(a) denotes the lowest stopband.

On the other hand, in the wavy structure the coupling of obliquely and vertically propagating modes opens additional PBGs as M₂, M₃, and M₄ [5, 13]. Particularly, the decay characteristic at the low frequency side of the M₂ PBG is extremely sharp as it inherits the steep decay property of the D₁ mode in Fig. 2(b). The proposed wavelength filters are designed to make use of such steep decay curves.

Note that three-dimensional periodic structures can be created by adding similar periodic perturbation in the y direction in Fig. 1(b) [14]. Although such structures are expected to exhibit deep stopbands due to the similar mode coupling, their widths are generally much narrower than the one- and two-dimensional PhCs. For this reason, in this study we do not deal with three dimensional ones.

The major drawback of employing the proposed kinds of structure is the polarization dependence. If the state of polarization of the signal light is unknown, we are forced to lose unwanted polarization component. However, this dependence might not be serious for the class of applications in which the polarization state of the signal is always fixed.

The decay characteristics in the wavy layer depend on the aspect ratio (a_x/a_z) of lattice. In the actual autocloning fabrication process, the slope of the wavy corrugation remains almost constant regardless of the aspect ratio [15]. We therefore numerically investigated the relation between the aspect ratio and the maximum decay constant (α_max) of the M₂ PBG while fixing the slope at a typical angle of 45 degrees. The result of calculation is shown in Fig. 4(a). The dashed line shows the maximum decay constant of a flat multilayer. As shown, the decay constant exceeds the unmodified flat structure over a wide range of the aspect ratio. The maximum possible decay rate for this material system is achieved at a_x/a_z ≈ 1.03 and is found to be about 12 dB per lattice in intensity, which is about three times larger than the flat layers.

Another important property as an edge filter is the steepness concerning the passband-stopband transition. As can be seen in Fig. 3(b), the decay curve of M₂ can become sharper on the low-frequency side.

 

Fig. 4. Calculated performances of the filters as a function of the aspect ratio of the unit cell. (a) Maximum decay constant (α_max) in the “M₂” PBG. The dashed line denotes the decay constant of the “N₁” PBG in the flat layer. (b) Bandwidth of the passband-stopband transition. For the definition of Δ, see Eq. (2) and Eq. (3).

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We tried to define a measure of the “steepness” at the cut-off by the bandwidth between the lower band edge frequency (F_c in Fig. 3(c)) and the frequency at which the decay constant reaches 3dB/lattice or 5dB/lattice in intensity (F_3dB and F_5dB in the same figure). The relation between the aspect ratio of the unit cell and the steepness factor is summarized in Fig. 4(b). The vertical axis denotes the relative transition bandwidth Δ defined by

where

The transition bandwidth of the displayed range of the aspect ratio was found always to be narrower than that of the flat layers. Especially, the transition width falls almost to zero at a_x/a_z = 1.25 . This means that the wavy structure of such aspect ratio exhibits ideal step-like transmission characteristics.

 

Fig. 5. Calculated transmission spectra (red lines). The number of layers is 20 (10 periods). (a) Flat multilayer. The stopband corresponds to the “N₁” PBG in Fig.3(a). Blue line shows expected transmittance, obtained by multiplying the decay constant (Im(k) of the complex dispersion relation) by the number of periods. (b) Wavy structure of a_x/a_z=1.03 (maximum decay configuration). The stopband corresponds to “M₂” in Fig. 3(b). Blue lines indicate the expected intensity suppression obtained by the decay constant (Im(k) of all the decaying modes) in the complex dispersion relation. (c) Wavy structure of a_x/a_z=1 .25 (steepest cut-off). The stopband corresponds to “M₂” in Fig. 3(b). Blue line indicates the expected intensity suppression obtained by the decay constant (Im(k) of all the decaying modes) in the complex dispersion relation.

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We calculated the transmission spectra of finite numbers of layers for the above two extreme cases: (a) a_x/a_z = 1 03 (maximum α_max) and (b) a_x/a_z = 1.25 (narrowest transition bandwidth), using the FDTD method [16]. Results are shown in Fig. 5 together with that of the flat multilayer. The number of layers were 20 (10 film pairs) and the assembly was assumed to be stacked on a silica substrate (n=1.45). The red curves represent the transmission spectrum. The blue ones are the theoretical decay rate, obtained by multiplying Im(k) of the dispersion relation in an infinitely extending PhC shown in Fig. 3 by the number of film pairs, and displayed on the dB scale.

The envelope of the spectra in the PBGs agrees well with the decay curves from the dispersion relation. The maximum decay of more than 100 dB is expected in (b). Step-like drop of transmission down to ~ -60dB at the low-frequency edge is seen in Fig. 5(c). Note that the comb-like dips seen on the bottom of the PBGs are unique to multi-dimensional structure, and their origin can be explained by a Fabry-Perot interference effect of decaying Bloch modes [13].

2.2. Example structure II. An array of rods embedded in a background material

Figure 1(c) shows another example of the layer modification. A periodic array of high index rods is embedded in a low index background in square lattice geometry. This kind of structure can be created by, for example, modifying the method for fabricating three-dimensional PhCs for near-infrared wavelengths, described in Ref. [17] and Ref. [18]; i.e., separating the individual rods by lithography and dry etching after the deposition of high index film, covering them with low index material and polishing its surface, and repeating them for a specified number of periods.

This type of PhC can be also regarded as a version of the laterally index-modulated alternating layers; i.e., a part of the high index layer is periodically replaced by the low index medium. To explain the nature of the dispersion curves that the embedded-rod structure inherits from the original flat multilayer, dispersion relations for various degree of modulation are calculated. He we assumed silicon (n₁=3.5) and silica (n₂=1.47) as sample materials. The results are displayed in Fig. 6. First, Fig. 6(a) represents the calculated complex photonic band diagram of a flat multilayer (no lateral index modulation). Note that virtual lateral periodicity is assumed for the convenience of the following explanation. Due to this horizontal periodic boundary condition, obliquely propagating modes with respect to the layers (indicated by “h1” and “h2” in the figure) are allowed to exist in addition to the ordinary vertically propagating ones (indicated by “v”). The obliquely propagating modes have 2Nπ’s phase shift (N: integer) per every a_x in the x-direction. In the flat layer structure, they do not couple each other because no diffraction occurs at layer interfaces.

Figure 6(b) shows a calculated complex photonic band diagram of the E-polarized modes (electric field parallel to y) in the modified structure, in which the refractive index of half of the high index film is slightly decreased (n₁=3.5→2.8). The pitch of the horizontal modification (a_x) is set the same value as a_z. The substantial refractive index modulation causes the coupling between the obliquely and vertically propagating modes. As a result of that, anti-crossings as indicated by circles on the figure occur and new PBGs as “M₂” open.

Figure 6(c) is a dispersion relation corresponding to the proposed “embedded-rod” structure, in which half of the silicon film is now completely replaced by silica. It is clearly seen that the “M₂” PBG that we are trying to utilize its sharp-cut characteristic, has the same origin as the anti-crossing as shown in Fig. 6(b).

As can be seen, the Γ point decaying curves of the similar shape as the autocloning structure appear (indicated by the thick arrow). Transmission spectrum of a finite number of layers of this type is calculated and displayed in Fig. 7. The number of layers were 16 (8 film pairs) and the structure was assumed to be stacked on a silica substrate. In the figure the transmittance of a quarter wave stack (n₁d₁=n₂d₂) of silicon/silica flat layers is also displayed as a dashed line, to see the maximum achievable PBG depth for this material system. The bottom of the PBG of the proposed structure is deeper than -125dB, which is more than twice deeper than that of flat layers.

 

Fig. 6. Calculated complex dispersion diagram of the embedded-rod type PhC, for various degrees of horizontal refractive index modification. (a) Flat multilayer consisting of silicon(n=3.5) and silica(n=1.47). No horizontal index modulation exists. “v” and “h” indicate the vertically and obliquely propagating modes with respect to the layers. As a virtual lateral periodicity is assumed, oblique modes are allowed to appear in this presentation of dispersion diagram. M₁,M₃ and M₄ denote the decay constants of decaying modes. (b) Structure with a slight index modulation. Half of the silicon layer is replaced by another material with n=2.8. Dispersion diagram is for the even symmetric E-polarized modes. Dashed circles in the diagram denote the anti-crossings caused as a result of the coupling of the “v” and “h” modes. (c) Proposed embedded-rods type PhC. Half of the silicon layers are completely replaced by silica. The decaying mode indicated by a thick arrow is useful for the sharp cut-off filtering function.

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One of the practical configurations of this type of PhC filter is to integrate multiple filter elements having different wavelength characteristics on a common substrate [5,6]. Such multi-channel filters will be useful for, for example, spectroscopy instruments and telecommunication devices [14]. To create multiple filters on a common substrate by a single fabrication process, it will be reasonable to change the horizontal spacing between the high-index rods for each filter region, while keeping the thickness of high and low index layers and the width of the rods constant, from a viewpoint of the simplicity of the fabrication procedure. Fig. 8 summarizes calculated filter performances under such condition (width, height of the rods and the thickness of the low index layer: constant, horizontal lattice constant: varied).

The solid and dotted lines show the maximum decay constant in the PBG, and the transition bandwidth (Δ_3dB) as a function the aspect ratio of the unit cell.

The width of the rods was determined so that it is just the half of the horizontal pitch if the unit cell is square (a_x=a_z). The thickness of the rods and the low index layer was fixed as the same ratio as the previous example (Fig. 6).

The maximum available decay rate in the middle of the PBG of the Si/SiO₂ quarter-wave stack of conventional flat layers is also plotted on Fig. 8 as a dashed line. To compare both performances, it is clarified that the decay constant of the proposed structure exceeded the flat one for the wide range of the aspect ratio (0.8<a_x/a_z<1.6). In addition, the transition bandwidth at the cut-off wavelength becomes minimum at a_x/a_z~1.25.

The similar performance curves will be expected for various rod dimensions.

 

Fig. 7. Calculated transmission spectra for E-polarization of the silicon rods/silica background structure (red line). The total number of layers is 16 (8 periods). Structural parameters are the same as Fig. 6. Dashed line represents the transmittance of a quarter wave stack of silicon/silica flat layers.

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Fig. 8. Calculated performances of the filters as a function of the aspect ratio (a_x/a_z) of the unit cell. Solid line: maximum decay constant (α_max) in the “M₂” PBG. Dashed line: maximum decay constant in the first PBG of a quarter-wave stack of a Si/SiO₂ flat multilayer. Dotted line: bandwidth of the passband-stopband transition. Δ_3dB is defined by Eq. (2).

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3. Conclusion

We proposed a novel configuration for enabling sharp cut-off characteristics in dielectric multilayer wavelength filters and demonstrated possible performances through numerical simulation. Two types of refractive index modulation, wavy autocloning layers and embedded high index rods, were considered.

The results shown here provide an insight to design and manufacture optical filters with fewer layers for a given specification. This will help us reduce the process time and the inner stress of the film, and then finally increase the product yield. The proposed class of filters will be applicable to a wide range of fields of research and industry where the state of polarization of signal light is predetermined. Fabrication and experimental verification of the filters will be conducted as our next study in the near future.