1. Introduction

The optical properties of dielectric multilayer stacks which have quasi-periodic and random structures have recently been studied from the viewpoint of one-dimensional photonic crystals. Among various layered structures, particular attention has been paid to Cantor fractal multilayers because they are found to have some interesting features; they exhibit a self-similar transmission spectrum[1, 2, 3, 4] and a very low group velocity at the edge of the bandgap[5]. The study on the temporal response from a Cantor filter has revealed that the filter can compress an input Gaussian pulse[6, 7]. The Cantor structure has also been applied to a corrugated waveguide[8, 9] and a fiber Bragg grating[10]. In both cases the results show that self-similarity properties can be observed in the transmission spectrum.

Almost all the studies on the spectral and temporal properties of Cantor multilayer and waveguide structures have concerned with transparent and linear materials. Although there are some exceptions, such as nonlinear corrugated waveguides[9] and Cantor multilayers with negative refractive index materials[11], gain properties of Cantor resonators have not been fully analyzed yet.

In a previous study, we have introduced optical gain into fractal multilayers in such a way that gain layers form a Cantor set[12]. In this paper, we study the amplification properties of light transmitted through periodic and Cantor multilayer structures which have the gain layer in the middle of the resonators to make the model of fractal light amplifiers more simple and realistic. We also consider an asymmetric resonator structure which has two multilayers with different unit thicknesses on either side of the gain layer. The density of modes[13] and the optical gain are evaluated, and similarities and differences of their characteristics are discussed.

2. Cantor and asymmetric structures

Figure 1 depicts the structures of multilayers considered in the present study. The triadic Cantor set is used for a fractal multilayer. First, the transparent layer of refractive index n ₁ with optical thickness L is prepared as the initiator. The generator is given by the operation that the central part of the initiator (thickness L/3) is replaced by the transparent layer of refractive index n ₂. By iterating this operation on each of the remaining index n ₁ layers for N times, we obtain a Cantor multilayer of level N. Finally, the central layer is replaced with a gain layer which has a negative imaginary part of the refractive index. A periodic multilayer also has a gain layer in its central part, and has the same total optical thickness as the Cantor multilayer. The periodic multilayer has, therefore, more layers than the Cantor one as shown in Fig. 1. We define the thicknesses l, c, and r of unit layers as the thinnest layer thicknesses of left, central, and right parts of the structures in Fig. 1, respectively. These values are normalized by the wavelength λ of light in the corresponding layers. We use three parameters for the analysis: unit layer thickness l, gain length c, and asymmetric parameter r/l. The gain of transmitted light is calculated using transfer matrices under the assumption that both the passive and active media are non-dispersive. We also neglect gain saturation and any other nonlinear phenomena which could occur in the active media to simplify the amplification phenomena.

 

Fig. 1. Multilayer structures used for the analysis. Refractive indices n ₁, n ₂, and n ₃ used for the analysis are 2.30, 1.45, and 1.45(1-0.0001i), respectively.

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3. Results

In order to elucidate the light amplification properties of Cantor and asymmetric multilayer resonators, we have employed two-dimensional plots of optical gain, which we call gain maps. In the map, the amplitude gain is plotted as functions of two different parameters chosen among the three parameters mentioned in the previous section. First, we examine the resonator which has the symmetric structure with r/l = 1. Figure 2 shows the amplitude gain of transmitted light for Cantor of level 4 and periodic multilayers as functions of the unit layer thickness and the gain length. In Fig. 2, yellow, orange, and red denote that the light is amplified, whereas the light is attenuated in blue regions. Two bandgaps shown as blue stripes around l =0.25 and 0.75 in Figs. 2(a) and (b) correspond to Bragg reflection, which exists independent of the thickness c of the gain layer. In the bandgaps, however, there are some spots where the transmittance is slightly larger than the other areas in the gaps (see blue diagonal lines around l =0.25 in Fig. 2(c)). Such ’leak’ of light occurs when the thickness of gain layer is around integer multiples of λ/2, the condition in which the photonic-defect state appears.

 

Fig. 2. Amplitude gain distributions for (a) Cantor and (b) periodic structures, both of which are symmetric about the central gain layer. (c) and (d) are partially magnified versions of (a) and (b), respectively, showing high gain spots.

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The one-dimensional Cantor structure has multiple bandgaps, the reason of which has been presented elsewhere[4]. This peculiar band structure can be seen in Figs. 2(a) and (c). These figures also illustrate how the bandgap distribution evolves as the thickness of the central layer increases. We can see diagonal branches which cross over bandgaps and appear every λ/2 periods of c. The gain map shows that how the defect of the central layer affects the position of sub-bandgaps located on ether side of the Bragg gap.

It is well known that gain enhancement occurs at the edges of bandgaps for periodic structures with gain. The gain distribution shown in Fig. 2(d) confirms this phenomenon, where high gain (red spots) appears at the band edge of the Bragg gap. This is also true for Cantor multilayers, which have gain peaks at band edges as shown in Fig. 2(c). The number of gain peaks, however, is much larger than that for periodic multilayers. It should be noted that all the bandgaps do not necessarily have high gain spots at their band edges.

The cross-sections of line A and line B in Fig. 2(a) are shown in Figs. 3(a) and (b), respectively. In Fig. 3(a), the gain length is fixed and only the thickness of transparent layers is changed, so that the same distribution appears every half period of l. On the other hand, Fig. 3(b) represents the situation where the gain length c increases proportionally with an increase of l. This means that if the actual thickness of all layers remains constant, the abscissa is proportional to the frequency of incident light. We usually analyze multilayer structures using a spectral representation such as shown in Fig. 3(b), and hence, it is difficult to explain the reason why the gain spectrum distributions become asymmetric about the bandgaps. Our gain maps such as shown in Fig. 2 give some insight into the origin of this asymmetry property. We can obtain high gain at a specific frequency by adjusting the gain length in such a way that line B in Fig. 2(a) overlap the corresponding high gain spot. The maximum amplitude gain for the periodic multilayer is 39.42 (l = 0.7858, c = 0.82) in the ranges of 0 ≤ l ≤ 1 and 0 ≤ c ≤ 10, and that for the Cantor multilayer is 325.6 (l = 0.3310, c = 8.36), much larger than that for the periodic one.

Next, we fix the gain length to unity and examine the effect of asymmetry in the layer thickness of periodic and Cantor multilayers. The results are shown in Fig. 4, where the amplitude gain is plotted against the unit thickness and the asymmetric parameter. We can recognize two groups of bandgaps: one is independent of the asymmetric parameter and the other follows hyperbolic curves. The latter bandgaps exist at $l=\frac{2m-1}{4p}\left(m=\mathrm{1,2},\cdots \right)$ with p = r/l. Considering that the Bragg condition for the left side of transparent multilayers is $l=\frac{2m-1}{4}\left(m=\mathrm{1,2},\cdots \right)$(L bandgaps), the latter bandgaps are the Bragg gap by the right side multilayers (R bandgaps). Figures 4(a) and (b) illustrate that the bandgaps of composite multilayers with different unit thicknesses appear independent of each other.

 

Fig. 3. (a) Cross-section A (c =5) and (b) Cross-section B (c/l =9) in Fig. 2(a).

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Fig. 4. Amplitude gain distributions for (a) Cantor and (b) periodic structures with the gain length of 1. (c) and (d) are partially magnified versions of (a) and (b), respectively. Circles in (c) and (d) denote the position of high gain spots.

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It is an adverse condition for light amplification that the gain length is integer multiples of λ/2. However, we can observe some gain peaks in Figs. 4(c) and (d), the position of which is indicated by circles. The peaks exist only in the regions where the edges of L and R bandgaps coincide. It is interesting to see, however, that when a high gain spot is located at the left side of L bandgap, it is at the right side of R bandgap. The side of the band edges in which gain peaks exist is opposite to each other for L and R bandgaps. This situation never occurs for symmetric multilayers, in which L and R bandgaps coincide. For c = 1, the maximum amplitude gain of the periodic multilayer is 5.45 (l = 0.7143, r/l = 0.40) in the ranges of 0 ≤ l ≤ 1 and 0 ≤ r/l ≤ 10, and that of the Cantor multilayer is 8.92 (l = 0.7365, r/l = 5.11).

It is expected that light is amplified more when it stays longer in the layered structure with gain. To verify this, we calculated the group velocity of light in the medium using a method presented in Ref. [13]. Figure 5 shows the reciprocal of the group velocity v_g (that is, the electromagnetic density of modes) and the amplitude gain as a function of the normalized angular frequency. In the figure, v_g is normalized by the reciprocal of the group velocity v ₀ for the transmitted light without reflections. We can see that the overall profiles of the mode density and the gain are correlated, but there exist some differences, particularly in the peak regions (see Fig. 5(b)). One reason for this is that the light does not necessarily stay long in an amplifying layer even if the ’average’ group velocity is low. Another reason is that even if the intensity of light is high in the amplifying layer, there is a possibility that the light cannot escape efficiently from transparent multilayers. Typical examples are shown in Figs. 6(a) and (b), which illustrate the electromagnetic energy density distributions in the medium at frequencies where the mode density has large values. Figure 6(a) shows the case where both the mode density and the gain are high, whereas Fig. 6(b) depicts the case where the gain is low. It can be seen that the energy distribution is symmetric about the center layer when the gain is high, whereas the distribution becomes asymmetric for low gain conditions. Therefore, to obtain large amplification of light using symmetric Cantor stratified media, we need an efficient resonance over the whole structure as well as a large density of modes.

 

Fig. 5. Amplitude gain (blue curve) and mode density (red curve) plotted against input frequency normalized by the midgap (Bragg) frequency for a Cantor structure of level 4. (a) is partially magnified in (b).

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Fig. 6. Energy density distributions inside a Cantor multilayer structure. (a) ω/ω ₀ = 2.6352. (b) ω/ω ₀ = 2.9008.

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4. Conclusion

It has been shown that the two-dimensional gain representations allow us to analyze systematically the light amplification properties of Cantor and asymmetric multilayers which have complex gain spectra. We have confirmed that light transmitted through a Cantor structure is amplified more than that through a periodic structure. Asymmetric structures can be utilized to obtain light amplification from a relatively low gain medium. Further study is needed to elucidate the relationship among the group velocity, the energy distribution, and the optical gain.