Introduction

Slowing electromagnetic waves is believed to be an attractive technique for enhanced nonlinear optics1, light harvesting2,3 and optical signal processing4,5,6. Recently, the concept of a “trapped rainbow” has generated considerable interest for potential use in optical data storage and processing7. It aims to trap different frequency components of the wave packet at different positions in space permanently7. Various waveguide structures are proposed to achieve the trapped rainbow effect, such as tapered waveguides made by negative or hyperbolic metamaterials7,8,9, tapered plasmonic waveguides10,11 and tapered photonic crystal waveguides12. Some experiments have also been conducted13,14,15. These structures can slow down the incident wave around a critical position where the group velocity of the wave is zero16,17. However, all these structures cannot truly achieve the trapped rainbow effect. As first discovered in ref. 18, because of the strong coupling between the forward and backward modes near the critical position, the entire incident wave will be reflected back before reaching the critical position and the wave is not completely standstill in these waveguides18. There are other types of structures supposed to completely trapping electromagnetic waves without reflection such as “electromagnetic black holes”19,20 and magneto-plasmonic waveguides with a block at one end21,22. However, these structures are unable to distinguish between different frequency components, because the wave is absorbed or blocked in one singular point for the whole frequency range, rather than trapped at different positions.

In this article, for the first time in the literature, we found a physical mechanism to achieve a truly “trapped rainbow” storage of electromagnetic wave by simultaneously overcoming these two difficulties, namely, suppressing the reflection of the incident wave and distinguishing different frequency components. We utilize a nonreciprocal waveguide under a tapered external magnetic field to achieve this and such a trapping effect is stable even under fabrication disorders. We also observe hot spots and relatively long duration time of the trapped wave around critical positions through frequency domain and time domain simulations with loss taken into consideration. In addition, we investigate the influences of the loss and the applied magnetic field gradient on the trapping effect.

Physical mechanism

First, we demonstrate the physical mechanism to achieve trapped rainbow effect by studying the dispersion diagram of the nonreciprocal waveguide and the effect of the applied magnetic field. The waveguide structure is a two-dimensional (2D) slab (infinite in z direction) with three layers (Fig. 1). The top layer is a perfect electric conductor (PEC), which can be a good metallic conductor such as copper in microwave frequencies. The middle layer is a dielectric layer with thickness d and permittivity εd. The bottom layer is a gyromagnetic material such as yttrium–iron–garnet (YIG) in microwave frequencies. The permittivity and permeability of the YIG are denoted by εm and μr. An external magnetic field H0 is applied on the YIG along the +z direction. In this configuration, μr takes the form23

Figure 1
figure 1

Waveguide structure.

The waveguide structure is a slab with three layers. The top layer is a perfect electric conductor (PEC). The middle layer is a dielectric layer with thickness d and permittivity εd. The bottom layer is an yttrium–iron–garnet (YIG) layer, which is gyromagnetic material in microwave frequencies. The permittivity and permeability of the YIG are denoted by εm and μr. An external magnetic field H0 is applied on the YIG along the +z direction. The wave of transverse electric (TE) mode (Ez, Hx, Hy) goes along x direction.

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with μ1 = 1 + (ω0 − iν)ωm/[(ω0 − iν)2 − ω2] and μ2 = ωωm/[(ω0 − iν)2 − ω2], where ωo = 2πγH0 is the precession frequency, ν = πγΔH indicating that the material loss is determined by the resonance linewidth ΔH, ωm = 2πγMs is determined by the saturation magnetization Ms and γ = 2.8E6 rad s−1G−1 is the gyromagnetic ratio. Within the bandgap of the bulk modes in the YIG, we only consider the surface wave of transverse electric (TE) mode (Ez, Hx, Hy) with wave vector k along x in the waveguide24. The dispersion relation ω(k) for this mode is governed by

with , , and k0 = ω/c (where c is the light speed in vacuum)25. The linear term (μ2/μ1)k with respect to k in equation (2), which originates from the off-diagonal element of μr, breaks the symmetry of dispersion relation (i.e., ω(k) ≠ ω(−k)). The propagation of the wave in this waveguide is nonreciprocal. In addition, the dispersion relation is also affected by the thickness of the dielectric layer d25. Therefore, we study two lossless cases (ΔH = 0) with very different d. In the first case, d = 0.13λm and the other, d = 0.013λm, where λm = 2π/km and km = ωm/c. The other parameters are chosen as follows: εd = 1, εm = 15 and Ms = 1780 G.

For the first case in which d = 0.13λm, ω(k) changes with H0. Figure 2a shows three curves of ω(k) with H0 = 0.5, 0.6 and 0.7 Ms, respectively. For each curve, there are two frequencies whose group velocity . One is at the asymptotic frequency when k → −∞ and the other one is at a lower frequency ωL. When ω > ω−∞, group velocity vg points in only one direction and the waveguide is a one-way waveguide. Since ω(k) can be tuned by H0, we can achieve vg = 0 by controlling H0. Figure 2b shows three relation curves of k(H0) for ω = ωm, 1.1ωm, 1.2ωm, respectively. For each curve, there are two critical magnetic fields, Hc1 and Hc2, where vg = 0. Here, Hc1 = (ω/ωm − 1/2)Ms and Hc2 > Hc1. We analyze the curve with ω = 1.2ωm as an example. For H0 < Hc1, the wave has only one finite k and propagates in only one direction with vg > 0 (① in Fig. 2b). For Hc1 < H0 < Hc2, the waveguide has two modes with opposite group velocity directions. For H0 > Hc2, the waveguide has no propagating mode. The property of the waveguide around Hc1 and Hc2 is very different. For , the two corresponding modes are separated. One has finite k with vg > 0 (② in Fig. 2b) and the other one has k → −∞ with vg → 0− (⑤ in Fig. 2b). These two modes cannot couple with each other. For , the two corresponding modes are nearly degenerated (③ and ④ in Fig. 2b). Although the group velocities of these two modes are opposite, the wave vectors and the modal fields are very close and consequently these two modes can couple with each other.

Figure 2
figure 2

Physical mechanism of trapped rainbow in nonreciprocal waveguide.

(a) Dispersion relation for d = 0.13λm, H0 = 0.5 Ms (black), 0.6 Ms (red) and 0.7 Ms (blue). (b) Relation curve of k(H0) for d = 0.13λm, ω = ωm (black), 1.1ωm (red) and 1.2ωm (blue). Hc1 and Hc2 indicate the critical fields for ω = 1.2ωm. ➀ ➁ ➂ ➃ ➄ denote five positions (white dots) on the relation curve for ω = 1.2ωm. At ➀ H0 < Hc1, at ➁ and ➄ and at ➂ and ➃ . (c) Schematic diagram for vg (yellow wide arrow) and k (blue thin arrow) at corresponding positions marked in (b). H0(x) continuously increases with x. xc1 and xc2 denote the corresponding critical positions for Hc1 and Hc2. ➀ denotes the incident wave at x < xc1. The yellow arrow with a red cross denotes that the wave cannot propagate in −x direction at x < xc1. The green straight arrow with a red cross denotes no coupling between the waves at ➁ and ➄. The green curved arrow denotes coupling between the waves at ➂ and ➃. (d) Dispersion relation for d = 0.013λm, H0 = 0.5 Ms (black), 0.6 Ms (red) and 0.7 Ms (blue). (e) Relation curve of k(H0) for d = 0.013λm, ω = ωm (black), 1.1ωm (red) and 1.2ωm (blue). Hc indicates the critical field for ω = 1.2ωm. ➀ ➁ ➂ denote three positions (white dots) on the relation curve for ω = 1.2ωm. (f) Schematic diagram for vg (yellow wide arrow) and k (blue thin arrow) at the corresponding positions marked in (e). H0(x) continuously increases with x. xc denotes the corresponding critical position for Hc. The yellow wide arrow with a red cross denotes that the wave cannot propagate in −x direction. The other parameters are as follows: εd = 1, εm = 15 and Ms = 1780 G.

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Next, we use a tapered H0(x) to show the physical mechanism for realizing the trapped rainbow in this waveguide (Fig. 2c). Firstly, we assume H0(x) continuously increases with x. xc1 and xc2 are the corresponding critical positions for Hc1 and Hc2 at one certain frequency. The wave is incident at x < xc1 along +x direction and it passes through xc1 in a unidirectional manner (① in Fig. 2c). The wave continues to propagate without coupling (② in Fig. 2c). When the wave approaches xc2 (③ in Fig. 2c), the wave can couple with the corresponding nearly degenerated mode (④ in Fig. 2c). In fact, all the electromagnetic energy can couple back to the −x direction before reaching xc218. The wave then travels along the −x direction and approaches xc1 where k → −∞ and vg → 0− (⑤ in Fig. 2c). There is no coupling between ② and ⑤ in Fig. 2c and the wave cannot penetrate xc1, so the wave is trapped at xc1. Secondly, we consider some disorders in the waveguide such as surface roughness or a non-homogeneous material. The disorders at x < xc1 will not generate backscattering, since the waveguide is a one-way waveguide in this region25. The disorders at xc1 < x < xc2 can generate a scattering wave. The scattering wave traveling in the −x direction will approach xc1 and be trapped there, while the scattering wave traveling in the +x direction will couple back to the −x direction and still become trapped at xc1. Note that, at the frequencies within the bandgap of the bulk mode in YIG, the scattering cannot propagate inside the YIG layer. Thirdly, the critical positions are different for different frequencies (Fig. 2b), so the different frequency components of a wave packet can be trapped at different positions. To summarize, this waveguide can cage the wave between the two critical positions and achieve the trapped rainbow effect. It is worth to note that the physical mechanism shown in Fig. 2c is different from those of the previously proposed reciprocal structures7,8,9,10,11,12,13,14,15. Firstly, this nonreciprocal structure has a one-way region (x < xc1) which can prevent the wave propagating in −x direction. This unique property (different from previously proposed structures) is essential for achieving a truly trapped rainbow effect. Secondly, it has two critical positions xc1 and xc2, where the group velocity is zero, so that it can achieve two hot spots. Such a case has never been discussed before. Thirdly, the previous reciprocal structures have symmetric dispersion relations. As a result, the previous work cannot suppress the scattering caused by disorders. Our structure has non-symmetric dispersion relations and it is robust against the disorders.

For the second case in which d = 0.013λm, ω(k) also changes with H0. Figure 2d shows three curves of ω(k) with H0 = 0.5, 0.6 and 0.7 Ms, respectively. For each curve, group velocity vg points in only one direction and at the asymptotic frequency when k → −∞. Figure 2e shows the relation curves of k(H0) for ω = ωm, 1.1ωm and 1.2ωm, respectively. The critical magnetic field is Hc = (ω/ωm − 1/2)Ms, where vg = 0. We analyze the curve with ω = 1.2ωm as an example. For H0 < Hc, the wave has only one finite k and propagates in only one direction with vg > 0 (① and ② in Fig. 2e). When , the wave has k → −∞ and vg → 0+ (③ Fig. 2e). For H0 > Hc, there is no propagating mode. We next use a continuously increasing H0(x) with x to achieve the trapped rainbow effect. The physical mechanism is quite simple in this case: Set xc as the corresponding critical position for Hc at one certain frequency. The wave is incident at x < xc along +x direction (① in Fig. 2f). Since there is no mode propagating in the −x direction, the coupling does not exist. Disorders cannot generate backscattering waves in this one-way waveguide or propagating waves in the YIG layer. Therefore, the wave continues to propagate along +x with decreasing vg (② in Fig. 2f). Finally, the wave approaches xc with vg → 0+. Since it cannot penetrate xc, the wave is trapped at xc (③ in Fig. 2f). As critical position xc is related to the frequency, different frequency components of a wave packet can be trapped at different positions. Therefore, this waveguide can achieve the trapped rainbow effect by trapping the wave at the critical position.

Trapped rainbow

We simulate the trapped rainbow effect in the nonreciprocal waveguide structures. Both frequency domain and time domain simulations for two cases d = 0.013λm and d = 0.13λm are conducted with taking loss (ΔH ≠ 0) into consideration. According to the analysis above, a continuously increasing H0(x) with x can trap the wave. We choose H0(x) changing linearly with x:

where α indicates the increasing rate of H0.

For the first case d = 0.13λm, we initially conduct the frequency domain simulation by the finite element method (COMSOL). The parameters are chosen as follows: α = 0.1 and ΔH = 2 Oe. The matched field is excited at x = 0. The amplitude Ez distributions for ω = ωm, 1.1ωm and 1.2ωm (Fig. 3a–c) show that the field is well confined at the dielectric-YIG interface (y = 0) and enhanced in a region between xc1 and xc2. Here xc1 = λm, 2λm and 3λm and xc2 = 2.03λm, 2.99λm and 3.95λm for ω = ωm, 1.1ωm and 1.2ωm, respectively. Figure 3d shows the normalized amplitude of Ez along the interface. The amplitude has no ripples on the left side of xc1, since the wave propagates only one-way in this region. On the right side of xc1, the ripples of the amplitude demonstrate an interface between the waves propagating in ±x direction. The wave propagating in the −x direction comes from the coupling when the wave in the +x direction approaches xc2. In addition, when the wave approaches xc2, the group velocity is slower, so the field is also enhanced in this process. Therefore, around these two positions, we can observe hot spots. Between these two positions, the intensity of the field oscillates and has several peaks.

Figure 3
figure 3

Simulated trapped rainbow for d = 0.13λm.

In the frequency domain, matched fields are excited at x = 0 in the waveguide with α = 0.1 and ΔH = 2 Oe. The distribution of the Ez amplitude is plotted for (a) ω = ωm, (b) 1.1ωm and (c) 1.2ωm, respectively. (d) Ez amplitude along the dielectric-YIG interface for ω = ωm (black), 1.1ωm (red) and 1.2ωm (blue). In the time domain, a Gaussian wave packet with center frequency ωc = 1.1ωm is injected into the waveguide with α = 0.1 and ΔH = 2 Oe. The Ez amplitude distribution of the wave packet is plotted at (e) 0 ns, (f) 4 ns, (g) 9 ns, (h) 13 ns and (i) 25 ns, respectively. (j) Normalized distribution of the Ez amplitude along the dielectric-YIG interface at different time. (k) Time evolution of the Ez amplitude at x = 0.50λm (blue), 2.87λm (green), 2.58λm (red) and 2.00λm (black). The other parameters are εd = 1, εm = 15 and Ms = 1780 G.

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We then study the propagation of a wave packet in the time domain for case d = 0.13λm (see Supplementary Information for details of the calculation method). A Gaussian wave packet with center frequency ωc = 1.1ωm is injected into the waveguide with α = 0.1 and ΔH = 2 Oe. Figure 3e–i shows the Ez amplitude distribution of the wave packet at different times (see also Supplementary Movie 1). Figure 3j shows the nomalized distribution of Ez amplitude along the dielectric-YIG interface at different times. They show that the wave packet initially propagates in +x direction and is compressed and enhanced when approaching xc2 = 2.99λm. The wave then goes back and antenuates gradually at xc1 = 2λm. Figure 3k shows the time evolution of the Ez amplitude at four positions: x = 0.50λm (blue), 2.87λm (green), 2.58λm (red) and 2.00λm (black) on the interface. Note that between xc1 and xc2, such as x = 2.58λm, the field can have two peaks in the time domain.

For the second case d = 0.013λm, we first simulate the propagation of the wave in the frequency domain. The parameters are chosen as follows: α = 0.2 and ∆H = 1 Oe. The matched field is excited at x = 0. The amplitude Ez distributions for ω = ωm, 1.1ωm and 1.2ωm (Fig. 4a–c) show that the field is well confined at the dielectric-YIG interface (y = 0) and enhanced around the corresponding critical positions xc = 0.5λm, 1.0λm and 1.5λm, respectively. Figure 4d shows the normalized amplitude of Ez along the interface. The amplitude increases monotonically from x = 0 to the critical positions, which demonstrates that the wave is trapped and extremely enhanced at the critical positions.

Figure 4
figure 4

Simulated trapped rainbow for d = 0.013λm.

In the frequency domain, matched fields are excited at x = 0 in the waveguide with α = 0.2 and ΔH = 1 Oe. The distribution of the Ez amplitude is plotted for (a) ω = ωm, (b) 1.1ωm and (c) 1.2ωm, respectively. (d), Ez amplitude along the dielectric-YIG interface for ω = ωm (black), 1.1ωm (red) and 1.2ωm (blue). In the time domain, a Gaussian wave packet with center frequency ωc = 1.1ωm is injected into the waveguide with α = 0.2 and ΔH = 3 Oe. The Ez amplitude distribution of the wave packet is plotted at (e) 2 ns, (f) 11 ns, (g) 20 ns, (h) 29 ns and (i) 38 ns, respectively. (j) Normalized distribution of the Ez amplitude along the dielectric-YIG interface at different times. (k) Time evolution of the Ez amplitude at x = 0.500λm (blue), 0.935λm (green), 1.000λm (red) and 1.050λm (black). (l) Duration time Δt for three cases: (1) α = 0.2 and ΔH = 3 Oe (black solid line), (2) α = 0.2 and ΔH = 2 Oe (red dot line) and (3) α = 0.3 and ΔH = 3 Oe (blue dot-dash line). The other parameters are as follows: εd = 1, εm = 15 and Ms = 1780 G.

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Next, we study the propagation of a wave packet in the time domain for d = 0.013λm. A Gaussian wave packet with center frequency ωc = 1.1ωm is injected into the waveguide with α = 0.2 and ∆H = 3 Oe. Figure 4e–i show the Ez amplitude distributions of the wave packet at different times (see also Supplementary Movie 2). Figure 4j shows the nomorlized distribution of Ez amplitude along the dielectric-YIG interface at different times. They show that the wave packet is compressed, enhanced and trapped around the critical position and then antenuates gradually. Figure 4k shows the time evolution of the Ez amplitude at x = 0.500λm (blue), 0.935λm (green), 1.000λm (red) and 1.050λm (black). We notice that the amplitude of the wave packet achieves the maximum value before it reaches the critical position. This means that, although slower group velocity can enhance the field, when the wave packet approaches the critical position, the loss increases substantially and the intensity of the wave decreases. We also notice that the wave packet can remain at the critical position in for a relatively long time. Define the duration time Δt as the time period when the amplitude of Ez is greater than half of the maximum value. Figure 5l shows Δt for three cases: (1) α = 0.2, ΔH = 3 Oe; (2) α = 0.2, ΔH = 2 Oe and (3) α = 0.3, ΔH = 3 Oe. It demonstrates that Δt has maximum value at the critical position. In addition, lower ΔH gives less loss and contributes to longer duration and α seems not to affect the duration time.

Figure 5
figure 5

Trapped rainbow effect under disorders in the nonreciprocal waveguides.

For d = 0.13λm, two dielectric slabs (white slabs) with permittivity ε = 10, width Δx = λm/20 and height Δy = λm/20 are inserted at x = λm and x = 1.5λm through the dielectric-YIG interface. The other parameters are the same as those in Fig. 3b. (a) Distribution of the Ez amplitude. (b) Ez amplitude along the dielectric-YIG interface. For d = 0.013λm, one dielectric slabs (white slabs) with permittivity ε = 10, width Δx = λm/50 and height Δy = λm/100 is inserted at x = 0.5λm through the dielectric-YIG interface. The other parameters are the same as those in Fig. 4b (c) Distribution of the Ez amplitude. (d) Ez amplitude along the dielectric-YIG interface.

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We next investigate the trapped rainbow effect in the waveguides with disorders. For d = 0.13λm, two dielectric slabs (white slabs in Fig. 5a) with permittivity ε = 10, width Δx = λm/20 and height Δy = λm/20 are inserted at x = λm and x = 1.5λm through the dielectric-YIG interface. The other parameters are the same as those in Fig. 3b. The 2D distribution of Ez (Fig. 5a) and the amplitude Ez along the dielectric-YIG interface (Fig. 5b) are almost identical to the results in Fig. 3b,d. This demonstrates that the waveguide can preserve the trapped rainbow effect under disorders. For d = 0.013λm, one dielectric slab (white slab in Fig. 5c) with permittivity ε = 10, width Δx = λm/50 and height Δy = λm/100 is inserted at x = 0.5λm through the dielectric-YIG interface. The other parameters are the same as those in Fig. 4b. The 2D distribution of Ez (Fig. 5c) and the amplitude Ez along the dielectric-YIG interface (Fig. 5d) are almost identical to the results in Fig. 4b,d. This demonstrates that this waveguide can also preserve the trapped rainbow effect under disorders. Compared with other slow-light structures, in which the disorder can generates reflection and destroys the trapped rainbow effect (see Supplementary Information), the nonreciprocal waveguides are immune to the disorders, which is an important advantage for achieving the trapped rainbow effect.

Finally, to get much stronger enhancement of the electromagnetic field, we investigate the influence of the increasing rate α and the loss on the enhancement at ω = 1.1ωm for d = 0.013λm in the frequency domain. The loss is mainly from the bottom layer of the magnetic material YIG. This loss is determined by ΔH. The metal loss of a good conductor such as copper in the upper layer can be included by using an impedance boundary condition in the modeling. The results (not presented here) show that the metal loss has quite small influence on the enhancement and thus we can treat the upper metallic layer as PEC in our modeling. We define an enhancement factor (EF) as ratio of the amplitude of Ez at the critical position to the amplitude at x = 0 on the dielectric-YIG interface. We sweep α from 0.13 to 0.40 and sweep ΔH from 1 Oe to 10 Oe to find the corresponding EF (Fig. 6a). Figure 6b shows that, for a fix α = 0.2, EF decreases as ΔH increases, because larger ΔH contributes more loss. Figure 6c shows that, for a fix ΔH = 3 Oe, EF increases with α. This is because in this one-way waveguide, the more rapidly increasing rate will contribute a shorter way to reach the critical position with less loss and higher EF, rather than generate reflection. Figure 6a also shows the contour lines for EF = 1 and EF = 10. To summarize, smaller ΔH and larger α can contribute to a higher EF.

Figure 6
figure 6

The influence of α and ΔH on the enhancement factor (EF).

(a) EF changes with α from 0.13 to 0.4 and ΔH from 1 Oe to 10 Oe. The white solid (dashed) line is the contour line for EF = 1 (EF = 10). (b) EF changes with ΔH for a fix α = 0.2, corresponding to the vertical black dash line in (a). (c) EF changes with α for a fix ΔH, corresponding to horizontal black dash line in (a). The other parameters are as follows: ω = 1.1ωm, d = 0.013λm, εd = 1 and εm = 15, Ms = 1780 G.

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In conclusion, we found a physical mechanism to achieve a truly “trapped rainbow” storage of waves by utilizing nonreciprocal waveguides. Two nonreciprocal waveguides with different thicknesses under tapered applied magnetic fields are investigated via frequency domain and time domain simulations. Results demonstrate that both can achieve the trapped rainbow effect even under disorders. One can cage the wave between two critical positions and the other can trap the wave at one critical position. The field can be enhanced at the trapped positions. In addition, low loss is essential to achieve strong enhancement of the field and long duration time and a more rapidly increasing rate of the tapered external magnetic field can produce stronger enhancement of the field. To verify the trapped rainbow effect in future microwave experiments, a very low loss YIG film is necessary and the tapered external magnetic field should have high gradients. Further research may make use of the physical mechanism proposed here to investigate light wave or acoustic wave trapping26. Applications ranging from wave harvesting and storage to nonlinearity enhancement might also benefit from the physical mechanism we suggest.

Additional Information

How to cite this article: Liu, K. and He, S. Truly trapped rainbow by utilizing nonreciprocal waveguides. Sci. Rep. 6, 30206; doi: 10.1038/srep30206 (2016).