1 Introduction

A recent nomenclature adapted in the literature is “rainbow trapping” [1]. The meaning of that term refers to electromagnetic wave localization at spatial positions (occurring particularly in the visible spectrum, and hence the name rainbow). Optical pulse trapping by various mechanisms has been actively researched in photonics, because it has innumerable potential applications in optical signal processing devices such as optical buffers/memories, data processors, and filters. The enhanced light–matter interaction may also provide unique advantages to other fields of photonics such as optical sensing. Rainbow trapping can be achieved by altering and reducing the group velocity of waves [2–13, 15]. Most recent studies have succeeded in trapping the visible range of the electromagnetic spectrum using different tools and methods. The early approaches adapted different ways to achieve rainbow trapping. For instance, metamaterial waveguides [1, 2], plasmonic graded metallic grating structures [3–5], chirped plasmonic waveguides [6–9], and chirped photonic crystals [10, 11] have been implemented to spatially separate the spectral content of a wave packet. A recent study reported the spectroscopy-on-a-chip concept with enhanced fluorescence emission [12, 14].

When categorizing the previous studies, we realize that most of the approaches employ all-metallic or metallic–dielectric compositions. Metallic structures are known to be lossy at optical wavelengths, and gain is usually introduced to compensate for the loss. Another important issue arises when a broadband “trapping rainbow” is targeted. Considering these two issues for metallic structures, we suppose that transparent dielectric materials may be a superior alternative for trapping electromagnetic waves. A search in the literature shows that very few such investigations have utilized all-dielectric materials. For example, Ref. [11] is restricted to only a one-dimensional periodic structure to explore rainbow trapping. To the best of our knowledge, no work has proposed two- or three-dimensional periodic dielectric structures to implement spatial localization and separation of the visible spectrum. Dielectric structures with periodic refractive index variation are designated as photonic crystals (PCs) [15].

In this study, two-dimensional PCs constitute the elementary structure, and we only used all-dielectric materials to trap light waves across a broadband spectrum. We designed different photonic configurations and numerically demonstrated the operating mechanism of light trapping. The corresponding frequency interval ranged from 450 to 800 nm. The prominent features of our approach are low material loss, wide wavelength range, and simple structural configuration.

2 Trapped rainbow structure: chirped PCs

A basic square-lattice PC structure was modified to trap optical pulses with different wavelengths. Dielectric cylinders are periodically distributed in two dimensions, and the background is assumed to be air. Transverse magnetic (TM)-polarized light (electric and magnetic field components are H x , H y , and E z , respectively) is only considered because of the favored dispersion features of the square-lattice PC to that polarization. If the periodic nature of a PC is kept intact, a forbidden frequency range exists, and an optical pulse at a forbidden frequency is reflected. A straightforward way to calculate a dispersion diagram that shows the forbidden and allowed modes is based on the plane-wave expansion method [16].

A waveguide can be formed from a PC structure, a schematic representation of which is shown in Fig. 1a. The lattice constant is represented by a. A guided mode may emerge if the periodic structure is perturbed sufficiently to guarantee the appearance of an artificial band within the forbidden frequency range. One way to accomplish this waveguide is the complete removal of the dielectric rods along the \( \Upgamma {\rm X} \) symmetry direction. Those rods are highlighted with dashed lines in Fig. 1a. Figure 1b is a dispersion diagram of a square-lattice PC waveguide with rods with radii of 0.22a, where a represents the lattice constant and a refractive index of 3.46. A photonic band-gap-guided mode appears in the dispersion diagram and spans a wide frequency range. The basic structural parameters of the PC and waveguide structures will be subject to chirping in the following sections. Upward and downward movement of the allowed bands will produce low group velocity regions at different propagation distances.

Fig. 1
figure 1

a Basic square-lattice photonic crystal and photonic crystal waveguide structure. b Dispersion diagram of the square-lattice photonic crystal waveguide

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The first proposed rainbow trapping structure is a chirped PC shown in Fig. 2a. The distance between each neighboring column of the dielectric rods has been linearly increased along the propagation direction (x axis) with a step size of 0.02a. The distance between each neighboring column of the dielectric rods starts at 0.50a and increases to 1.80a. The right-hand side of Fig. 2a shows the detail of the front and end sections of the chirped PC. The longitudinal length of the entire structure w 1 is 75a, and the transverse length of this structure is h = 12a. The periodicity of the structure along the y direction is kept constant at a. As mentioned previously, light waves with TM polarization are used in the numerical analysis. The chirping occurs only along the propagation direction, as can be seen in Fig. 2a.

Fig. 2
figure 2

a Schematic of a chirped photonic crystal with 0.02a distance increments between the columns of adjacent dielectric rods. b Schematic of a chirped photonic crystal waveguide. The width of the waveguide is 2.0a

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The plane-wave method associated with the super-cell technique is a versatile approach for calculating band diagrams of photonic structures [16]. However, monitoring optical-pulse propagation in the time domain can be accomplished by the finite-difference time domain (FDTD) method [17]. An appropriate absorbing boundary condition such as a perfectly matched layer should surround the computational domain to inhibit back reflections occurring because of the finite size of the studied structure. In the FDTD method, the spatial discretization (grid size) is adjusted to ∆x = ∆y = a/25. Waves with different wavelengths are sent to the chirped photonic structure. The range of trapped frequencies lies between \( {{\omega a} \mathord{\left/ {\vphantom {{\omega a} {2\pi c}}} \right. \kern-0pt} {2\pi c}} \) = 0.16 and 0.30. When we inspect the spatial intensity distribution of each input signal depicted in Fig. 3, it is seen that different frequencies localize at different spatial positions. If the input source has a normalized frequency below 0.16, then it can propagate through the chirped PC without slowing and stopping. When light with a normalized frequency of 0.16 is injected into the structure, it stops and is trapped near the end of the structure, as shown in the top frame of Fig. 3. The intensity distribution indicates the light propagation inside the photonic structure. The field intensity increases at the localization point. After that, backward propagation occurs owing to inhibited wave propagation in the forward direction. As we increase the normalized frequency (or reduce the wavelength), the locations where light is trapped moves to the left side (−x direction) of the structure. This is expected because the structure becomes dense because of closely placed dielectric rods at the front side of the structure. The slowing and stopping of light occurs in close proximity to the band gap edge regions. The band gap property of a PC arises when wavelength of the incident light is in the order of the periodicity of the structure. The front/back side of the structure is composed of smaller/larger periodicity. This in turn affects shorter/longer wavelengths to be trapped at appropriate positions as expected. We can spatially distinguish the localized position of each wavelength closely enough inside the chirped PC structure. Light cannot penetrate the structure if its normalized frequency is above 0.30. For this structure, the lattice constant a is chosen as 128 nm so that the wavelength is within the visible spectrum. When a is kept at 128 nm, the trapped wavelengths vary from 800 to 427 nm, as shown in Fig. 3. When the wavelength is decreased, it is seen that the light waves are trapped closer to the front end of the structure, which has a smaller periodicity than the back end of the structure.

Fig. 3
figure 3

The intensity distributions of light waves with normalized frequencies between \( {{\omega a} \mathord{\left/ {\vphantom {{\omega a} {2\pi c}}} \right. \kern-0pt} {2\pi c}} \) = 0.16 and 0.30 are sent to the chirped photonic crystal structure. The wavelengths corresponding to normalized frequencies trapped at different locations are stated in the top right corner of the structure. The color bar at the right designates the minimum and maximum values of the intensity variations

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When a waveguide of width 2a is constituted in the chirped PC structure, we obtain the configuration indicated in Fig. 2b. We scanned different wavelengths and inspected their propagation characteristics. Figure 4 shows light propagation in a chirped PC waveguide with 0.02a increments between the columns of the dielectric rods. The other parameters are the same as in Fig. 2a. The results, such as the shift in the locations of trapped light in the −x direction as frequency increases, are similar to those presented in Fig. 3. However, there are also differences when compared to the previous result. For example, injected light travels mainly within the waveguide region. The bandwidth for light trapping changes from \( {{\omega a} \mathord{\left/ {\vphantom {{\omega a} {2\pi c}}} \right. \kern-0pt} {2\pi c}} \) = 0.16–0.30 to 0.31–0.39. As we can see, the bandwidth shifts to higher frequencies, and it becomes narrower compared to Fig. 3. Different wavelengths experience distinct dispersive effects. For example, a normalized frequency of 0.31 produces a field distribution with the signature of slow light wave propagation, so higher frequencies propagate shorter distances. For this structure, the lattice constant a is chosen as 250 nm to adjust the trapped wavelengths to the visible spectrum.

Fig. 4
figure 4

Light waves of different normalized frequencies within the normalized frequency interval of 0.31–0.39 propagate in the chirped photonic crystal waveguide. The color bar at the right designates the minimum and maximum values of the intensity variations

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Figure 5 represents the spectral region of trapped signals with respect to the chirping parameter for two chirped photonic structures. Normalized wavelengths in nanometer units are shown in the figure. In Fig. 5a, the trapped electromagnetic spectrum covers 800–427 nm for the former configuration. The bandwidth of the trapped signals is reduced to 806.5–641.03 nm in the second case as shown in Fig. 5b. The chirped PC structure allows for tuning the chirping parameter within the range 64–230.4 nm. It is different for the chirped waveguide configuration. The chirping parameter covers the range from 128 to 450 nm. The nearly linear dependencies of light localization on the chirping parameter for the two cases are apparent in Fig. 5a, b.

Fig. 5
figure 5

The locations of different trapped wavelengths with respect to chirping lattice distance along the x direction. a and b are chirped PCs without and with a waveguide, respectively

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3 An alternative chirping approach

In another proposed design, the radii of rods are increased linearly in the propagation direction from 0.10a up to 0.34a in increments of 0.04a as shown in Fig. 6. The distance between the columns of the dielectric rods is kept constant at a. Increasing the radii of the cylinders causes an incremental increase in dielectric amount in each unit cell. The two insets in the right side correspond to a magnified view of the two selected parts at the beginning and end of the structure.

Fig. 6
figure 6

a Schematic of a square-lattice photonic crystal with the radii of the rods increasing linearly in the propagation direction. b Schematic of a photonic crystal waveguide having the same parameters as in a, except the removal of the central row of rods

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In this second stage of the design, cases both without and with a waveguide are again considered in Fig. 6a and b, respectively. Similar procedures to those presented in Sect. 2 were followed. The trapped region of the electromagnetic spectrum was investigated, and a 0.18–0.32 interval in terms of normalized units becomes a candidate for light localization. Optical pulses of different frequencies are trapped at different spatial positions as shown in Fig. 7a, b. The lattice constant a is chosen as 144 nm, therefore, the wavelength varies from 800 to 450 nm for both cases. When we reduce the dielectric filling factor of a unit cell, the bands move to higher frequencies. The slowing and localizing of shorter wavelengths occur at locations where the dielectric filling factor is smaller. Figure 7 shows the spatial intensity distributions of various wavelengths. The locations of localized waves are extracted by inspecting the field distributions in Fig. 7a, b. The results for the two different chirped photonic structures are shown in Fig. 8. The trapped bandwidth is broader than that of the proposed structure in Sect. 2. The data presented in Fig. 8a and b are extracted from Fig. 7a and b, respectively. The approximate linear relation between the chirping parameter and the trapped wavelength is also obvious in Fig. 8.

Fig. 7
figure 7

a Light waves within the normalized frequency interval 0.18–0.32 propagate in a chirped photonic crystal. b Similar light waves at 0.18–0.32 normalized frequencies propagating in a chirped photonic crystal waveguide

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Fig. 8
figure 8

Different trapped frequencies versus corresponding radii of rods. a and b show the alternative chirped photonic crystal without and with waveguide, respectively

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The governing principle of optical signal propagation in photonic structures is based on the dispersion relation between the radial frequency \( \omega \) and the wave vector \( k. \) Light speed can be traced by controlling the group velocity. There are special regions in the dispersion diagram where group velocity becomes small. The chirping parameters modify the photonic band structure such that different sections reduce light speed and localize it at certain wavelengths. Figure 9 shows the continuum dielectric band movement of the second design. Cases without and with a waveguide correspond to Fig. 9a and b, respectively. A nearly similar dispersion characteristic is observed for both structures. If the radii of the rods in the super-cell are increased, the bands move down. This means that the super-cell with the greater rod radius allows the propagation of waves with longer wavelengths. Figure 10 schematically shows the stopping and reflecting of different colors (wavelengths) at certain locations. The longest wavelength represented by red is located at the farthest position with respect to the input, and the shortest wavelength shown by blue is localized at the nearest position. As expected, the middle wavelength shown by green slows and returns back to a position somewhere between where the two other waves are localized.

Fig. 9
figure 9

The dispersion diagrams of the second proposed design in which radii of the rods are 0.10a and 0.34a. a and b show chirped photonic crystal without and with a waveguide, respectively

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Fig. 10
figure 10

Schematic showing the different wavelengths of light being trapped at different locations in a chirped photonic structure

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In this paper, two different structures are proposed to trap light with different wavelengths at different locations. In the first type of the proposed chirped PC structure, wavelengths varying from 800 to 427 nm are trapped. In this first kind of chirped PC, when a waveguide is formed, wavelengths varying from 806.5 to 641.03 nm are successfully trapped. In the second type of chirped PC structure, the trapped wavelength range is 800–450 nm for two configurations, i.e., with and without a waveguide.

The advantage of using a dielectric PC for rainbow trapping is attributed to the fact that dielectric structures are transparent in a wider bandwidth range and less lossy (except for insertion and scattering losses) than metallic structures. Another advantage of dielectric PC is that it is less complex in terms of structural form than the other structures used for trapping light. Moreover, the wavelength range trapped by the present method can be expanded by further increasing the length of the deployed structure.

To demonstrate the trapping of different wavelengths for each designed photonic structure, we used continuous source excitation. Instead, one can also use Gaussian pulses with different central frequencies. Since different wavelengths are excited at the same time with different weightings, it can be assumed that the total field intensity is the sum of the field intensities at all wavelengths. In this case, it is rather difficult to designate the exact location of each wavelength except for the one that travels longer propagation distances and ends up close to the end of the structure. Due to the superposition principle, the intensity map collects the contributions of the all wavelengths.

4 Conclusion

In conclusion, the aim of this work is to trap light with a wide spectrum using all-dielectric chirped photonic crystals. To achieve this goal, the distances between neighboring columns of rods in square-lattice photonic crystal are modified to obtain a chirped device. In addition, along the propagation direction the radii of the cylinders are increased from 0.10a to 0.34a. In the proposed designs, light waves of different frequencies localize at different spatial positions. The broadband feature of the electromagnetic wave trapping structure is an important advantage over previously reported approaches. The nearly linear dependency between the light-trapping location and the chirped parameter may allow for correct prediction of the positions where various wavelengths can be trapped. The presented design based on a linear optics concept is a practical choice and may be utilized in optical buffers, spectrometers, data processors, and optical memory.