1. Introduction

To achieve lasing a gain medium and a feedback mechanism are required. Conventional lasers employ external mirrors or reflecting surfaces to form optical cavities that provide feedback and enhance the dwell time as well as the population of excited states. Recently two new approaches have been pursued—the formation of standing Bloch waves in photonic crystals [1] and Anderson localization in random media [2]. In the first approach, feedback is provided by standing Bloch waves with enhanced density of states at the bandedges of photonic bandgaps—frequency ranges in which light propagation is forbidden—in photonic crystals consisting of periodic dielectric materials [3-5]. This feedback mechanism has been well demonstrated in simple 1-dimensional (1D) [6-10] as well as two-dimensional (2D) [11-13] photonic crystals. In the second approach, localization of light in strongly scattering random media is used to provide the feedback [14-15].

Quasicrystalline structures (quasicrystals) [16-17] have a long-range quasiperiodic order and higher point-group symmetry than ordinary periodic crystals [18-19] and are more favorable for the formation of complete bandgaps. This has been demonstrated in 1D [20] and 2D quasicrystals [21-24]. In addition, the long-range quasiperiodicity also provides confinement for extended waves. The feedback mechanism in quasicrystals can be regarded as something in between the one provided by the Bloch waves of periodic crystals and that provided by the localization of light waves in random media. Efficient light emission as well as lasing has been observed in 1D and 2D quasicrystals embedded with active gain materials [25-28]. However, there is so far no report on lasing from 3D photonic quasicrystals due presumably to the difficulties involved in their fabrication. As a result of new advances in micro-fabrication techniques, 3D icosahedral quasicrystals in the microwave and infra-red ranges have been fabricated recently by the use of state-of-the-art stereo lithography [29] and direct laser writing [30] methods. More recently, we have successfully fabricated 3D icosahedral quasicrystals using a novel 7-beam optical holographic lithography (HL) method [31-32]. Here we report on observation of optically pumped lasing from dye-doped 3D icosahedral quasicrystals fabricated in dichromate gelatin (DCG) emulsions using the HL method we developed. The observed lasing pattern exhibits icosahedral symmetry and consists of many directional lasing modes, with lasing occurring at wavelengths in the range of 580-600 nm. We show that the lasing modes and pattern agree quantitatively with calculations obtained by using the lasing condition expressed in terms of the reciprocal vectors of the icosahedral quasicrystal.

2. Fabrication

 

Fig. 1. (a) Icosahedral quasicrystal lattice. (b) 7-beam configuration for the icosahedral quasicrystal. (c) Actual 7-beam arrangement using a truncated pentagonal pyramid.

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Figure 1(a) shows the lattice of a regular icosahedral quasicrystal represented by six lattice basis vectors {a⃗_i}from the origin O to the vertices A, B, …., F. To design and fabricate icosahedral quasicrystals using the HL method it is better to work in the reciprocal space. To do that, one has to identify a set of reciprocal basis vectors {q⃗_i} that can be generated from a set of interfering beams {k⃗_i} . It turns out that six reciprocal basis vectors are needed to construct an icosahedral quasicrystal and hence seven interfering wave vectors are required [33]. Table 1 of Ref. 32 shows the structural parameters: {a⃗_i} and{q⃗_i}, for the icosahedral quasicrystal and the relation with the interfering beams {k⃗_i} . Figure 1(b) shows the configuration of the 7-beam HL setup, which consists of 5 side beams {k⃗_i,i = 1−5} and 2 oppositely directed central beams ( k⃗₀ and k⃗₆) along the OF direction. The 7-beam configuration can be realized by using a truncated pentagonal pyramid as shown in Fig. 1(c) [32]. For our experiment, the five side beams k⃗_i and the central beam k⃗₀ needed for the HL were obtained by passing an expanded beam from an argon ion laser of wavelength 488 nm through a template with five holes distributed evenly around a central hole as shown in Fig. 1(c). The resulting beams, with diameter 9 mm and power 1.2 mW, could be adjusted individually by wave plates mounted at the holes of the template to obtain the required beam polarizations. The side beams then entered the truncated pentagonal pyramid from the base and then reflected internally at the slanted surfaces of the pyramid to achieve an incidence angle of ψ = 58°, while the central beam went straight up the pyramid. (To form regular icosahedral quasicrystal, the incidence angle φ should be 63.4°.) The sample (i.e., the DCG holographic plate) was placed with the gelatin side facing down on the top of the truncated pyramid and a mirror attached to the substrate side so as to generate the seventh beam k⃗₆ from the reflection of k⃗₀. The side beams then interfered with the two central beams to generate the icosahedral pattern inside the sample. Index-matching fluids were added to all interfaces to reduce multiple reflections.

The DCG plate was exposed to the icosahedral pattern for 15 seconds using the beam polarizations stated in Ref. 32. It was then rinsed gently in a 25 °C isopropanol bath for 20 s before being hardened by baking in an oven at 100 °C for 60 min. The plate, after cooling down to room temperature, was soaked in a water bath at 15 °C containing a 2.5 × 10^-4 g/ml organic dye Rhodamine 590 for 30 min to obtain the dye-doped 3D icosahedral quasicrystals used in this study. After the swelling step, the DCG plate was rinsed under running tap water for 3 min to remove any residual dichromate. Then the plate was dehydrated by soaking it in 50%, 75%, and 100% isopropanol baths containing the same Rhodamine 590 dye concentration as in the swelling water bath at 16 °C for 3 min each. After dehydration, the plate was baked again at 100 °C for 60 min. Finally, a 0.2 mm thick cover glass was placed on top of the DCG emulsion and sealed with wax to protect it from air moisture. The icosahedral quasicrystal fabricated has “lattice spacings” scaled to that reported in Ref. 32 by the refractive indexes of SU8 photoresist and DCG emulsion. (See the SEM images in Figs. 2 and 3 of Ref. 32.)

3. Results

3.1 Measurement of photonic bandgap

The dye-doped DCG icosahedral quasicrystal sample was first characterized by measuring the photonic bandgap. White light from a 150 W halogen lamp was collimated down to a 0.5 mm spot and shone onto the DCG sample mounted vertically on a rotation platform. An optical fiber was used to couple the transmitted light to a UV-visible spectrometer (Ocean Optics USB2000) for spectral measurements. To obtain the angular dependent transmission spectrum, the sample was rotated as shown schematically in the inset of Fig. 2(b) to vary the incidence angle from θ = -65° to 65°, at intervals of 2.5°, corresponding to θ = -40° to 40° in the DCG (calculated by using an effective dielectric constant of 1.4 for the DCG).

Figure 2(a) shows the normal transmittance of a dye-doped DCG icosahedral quasicrystal sample. Multiple bandgaps, which manifest as dips in the transmittance, were observed. One of the bandgaps turns out to be hidden in the absorption band of the Rhodamine dye. We chose Rhodamine 590 as the gain medium because its emission (PL) spectrum, shown as violet line in Fig. 2(a), peaks at the bandedge of the bandgaps of our icosahedral quasicrystal, making for more efficient light emission and lasing in our samples. The inset in Fig. 2(a) shows the diffraction pattern of our icosahedral quasicrystal, which clearly exhibits, after spectrally enhanced, the characteristic 10-fold symmetry (with 5 strong and 5 weak spots). The strong blue diffraction spots are due to the bandgap at ~ 450 nm. The weak spots, with wavelengths at ~ 490 nm, were only partially visible before the spectral enhancement due to absorption of the Rhodamine dye. We also measured the angular transmittance shown in Fig. 2(b), which is in good agreement with our previous results [32] except for the appearance of a broad horizontal band due to dye absorption and a shift in the wavelengths of the bandgaps due to differences in the degree of swelling that occurred in the DCG emulsions during the development processes [34]. (See Figs. 4(b) - 4(c) of Ref. 32 and Figs. 2(c) - 2(d) below for the angular transmittance of un-doped DCG icosahedral quasicrystal samples.) Also shown are the bandgaps (color lines) calculated by applying the Bragg diffraction condition on planes inside the icosahedral quasicrystal obtained from reciprocal vectors {∆k⃗_i,j =k⃗_i-k⃗_j} with a 6.6% swelling and 2.5% shrinkage in the z- and xy- directions to account for the deformations. (See Refs. 32 and 35 for details.) The bandgaps calculated from this model are in excellent agreement with the experimental results.

 

Fig. 2. (a) Normal transmittance of dye-doped DCG icosahedral quasicrystal (green) and dye transmission back ground (red). Violet line is the PL of the dye. Inset is an enhanced diffraction pattern of the quasicrystal under white light. (b) Angular transmission spectrum for rotation about an axis perpendicular to a 5-fold axis for the dye-doped sample in 2(a). The scale bar on the right is the normalized (in percentage) transmittance intensity. The color lines are bandgaps from diffraction planes inside the icosahedral quasicrystal obtained by the reciprocal vectors ∆k⃗_0-6 (orange), ∆k⃗_i-6 (green), ∆k⃗_0-j (red), and ∆k⃗_i-j (yellow), for i,j = 1-5. (c) Normal transmittance of un-doped DCG icosahedral quasicrystal. Inset is a diffraction pattern of the quasicrystal under white light. (d) Angular transmission spectrum for rotation about a 5-fold axis for the un-doped sample in 2(c). The color lines, following the same color scheme as in 2(b), are calculated bandgaps for a regular icosahedral quasicrystal. Note that the discreteness in the transmittance in both 2(b) and 2(d) is due to the finite angular steps taken in the experiment.

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Since the incidence angle for the DCG sample used for Figs. 2(a) – 2(b) was smaller than 63.4° the resulted structure corresponds to that of slightly flattened icosahedrons (See Fig. 3(a) in Ref. 32), despite the elongation effect in the z-direction due to the swelling and the shrinkage of the gelatin in the z- and xy-directions, respectively. For a regular icosahedral quasicrystal, the normal transmittance bandgaps will overlap to form one single bandgap. Figures 2(c) - 2(d) show the normal and angular transmittance of an un-doped DCG icosahedral quasicrystal sample with a proper swelling such that the normal transmittance bandgaps overlap within the experimental resolution. The inset in Fig. 2(c) shows clearly the 10-fold symmetry of the DCG icosahedral sample with diffraction wavelengths at ~ 470 nm. Note that in Fig. 2(d) there is no horizontal band from dye absorption. The solid curves in Fig. 2(d) are calculated bandgaps for a regular icosahedral quasicrystal with the normal incidence bandgap wavelength as the only adjustable parameter. The agreement with the experiment is very good, indicating that the DCG sample is very close to the regular icosahedral quasicrystal. Despite that the dye-doped DCG icosahedral quasicrystal samples differ slightly from that of regular icosahedrons, we will still consider them as members of the icosahedral family based on the findings from the SU8 icosahedral samples reported earlier using the same 7-beam interference method [32].

3.2 Lasing pattern

 

Fig. 3. (a) Icosahedral quasicrystal lasing pattern projected on the back side of the glass substrate (see inset) pumped with a 532 nm 0.4 μJ/pulse and polarization ϕ = 30°. (b) Higher resolution projection of the icosahedral quasicrystal lasing for inner region. The lines are guides to the eyes. The circled spot was used for lasing onset measurement in Fig. 4(a). The white bar is 0.5 unit in the scale of Fig. 5(b) below. (c) Lasing patterns of the icosahedral quasicrystal with different pumping polarization ϕ at energy 0.4 μJ/pulse. Top-left inset is the schematic for the sample orientation with a 5-fold axis aligned vertically and pointing downward.

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The dye-doped DCG icosahedral quasicrystal was optically pumped by the second harmonic from a mode-locked Nd₃₊-doped yttrium aluminum garnet (Nd-YAG) laser at 532 nm, with 35 ps wide pulses and a 10 Hz repetition rate, to study the PL of the dye-doped icosahedral quasicrystals. After passing through a set of polarizers and a half-wave plate to control the beam intensity and polarization, the pumping beam was focused to a ~25 μm spot on the sample to excite the dye molecules embedded inside the DCG. The sample was excited with the gelatin side facing the pumping beam normally and the PL was measured from the substrate side. To display the lasing pattern, a piece of white paper was attached to the glass substrate of the DCG plate using a matching fluid as shown in the inset of Figs. 3(a) - 3(b). The lasing pattern was captured by using a CCD camera with a 570 nm high-pass filter to filter out the pumping light. For spectral measurements a half-spherical lens was attached to the glass substrate so that the lasing light, including those at large angles, could emerge at the curved side of the spherical lens. This arrangement facilitates our measurement of the lasing spectrum from the different lasing spots. The emerging PL from different regions was then collimated individually by a lens and focused onto an optical fiber that was coupled to a linear CCD spectrometer (Ocean Optics ADC1000) that has a spectral range of 536 to 697 nm and a resolution of 0.1 nm. Note that the pumping wavelength is near the absorption peak of Rhodamine 590 dye. Thus only laser emission at wavelengths longer than the absorption edge of the dye, ~560 nm, is possible.

 

Fig. 4. (a) Lasing spectrum taken at the spot circled in white in Fig. 3(b) for different pumping energies with ϕ = 30° The inset is the integrated PL intensity. (b) Normalized spectra obtained from lasing areas 1- 4 as shown in the inset for the DCG sample in Fig. 2 using 0.5μJ/pulse pumping energy. (The overall lasing pattern of this sample is by and large the same as Fig. 3, except for a slightly different intensity contrast.)

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Figure 3(a) shows the entire image of the lasing pattern which consists of many overlapping lasing spots. The pattern can be divided into a star-like inner region (with lasing angle ψ < 40°), also shown in higher resolution in Fig. 3(b); a pin-wheel like middle region (with 45° < ψ < ~70°), with lasing spots piling up like the spokes of bicycle wheel; and an in-plane lasing outer region with diffused spots at the edges of the DCG plate. The lasing pattern exhibits clearly the 10-fold symmetry of icosahedral quasicrystal. Furthermore, the lasing spots in the inner and middle regions are highly directional with a ~1° divergence as compared to the ~14° diverging lasing beam observed in 1D DCG samples [10]. In addition, the lasing intensity depends on the polarization direction of the pumping beam, indicated as blue arrows in Fig. 3(c). For the middle region lasing spots, they are more intense in the directions close to the direction of polarization of the pumping beam. However, the in-plane lasing spots—10 of them by counting all the images in Fig. 3(c)—are only visible near the perpendicular direction of the polarization of the pumping beam. As for the inner region, the polarization dependence is not clear due to overcrowding of the many lasing spots. All lasing spots, including the diffused in-plane lasing, show similar orange color with wavelengths in the long-wavelength region of the dye emission spectrum. (See Fig. 2(b).)

Figure 4(a) shows the lasing spectrum of the spot circled in white in Fig. 3(b) as a function of pumping energy. Lasing started to appear at a pumping energy about 38 nJ/pulse. More modes were observed at higher pumping energies. The inset shows the wavelength integrated PL intensity as a function of the pumping energy, showing a threshold comparable to that observed in periodic layer systems [6-10]. The spectra taken at 4 other areas of the lasing pattern, shown in the inset of Fig. 4(b), show many lasing modes with wavelengths spanning from 580 – 600 nm and spectral widths narrower than 0.5 nm. Furthermore, some of the lasing modes from different areas in the lasing pattern have almost the same wavelengths as shown in Fig. 4(b). Similar spectra were also obtained from the in-plane lasing spots in the outer region.

4. Model

In conventional periodic photonic crystals lasing modes can be calculated using the standing Bloch wave approach. However, it is more convenient to analyze the lasing modes in terms of the reciprocal lattice space {G⃗_i} [13]. A wave with wave vector k⃗ inside a crystal can be scattered many times, hence it is better to express it as a linear combination of plane waves diffracted by the reciprocal lattice points. Thus the lasing condition is equivalent to the standing wave condition written as $\sum _i\left(\overrightarrow{k}-{\overrightarrow{G}}_i\right)=0$. This approach applies to quasiperiodic crystals as well [13, 26]. For the icosahedral quasicrystal, G⃗_i, can be written as ${\overrightarrow{G}}_i=\sum _j{a}_{ij}{\overrightarrow{q}}_j$ where a_ij =0,±1,±2,…..and q⃗_j is given by Eq. (3) in Ref. 32. Thus the lasing condition can be rewritten as $\sum _i\overrightarrow{k}=n\overrightarrow{k}=n\frac{\mathrm{2\pi }}{{\lambda }_l}\hat{k}=\sum _i\sum _j{a}_{ij}{\overrightarrow{q}}_j=\sum _j{n}_j{\overrightarrow{q}}_j\mathrm{for}n=\mathrm{1,2,3},\dots ,$ for n = 1,2,3,…, n_j= 0,±1,±2,….., where λ_l is the lasing wavelength. In principle the whole reciprocal space is densely filled because of the quasiperiodicity of the icosahedral symmetry. This leads to an infinite number of lasing modes, in contrast to the case of periodic crystals. However, many of them are not excited because of the presence of bandgaps and the emission/absorption spectrum of the dye. Note that since the lasing condition is evaluated in 3D, both lasing wavelengths and directions can be calculated directly without the additional phase-matching condition required for out-of-plane emission in 2D crystals [26].

Taking the above approach, we first calculated the reciprocal basis vectors {q⃗_i} using the effective wave vectors {k⃗_i} to account for deformations in the DCG in fitting the angular transmission spectra in Fig. 2(b); then, we calculated the lasing modes and wavelengths using the lasing condition with 1 ≤ n ≤ 6 and n_j = 0,±1,±2,±3. Furthermore, we selected only lasing wavelengths (in air) in the range of 570 - 610 nm by considering a ~10 nm bandgap width that we estimated from Fig. 2 in view of the fact that it is more likely to emit at the bandedges of bandgaps. Figure 5(a) shows the calculated lasing pattern with lasing angles corrected for refraction by the glass substrate, which is on the back side of the DCG sample. Note that because of total internal reflection due to the mismatch in the refractive indexes in DCG (1.4) and the glass substrate (1.52), the maximum lasing angle ψ that can be observed in the forward pattern is 67°. This is in good agreement with the experimental results shown in Fig. 3.

Table 1. Possible lasing processes for the icosahedral quasicrystal with wavelengths in the range of 570 - 610 nm. i = 1 - 5 and q⃗_i = q⃗_i+5.

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The lasing modes listed in Table 1 were those with the smallest values for n and n_j. The modes are listed according to the lasing angle ψ, as groups A, B and C, group D, and group E (omitted in Fig. 5(a)), and they correspond to the inner, middle, and outer lasing regions observed in the experiment, respectively. The inner lasing modes are also displayed on an expanded scale in Fig. 5(b). The resemblance between Figs. 5(a) - 5(b) and Figs. 3(a) - 3(b) is striking, despite the fact that not all of the predicted lasing spots were observed in the experiment, due possibly to imperfections in our DCG samples. Note that some of the lasing modes are spatially close to each other, i.e. the modes lase in almost the same direction (e.g. groups A and B modes in Table 1). Thus they could not be resolved by our detecting system, resulting in the appearance of “multimode” lasing as observed in the experiment. Furthermore, the number of overlapping modes increases with larger n or n_j. This “multimode” lasing feature is quite interesting and is unique to 3D icosahedral quasicrystals unlike the case of 2D icosahedral quasicrystals. While the lasing modes calculated from the lasing condition are in good agreement with our observations, the lasing condition alone does not give predictions of the amplitudes of the excited modes as well as their dependence on the polarization of the pumping beam. A more elaborate calculation is needed to explain these additional features, which could be challenging because of the 3D quasiperiodic nature of the DCG samples.

 

Fig. 5. (a) Lasing pattern selected with wavelengths from 570 - 610 nm, including the results in Table 1. Circles (groups A, B, and C) and pentagons (lasing group D) corresponding to the inner and middle lasing regions observed in the experiment, respectively. (b) Inner lasing region from Fig. 5(a) in expanded scale with groups A (red, blue, green, magenta), B (dark yellow, cyan), and C (violet), corresponding to the sequence listed in Table 1. The circled lasing spot (in black) corresponds to that circled in white in Fig. 3(b).

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5. Conclusion

To conclude, we have observed multi-directional lasing modes in the visible range from dye-doped 3D icosahedral quasicrystals fabricated in DCG by using a novel 7-beam optical interference holographic method. The DCG icosahedral quasicrystal samples displayed beautiful lasing patterns characterized by icosahedral symmetry. We have also calculated the lasing modes and wavelengths using the lasing condition expressed in reciprocal lattice space. Quantitative agreement between the model and experiment was obtained. This work opens up new degrees of freedom in the design of laser devices using 3D photonic quasicrystals.