1. Introduction

Berry coined the term “diffractals” in 1979 to describe the interaction of electromagnetic waves with fractal structures such as a random phase screen. His intent to “point out that fractals cause waves to adopt unfamiliar forms that should be studied in their own right” [1] stimulated subsequent work in electromagnetics and optics that has culminated in the introduction of fractal plasmonics, antennas, and metamaterials [2–4], optical confinement in fractal-shaped cavities and waveguides [5,6], and fractal photonic lattices [7,8], as well as insight into the optical properties of fractal aggregates [9,10]. Following Berry’s pioneering publication, Karman et al. predicted, and subsequently demonstrated, the production of fractal light intensity patterns by the injection of laser radiation into canonical unstable laser resonators having a hard-edged output coupler [11–13]. Multiple theoretical studies of two- and three-dimensional, unstable optical resonators with a polygonal mirror or binary aperture later corroborated the attribution of the fractality of the predicted modes to edge-wave interference (arising from Fresnel diffraction) at the cavity aperture, combined with the magnification provided by the unstable resonator [14–21].

Despite these advances in predicting and applying the production of fractal light patterns, the generation of fractal transverse eigenmodes within a laser itself was not reported until 2018 when Rivera et al. [22] observed fractal laser modes generated within a hybrid optical resonator. The cavity mirrors were planar and the primary mechanism responsible for fractal mode generation from the noise, as opposed to shaping an existing laser beam, was suggested to be Fresnel diffraction from intracavity microrefractive components. Subsequently, laser fractal modes were observed in a conventional unstable cavity [23], but both previous experimental studies have reported only fundamental (i.e., lowest order) eigenmodes.

We report the computational prediction and experimental observation of two distinct groups of higher-order, transverse laser modes having fractal electric field profiles with order number m as large as 5, generated by a microresonator similar to that of Ref.[22]. Although the fundamental laser fractal mode associated with a triangular intracavity aperture was reported in 2018 [22], higher-order fractal modes have not been observed previously. All of the modes are non-Gaussian and, in contrast to conventional laser resonators, their transverse intensity profiles evolve continuously and rapidly within the resonator. Specific arrangements of microspheres located on or near one mirror of a Fabry-Pérot microcavity act as a refractive, intracavity aperture which defines the geometry of an entire class of laser fractal eigenmodes. In particular, the equatorial plane of a close-packed array of microspheres, lying on one mirror within a Fabry-Pérot resonator and immersed in the laser gain medium, acts as a refractive slit array in a plane transverse to the optical axis. Edge-diffraction from the slit array generates the high spatial frequencies (>10⁴cm^-1) required for the formation of high-order laser fractal modes, and fractal transverse modes are generated, amplified, and evolve within the active medium. Because the active medium, a quantum dot solution, fills the resonator, the fractal transverse mode profile at any position within the resonator is also influenced by the spatial variation of the pump intensity and local gain. Experimental observations of the fractal transverse modes produced by the laser closely resemble the predictions of numerical simulations for distended-rectangular (4-microsphere) and quasi-triangular (3-sphere) apertures. Simulations and experiments demonstrate that, with the rectangular aperture, the m = 2,4, and 5 higher-order eigenmodes are observed, all of which have calculated round-trip losses <18%. Furthermore, the m = 1 and 2 modes are found from computations to be of even and odd parity, respectively, and nearly degenerate in frequency—their resonant wavelengths differ by ∼300 MHz. In contrast, the modes formed by a 3-microsphere (triangular) aperture are considerably more lossy (round-trip loss of ∼30% for m = 2) and, therefore, only the m = 1 and 2 modes are observed reliably by experiments for this aperture topology. Simulations show that the fractal dimension (D) of the electric field generally varies slowly with position in the cavity, but rises rapidly on the downstream side of the intracavity aperture as a result of edge diffraction. For 4-microsphere apertures, 2.25 < D < 2.40 but D varies over a wider range (2.19–2.48) when the refractive aperture is triangular. For all of the observed eigenmodes, the fractality increases abruptly at, and just after, the aperture because of the high spatial frequencies (>10⁴cm^-1) introduced to the transverse mode profile by near-field diffraction. Calculations demonstrate that D increases by as much as 0.15 downstream of the aperture, and experiments confirm that the fractal mode profile changes continuously along the longitudinal axis of the resonator. Angular power spectra for the fractal modes, from which D is calculated, have also been computed and the corresponding spectral phases are found to be δ-correlated. These results demonstrate that the fractal modes transition smoothly from fractal to non-fractal behavior while propagating in the microresonator. Although the discovery of high-order fractal laser modes is itself of fundamental significance, the variable fractal dimension and extraordinary spatial frequencies of these non-Gaussian modes suggest their applicability to photonic devices, such as sensors and switches, and optical imaging. A new form of optical microscopy is also proposed in which highly structured objects such as cells in vivo are analyzed by forward or back scattering of high spatial-frequency fractal radiation from the object, in combination with electromagnetic inverse scattering algorithms.

2. Microsphere resonator

A generalized illustration of the microresonator is shown by the inset of Fig. 1(a). As discussed in detail in Materials and Methods, the critically stable, Fabry-Pérot cavity comprises two high reflectors (R > 99% at 650 nm) separated by 250–300 µm. The 200 µm diameter, transparent microspheres located on or near the lower mirror behave as thick lenses and, if arranged properly, form a refractive aperture that may be computationally approximated as a hard-edged aperture. In order to preserve the high spatial frequencies produced by the interaction of an optical wave with an aperture on each pass, near-field diffraction should continue to be dominant after multiple round trips through the cavity because far-field (i.e., Fraunhofer) diffraction necessarily lacks the high spatial frequencies required for generating the fine structure inherent to fractal modes. Consequently, the cavity Fresnel number F = a²/(λL), where 2a is the maximum aperture width and the single pass length of the cavity is L, must (at a minimum) be ∼5 for a cavity in which the aperture resides on one mirror. For the cavities presented here, ∼9 < F < 35. Because the gain medium (a solution of CdS/ZnSe quantum dots) occupies the interior of the resonator, fractal transverse modes form and evolve while propagating through the active medium. Pump radiation (not shown in Fig. 1) is provided by a frequency-doubled, pulsed Nd:YAG laser and enters the resonator through the upper mirror. A camera coupled to an optical microscope provides depth-resolved images of the microlaser transverse modes within the resonator.

 

Fig. 1. Optical micrographs of several laser eigenmodes, frequently observed in hybrid microsphere resonators and recorded by imaging laser emission through the upper mirror: (a) Examples of modes generated within interstices between 4 close-packed spheres. This image was recorded with near-infrared backlighting in order to highlight the periphery of each microsphere. The inset is a generalized diagram (not to scale) of the optical resonator and experimental arrangement in which the gain medium, a colloidal quantum dot solution, fills the entirety of the Fabry-Pérot resonator internal volume. The upper and lower mirrors for the microlaser are shown in gray; (b) Magnified image of a transverse laser eigenmode, generated in a critically stable (interstitial) region of the cavity by edge-wave diffraction originating from the equatorial plane of the microspheres, exhibits a nested triangle pattern similar to those of Ref. [22]; (c) Another, frequently observed laser transverse intensity profile exhibiting high spatial frequency structure.

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Images (a)–(c) of Fig. 1 provide examples of laser modes observed experimentally when three microspheres are arranged so as to be close-packed. In panel (a), four 200 µm diameter spheres in contact form two quasi-triangular regions lying between the spheres, regions in which the microresonator is critically stable. Within these interstitial regions, which were backlit by near-infrared radiation (λ∼780 nm) from a light-emitting diode in order to accentuate the periphery of each microsphere, complex modes are observed. Two examples of the modes most frequently observed in experiments are presented by the magnified optical micrographs of Figs. 1(b) and 1(c). The nested-triangle structure of panel (b) is representative of the fundamental fractal mode reported in Ref. 22. In contrast, obvious triangular structure is absent in the mode of panel (c) but multiple, high-spatial frequency features, arising from diffraction occurring at the equatorial plane of the spheres, are evident.

3. Eigenmode simulations

Resonant modes of the optical cavity were calculated by the method introduced by Fox and Li in 1960 [24], a process that injects into the resonator an electric field (E) having an arbitrary initial distribution, and numerically propagates the light through the cavity repeatedly until a steady-state solution emerges. We adopt the first Rayleigh-Sommerfeld integral as the propagation kernel by which E is mapped from a surface S in a plane (ξ,η,z) at longitudinal position z onto the parallel plane (x,y,z+Δz):

As illustrated in Fig. 2(a), the round-trip computational sequence begins with initialization of the field (step (0) in Fig. 2(a)), which is followed by applying a transmission function representing the refractive aperture (denoted by the dashed horizontal line). After the field is propagated through the upper half of the resonator, the aperture function is again applied. Propagation of the field through the lower section of the optical cavity then completes one round-trip. In order to avoid 3D discretization of the entire resonator volume, the refractive aperture formed by 3 or 4 close-packed microspheres was computationally replaced by one of two binary transmission masks which are shown in Figs. 2(b) and 2(c), respectively. This approximation is justified because ray tracing demonstrates that most of the light incident on the microspheres is refracted out of the cavity volume defined by the interstices between the microspheres and, therefore, microsphere refraction does not appear to contribute significantly to the mode profiles studied here. Close packing of three microspheres yields a quasi-triangular aperture (Fig. 2(b)) whereas the 4-sphere arrangement, in which one pair of opposing spheres is spaced by a distance a factor of ∼5 larger than that for the other pair of opposing spheres, results in a distended rectangular aperture having two axes of symmetry (Fig. 2(c)). In the simulations, the transmission mask is situated in the equatorial plane of the microspheres, i.e., 100 µm above, and parallel to, the bottom mirror for 200 µm diameter spheres.

 

Fig. 2. Illustrations of the microresonator structure, the optical path for calculating resonator eigenmodes, and outlines for two intracavity apertures: (a) Cross-sectional schematic of the optical cavity, depicting the replacement of the microsphere aperture with a binary transmission aperture (represented by the horizontal, dashed line) for computational purposes. After initialization of the field (0), the aperture transmission function is applied (1). The field is then propagated through the upper portion of the cavity (2), and the aperture function is again invoked (3). One round trip is completed by propagating the field through the lower portion of the cavity (4); (b),(c) Binary transmission masks implemented numerically to represent the apertures formed in the equatorial plane of close-packed arrangements of three and four spheres, respectively. Steps 1–4 constitute one cavity round trip, and this computational sequence was iterated until the field distribution converged. See Materials and Methods for details.

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Further details concerning the computational procedure itself can be found in Materials and Methods.

4. “Triangular” fundamental mode properties: power and phase spectra, and fractal dimension

Figure 3 shows two images of the calculated lowest-order (m = 1) mode generated in a resonator having the close-packed, 3-sphere topology. Designating the longitudinal coordinate of the resonator as z, the mode of Fig. 3(a) is that calculated at the surface of the lower mirror (z = 0). Panel (a) of Fig. 3 also indicates in white the perimeters of the aperture’s three spheres, whereas Fig. 3(b) presents the magnified and gray-scale spatial map of Re(E) for the same m = 1 mode, but calculated at z = 150 µm (i.e., 50 µm above the aperture). This transverse intensity map vividly illustrates the high spatial-frequency structure that was acquired by the mode as it encountered the 3-sphere aperture. A computer-generated video, illustrating the calculated spatial evolution of the fundamental mode intensity profile as the optical field propagates upwards between the lower and upper mirrors (at z = 0 and 300 µm, respectively), is shown in Visualization 1. This video comprises a sequence of transverse mode intensity maps, calculated in increments of 5 µm along the entire longitudinal axis of the resonator.

 

Fig. 3. Images of the calculated fundamental (m = 1) fractal mode at the surface of the lower mirror (z = 0), and 50 µm above the sphere equator (z = 150 µm), for the close-packed 3-sphere topology: (a) Calculated transverse intensity map at z = 0, and the white curves indicate the positions of the three microsphere perimeters; (b) Spatial profile of Re(E) for the fundamental mode at z = 150 µm, shown in grayscale to accentuate the underlying high spatial-frequency structure. The color of the map in (a) reflects the emission spectrum of the quantum dots which peaks at ∼650 nm. Note that ${|\gamma |^2}$ for this fundamental mode is calculated to be 0.861.

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Because of the high spatial frequencies evident in Fig. 3, one would expect the propagation of the mode over even relatively small distances to alter the transverse spatial intensity profile considerably. This is, indeed, the case as observed in Fig. 4 which compares simulated and experimental mode intensity maps for six selected positions within the resonator. The image at upper left in both sets of images is associated with z = 0, and the remaining images correspond to selected planes located successively further from the lower mirror. That is, z progresses from left to right in a row, as well as downward from row-to-row, and the final image in each set is associated with a plane near the sphere equator. Emphasis is placed here on the upward-propagating mode because experiments view the resonator through the output coupling mirror at top in Fig. 2(a), and the modes of standing-wave unstable resonators exhibit, in general, different transverse profiles for the two counter-propagating waves. Each experimental image was recorded with a single pulse of the pump laser and a microscope having an estimated longitudinal resolution (i.e., coordinate associated with the optical axis) of 1.6 µm. For clarity, white curves superimposed onto the simulated fractal mode images at right represent the periphery of each of the three spheres forming the triangular aperture.

 

Fig. 4. Simulated and experimental intensity images for the fundamental triangular mode at different locations within the resonator: Comparison of (left) experimental and (right) calculated fundamental mode intensity maps for the close-packed, 3-sphere topology. Both sets of six images begin at upper left with z ∼ 0, and each successive image is recorded (or calculated) at a plane progressively further from the bottom mirror. The edges of the three close-packed spheres are shown in white for each of the simulated images (at right). The final image in each set is associated with a plane near the equator of the microspheres. The experimental images were recorded with an estimated axial (depth) resolution of <2 µm, and each image is that produced by only a single pulse of the pump laser.

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In contrast with conventional Gaussian modes, the fractal spatial profile changes markedly and continuously as the mode traverses the cavity and the nested triangle structure observed in Ref. 22, as well as other self-similar patterns, are evident in both the simulated and experimental images. Several of these closely resemble a finite Sierpinski triangle which provides support for the presumption that these modes are fractal in character. Furthermore, a close examination of the static images of Figs. 1(c) and 4, as well as the video associated with Fig. 3 (comprising a sequence of calculated transverse mode profiles; see Visualization 1), shows the inward propagation of edge waves from the perimeter of the refractive aperture. Although it is clear that similarities exist between the simulated and experimental images, the limited numerical aperture of the optical microscope (NA = 0.42) acts as a spatial-frequency low-pass filter which sets an upper limit to the spatial frequencies imaged onto, and captured by, the camera.

An asset of the simulations is the ability to calculate fractal optical field characteristics that are difficult to determine experimentally. One example is the real part of the electric field, which is presented in false color in Fig. 5(a) for z = 0. The dashed vertical line indicates the path of the intensity lineout shown in panel (b) of Fig. 5. Both the Re(E) surface and the lineout exhibit the self-similar scaling characteristic of fractals. Of greater significance in terms of confirming the fractality of the observed modes are the calculated angular power and phase spectra for the fundamental mode which are shown in Figs. 5(c) and 5(d), respectively. Both display third-order rotational symmetry near the origin which matches the symmetry observed in the spatial domain. Although the power spectrum exhibits a prominent peak at zero frequency (i.e., a DC component), a significant fraction of the power lies in the extended tails of the distribution function. The spectral phase oscillates rapidly with increasing spatial frequency and was determined to be nearly δ-correlated, which satisfies the condition that the spectral phase of a fractal must be effectively random for deterministic fractals such as these.

 

Fig. 5. Simulations of the electric field, power spectrum, and angular spectrum phase for the lowest-order mode associated with a quasi-triangular refractive aperture: (a) false-color map of Re(E); (b) Lineout obtained along the dotted white line of (a), illustrating fractality in one spatial dimension; (c) Calculated angular power spectrum for the fundamental mode, presented as a log-log false-color map. The units of the ordinate (k_y) are the same as those for the abscissa (k_x). A diagram illustrating an exemplary annular region in which the angular power spectrum is integrated in order to obtain the radial power spectrum is also shown and labeled in black; (d) Angular phase spectrum for which the units of the ordinate are again the same as those for the abscissa. The units for the false color scale at right are radians; (e), (f) Radial power spectra (shown in blue) calculated for z = 0 and z = 110 µm, respectively, in the microresonator of Fig. 1(a), presented in log-log format. The linear, least-squares fits to the spectra are denoted by the red lines, and the values of D deduced from the fittings are also indicated.

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Because the power spectrum of Fig. 5(c) is far from isotropic, the fractal dimension of the mode cannot be determined from a single lineout. Rather, the total radial power spectrum, F(k_r), must be calculated. Specifically, for each value of F(k_r), the two-dimensional power spectrum was integrated within an annulus defined by $({{k_r} - dk/2} )< \sqrt {k_x^2 + k_y^2} \le ({{k_r} + dk/2} )$, where dk is the sampling rate in the Fourier domain (cf. Figure 5(c)). From this integration, the fractal dimension D can be determined. Specifically, it has been shown previously [27] that a fractal characterized by a power spectrum scaling as $F(k )\propto {k^{ - \beta }}$ is given by: $D = {D_T} + ({{D_S} + 2 - \beta } )/2\; $, where ${D_T}$ is the topological dimension of the fractal and ${D_S}$ is the dimension of the power spectrum [27]. For the optical images of Figs. 3 and 4, the dimension of the power spectrum is effectively reduced by the azimuthal integration process to unity. The total radial power spectra for the fundamental optical mode at the bottom mirror (z = 0 µm) and 10 µm above the aperture (z = 110 µm) are shown in Figs. 5(e) and 5(f), respectively. Note that these graphs are also presented in log-log format, and both exhibit a linear segment that spans more than two decades in k_r. The linear least-squares fits of the relation: $D = 2 + ({3 - \beta } )/2$ to the spectra of Figs. 5(e) and 5(f), represented by the red lines, yield fractal dimension values of 2.28 and 2.36, respectively. Thus, D is found to be strictly greater than the topological dimension (2) of the generated electric field surfaces, but less than the embedded-space Euclidean dimension (3) for all axial positions in the optical cavity. Furthermore, the sharp cutoffs in panels (e) and (f) of Fig. 5 occur near k_r = k_crit, the critical value of k_r for diffraction from a one-dimensional slit. For Fresnel diffraction, k_crit = 4πa/(λΔz) [17], where a is the aperture half-width adopted for the simulations (∼37 µm) and Δz is the distance that the optical field has propagated since its last encounter with the refractive aperture. A cosine correction factor is also included in the calculation of k_crit in order to account for Rayleigh-Sommerfeld diffraction. In summary, power law scaling of the radial power spectrum with spatial frequency has been observed, and the fractal dimension D was found to be between ∼2.25 and 2.40. In combination with the qualitative self-similarity observed in the mode images and the δ-correlated spectral phase, these results lend support to the conclusion that the lowest-loss mode produced by a quasi-triangular aperture is fractal.

5. Higher-order fractal modes

Extensive simulations of the fundamental and high-order laser fractal modes formed by a 4-microsphere refractive aperture, conducted in a manner similar to that adopted for determining the fundamental mode associated with the 3-sphere aperture topology, also confirm experimental observations. If the four spheres are close-packed, two isolated apertures are formed in a plane transverse to the optical axis. However, if a sufficiently wide channel is established by separating two of the spheres, the aperture becomes quasi-rectangular with two axes of symmetry, and previously unobserved spatial laser modes are generated. The upper row of images in Fig. 6, for example, are those calculated for the fundamental laser fractal mode appearing at z = 0 µm when the channel width (i.e., the gap between the two spheres bordering the aperture at left and right) is varied from 20 µm to 32 µm. If the width of the aperture (d) is maintained below ∼20 µm (as shown by Fig. 6(c)), two separate and essentially independent laser modes are observed. Both of these are typically the triangular fundamental mode discussed in the last section. However, as the channel width at its narrowest point is increased to ∼26 µm and beyond, strong coupling between the two lasing regions occurs and weak, extended “tendrils” appear. Further increases in d result in laser modes that occupy much of the interstice defined by the 4-sphere ensemble, as shown by the 32 µm gap image in Fig. 6(a). It should be emphasized that simulations and experiments in which the horizontal gap of the rectangular aperture was varied continuously over the d = 0–40 µm range showed that distinct and unique modes were observed for each value of d. Accordingly, all simulations to follow for the 4-sphere aperture assume d to be 26 µm and the distance between the facing-surfaces of the top and bottom spheres is 130 µm, which defines the transverse aspect ratio of the distended-rectangular aperture to be 5:1. The middle image in the top row of Fig. 6 (panel (d)) is that for the lowest-order mode when d = 26 µm, for which ${|{{\gamma_1}} |^2}$ is calculated to be 0.938. The lower row of intensity maps in Fig. 6 displays several examples of experimentally recorded images that are observed routinely. As mentioned earlier, all of the experimental images of Fig. 6 (and throughout this article) were recorded with a single pulse of the pump laser. We wish to point out the high spatial frequency structure observed along both the longitudinal and transverse coordinates of the aperture’s equatorial plane. Of particular interest is the fine, “sawtooth” structure bordering the sphere at upper right in Fig. 6(d), structure similar to that evident in Fig. 3(b). It should also be mentioned that this particular set of experimental images was chosen because the channel widths are the closest to those for the images in the upper row.

 

Fig. 6. Simulated and representative, experimental fundamental laser fractal modes generated in 4-sphere interstices as the gap between two spheres is varied: (upper row) images calculated for gap widths of 20, 26, and 32 µm; (lower row) experimentally recorded images for microsphere arrangements similar to those in the top row. The image of panel (f) has intentionally been saturated at lower left so as to render visible the weak structure at upper right. Tendrils linking the mode patterns at both ends of the distended rectangular aperture are also noticeable in panels (e) and (f).

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The upper half of Fig. 7 shows simulations for the first four higher-order fractal modes (m = 2–5) associated with the 4-sphere aperture topology. Calculated values of ${|{{\gamma_m}} |^2}$ associated with each mode are also given, and the computational process for identifying higher-order fractal modes can be found in Methods and Materials. Note that, because the aperture for the four-sphere resonator is substantially larger than that for three close-packed spheres, the round-trip losses for the m = 2–4 modes of Fig. 7 are actually lower than that for the fundamental mode associated with the close-packed, 3-sphere aperture . Not surprisingly, therefore, experiments repeatedly observed quasi-rectangular fractal laser modes up to 5^th order, as shown by the images in the lower row of Fig. 7 (panels (e)-(g)). A clear trend is evident in the high-order modes for the 4-sphere aperture—namely, the number of intensity nulls along the vertical coordinate (y) increases with mode order m. This is also true of the TEM family of Gaussian modes, which suggests that the number of nulls along the y coordinate may provide a reasonable metric for assigning fractal mode numbers when the mode order is low.

 

Fig. 7. Simulated and experimental mode intensity maps for the first four high-order fractal modes associated with the 4-sphere (quasi-rectangular) refractive aperture: (top row) simulated images along with the calculated values of |γ_m|². The perimeters of the four spheres defining the aperture are again represented by the white curves; (bottom row) experimental images representative of the modes observed reproducibly. The eigenmode in panel (g) appears to be m = 4 and is coupled to a continuum because the 4^th sphere is absent. Note, too, the periodic structure occurring parallel, and orthogonal, to the long axis of the aperture.

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At first glance, the m = 2 mode at upper left in Fig. 7 does not appear to be a higher-order mode because of its similarity to the fundamental mode of Fig. 6. However, calculations demonstrate that the electric field in the mode identified as second-order in Fig. 7 switches in polarity in passing through the null. Therefore, the mode is odd in parity whereas the fundamental (m = 1) mode of Fig. 6 exhibits even parity, which is similar to the relationship between the TEM₀₀ and TEM₀₁ Gaussian modes. Interestingly, the round-trip loss for the m = 2 mode is actually slightly lower (by 0.3%) than that for the mode of Fig. 6 identified as the “fundamental”. Also, the difference in the eigenvalue phase (for the same wavelength) between the two modes is 5 mrad which corresponds to a resonant wavelength shift of ∼300 MHz, considerably smaller than the 375 GHz free spectral range of the microresonator. Therefore, these two fractal modes are nearly degenerate in frequency, and we conclude that the resonator behavior for this mode is analogous to that of two coupled oscillators having a lowest-order state in which the oscillators are out-of-phase by π radians. The simulated m = 6 mode (not shown in Fig. 7) also has odd parity, but the other modes of Fig. 7 are all even in parity.

Several selected higher-order eigenmodes, representative of those observed experimentally, are shown in the lower-half of Fig. 7. The most frequently produced modes are m = 4,5 and one of the experimental images, shown in Fig. 7(g), is that of the 4^th-order mode coupled to a continuum because the fourth sphere that would normally define the aperture has been removed. It should be emphasized that simulations predict periodic structure along both orthogonal coordinates in the m = 4,5 transverse intensity maps, in particular. A careful comparison of the simulations and experimental images shows that several of the intensity oscillations predicted by calculations that exist along the coordinate orthogonal to the long axis of the aperture are observed in the experiments. However, the micrograph in Fig. 7(g) also shows the existence of fine oscillatory structure along this transverse coordinate—structure not captured by the simulations. Finally, it should be mentioned that although several experimental images show evidence of the m = 3 mode, we are able to report that only the m = 2,4, and 5 higher-order modes have been observed unambiguously.

Calculations of the lowest 10 of the higher-order modes associated with the three-sphere aperture topology have been conducted, and the first four of these are shown in the upper row of Fig. 8, all of which were calculated for z = 0. Once again, the ${|{{\gamma_m}} |^2}$ values associated with each mode are given, and it is immediately evident that the round-trip loss for this resonator geometry is considerably higher than that for modes of the same order in the 4-sphere resonator configuration. All of the modes exhibit one axis of symmetry and all but two also have three-fold rotational symmetry. Because these modes are generated in a resonator having spatially non-uniform loss, calculations confirmed the modes to not be orthogonal but rather biorthogonal, as theory predicts for non-unitary resonators [28]. As noted above, the round-trip losses for this refractive aperture/resonator geometry are prohibitive (i.e., ∼30% and ∼45% for m = 2 and 3, respectively) and, accordingly, observing higher-order modes was more challenging than was the case for the 4-sphere resonator. Nevertheless, the 2^nd-order simulated mode of Fig. 8(a) has been observed. One example is shown in Fig. 8(f) in which the transverse mode profile is slightly distorted as a result of the separation between two of the microspheres. Other representative, experimentally observed images are given in Figs. 8(g) and 8(h). The easily identifiable portions of the 2^nd and 3^rd- order fractal modes (at z = 0) are the distinct patterns formed in the apex of the m = 2 image in Fig. 8(a) and in all three “corners” of the m = 3 simulation. The intensity pattern repeated at the perimeter of the calculated mode of Fig. 8(b), for example, can be seen in the experimental images of panel (g) of Fig. 8, and Fig. 6(f) as well. Similarly, the “M” pattern predicted in Fig. 8(a) appears in the upper portion of the experimental image of Fig. 6(e) shown earlier. Variations of both of the intensity patterns in Fig. 8(a) and 8(b) have been observed multiple times in experimental images. Further corroboration of the computed mode patterns is provided by experimental images recorded at other positions within the resonator. The fractal mode calculated for z = 120 µm (Fig. 8(e)), for example, exhibits patterns at the mode corners that are frequently observed in images such as that of Fig. 8(h). Once again, it should be pointed out that the ultrafine detail of fractal mode patterns can be seen in Figs. 8(f), 8(g), and 8(h) but in panel (f), in particular.

 

Fig. 8. Simulations and experimental images for higher-order laser fractal modes associated with the 3-sphere, triangular aperture geometry: (a)–(d) mode patterns for four of the lowest higher-order modes at z = 0, along with the corresponding values of |γ_m|²; (e) the same mode as shown in (b) but calculated for the z = 120 µm transverse plane; (f),(g) experimentally recorded images obtained with a single pump pulse. The null extending across the mode observed in (f) matches the profile predicted by the simulation shown in (a). The most intense portions of both the calculated m = 2,3 fractal modes appear in experimental images, whereas the faint central feature in the m = 3,5, and 6 patterns has not been detected experimentally.

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The fractal dimension for the higher-order modes of Figs. 7 and 8 has been calculated for the upward-propagating wave and along the resonator at intervals of 5 µm. The results are presented in Figs. 9(a) and 9(b) for the three- and four-sphere aperture resonators, respectively. Both show that the transverse profile of any fractal mode cannot be adequately described by a single value of D. To the contrary, the fractal dimension varies considerably during the single-pass propagation of the mode to the upper mirror, and the abrupt increase in D at the aperture (lying at z = 100 µm) is the result of the high spatial frequencies introduced by edge diffraction when the propagating mode encounters the aperture. The magnitude of the increase in D is greatest for the 3-sphere aperture and, for the fundamental mode, is ∼0.15. Aside from this general observation, however, the behavior of D differs considerably between the two resonators. For example, consider the trend in the fractal dimension when the optical field is far from the diffracting aperture (i.e., z < 100 µm and near 300 µm). For the 3-sphere aperture data of Fig. 9(a), it is found that D increases monotonically with decreasing ${|\gamma |^2}$ which is reasonable since the lower-loss modes have a smaller degree of overlap with the aperture. In contrast, D evolves quite differently in the 4-sphere resonator. Although the two lowest-loss modes of the optical cavity are responsible for the smallest values of D throughout the cavity, one observes that the largest values of the fractal dimension are exhibited by the m = 3 and 5 modes. It is also evident that the ordering of the curves correlated with the six modes for the 4-sphere aperture is far from monotonic at every position in the resonator. This unexpected behavior is likely due to the large aspect ratio of the rectangular aperture and the spatial overlap of these fractal laser modes with the aperture geometry, in particular, but this presumption has not been confirmed.

 

Fig. 9. Spatial evolution of the fractal dimension D of the modes generated with the triangular and quasi-rectangular apertures, and predicted wavelengths for several fractal laser eigenmodes: (a), (b) Calculated variation of D as the optical field propagates upwards in the resonator of Fig. 1(a). Results are given in (a) and (b) for the 3-sphere and 4-sphere apertures, respectively, where the binary transmission masks are situated numerically at the microsphere equatorial plane (z = 100 µm, denoted by the dashed vertical lines). Note the abrupt rise in fractal dimension at, and downstream of, the aperture; (c) Calculated resonant wavelengths for several high-order laser fractal modes, and the dependence of the value of |γ_m|² on mode wavelength. Observed and unobserved modes are denoted by solid and dashed lines, respectively. Note that the m = 1 and 2 modes for the four-sphere aperture are degenerate.

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Panel (c) of Fig. 9 is a spectrum summarizing the predicted resonant wavelengths of both experimentally observed and unobserved laser fractal modes for both the triangular and distended-rectangular apertures. Observed modes are represented by solid lines, whereas those not observed to date are identified by dashed vertical lines, and the 3-sphere and 4-sphere modes are shown in red and blue, respectively. Note, too, the dependence of |γ_m|² on the laser mode wavelength and, specifically, the rapid rise of the loss of 3-sphere modes with increasing m. The resonant wavelengths for each eigenmode were determined by first calculating the phase ϕ of the eigenvalue corresponding to each mode at a specific wavelength (650 nm) and shifting the wavelength by $\phi {\lambda ^2}/({4\pi L} )$. For each higher-order mode, the resonant wavelength was found to lie within the free spectral range (0.53 nm) of the fundamental mode.

6. Conclusions

Although the fundamental laser fractal mode associated with a triangular intracavity aperture was reported in 2018, higher-order fractal modes have not been observed previously. In this work, two classes of high-order fractal laser eigenmodes have been predicted computationally and observed experimentally in microresonators in which the nascent mode is shaped by a refractive aperture comprising microspheres and the gain medium in which it propagates. Both three- and four-sphere topologies were examined for the refractive aperture and the fundamental mode, as well as several higher-order eigenmodes (m = 2,4,5), are observed with the quasi-rectangular (4-microsphere) aperture. This fundamental (m = 1) mode had also not been observed previously. The 5^th order mode associated with the 4-sphere aperture, for example, has a calculated round-trip loss of ∼18% and the simulated transverse intensity maps bear a close resemblance to the experimental images. In contrast, the calculated round-trip losses for the quasi-triangular aperture are much larger (∼30% and ∼45% for m = 2 and 3, respectively) and, accordingly, only the m = 1 and 2 modes and portions of the third-order mode are observed reproducibly. For both the quasi-triangular and quasi-rectangular apertures, the fractal dimension D rises rapidly beyond the aperture (within ∼10 µm) as a result of the high spatial frequencies introduced into the mode’s transverse intensity pattern by near-field diffraction. For the triangular m = 1 mode, the rise in D is ∼0.15 beyond the aperture and D rises monotonically with mode number, except for m = 3 and 4 for which the fractal dimension is essentially the same near the aperture. Owing to the aspect ratio for the 4-sphere aperture, the fractality of the m = 3 and 5 modes is the largest of the modes investigated and the m = 1 and 2 modes are degenerate.

Although detailed calculations of higher-order fractal modes were only performed for the two aperture geometries shapes described here, it should be noted that the presence and structure of periodic nulls that are indicative of higher-order mode oscillation can be readily observed experimentally in various other configurations of microspheres. In fact, higher-order mode operation appears to be dominant in 4-sphere arrangements where the minimum sphere spacing (as shown in Fig. 6) is greater than ∼40 µm. The same statement is true for any microresonator bounded by 5 or more microspheres with a sufficiently wide effective aperture.

The interaction between the refractive aperture, the spatially varying pump intensity, and the gain medium in the present experiments suggest that this dynamic process of forming fundamental and higher-order fractal laser modes can be controlled with more than one spatial aperture of relatively arbitrary shape, each of which can be precisely defined by microfabrication. It is also expected that similar mode profiles can be generated in optical or microwave waveguides, containing discrete “apertures” such as those described here, at regular longitudinal intervals which is equivalent to simply “unwrapping” the resonators of the present work.

Finally, a few comments regarding the applicability of these unique, non-Gaussian modes are in order. Insofar as optical applications are concerned, the characteristics of laser fractal modes that appear to have the greatest potential are their extraordinary spatial mode frequencies (current limit of ∼10 µm^-1) and variable fractality, as evidenced by the variation of D as the mode propagates. These suggest that a new form of microscopy, in which a low optical loss sample is situated downstream of, or within, the resonator, is feasible. Recording two-dimensional (2D) maps of diffracted intensity (in reflection or transmission) as the object under study is translated along the axis of the resonator will expose the target to continuously changing fractal radiation patterns, thereby yielding 3D data sets through which the spatial structure of an object can be determined through electromagnetic inverse scattering algorithms. Recording the fractal laser mode pattern simultaneously with the diffracted pattern will be necessary. Another promising direction is the development of equivalent optical waveguide versions of the microresonator of the present work so as to realize optical sensors for particulates, for example.

7. Methods and materials

7.1 Numerical calculations of eigenmodes and eigenvalues

Approximating the sphere/interstitial resonators by the binary transmission apertures of Fig. 2(b),(c) circumvents the normal requirement for 3D discretization of the entire resonator volume. Instead, 2D discretization can be implemented at specific transverse planes within the cavity, and the optical field at these planes can be numerically propagated from one plane to the next. The binary transmission aperture of interest was situated at the equatorial plane (z = 100 µm) and the optical field was calculated along the resonator axis in increments of 5 µm. A transverse sampling interval of λ/10 – λ/5 was adopted for all of the results presented here. After each round trip of the electric field through the resonator, the ratio ${E^{(N )}}/{E^{({N - 1} )}}$ was calculated, where N is the cumulative number of round trips, and the field was considered to have converged if 99% of the pixels in the computational window experienced the same loss during the most recent round trip, to within ±0.1% of the mean. Although this convergence criterion is strict, it ensures that the solutions have converged. With increasing N, all ${({{\gamma_m}/{\gamma_1}} )^N}$ terms in Eq. (4) will asymptotically approach zero and the fundamental mode will predominate, but defining convergence for each eigenmode in terms of the electric field at each pixel in the computational window was found to be essential for precisely determining both ${\gamma _m}$ and the electric field distribution.

The number of iterations (round-trips) necessary to establish a consistent cavity mode corresponds to an intracavity intensity buildup time between 50 ps and a few ns, depending on the mode, which is less than the pulse duration (∼8 ns) of the pump and emission beams but considerably greater than the ∼3 ps cavity round-trip time. All of the numerical results presented here assume L (mirror separation) = 300 µm, a microsphere diameter of 200 µm, and a quantum dot solution having a refractive index of 1.33.

7.2 Experimental details

The design of the vertically oriented resonator is similar to that described by Refs. [22] and [29] in that flat dielectric mirrors having reflectivities >99.9% at 650 nm and <10% at the pump wavelength (532 nm) are interferometrically aligned and separated by 300 µm. Polymer spheres 200 µm in diameter are arranged on the surface of the lower mirror and the remainder of the resonator is filled with a colloidal suspension of CdS/ZnSe quantum dots. This gain medium exhibits peak fluorescence at ∼650 nm, and was chosen because observing high-order fractal modes requires a substantial gain coefficient so as to overcome diffractive losses. The optical pump for the microlaser was provided by ∼8 ns pulses from a frequency-doubled, Nd:YAG laser. A 20x microscope objective (numerical aperture of 0.42) and a dichroic beamsplitter permit the pump pulses to enter the resonator through the upper mirror while simultaneously allowing laser emission to be observed within the resonator (as opposed to only at the exit aperture of the laser) with an imaging camera. Obtaining images of maximum spatial resolution while viewing the interior of the microresonator with a camera and imaging optics required that the substrate of the upper mirror be thin (1.5 mm). Also, the nominal longitudinal (depth) resolution of the detection system is <2 µm. Each experimental image presented here was recorded with a single pulse of the pump laser.