1. Introduction

Photonic crystals (PCs) are lattices that have a spatially periodic modulation in the dielectric function [1]. PCs exhibit novel optical properties, and devices based on engineered PCs have been widely studied for applications in integrated photonics [2–4]. The spatial dispersion and propagation of light within a PC can be controlled by judiciously selecting the materials, symmetry of the unit cell, and the pattern residing within the unit cell [5,6]. Optics based on PCs can be classified as (1) bandgap devices and (2) in-band devices.

PC-bandgap devices operate by using forbidden modes to control the flow of light. A periodic lattice having a complete photonic bandgap can be modified by adding defect sites or defect channels. Light oscillating at frequencies within the bandgap cannot exist in regions where the lattice is uniform, but it can concentrate at defect sites and propagate along continuous defect paths within the lattice. This approach has been used to make a wide range of waveguides [7], PC-fibers [8], lasers [9], and sensors [10].

In contrast, PC in-band devices are designed to control the flow of light by leveraging the spatial dispersion of allowed modes [11]. Light can exist in a PC when the fields oscillate at frequencies lying within a band of allowed modes, and the shape of the band determines the dispersion experienced by the light. The spatial dispersion of a PC can be visualized using an iso-frequency surface (IFS), or projected slices through the IFS, called iso-frequency contours (IFCs). The link between power flow and the shape of an IFC is ${\vec{v}_\textrm{g}}({\vec{k}} )= \nabla {\omega _\textrm{n}}({\vec{k}} )$, where ${\omega _\textrm{n}} = a/{\lambda _0}$ is the normalized frequency of the light, ${\lambda _0}$ is the wavelength in vacuum, ${\vec{v}_\textrm{g}}$ is the group velocity, and $\vec{k}$ is the wave vector. This equation leads to the conclusion that optical power will flow in the direction normal to the IFC.

A PC and its corresponding IFSs (or IFCs) can be engineered to introduce novel dispersive properties, such as slow-light [12], the super-prism effect [11], negative refraction [13], and self-collimation (SC) [14,15]. When a device exhibits SC at a given frequency, the IFCs are flat, so light propagates normal to the dispersive surface without diverging. SC thereby offers a novel and versatile means for controlling the flow of light in integrated photonics.

The unit cells comprising a PC can be spatially varied throughout the lattice to further control how light flows within the device. Structural parameters that can be varied include the lattice spacing, material composition, unit-cell orientation, and unit-cell shape and anisotropy. The fill-factor can also be spatially varied to create graded-refractive-index (GRIN) devices. A wide range of GRIN-PCs having a complete bandgap have been reported, including a beam-bender [16], a lens [17], and a concentrator [18].

To date, GRIN-PCs based on in-band properties have been designed to operate at low frequency where IFCs are circular and ${\lambda _0} \gg a$. The fill-factors of the unit cells are modulated to control the local refractive index, which can be estimated using the effective medium theory [19,20]. Requiring IFCs to be circular limits the types and symmetries of lattices, and it makes it challenging to fabricate devices because the lattice period must be small compared to the wavelength. Consequently, GRIN-PCs operating at microwave and infrared wavelengths have been reported [18,21–23], but devices operating at optical wavelengths are lacking.

Designing a GRIN-PC to operate using SC relaxes the restriction ${\lambda _0} \gg a$. This makes it more practical to fabricate devices that operate at optical frequencies and over a wider range of polarization states [24]. PCs having $a \sim {\lambda _0}$ can be fabricated by multiphoton lithography (MPL) [25] and nano-imprint lithography [26].

Rumpf et al. [27] reported an algorithm that can be used to design spatially-variant photonic crystals (SVPCs) in which one or more parameters of the lattice are varied globally while maintaining the shape of the unit cells locally so that SC is preserved throughout the device. The orientation of unit cells within a self-collimating lattice can be spatially-varied to create devices that direct the flow of optical power through a tight turn [28]. Simulations suggest that other structural features can be varied to control phase, polarization, or wavelength, in addition to power flow. Spatially varying two or more structural parameters opens a route to multi-functional devices that control power-flow, polarization, phase, or other characteristics of the light, all in a single device [29]. But this had yet to be demonstrated experimentally, until now.

The present work reports a new class of multi-function SVPCs, illustrated in Fig. 1, that alter the phase of an optical field while independently controlling power flow. The example reported here is a lens-embedded SVPC (LE-SVPC). The LE-SVPC performs two fundamental functions in a single 3D nanophotonic device: (1) it directs the flow of optical power through SC, and thereby suppresses beam divergence; and (2) it reshapes the phase front along one axis, so the light cylindrically focuses after exiting the device. The LE-SVPCs are fabricated by MPL using the cross-linkable photopolymer IP-Dip (Nanoscribe), structurally characterized by scanning electron microscopy (SEM), and optically characterized at ${\lambda _0} = 1550\; \textrm{nm}.$ Interestingly, although the LE-SVPC has a thickness several times the wavelength ${\lambda _0}$, SC prevents focusing from occurring within the device, so the optical behavior can be modeled as focusing by a perfectly thin cylindrical lens.

 

Fig. 1. (a) Cubic unit cell upon which the LE-SVPC is based. The unit cell has side-length a and is comprised of a vertical wall having thickness t intersected by a horizontal rod with elliptical cross-section. The red arrow indicates the direction light is incident. (b) Diagram illustrating the behavior of a lens-embedded spatially-variant photonic crystal (LE-SVPC). Light couples into the LE-SVPC and accumulates spatially-variant phase as it propagates through the device. Self-collimation (SC) prevents the beam from focusing or diverging as it moves through the lattice. Upon exiting, the sculpted phase front leads to focusing as the light propagates in free space. The lattice is comprised of unit cells whose wall thicknesses are varied parabolically from the center to the side. (c1, d1) SEM images of two LE-SVPCs fabricated by multi-photon lithography (MPL). Each has a transverse-area of 71 × 71 unit cells, but they differ in length. The long LE-SVPC (left) has a length along $\hat{x}$ of ${L_x} = 71$ unit cells. The short LE-SVPC (right) has ${L_x} = 35$ unit cells. Panels c2 - c3 and d2 - d3 show zoomed-in views of the two LE-SVPCs from the top and side with the sample tilted by 65°.

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2. Design of the lens-embedded spatially-variant photonic crystal

2.1 Design concept

The LE-SVPC is based on the cubic unit cell shown in Fig. 1(a). The unit cell consists of a vertical wall of thickness t intersected by a transverse rod having an elliptical cross-section, with ${r_1}$ and ${r_2}$ giving the radii of the minor- and major-axes, respectively. The LE-SVPC is designed so light enters at a face parallel to the yz-plane and propagates along $\hat{x}$ (optical axis). The unit cell is engineered to exhibit strong SC that forces light to propagate along one of the principal axes of the unit cell. Unit cells having period a repeat along $\hat{x}$, $\hat{y}$, and $\hat{z}$ forming an LE-SVPC of lengths ${L_x}$, ${L_y}$, and ${L_z}$. Across the beam (parallel to $\hat{y}$), t varies from thick in the center to thin at the sides, creating an effective refractive index profile ${n_{\textrm{eff}}}(y )$ that impresses a quadratic phase profile onto the beam. SC prevents focusing from occurring within the lattice, so as power flows forward the wave accumulates a total phase delay determined by ${n_{\textrm{eff}}}{L_x}$.

2.2 Structure of the unit cell

The unit cell in Fig. 1(a) is inspired by a study reported by Hamam et al. [30]. They found that a 2D lattice of alternating rods and walls offers broadband SC across a wide range of input angles. That particular unit cell is not suitable for the LE-SVPC because it is not 3D, and the rods and walls are disconnected, so it cannot form a free-standing, self-supporting structure. To arrive at the unit cell shown in Fig. 1(a), the 2D design of Haman et al. is modified as follows. The walls are extruded vertically (along $\hat{z}$). The rods, on the other hand, are elongated horizontally (along $\hat{y}$), so they penetrate the walls and connect to one another forming a connected lattice. Layers of horizontally oriented rods are then repeated vertically, separated by unit cell spacing a, to introduce periodicity and resulting SC that prevents the beam from spreading in the vertical direction as it propagates down the optical axis. The rods have an elliptical cross-section that can be adjusted to tune SC.

2.3 Optimizing self-collimation

The geometric parameters of the unit cell shown in Fig. 1(a) were varied to find those which optimize SC. The ratio $t/a = 0.24$ was fixed while ${r_1}/a$ and ${r_2}/a$ were individually swept from 0 to 0.5, and the band diagram for each given unit cell was obtained using the plane wave expansion method (PWEM) [31]. In the calculations, the refractive index of the material was set to n = 1.525, for cross-linked IP-Dip at λ₀ = 1550 nm [32]. The quality of SC was judged using a figure of merit (FOM) described in [33]. The FOM consists of three performance metrics: the frequency-bandwidth, angle of acceptance ${\theta _\textrm{c}}$, and location of the inflection point of SC. The best SC was achieved when ${r_1}/a = 0.209$ and ${r_2}/a = 0.416$.

The frequency of operation is chosen where the IFC is flattest. IFCs for the second TM band (E-field polarized along $\hat{z}$) of the optimized unit cell are shown in Fig. 2. Figure 2(a) shows only the first quadrant because the unit cell is symmetric about reflection through the xy-, xz-, and yz-planes. One complete IFS is drawn at ${\omega _\textrm{n}} = 0.64$. For other frequencies, only in-plane IFCs are shown. The IFS is flat where it is crossed by $\vec{k}$ parallel to $\hat{x}$, $\hat{y}$, or $\hat{z}$, which indicates that light having ${\omega _\textrm{n}}$ corresponding to those values of $\vec{k}$ will self-collimate and propagate along the principal axes. Light is chosen to be incident along $\hat{x}$ (red arrow in Fig. 2(b)), where ${\theta _\textrm{c}}$ is the largest. In this work, light is only introduced within the xy-plane, so ${k_z}$ vanishes, and the allowed modes can be estimated well by 2D IFCs. The first two quadrants of IFCs are shown in Fig. 2(b) because potentially two modes can be excited. At high frequency, the IFCs are concave. As ${\omega _\textrm{n}}$ decreases, the IFCs shrink towards the center, flatten at ${\omega _\textrm{n}} = 0.64$, and become convex at low frequency. The LE-SVPCs were designed to operate at the inflection point, where ${\omega _\textrm{n}} = 0.64$, as indicated by red and purple dashed lines in Figs. 2(a) and 2(b), respectively. The second TE band also exhibits SC at ${\omega _\textrm{n}} = 0.64$ with a shape like that of the TM band. It is worth noting that the 3D unit cell employed in [28] also provides SC, but only for the TM mode.

 

Fig. 2. (a) A quadrant of the second TM band shown as one IFS at ω_n = 0.64 and in-plane IFCs. (b) IFCs of the second TM band having k_z = 0. The IFC for ω_n = 0.64 is indicated with the purple dashed line.

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To embed a lens within the PC, a range of t was identified over which SC remains strong. Figure 3(a) shows how IFCs change for the second TM band at ${\omega _\textrm{n}} = 0.64$ when t/a increases from 0.24 to 0.40. The IFCs move towards the origin, becoming flatter, and ${\theta _\textrm{c}}$ becomes wider. The corresponding electric field distributions and eigenmodes $\vec{u}({\vec{r}} )$ were obtained from PWEM. Figures 3(b)–3(f) show the real part of $\vec{u}({\vec{r}} )$ within the xy-plane indicated by a red dashed line and superposed on a grey background that shows the profile of the underlying lattice. Along $\hat{x}$, $\vec{u}({\vec{r}} )$ peaks at the boundaries, is minimum in the center, and varies only slightly as the walls are thickened. Because the profile of $\vec{u}({\vec{r}} )$ is similar for all values of t, phase can vary smoothly between adjacent unit cells that have walls of slightly different thickness. Within t/a = 0.24 to 0.40, wall thickness can be spatially varied to increase fill-fraction and introduce phase delay that reshapes the wavefront without losing SC.

 

Fig. 3. The shift of IFCs and change of mode profile for the second TM band at ω_n = 0.64 as t/a varies from 0.24 to 0.40. (a) IFCs of the second TM band. Due to symmetry, only the second quadrant is drawn. The corresponding unit cells are shown at the right. The red dashed line indicates an xy-plane in which E_z is evaluated. (b-f) The real part of the eigenmodes calculated using PWEM. (g-k) Propagation of Bloch modes over four periods, calculated using FDTD. The light grey background shows the structure of the lattice within the xy-plane.

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Which mode is excited depends upon the properties of the incident light, such as frequency and direction of propagation, and the spatial dispersion of the lattice [34,35]. The incident light only excites modes that have the same frequency. Maxwell’s equations further require that the wavevector parallel to the boundary ${k_{||}}$ must be continuous. For example, consider a 2D unit cell having rotational symmetry for which the incident boundary is parallel to $\hat{y}$. Two mode-pairs can be identified, (${k_x}$, ${k_{||}}$) and ($- {k_x}$, ${k_{||}}$), but only the one for which energy flows in the same direction as the incident beam can be excited. We apply these principles to a TM-polarized beam incident along the optical axis having ${\mathrm{\omega _n}}$= 0.64. Only the modes having the same ${k_{||}}$ (${k_y} = 0$) remain. If we look at IFCs of the second TM band at ${\mathrm{\omega _n}}$ = 0.64 shown in Fig. 2(b), we see that for the ($- {k_x}$, 0) mode, ${\omega _\textrm{n}}$ increases as ${k_x}$ increases. This implies dispersion is normal and power flows along the same direction as the incident light. On the other hand, for the ($+ {k_x}$, 0) mode, ${\mathrm{\omega _n}}$ decreases as ${k_x}$ increases, so dispersion is anomalous, and power would have to flow opposite to the incident beam. As such, only modes having ${k_x}\; <\;0$ can be excited. The modes which can be excited when t is varied are indicated in Fig. 3(a) by the black dots placed along the ${k_x}$-axis.

2.4 Engineering optical phase delay

The finite-difference time-domain (FDTD) method and PWEM were used to calculate how light propagates within a uniform lattice based on the unit cell in Fig. 1(a), for a range of wall thicknesses t. These data were used to obtain a relationship between t and ${n_{\textrm{eff}}}(t )$ with which the LE-SVPC could be designed to generate a quadratic phase profile.

The wave vector $\vec{k}$ and effective refractive index ${n_{\textrm{eff}}}$ of a uniform lattice having a fixed value of t can be uniquely determined with FDTD. These simulations give phase information of an electromagnetic field propagating in a lattice. For a plane wave that takes the form of $\textrm{exp} ({i\vec{k} \cdot \vec{r}} )$, its wavefront is well defined, and its phase velocity is given by $\omega /|{\vec{k}} |$. Once $\vec{k}$ is known, the refractive index n can be calculated with

The FDTD simulations were performed using the open-source software MEEP [37]. The light source was a vertically polarized plane wave (electric field parallel to $\hat{z}$) launched parallel to $\hat{x}$ with ${\lambda _0} = 1550\; \textrm{nm}$. The corresponding lattice constant $a = {\omega _\textrm{n}}{\lambda _0} = 992\; \textrm{nm}$. The grid was periodic along $\hat{y}$ and $\hat{z}$ to mimic an infinite lattice. Parallel to $\hat{x}$, the boundaries were set to perfectly matched layers (PMLs) to avoid non-physical reflections. Field propagation was calculated with lattices having the parameter t varied as shown in Figs. 3(b)–3(f). Convergence was verified for grid resolution, PML thickness, and the total time steps. The converged results are shown in Figs. 3(g)–3(k) as the real part of ${E_z}$, sampled within the plane identified in Figs. 3(b)–3(f), and propagated along $\hat{x}$ over four lattice periods. The dashed white lines identify positions of equal phase. Phase advances slower in a lattice with thick walls, indicating the effective refractive ${n_{\textrm{eff}}}$ increases as t increases. With care for phase wrapping, ${n_{\textrm{eff}}}$ was calculated by evaluating phase difference between a pair of points along $\hat{x}$ and separated by an integer multiple of the lattice spacing a. It is worth noting that the ${n_{\textrm{eff}}}$ obtained does not depend on the choice of point pairs with the yz-plane. Initially, ${n_{\textrm{eff}}}$ increases and then stabilizes after roughly three unit cells as the input plane wave transforms to a Bloch mode. The stabilized ${n_{\textrm{eff}}}$ obtained for a range of $t/a$ is plotted in Fig. 4. As $t/a$ increases from 0.24 to 0.40, ${n_{\textrm{eff}}}$ increases linearly from 1.266 to 1.351. The values of ${n_{\textrm{eff}}}$ lie between that of vacuum and the bulk polymer, and the maximum $\mathrm{\Delta }{n_{\textrm{eff}}}$ is 0.085.

 

Fig. 4. Relationship between t/a and n_eff extracted from FDTD (yellow cross) and PWEM (red dot) simulations for the TM mode.

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PWEM can be used to confirm ${n_{\textrm{eff}}}$ if information lost from band folding is recovered with help from the FDTD simulations. Band diagrams are typically shown for the first Brillouin zone (FBZ). For modes in the first band, $\vec{k}$ can be read off directly. Folding must be considered for obtaining $\vec{k}$ in higher-order bands. The unfolded $\vec{k}$ can be determined by first shifting $\vec{k}$ by integers of the width of the FBZ, calculating ${n_{\textrm{eff}}}$ using Eq. (1), and identifying which value of ${n_{\textrm{eff}}}$ agrees with that from FDTD simulations. It was found that $\vec{k}$ read directly from Fig. 3(a) must be shifted up by one FBZ-width to obtain the unfolded $\vec{k}$. Values of ${n_{\textrm{eff}}}$ obtained from PWEM after unfolding are plotted in Fig. 4 and found to agree to within 0.8% with those obtained from FDTD calculations. The relationship in Fig. 4 was used to determine how thick the walls of the unit cells must be to achieve ${n_{\textrm{eff}}}$ at a given horizontal position y across the LE-SVPC.

The LE-SVPC is designed to have a phase distribution given by

Substituting Eq. (3) to Eq. (4) and solving for ${n_{\textrm{eff}}}(y )$ gives

Knowing how ${n_{\textrm{eff}}}(y )$ must change across the device, the relationship in Fig. 4 can be used to obtain $t(y )$ that defines the physical profile of the LE-SVPC for a given $f.$ Similarly, when ${n_{\textrm{eff}}}(y )$, ${n_{\textrm{eff}}}(0 )$, and ${L_x}$ are known, f can be calculated with Eq. (5) rearranged as

3. Fabrication and characterization

3.1 Fabrication method

The LE-SVPCs were configured to function at ${\omega _\textrm{n}} = 0.64$ and ${\lambda _0} = 1550\; \textrm{nm}.$ The corresponding targeted dimensions are then $a = {\omega _\textrm{n}}{\lambda _0} = 992\; \textrm{nm},$ ${r_1} = 208\; \textrm{nm},$ ${r_2} = 413\; \textrm{nm}$, and t parabolically decreases in width from $t/a = 0.40$ at the center $({{t_{\textrm{thick}}} = 397\; \textrm{nm}} )$ to $t/a = 0.24$ $({{t_{\textrm{thin}}} = 238\; \textrm{nm}} )$ at the side.

LE-SVPCs were fabricated using a home-built MPL system. The laser source is a mode-locked Ti:sapphire laser (Coherent) producing 120 fs pulses at a repetition rate of 76 MHz and a center-wavelength of 800 nm. The linearly polarized beam is passed through an acousto-optic modulator (Gooch and Housego) to adjust the average exposure power P and then expanded by a telescope to overfill an objective lens (Nikon, 60×, NA = 1.4). The lens focuses light into the photopolymer to activate polymerization at the focal spot. A calibrated integrating sphere (Optronic Laboratory 731) is used to measure P after the lens. Borosilicate glass microscope slides were used as substrates for fabrication. An adhesion layer was added by spin-coating 1 vol-% (3-acryloxypropyl)trimethoxysilane (CAS# 4369-14-6) in methanol (CAS# 67-56-1) onto the substrate (2500 rpm, 30 s), baking on a hotplate (90 °C, 30 s), then allowing it to cool in air. The substrate was mounted on a three-axis nanopositioner (Physik Instrumente 563.3CD), a drop of IP-Dip was added, and the objective was lowered into the photopolymer. The pattern of the LE-SVPC was exposed under control of a microcomputer that coordinates the movement of the stage (50 µm s⁻¹) with adjustment of P. Following exposure, structures were developed by immersing sequentially in propylene glycol methyl ether acetate (PGMEA, 30 min, 3×), isopropyl alcohol (IPA, 5 min), and deionized water (5 min). Afterwards, the samples were drained and left to dry in the air.

The devices were fabricated along with a tower that elevates the structure off the supporting substrate to facilitate optical characterization. P was varied as needed to create walls having the targeted thickness. Rods with elliptical cross-section were created with three partially overlapping and adjacent exposure lines. The central line was exposed at high power $({P = 1.87\; \textrm{mW}} )$. The outer lines were exposed at low power $({P = 1.48\; \textrm{mW}} )$ and laterally offset from the center line by +97 nm and -97 nm, respectively. The fabrication time for the PCs shown in Fig. 1(c1) and 1(d1), excluding the supporting towers, was 27 hours and 20 hours, respectively.

3.2 Structural characterization

LE-SVPCs were structurally characterized using SEM images. Figures 1(c1) and 1(d1) show two typical devices. Each was fabricated to have the same $t$-profile and transverse dimensions ${L_y} = {L_z} = 71$ unit cells, but their lengths differ. A short device having ${L_x} = 35$ unit cells and a long device having ${L_x} = 71$ unit cells were created to study how length effects focal power.

Measurements of the periodicity and the dimensions of walls and elliptical rods were obtained from the SEM images. The periodicities of the LE-SVPC shown in Fig. 1(c1) are 1010 nm, 945 nm, and 987 nm along $\hat{x}$, $\hat{y}$, and $\hat{z}$, which differ from the targeted periodicity by 1.7%, 4.8%, and 0.5%, respectively. For the rod geometry, ${r_1}$ and ${r_2}$ are 201 nm and 414 nm, which differs from the targeted dimensions by 3.3% and 0.1%, respectively. The LE-SVPCs shown in Figs. 1(c1) and 1(d1) were fabricated back to back and their measured structural parameters are similar.

To verify that t follows the targeted quadratic profile, wall thicknesses of the LE-SVPC shown in Fig. 1(c1) were measured on the input face, where walls intersect rods. The values obtained are shown in Fig. 5 and are fitted to a second-order polynomial. The variation in wall thickness is visibly parabolic, differing from the fitted curve by no more than 4%. The $t$-profile of the LE-SVPC in Fig. 1(d1) varies similarly (not shown) but has an overall offset of 17 nm, which cannot adversely affect the wavefront because ${n_{\textrm{eff}}}$ varies linearly with t.

 

Fig. 5. Profile of wall-thickness t across the LE-SVPC (parallel to $\hat{y}$) shown in Fig. 1(c1). Measured values of t are plotted as black dots. The solid red line shows a fit of the measurements to a second-order polynomial. The error bars represent the ±1 standard deviation of five measurements.

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The PWEM calculations were repeated using the experimentally measured dimensions to determine how deviation from the targeted form affects the optical properties of the LE-SVPC. The maximum $\mathrm{\Delta }{n_{\textrm{eff}}}$ was found to be 0.0965 and 0.0777 for vertically and horizontally polarized light, respectively, which compares well with the targeted $\mathrm{\Delta }{n_{\textrm{eff}}}$. The PWEM simulations also show that deviations from the targeted design do not significantly alter the shape of the IFCs nor the resulting strength of SC. The values of $\mathrm{\Delta }{n_{\textrm{eff}}}$ obtained for the fabricated LE-SVPC in Fig. 1(c1) were substituted into Eq. (6) to obtain the theoretical focal length as $f = 81$ µm and 99 µm for vertically and horizontally polarized light, respectively.

3.3 Method of optical characterization

LE-SVPCs were optically characterized at the wavelength ${\lambda _0} = 1550\; \textrm{nm}$ using a scanned-optical-fiber system (Fig. 1), the details of which are reported elsewhere [15]. In brief, single-mode optical fibers (ThorLabs 1550 BHP) were used to couple laser light into the device and observe how light propagated upon exiting. Light from the source fiber (SF) propagated distance ${d_1}$ along $\hat{x}$ as a gently diverging Gaussian beam [38] before illuminating the entrance face of the LE-SVPC. Light exiting the device traveled distance x before being sampled by a detection fiber (DF) coupled to a photodiode. The output was referenced to the reading from an external detector to eliminate fluctuations due to shot-to-shot variation of the laser-pulse energy. The referenced signal $S({x,y,z} )$ was recorded as the DF scanned through the beam to obtain spatial intensity maps. The coordinate system is shown in Fig. 1, and position $x = 0$ is located at the exit-face of the LE-SVPC. The positions of the SF, LE-SVPC, and DF were observed from above with an optical microscope and corresponding images were used to confirm values of ${d_1}$ and x.

Scans of S within a $yz$-plane were used to observe the transverse profile of the beam at a given x along the propagation direction. Scans within an $xy$-plane were used to observe how the transverse profile changed with x, and the horizontal width of $S(x )$ was taken as its full-width at half-maximum (FWHM). Scans were repeated multiple times, with SF shifted to different distances ${d_1}$, to explore how curvature and size of the incident beam affected focusing by the LE-SVPC.

The referenced signal S is actually a convolution of the intensity of the propagating beam with the input-response function of DF, so the FWHM of $S(x )$ is not the true beam width. The FWHM converges to the beam width when it is large relative to the mode-field diameter of the DF, and both the beam width and the FWHM of $S(x )$ are minimum at the focal point. The FWHM of the beam itself can be extracted by deconvolution, and the spatial profile and FWHM of $S(x )$ can be modeled when the distribution of intensity and phase of the beam are specified.

3.4 Example of optical characterization

Figures 6(a1)–6(a4) and 6(b1)–6(b4) show results from the optical characterization of a long- and short LE-SVPC, like those in Figs. 1(c1) and 1(d1). The source light was vertically polarized. The scans were started at $x = 10$ µm to avoid colliding the DF with the LE-SVPC. Figures 6(a3) and 6(b3) are examples of $xy$-scans, obtained with ${d_1} = 150$ µm. Measures of FWHM obtained from $xy$-scans are plotted in Figs. 6(a4) and 6(b4). The data were fitted to a fourth-order polynomial, then the point where FWHM was minimized was identified, and the corresponding value ${d_2} = x$ was recorded as the focal distance. When a fit yielded ${R^2}\;<\;0.9$, a higher-order polynomial was used. Values of ${d_2}$ obtained from fourth- and higher-order polynomials differed by less than 5 µm. This value is comparable to the minimum step-size in x used to record the scans and is therefore taken as a reasonable estimate for the uncertainty of ${d_2}$. To measure the minimum beamwidth FWHM_min more accurately, the DF was moved to the focal spot ${d_2}$ and line-scans parallel to $\hat{y}$ were recorded three times. The final value of FWHM_min was obtained as an average from three scans.

 

Fig. 6. Optical characterization of two LE-SVPCs like those shown in Figs. 1(c1) and 1(d1), having (left) L_x = 71 unit cells and (column) L_x = 35 unit cells, and measured with d₁ = 150 µm. Panels (a3) and (b3) are xy-scans that show how the transverse beam profile changes along $\hat{y}$ when light exits the LE-SVPC and propagates distance x. These data were used to obtain the width of the scan-profiles as FWHM versus x, shown in (a4) and (b4). Green dashed lines identify focal points d₂ = x, where the beam is narrowest along $\hat{y}$ and where FHWM_min is located. At these points, full transverse scans of the beam profile appear as seen in (a1) and (b1). For comparison (a2) and (b2) show transverse profiles recorded without the LE-SVPC present. When the LE-SVPC is present, the beam narrows in the horizontal plane, showing the device behaves like a cylindrical lens, and the focal power is stronger for the longer LE-SVPC.

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The plots of FWHM show that beams exiting the LE-SVPCs become narrower along $\hat{y}$, then reach a minimum width and begin to diverge. The green dashed lines indicate the point ${d_2} = x$, where FWHM_min is located and where the transverse profiles were recorded. For comparison, profiles obtained without the LE-SVPC are also shown. When the device is present, the profile narrows in the horizontal axis and becomes elliptical, which is consistent with the LE-SVPC functioning like a cylindrical lens. The long LE-SVPC produces the narrower profile and therefore focuses stronger. It is worth noting that the point of maximum intensity appears before FWHM_min. If a beam is focused by a rotationally symmetric lens, then the FWHM_min and point of maximum intensity should appear at the same ${d_2}$. But because the LE-SVPC focuses only along $\hat{y}$, the beam exiting the LE-SVPCs converges along $\hat{y}$ and diverge along $\hat{z}$, causing the point of maximum intensity to appear before FWHM_min.

Measurements were performed on multiple, separately fabricated LE-SVPCs, like those in Fig. 1. The uncertainties in FWHM_min and the corresponding focal distance were estimated from the analysis of $xy$-scans obtained with three separate long- and short devices. The PCs were characterized with SF positioned at several distances ${d_1}.$ When ${d_1} = 100$ µm, the FWHM_min and ${d_2}$ differed between scans by less than 0.33 µm and 5 µm, respectively, for both polarizations. When ${d_1} = 150$ µm, FWHM_min and ${d_2}$ differed by less than 0.16 µm and 6 µm, respectively.

3.5 Modeling as a thin lens

The performance of the LE-SVPC can be modeled as illustrated in Fig. 7(a). If the LE-SVPC behaves as designed, then SC should force light to propagate through the device without diverging or focusing, even as the beam accumulates phase across its transverse profile. Light exiting the SF propagates as a diverging, rotationally symmetric Gaussian beam [38], then couples into the LE-SVPC. Geometrically, rays should enter and exit the device at the same position, and light should focus only after exiting the device. It follows then that it should be possible to model the LE-SVPC as a thin lens, even though it is physically thick.

 

Fig. 7. Optical modeling and characterization of focusing by LE-SVPCs. (a) Schematic of the model. Light emanates from a single-mode optical fiber (SF) as a Gaussian beam of width w₀₁, propagates distance d₁, then couples into the LE-SVPC, which is treated as a thin lens. SC forces light to travel the length of the device without spreading or focusing, even as it accumulates a quadratic phase profile. Upon exiting, the light focuses to a Gaussian beam of width w₀₂ after propagating distance d₂. Because SC prevents the beam width from changing within the LE-SVPC, it can be treated as a thin lens. (b) Experimental measurements of FWHM as a function of distance x when d₁ = 50 µm, 75 µm, 100 µm, and 150 µm, after passing through the long LE-SVPC, like that in Fig. 1(c1). (c) Experimental and simulated change of d₂ and FWHM_min for a long LE-SVPC as a function of d₁, with vertically ($\hat{z}$) and horizontally ($\hat{y}$) polarized light. The shaded region shows how d₂ and FWHM_min vary in simulation when f changes by ±5 µm.

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It is well known that a Gaussian beam incident on a thin lens focuses also to a Gaussian beam [39]. The waist of source beam ${w_{01}}$ at distance ${d_1}$ can be related to the waist of focused beam ${w_{02}}$ at distance ${d_2}$ through the focal length f using Eqs. S(1) and S2 described in the Supplement 1. The equations can be solved to find ${d_2}$ and ${w_{02}}$ when ${d_1}$, ${w_{01}}$, and f are known, or they can be used to find f when ${w_{01}}$, ${d_1}$, ${w_{02}}$, and ${d_2}$ are known. In this work, solutions were found numerically using the Equations and Systems Solver in MATLAB (R2018a). The waist of the source beam ${w_{01}}$ was measured experimentally with the process described in [15] and found to be 5.7 µm.

To compare with experiment, values of ${w_{02}}$ predicted from the thin-lens model must be converted to FWHM obtained from a line-scan of detection fiber DF. The conversion was obtained by simulating the line-scan process. The energy collected by DF at each point in the scan is calculated as the overlap integral between its response function and the complex field of the beam (see Supplement 1). The calculation is repeated for all points along the line-scan and the resulting scan-profile is used to obtain a simulated FWHM for comparison to experiment.

3.6 Results and analysis

The width and curvature of a Gaussian beam affect where and how tightly it is focused by a given lens. To determine f of the LE-SVPC properly, FWHM and ${d_2}$ were measured with the source beam positioned at several distances ${d_1}$ from the device. Measurements like those shown in Figs. 6(a3) and 6(a4) were repeated for the long LE-SVPC at ${d_1}$ = 50 µm, 75 µm, 100 µm, 150 µm, 200 µm, 250 µm, and 300 µm, and with the electric field of the source beam vertically or horizontally polarized. Figure 7(c) shows how ${d_2}$ and FWHM_min change versus ${d_1}$. The solid lines in Fig. 7(c) are theoretical values of ${d_2}$ and FWHM_min, calculated using the thin-lens model with $f$ = 74 µm. The values of FWHM_min were obtained by convoluting the beam's electric field with the response function of the DF using the integral-overlap method. The blue and orange shaded regions show how widely ${d_2}$ and the FWHM_min vary when f is varied by ±5 µm. The experimentally measured ${d_2}$ and FWHM_min follow the same trend as the simulated curves, up to ${d_1}$ = 200 µm. Disagreement at ${d_1} \ge 200$ µm is expected because the incident beam has spread to the point t $f\; $ hat it overfills the LE-SVPC, so the Gaussian-focusing model no longer applies. The analysis shows that the focusing behavior of the LE-SVPC is well described by the thin-lens model.

The focal length was estimated by regression analysis of the experimental data against simulations using a least-squares method. Regressions were performed on ${d_2}$ and FWHM_min separately for all ${d_1}\;<\;250$ µm, yielding two values of f that could be compared. For the long LE-SVPC, f was found to be 68 µm and 70 µm for vertically polarized light, and 72 µm and 78 µm for horizontally polarized light. Weighting the two methods equally, the values can be averaged giving $f$ = 69 µm ± 2 µm and $f$ = 75 µm ± 3 µm for vertically and horizontally polarized light, respectively. Applying the same method to the short LE-SVPC gives $f$ = 138.5 µm ± 8.5  µm and $f$ = 144 µm ± 8.5 µm for vertically and horizontally polarized light. The uncertainty in f is larger for the short LE-SVPC because it focuses more gently, resulting in lower intensity at the focus and a poorer signal-to-noise ratio. The data show that the focal length decreases by half when the length of the LE-SVPC is doubled, as expected. The experimentally determined f is smaller than that obtained from theory. Given that the PWEM calculations on uniform lattices were used to obtain ${n_{\textrm{eff}}}$ and f, the gradient of t in the actual LE-SVPC and its effect on local ${n_{\textrm{eff}}}$ is not considered. The stronger focusing observed experimentally suggests that spatially varying the unit cells produces a larger change in $\mathrm{\Delta }{n_{\textrm{eff}}}$.

The power throughput $\eta $ can be obtained from the integrated power transmitted through the LE-SVPC divided by the total power incident on the device. Directly integrating beam-profiles like those in Fig. 6 and using these to calculate $\eta $ overestimates throughput because the DF collects less power when the LE-SVPC is not present, and the beam is diverging. Loss due to the angular sensitivity of the DF can be compensated by calculating the integrated signal scaled by the response function of the DF using the overlap-integral method. In this way, the scaled throughputs were found to be $\eta $ = 85% and $\eta $ = 90% for the long- and short LE-SVPC, respectively.

4. Discussion

The focusing behavior of the LE-SVPC was maintained even as the angle of incidence was varied for the input beam; however, the throughput dropped as the angle of incidence became more oblique. For example, the throughput of the long LE-SVPC dropped by 10%, 15%, 45%, and 78%, when the input beam was incident at 5°, 10°, 15°, and 20°, for vertically polarized light at ${d_1} = 150$ µm. Yet, $xy$-scans at these angles show no significant change in ${d_2}$ or FWHM_min.

The unit cell of the LE-SVPC is asymmetric relative to the input polarization, so we should expect a birefringence that changes how vertically and horizontally polarized light focus. Unlike a conventional lens, for which the focal power depends on ${n_{\textrm{eff}}}$, the focal power of an LE-SVPC is determined by $\mathrm{\Delta }{n_{\textrm{eff}}}$. The PWEM calculations show that $\mathrm{\Delta }{n_{\textrm{eff},\; \textrm{vert}}}$ > $\mathrm{\Delta }{n_{\textrm{eff},\; \textrm{horz}}}$, so vertically polarized light is expected to focus more strongly, which is indeed observed experimentally.

If the device were a uniform lattice, then the birefringence would cause it to behave like a waveplate. But the LE-SVPC does not behave simply as a waveplate because ${n_{\textrm{eff}}}$ varies with y causing focusing. If $\mathrm{\Delta }{n_{\textrm{eff},\; \textrm{vert}}}$ equaled $\mathrm{\Delta }{n_{\textrm{eff},\; \textrm{horz}}}$, then both polarizations would focus to the same point, and their superposition could yield another polarization state. Such a device would behave like a lens and a waveplate combined, similar to the meta-lens reported in [40]. For the LE-SVPCs reported here, the two polarizations focus separately, so they do not mix. Clearly, spatially-varied birefringent PCs can have rich polarization dependence, which itself may be exploited for new kinds of photonic devices.

The experiments discussed here do not provide direct evidence that light is self-collimated as it travels through the LE-SVPCs. However, the fact that f is halved when ${L_x}$ is doubled indicates that the beam width does not change within the device, which is consistent with SC. It is worth noting that if ${d_1}$ is held constant, the beamwidth should be the same regardless of the length of the LE-SVPC through which it travels. Yet, this is not directly reflected in the experimental scans because the response function of the DF decreases as the interrogated beam becomes more curved, which makes the FWHM at $x = 0$ artificially smaller. The wavefront of light exiting the long LE-SVPC is more strongly curved than that exiting the short LE-SVPC. Consequently, the FWHM measured at $x = 0$ is about 7 µm smaller for the long LE-SVPC. To verify the source of this difference, the thin-lens model and overlap-integral methods were used to calculate FWHM at $x = 0$, assuming the values of f found by experiment, and indeed the more tightly focused beam yields a FWHM that is smaller by 6.4 µm.

The chromatic aberration of an LE-SVPC depends on the frequency dispersion of ${n_{\textrm{eff}}}$. Calculations performed versus wavelength suggest f varies by 0.8% per 100 nm around ${\lambda _0} = 1550\; \textrm{nm}$. Tighter focusing could be realized by either increasing the range over which t varies or increasing the length ${L_x}$ of the lens. A tighter-focusing lens could still be compact by wrapping the phase, creating a Fresnel lens within the PC. When the device operates at 1550 nm and has $t/a$ in the range of 0.24 to 0.40, $\mathrm{\Delta }{n_{\textrm{eff}}} = 0.08$, which requires ${L_x} = 20$ µm to accumulate 2π phase.

A wide range of optical behaviors can be accessed by engineering the spatial dispersion of a PC, and subtle changes in structure can result in remarkably different properties. For example, Trull et al. [41] reported the optical properties of a polymeric woodpile PC that also exhibits focusing under certain conditions. The device is a uniform PC, but the structure of the unit cell produces concave IFCs that introduce anomalous dispersion at the operating wavelength. Because the IFCs are not flat, the PC does not self-collimate light internally. But under the conditions tested, anomalous dispersion of the lattice offsets the divergence of the input beam producing an on-axis output that is collimated, along with a complex series of off-axis diffracted beams.

The work reported here is distinct from that of Trull et al. in several ways. First, the unit-cell of the LE-SVPC is engineered to produce true SC within the lattice, and the spatial dispersion is sufficiently strong that the fill-factor can be varied to modulate phase and cause focusing without losing SC. Second, focusing does not occur at the input of the lattice, due to anomalous dispersion. Instead, the spatially-varied profile of fill factor and the resulting modulation in ${n_{\textrm{eff}}}$ alters the phase front of the beam causing it to focus after exiting the device. Third, the method used to create LE-SVPCs is versatile. Focusing can be achieved without trying to match anomalous dispersion of a uniform lattice to the divergence of an input beam. With the LE-SVPCs, the targeted phase profile is achieved by simply tuning fill-factor, using a single unit cell design. Many other phase profiles could in principle be achieved, including those that generate diverging beams at the output, or other more complex mode-profiles. The versatility results from spatial variation of the lattice.

5. Conclusion

This work demonstrates a path toward a new class of integrated photonic device – the LE-SVPC. The device is based on a strongly self-collimating lattice that forces light to propagate along principles axes without diverging (or focusing). The thickness of walls comprising the lattice are spatially varied to alter local ${n_{\textrm{eff}}}$ and effect a change in phase across the beam. Upon exiting the device, the curved phase front causes the beam to focus. Mapping and analysis of the beam profile shows that the LE-SVPC behaves like an infinitely thin cylindrical lens. Processes are described for engineering the form of the unit cell and the structure of the LE-SVPC. PWEM simulations were used to design the unit cell so that it is tolerant to spatial variation and retains SC even when walls are thickened to alter the fill factor and resulting ${n_{\textrm{eff}}}$. FDTD and PWEM simulations were used in combination to calculate how ${n_{\textrm{eff}}}$ varies with feature size in the unit cell.

In a sense, the LE-SVPC “programs” the light wave to focus so that it converges only after exiting the self-collimating lattice. The device could be thought of as a graded-refractive-index (GRIN) lens embedded in a self-collimating lattice. However, its performance is distinct from and simpler than conventional GRIN lenses, such as Luneburg lenses [23], because focusing does not occur within the LE-SVPC. The phase-profile is engineered, and it can be simple parabolic, as in the present example, or more complex, such as an aberration-correcting profile. The lensing function of the LE-SVPC could be used to improve fiber-to-chip and chip-to-chip level interconnects. LE-SVPCs could also be integrated onto the tip of an optical fiber or microfluid channels for sensing and imaging applications. The orientation of the unit cells could in principle be spatially varied so the device bends power through a turn while generating a focusing wavefront. Embedding spatially programmed phase-shifting in a SC lattice is a general approach that could be used to create other types of integrated photonic devices such as mode converters.